Properties

Label 1134.2.d.b.1133.9
Level $1134$
Weight $2$
Character 1134.1133
Analytic conductor $9.055$
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] = [1134,2,Mod(1133,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.1133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.d (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: \(9.05503558921\)
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: \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + 26533 x^{8} - 38796 x^{7} + 47500 x^{6} - 47396 x^{5} + 38144 x^{4} + \cdots + 457 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1133.9
Root \(0.500000 + 2.18553i\) of defining polynomial
Character \(\chi\) \(=\) 1134.1133
Dual form 1134.2.d.b.1133.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -3.79792 q^{5} +(2.59077 - 0.536572i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -3.79792 q^{5} +(2.59077 - 0.536572i) q^{7} -1.00000i q^{8} -3.79792i q^{10} +1.61401i q^{11} -5.11225i q^{13} +(0.536572 + 2.59077i) q^{14} +1.00000 q^{16} +2.33198 q^{17} +4.24621i q^{19} +3.79792 q^{20} -1.61401 q^{22} +2.62863i q^{23} +9.42418 q^{25} +5.11225 q^{26} +(-2.59077 + 0.536572i) q^{28} +4.01461i q^{29} +7.98863i q^{31} +1.00000i q^{32} +2.33198i q^{34} +(-9.83953 + 2.03786i) q^{35} -5.74907 q^{37} -4.24621 q^{38} +3.79792i q^{40} +8.51737 q^{41} +9.25725 q^{43} -1.61401i q^{44} -2.62863 q^{46} -9.69576 q^{47} +(6.42418 - 2.78027i) q^{49} +9.42418i q^{50} +5.11225i q^{52} -2.19615i q^{53} -6.12989i q^{55} +(-0.536572 - 2.59077i) q^{56} -4.01461 q^{58} -1.46594 q^{59} +14.4152i q^{61} -7.98863 q^{62} -1.00000 q^{64} +19.4159i q^{65} -5.78094 q^{67} -2.33198 q^{68} +(-2.03786 - 9.83953i) q^{70} +15.6668i q^{71} +7.65133i q^{73} -5.74907i q^{74} -4.24621i q^{76} +(0.866035 + 4.18154i) q^{77} -1.64324 q^{79} -3.79792 q^{80} +8.51737i q^{82} +3.02145 q^{83} -8.85666 q^{85} +9.25725i q^{86} +1.61401 q^{88} +17.5236 q^{89} +(-2.74309 - 13.2447i) q^{91} -2.62863i q^{92} -9.69576i q^{94} -16.1268i q^{95} -10.3761i q^{97} +(2.78027 + 6.42418i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 8 q^{7} + 16 q^{16} + 16 q^{25} - 8 q^{28} + 16 q^{37} + 64 q^{43} - 32 q^{49} - 48 q^{58} - 16 q^{64} - 16 q^{67} - 24 q^{70} + 32 q^{79} - 48 q^{85} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −3.79792 −1.69848 −0.849240 0.528007i \(-0.822940\pi\)
−0.849240 + 0.528007i \(0.822940\pi\)
\(6\) 0 0
\(7\) 2.59077 0.536572i 0.979219 0.202805i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 3.79792i 1.20101i
\(11\) 1.61401i 0.486644i 0.969946 + 0.243322i \(0.0782371\pi\)
−0.969946 + 0.243322i \(0.921763\pi\)
\(12\) 0 0
\(13\) 5.11225i 1.41788i −0.705268 0.708941i \(-0.749173\pi\)
0.705268 0.708941i \(-0.250827\pi\)
\(14\) 0.536572 + 2.59077i 0.143405 + 0.692412i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.33198 0.565587 0.282794 0.959181i \(-0.408739\pi\)
0.282794 + 0.959181i \(0.408739\pi\)
\(18\) 0 0
\(19\) 4.24621i 0.974148i 0.873361 + 0.487074i \(0.161936\pi\)
−0.873361 + 0.487074i \(0.838064\pi\)
\(20\) 3.79792 0.849240
\(21\) 0 0
\(22\) −1.61401 −0.344109
\(23\) 2.62863i 0.548106i 0.961715 + 0.274053i \(0.0883645\pi\)
−0.961715 + 0.274053i \(0.911636\pi\)
\(24\) 0 0
\(25\) 9.42418 1.88484
\(26\) 5.11225 1.00259
\(27\) 0 0
\(28\) −2.59077 + 0.536572i −0.489610 + 0.101403i
\(29\) 4.01461i 0.745495i 0.927933 + 0.372747i \(0.121584\pi\)
−0.927933 + 0.372747i \(0.878416\pi\)
\(30\) 0 0
\(31\) 7.98863i 1.43480i 0.696661 + 0.717401i \(0.254668\pi\)
−0.696661 + 0.717401i \(0.745332\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.33198i 0.399931i
\(35\) −9.83953 + 2.03786i −1.66318 + 0.344461i
\(36\) 0 0
\(37\) −5.74907 −0.945141 −0.472570 0.881293i \(-0.656674\pi\)
−0.472570 + 0.881293i \(0.656674\pi\)
\(38\) −4.24621 −0.688826
\(39\) 0 0
\(40\) 3.79792i 0.600504i
\(41\) 8.51737 1.33019 0.665095 0.746759i \(-0.268391\pi\)
0.665095 + 0.746759i \(0.268391\pi\)
\(42\) 0 0
\(43\) 9.25725 1.41172 0.705859 0.708352i \(-0.250561\pi\)
0.705859 + 0.708352i \(0.250561\pi\)
\(44\) 1.61401i 0.243322i
\(45\) 0 0
\(46\) −2.62863 −0.387570
\(47\) −9.69576 −1.41427 −0.707136 0.707078i \(-0.750013\pi\)
−0.707136 + 0.707078i \(0.750013\pi\)
\(48\) 0 0
\(49\) 6.42418 2.78027i 0.917740 0.397181i
\(50\) 9.42418i 1.33278i
\(51\) 0 0
\(52\) 5.11225i 0.708941i
\(53\) 2.19615i 0.301665i −0.988559 0.150832i \(-0.951805\pi\)
0.988559 0.150832i \(-0.0481954\pi\)
\(54\) 0 0
\(55\) 6.12989i 0.826555i
\(56\) −0.536572 2.59077i −0.0717024 0.346206i
\(57\) 0 0
\(58\) −4.01461 −0.527144
\(59\) −1.46594 −0.190849 −0.0954247 0.995437i \(-0.530421\pi\)
−0.0954247 + 0.995437i \(0.530421\pi\)
\(60\) 0 0
\(61\) 14.4152i 1.84568i 0.385186 + 0.922839i \(0.374137\pi\)
−0.385186 + 0.922839i \(0.625863\pi\)
\(62\) −7.98863 −1.01456
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 19.4159i 2.40824i
\(66\) 0 0
\(67\) −5.78094 −0.706255 −0.353127 0.935575i \(-0.614882\pi\)
−0.353127 + 0.935575i \(0.614882\pi\)
\(68\) −2.33198 −0.282794
\(69\) 0 0
\(70\) −2.03786 9.83953i −0.243570 1.17605i
\(71\) 15.6668i 1.85931i 0.368432 + 0.929655i \(0.379895\pi\)
−0.368432 + 0.929655i \(0.620105\pi\)
\(72\) 0 0
\(73\) 7.65133i 0.895521i 0.894154 + 0.447760i \(0.147778\pi\)
−0.894154 + 0.447760i \(0.852222\pi\)
\(74\) 5.74907i 0.668315i
\(75\) 0 0
\(76\) 4.24621i 0.487074i
\(77\) 0.866035 + 4.18154i 0.0986938 + 0.476531i
\(78\) 0 0
\(79\) −1.64324 −0.184879 −0.0924394 0.995718i \(-0.529466\pi\)
−0.0924394 + 0.995718i \(0.529466\pi\)
\(80\) −3.79792 −0.424620
\(81\) 0 0
\(82\) 8.51737i 0.940586i
\(83\) 3.02145 0.331647 0.165824 0.986155i \(-0.446972\pi\)
0.165824 + 0.986155i \(0.446972\pi\)
\(84\) 0 0
\(85\) −8.85666 −0.960639
\(86\) 9.25725i 0.998235i
\(87\) 0 0
\(88\) 1.61401 0.172055
\(89\) 17.5236 1.85750 0.928752 0.370703i \(-0.120883\pi\)
0.928752 + 0.370703i \(0.120883\pi\)
\(90\) 0 0
\(91\) −2.74309 13.2447i −0.287554 1.38842i
\(92\) 2.62863i 0.274053i
\(93\) 0 0
\(94\) 9.69576i 1.00004i
\(95\) 16.1268i 1.65457i
\(96\) 0 0
\(97\) 10.3761i 1.05353i −0.850010 0.526767i \(-0.823404\pi\)
0.850010 0.526767i \(-0.176596\pi\)
\(98\) 2.78027 + 6.42418i 0.280850 + 0.648940i
\(99\) 0 0
\(100\) −9.42418 −0.942418
\(101\) 7.70683 0.766858 0.383429 0.923570i \(-0.374743\pi\)
0.383429 + 0.923570i \(0.374743\pi\)
\(102\) 0 0
\(103\) 2.93188i 0.288887i 0.989513 + 0.144444i \(0.0461392\pi\)
−0.989513 + 0.144444i \(0.953861\pi\)
\(104\) −5.11225 −0.501297
\(105\) 0 0
\(106\) 2.19615 0.213309
\(107\) 12.3166i 1.19069i 0.803470 + 0.595345i \(0.202985\pi\)
−0.803470 + 0.595345i \(0.797015\pi\)
\(108\) 0 0
\(109\) 1.47720 0.141490 0.0707452 0.997494i \(-0.477462\pi\)
0.0707452 + 0.997494i \(0.477462\pi\)
\(110\) 6.12989 0.584462
\(111\) 0 0
\(112\) 2.59077 0.536572i 0.244805 0.0507013i
\(113\) 6.93237i 0.652142i −0.945345 0.326071i \(-0.894275\pi\)
0.945345 0.326071i \(-0.105725\pi\)
\(114\) 0 0
\(115\) 9.98331i 0.930948i
\(116\) 4.01461i 0.372747i
\(117\) 0 0
\(118\) 1.46594i 0.134951i
\(119\) 6.04162 1.25127i 0.553834 0.114704i
\(120\) 0 0
\(121\) 8.39496 0.763178
\(122\) −14.4152 −1.30509
\(123\) 0 0
\(124\) 7.98863i 0.717401i
\(125\) −16.8027 −1.50288
\(126\) 0 0
\(127\) 14.7727 1.31086 0.655430 0.755256i \(-0.272487\pi\)
0.655430 + 0.755256i \(0.272487\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −19.4159 −1.70289
\(131\) −7.70683 −0.673348 −0.336674 0.941621i \(-0.609302\pi\)
−0.336674 + 0.941621i \(0.609302\pi\)
\(132\) 0 0
\(133\) 2.27840 + 11.0010i 0.197562 + 0.953904i
\(134\) 5.78094i 0.499397i
\(135\) 0 0
\(136\) 2.33198i 0.199965i
\(137\) 0.476309i 0.0406939i −0.999793 0.0203469i \(-0.993523\pi\)
0.999793 0.0203469i \(-0.00647708\pi\)
\(138\) 0 0
\(139\) 7.83702i 0.664727i 0.943151 + 0.332364i \(0.107846\pi\)
−0.943151 + 0.332364i \(0.892154\pi\)
\(140\) 9.83953 2.03786i 0.831592 0.172230i
\(141\) 0 0
\(142\) −15.6668 −1.31473
\(143\) 8.25124 0.690003
\(144\) 0 0
\(145\) 15.2472i 1.26621i
\(146\) −7.65133 −0.633229
\(147\) 0 0
\(148\) 5.74907 0.472570
\(149\) 19.7434i 1.61744i −0.588191 0.808722i \(-0.700160\pi\)
0.588191 0.808722i \(-0.299840\pi\)
\(150\) 0 0
\(151\) 12.1750 0.990788 0.495394 0.868668i \(-0.335024\pi\)
0.495394 + 0.868668i \(0.335024\pi\)
\(152\) 4.24621 0.344413
\(153\) 0 0
\(154\) −4.18154 + 0.866035i −0.336958 + 0.0697871i
\(155\) 30.3402i 2.43698i
\(156\) 0 0
\(157\) 10.0794i 0.804426i −0.915546 0.402213i \(-0.868241\pi\)
0.915546 0.402213i \(-0.131759\pi\)
\(158\) 1.64324i 0.130729i
\(159\) 0 0
\(160\) 3.79792i 0.300252i
\(161\) 1.41045 + 6.81017i 0.111159 + 0.536716i
\(162\) 0 0
\(163\) −5.05934 −0.396278 −0.198139 0.980174i \(-0.563490\pi\)
−0.198139 + 0.980174i \(0.563490\pi\)
\(164\) −8.51737 −0.665095
\(165\) 0 0
\(166\) 3.02145i 0.234510i
\(167\) −2.38747 −0.184748 −0.0923740 0.995724i \(-0.529446\pi\)
−0.0923740 + 0.995724i \(0.529446\pi\)
\(168\) 0 0
\(169\) −13.1351 −1.01039
\(170\) 8.85666i 0.679274i
\(171\) 0 0
\(172\) −9.25725 −0.705859
\(173\) −0.232053 −0.0176427 −0.00882134 0.999961i \(-0.502808\pi\)
−0.00882134 + 0.999961i \(0.502808\pi\)
\(174\) 0 0
\(175\) 24.4159 5.05675i 1.84567 0.382254i
\(176\) 1.61401i 0.121661i
\(177\) 0 0
\(178\) 17.5236i 1.31345i
\(179\) 24.4451i 1.82711i −0.406711 0.913557i \(-0.633325\pi\)
0.406711 0.913557i \(-0.366675\pi\)
\(180\) 0 0
\(181\) 10.5527i 0.784373i 0.919886 + 0.392187i \(0.128281\pi\)
−0.919886 + 0.392187i \(0.871719\pi\)
\(182\) 13.2447 2.74309i 0.981759 0.203331i
\(183\) 0 0
\(184\) 2.62863 0.193785
\(185\) 21.8345 1.60530
\(186\) 0 0
\(187\) 3.76384i 0.275239i
\(188\) 9.69576 0.707136
\(189\) 0 0
\(190\) 16.1268 1.16996
\(191\) 12.1732i 0.880825i 0.897795 + 0.440413i \(0.145168\pi\)
−0.897795 + 0.440413i \(0.854832\pi\)
\(192\) 0 0
\(193\) −24.3784 −1.75479 −0.877397 0.479765i \(-0.840722\pi\)
−0.877397 + 0.479765i \(0.840722\pi\)
\(194\) 10.3761 0.744961
\(195\) 0 0
\(196\) −6.42418 + 2.78027i −0.458870 + 0.198591i
\(197\) 2.10759i 0.150159i 0.997178 + 0.0750797i \(0.0239211\pi\)
−0.997178 + 0.0750797i \(0.976079\pi\)
\(198\) 0 0
\(199\) 19.1753i 1.35930i 0.733537 + 0.679650i \(0.237868\pi\)
−0.733537 + 0.679650i \(0.762132\pi\)
\(200\) 9.42418i 0.666390i
\(201\) 0 0
\(202\) 7.70683i 0.542250i
\(203\) 2.15413 + 10.4009i 0.151190 + 0.730003i
\(204\) 0 0
\(205\) −32.3483 −2.25930
\(206\) −2.93188 −0.204274
\(207\) 0 0
\(208\) 5.11225i 0.354470i
\(209\) −6.85345 −0.474063
\(210\) 0 0
\(211\) 8.67511 0.597220 0.298610 0.954375i \(-0.403477\pi\)
0.298610 + 0.954375i \(0.403477\pi\)
\(212\) 2.19615i 0.150832i
\(213\) 0 0
\(214\) −12.3166 −0.841945
\(215\) −35.1583 −2.39778
\(216\) 0 0
\(217\) 4.28648 + 20.6967i 0.290985 + 1.40498i
\(218\) 1.47720i 0.100049i
\(219\) 0 0
\(220\) 6.12989i 0.413277i
\(221\) 11.9216i 0.801936i
\(222\) 0 0
\(223\) 3.71748i 0.248941i 0.992223 + 0.124470i \(0.0397232\pi\)
−0.992223 + 0.124470i \(0.960277\pi\)
\(224\) 0.536572 + 2.59077i 0.0358512 + 0.173103i
\(225\) 0 0
\(226\) 6.93237 0.461134
\(227\) −3.21943 −0.213681 −0.106841 0.994276i \(-0.534073\pi\)
−0.106841 + 0.994276i \(0.534073\pi\)
\(228\) 0 0
\(229\) 8.99060i 0.594116i 0.954859 + 0.297058i \(0.0960054\pi\)
−0.954859 + 0.297058i \(0.903995\pi\)
\(230\) 9.98331 0.658280
\(231\) 0 0
\(232\) 4.01461 0.263572
\(233\) 10.1604i 0.665630i −0.942992 0.332815i \(-0.892002\pi\)
0.942992 0.332815i \(-0.107998\pi\)
\(234\) 0 0
\(235\) 36.8237 2.40211
\(236\) 1.46594 0.0954247
\(237\) 0 0
\(238\) 1.25127 + 6.04162i 0.0811080 + 0.391620i
\(239\) 3.59851i 0.232768i −0.993204 0.116384i \(-0.962870\pi\)
0.993204 0.116384i \(-0.0371304\pi\)
\(240\) 0 0
\(241\) 23.2144i 1.49537i −0.664054 0.747685i \(-0.731165\pi\)
0.664054 0.747685i \(-0.268835\pi\)
\(242\) 8.39496i 0.539648i
\(243\) 0 0
\(244\) 14.4152i 0.922839i
\(245\) −24.3985 + 10.5592i −1.55876 + 0.674605i
\(246\) 0 0
\(247\) 21.7077 1.38123
\(248\) 7.98863 0.507279
\(249\) 0 0
\(250\) 16.8027i 1.06269i
\(251\) 25.9199 1.63605 0.818026 0.575181i \(-0.195068\pi\)
0.818026 + 0.575181i \(0.195068\pi\)
\(252\) 0 0
\(253\) −4.24264 −0.266733
\(254\) 14.7727i 0.926919i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.29638 0.392757 0.196379 0.980528i \(-0.437082\pi\)
0.196379 + 0.980528i \(0.437082\pi\)
\(258\) 0 0
\(259\) −14.8945 + 3.08479i −0.925500 + 0.191679i
\(260\) 19.4159i 1.20412i
\(261\) 0 0
\(262\) 7.70683i 0.476129i
\(263\) 26.6413i 1.64277i −0.570374 0.821385i \(-0.693202\pi\)
0.570374 0.821385i \(-0.306798\pi\)
\(264\) 0 0
\(265\) 8.34081i 0.512372i
\(266\) −11.0010 + 2.27840i −0.674512 + 0.139697i
\(267\) 0 0
\(268\) 5.78094 0.353127
\(269\) −3.27493 −0.199676 −0.0998379 0.995004i \(-0.531832\pi\)
−0.0998379 + 0.995004i \(0.531832\pi\)
\(270\) 0 0
\(271\) 16.5466i 1.00514i 0.864538 + 0.502568i \(0.167611\pi\)
−0.864538 + 0.502568i \(0.832389\pi\)
\(272\) 2.33198 0.141397
\(273\) 0 0
\(274\) 0.476309 0.0287749
\(275\) 15.2108i 0.917244i
\(276\) 0 0
\(277\) −6.98363 −0.419606 −0.209803 0.977744i \(-0.567282\pi\)
−0.209803 + 0.977744i \(0.567282\pi\)
\(278\) −7.83702 −0.470033
\(279\) 0 0
\(280\) 2.03786 + 9.83953i 0.121785 + 0.588025i
\(281\) 2.15864i 0.128773i −0.997925 0.0643867i \(-0.979491\pi\)
0.997925 0.0643867i \(-0.0205091\pi\)
\(282\) 0 0
\(283\) 6.34613i 0.377238i 0.982050 + 0.188619i \(0.0604012\pi\)
−0.982050 + 0.188619i \(0.939599\pi\)
\(284\) 15.6668i 0.929655i
\(285\) 0 0
\(286\) 8.25124i 0.487906i
\(287\) 22.0665 4.57018i 1.30255 0.269769i
\(288\) 0 0
\(289\) −11.5619 −0.680111
\(290\) 15.2472 0.895344
\(291\) 0 0
\(292\) 7.65133i 0.447760i
\(293\) −27.7945 −1.62377 −0.811886 0.583816i \(-0.801559\pi\)
−0.811886 + 0.583816i \(0.801559\pi\)
\(294\) 0 0
\(295\) 5.56753 0.324154
\(296\) 5.74907i 0.334158i
\(297\) 0 0
\(298\) 19.7434 1.14371
\(299\) 13.4382 0.777150
\(300\) 0 0
\(301\) 23.9834 4.96718i 1.38238 0.286304i
\(302\) 12.1750i 0.700593i
\(303\) 0 0
\(304\) 4.24621i 0.243537i
\(305\) 54.7478i 3.13485i
\(306\) 0 0
\(307\) 7.30829i 0.417106i 0.978011 + 0.208553i \(0.0668754\pi\)
−0.978011 + 0.208553i \(0.933125\pi\)
\(308\) −0.866035 4.18154i −0.0493469 0.238265i
\(309\) 0 0
\(310\) 30.3402 1.72321
\(311\) −3.65543 −0.207281 −0.103640 0.994615i \(-0.533049\pi\)
−0.103640 + 0.994615i \(0.533049\pi\)
\(312\) 0 0
\(313\) 20.1737i 1.14029i −0.821545 0.570143i \(-0.806888\pi\)
0.821545 0.570143i \(-0.193112\pi\)
\(314\) 10.0794 0.568815
\(315\) 0 0
\(316\) 1.64324 0.0924394
\(317\) 12.3330i 0.692688i 0.938108 + 0.346344i \(0.112577\pi\)
−0.938108 + 0.346344i \(0.887423\pi\)
\(318\) 0 0
\(319\) −6.47964 −0.362790
\(320\) 3.79792 0.212310
\(321\) 0 0
\(322\) −6.81017 + 1.41045i −0.379516 + 0.0786011i
\(323\) 9.90206i 0.550966i
\(324\) 0 0
\(325\) 48.1787i 2.67247i
\(326\) 5.05934i 0.280211i
\(327\) 0 0
\(328\) 8.51737i 0.470293i
\(329\) −25.1195 + 5.20247i −1.38488 + 0.286822i
\(330\) 0 0
\(331\) −22.5152 −1.23755 −0.618773 0.785570i \(-0.712370\pi\)
−0.618773 + 0.785570i \(0.712370\pi\)
\(332\) −3.02145 −0.165824
\(333\) 0 0
\(334\) 2.38747i 0.130637i
\(335\) 21.9555 1.19956
\(336\) 0 0
\(337\) −9.78726 −0.533146 −0.266573 0.963815i \(-0.585891\pi\)
−0.266573 + 0.963815i \(0.585891\pi\)
\(338\) 13.1351i 0.714453i
\(339\) 0 0
\(340\) 8.85666 0.480320
\(341\) −12.8938 −0.698237
\(342\) 0 0
\(343\) 15.1518 10.6501i 0.818118 0.575050i
\(344\) 9.25725i 0.499118i
\(345\) 0 0
\(346\) 0.232053i 0.0124753i
\(347\) 10.8185i 0.580765i 0.956911 + 0.290383i \(0.0937826\pi\)
−0.956911 + 0.290383i \(0.906217\pi\)
\(348\) 0 0
\(349\) 9.87806i 0.528760i 0.964419 + 0.264380i \(0.0851674\pi\)
−0.964419 + 0.264380i \(0.914833\pi\)
\(350\) 5.05675 + 24.4159i 0.270295 + 1.30508i
\(351\) 0 0
\(352\) −1.61401 −0.0860273
\(353\) 10.4167 0.554427 0.277213 0.960808i \(-0.410589\pi\)
0.277213 + 0.960808i \(0.410589\pi\)
\(354\) 0 0
\(355\) 59.5013i 3.15800i
\(356\) −17.5236 −0.928752
\(357\) 0 0
\(358\) 24.4451 1.29196
\(359\) 9.97078i 0.526237i −0.964763 0.263119i \(-0.915249\pi\)
0.964763 0.263119i \(-0.0847511\pi\)
\(360\) 0 0
\(361\) 0.969696 0.0510366
\(362\) −10.5527 −0.554636
\(363\) 0 0
\(364\) 2.74309 + 13.2447i 0.143777 + 0.694208i
\(365\) 29.0591i 1.52102i
\(366\) 0 0
\(367\) 24.2534i 1.26602i 0.774144 + 0.633010i \(0.218181\pi\)
−0.774144 + 0.633010i \(0.781819\pi\)
\(368\) 2.62863i 0.137027i
\(369\) 0 0
\(370\) 21.8345i 1.13512i
\(371\) −1.17839 5.68973i −0.0611791 0.295396i
\(372\) 0 0
\(373\) −2.65153 −0.137291 −0.0686455 0.997641i \(-0.521868\pi\)
−0.0686455 + 0.997641i \(0.521868\pi\)
\(374\) −3.76384 −0.194624
\(375\) 0 0
\(376\) 9.69576i 0.500021i
\(377\) 20.5237 1.05702
\(378\) 0 0
\(379\) −19.4305 −0.998078 −0.499039 0.866580i \(-0.666314\pi\)
−0.499039 + 0.866580i \(0.666314\pi\)
\(380\) 16.1268i 0.827285i
\(381\) 0 0
\(382\) −12.1732 −0.622838
\(383\) 21.2320 1.08490 0.542452 0.840087i \(-0.317496\pi\)
0.542452 + 0.840087i \(0.317496\pi\)
\(384\) 0 0
\(385\) −3.28913 15.8811i −0.167630 0.809378i
\(386\) 24.3784i 1.24083i
\(387\) 0 0
\(388\) 10.3761i 0.526767i
\(389\) 4.47156i 0.226717i 0.993554 + 0.113359i \(0.0361609\pi\)
−0.993554 + 0.113359i \(0.963839\pi\)
\(390\) 0 0
\(391\) 6.12989i 0.310002i
\(392\) −2.78027 6.42418i −0.140425 0.324470i
\(393\) 0 0
\(394\) −2.10759 −0.106179
\(395\) 6.24088 0.314013
\(396\) 0 0
\(397\) 35.0315i 1.75818i −0.476656 0.879090i \(-0.658151\pi\)
0.476656 0.879090i \(-0.341849\pi\)
\(398\) −19.1753 −0.961170
\(399\) 0 0
\(400\) 9.42418 0.471209
\(401\) 21.3229i 1.06482i 0.846488 + 0.532408i \(0.178713\pi\)
−0.846488 + 0.532408i \(0.821287\pi\)
\(402\) 0 0
\(403\) 40.8399 2.03438
\(404\) −7.70683 −0.383429
\(405\) 0 0
\(406\) −10.4009 + 2.15413i −0.516190 + 0.106908i
\(407\) 9.27908i 0.459947i
\(408\) 0 0
\(409\) 24.0927i 1.19131i 0.803241 + 0.595654i \(0.203107\pi\)
−0.803241 + 0.595654i \(0.796893\pi\)
\(410\) 32.3483i 1.59757i
\(411\) 0 0
\(412\) 2.93188i 0.144444i
\(413\) −3.79792 + 0.786583i −0.186883 + 0.0387052i
\(414\) 0 0
\(415\) −11.4752 −0.563297
\(416\) 5.11225 0.250648
\(417\) 0 0
\(418\) 6.85345i 0.335213i
\(419\) 25.8304 1.26190 0.630948 0.775825i \(-0.282666\pi\)
0.630948 + 0.775825i \(0.282666\pi\)
\(420\) 0 0
\(421\) −17.0648 −0.831686 −0.415843 0.909436i \(-0.636513\pi\)
−0.415843 + 0.909436i \(0.636513\pi\)
\(422\) 8.67511i 0.422298i
\(423\) 0 0
\(424\) −2.19615 −0.106655
\(425\) 21.9770 1.06604
\(426\) 0 0
\(427\) 7.73479 + 37.3465i 0.374313 + 1.80732i
\(428\) 12.3166i 0.595345i
\(429\) 0 0
\(430\) 35.1583i 1.69548i
\(431\) 27.4224i 1.32089i 0.750874 + 0.660446i \(0.229633\pi\)
−0.750874 + 0.660446i \(0.770367\pi\)
\(432\) 0 0
\(433\) 21.1385i 1.01585i 0.861401 + 0.507925i \(0.169587\pi\)
−0.861401 + 0.507925i \(0.830413\pi\)
\(434\) −20.6967 + 4.28648i −0.993474 + 0.205757i
\(435\) 0 0
\(436\) −1.47720 −0.0707452
\(437\) −11.1617 −0.533937
\(438\) 0 0
\(439\) 25.5428i 1.21909i −0.792750 0.609546i \(-0.791352\pi\)
0.792750 0.609546i \(-0.208648\pi\)
\(440\) −6.12989 −0.292231
\(441\) 0 0
\(442\) 11.9216 0.567054
\(443\) 30.9184i 1.46898i −0.678620 0.734490i \(-0.737422\pi\)
0.678620 0.734490i \(-0.262578\pi\)
\(444\) 0 0
\(445\) −66.5534 −3.15493
\(446\) −3.71748 −0.176028
\(447\) 0 0
\(448\) −2.59077 + 0.536572i −0.122402 + 0.0253506i
\(449\) 1.16428i 0.0549456i −0.999623 0.0274728i \(-0.991254\pi\)
0.999623 0.0274728i \(-0.00874596\pi\)
\(450\) 0 0
\(451\) 13.7472i 0.647328i
\(452\) 6.93237i 0.326071i
\(453\) 0 0
\(454\) 3.21943i 0.151095i
\(455\) 10.4180 + 50.3021i 0.488404 + 2.35820i
\(456\) 0 0
\(457\) 7.21166 0.337347 0.168674 0.985672i \(-0.446052\pi\)
0.168674 + 0.985672i \(0.446052\pi\)
\(458\) −8.99060 −0.420103
\(459\) 0 0
\(460\) 9.98331i 0.465474i
\(461\) −28.6513 −1.33442 −0.667211 0.744868i \(-0.732512\pi\)
−0.667211 + 0.744868i \(0.732512\pi\)
\(462\) 0 0
\(463\) −28.3402 −1.31708 −0.658540 0.752546i \(-0.728826\pi\)
−0.658540 + 0.752546i \(0.728826\pi\)
\(464\) 4.01461i 0.186374i
\(465\) 0 0
\(466\) 10.1604 0.470671
\(467\) −10.2188 −0.472867 −0.236434 0.971648i \(-0.575979\pi\)
−0.236434 + 0.971648i \(0.575979\pi\)
\(468\) 0 0
\(469\) −14.9771 + 3.10189i −0.691578 + 0.143232i
\(470\) 36.8237i 1.69855i
\(471\) 0 0
\(472\) 1.46594i 0.0674754i
\(473\) 14.9413i 0.687003i
\(474\) 0 0
\(475\) 40.0171i 1.83611i
\(476\) −6.04162 + 1.25127i −0.276917 + 0.0573520i
\(477\) 0 0
\(478\) 3.59851 0.164592
\(479\) −12.6276 −0.576972 −0.288486 0.957484i \(-0.593152\pi\)
−0.288486 + 0.957484i \(0.593152\pi\)
\(480\) 0 0
\(481\) 29.3906i 1.34010i
\(482\) 23.2144 1.05739
\(483\) 0 0
\(484\) −8.39496 −0.381589
\(485\) 39.4076i 1.78941i
\(486\) 0 0
\(487\) 21.8841 0.991664 0.495832 0.868418i \(-0.334863\pi\)
0.495832 + 0.868418i \(0.334863\pi\)
\(488\) 14.4152 0.652546
\(489\) 0 0
\(490\) −10.5592 24.3985i −0.477018 1.10221i
\(491\) 7.60770i 0.343330i 0.985155 + 0.171665i \(0.0549147\pi\)
−0.985155 + 0.171665i \(0.945085\pi\)
\(492\) 0 0
\(493\) 9.36198i 0.421642i
\(494\) 21.7077i 0.976674i
\(495\) 0 0
\(496\) 7.98863i 0.358700i
\(497\) 8.40638 + 40.5891i 0.377078 + 1.82067i
\(498\) 0 0
\(499\) 3.85666 0.172648 0.0863238 0.996267i \(-0.472488\pi\)
0.0863238 + 0.996267i \(0.472488\pi\)
\(500\) 16.8027 0.751439
\(501\) 0 0
\(502\) 25.9199i 1.15686i
\(503\) −2.16549 −0.0965545 −0.0482773 0.998834i \(-0.515373\pi\)
−0.0482773 + 0.998834i \(0.515373\pi\)
\(504\) 0 0
\(505\) −29.2699 −1.30249
\(506\) 4.24264i 0.188608i
\(507\) 0 0
\(508\) −14.7727 −0.655430
\(509\) 25.8985 1.14793 0.573966 0.818879i \(-0.305404\pi\)
0.573966 + 0.818879i \(0.305404\pi\)
\(510\) 0 0
\(511\) 4.10549 + 19.8228i 0.181616 + 0.876911i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 6.29638i 0.277721i
\(515\) 11.1351i 0.490669i
\(516\) 0 0
\(517\) 15.6491i 0.688246i
\(518\) −3.08479 14.8945i −0.135538 0.654427i
\(519\) 0 0
\(520\) 19.4159 0.851443
\(521\) −29.9192 −1.31079 −0.655393 0.755288i \(-0.727497\pi\)
−0.655393 + 0.755288i \(0.727497\pi\)
\(522\) 0 0
\(523\) 13.7497i 0.601234i 0.953745 + 0.300617i \(0.0971925\pi\)
−0.953745 + 0.300617i \(0.902807\pi\)
\(524\) 7.70683 0.336674
\(525\) 0 0
\(526\) 26.6413 1.16161
\(527\) 18.6293i 0.811505i
\(528\) 0 0
\(529\) 16.0903 0.699579
\(530\) −8.34081 −0.362301
\(531\) 0 0
\(532\) −2.27840 11.0010i −0.0987810 0.476952i
\(533\) 43.5429i 1.88605i
\(534\) 0 0
\(535\) 46.7774i 2.02236i
\(536\) 5.78094i 0.249699i
\(537\) 0 0
\(538\) 3.27493i 0.141192i
\(539\) 4.48739 + 10.3687i 0.193286 + 0.446612i
\(540\) 0 0
\(541\) −37.3402 −1.60538 −0.802690 0.596397i \(-0.796599\pi\)
−0.802690 + 0.596397i \(0.796599\pi\)
\(542\) −16.5466 −0.710738
\(543\) 0 0
\(544\) 2.33198i 0.0999827i
\(545\) −5.61029 −0.240319
\(546\) 0 0
\(547\) −1.35411 −0.0578975 −0.0289488 0.999581i \(-0.509216\pi\)
−0.0289488 + 0.999581i \(0.509216\pi\)
\(548\) 0.476309i 0.0203469i
\(549\) 0 0
\(550\) −15.2108 −0.648589
\(551\) −17.0469 −0.726222
\(552\) 0 0
\(553\) −4.25725 + 0.881715i −0.181037 + 0.0374944i
\(554\) 6.98363i 0.296706i
\(555\) 0 0
\(556\) 7.83702i 0.332364i
\(557\) 21.1944i 0.898035i −0.893523 0.449018i \(-0.851774\pi\)
0.893523 0.449018i \(-0.148226\pi\)
\(558\) 0 0
\(559\) 47.3253i 2.00165i
\(560\) −9.83953 + 2.03786i −0.415796 + 0.0861151i
\(561\) 0 0
\(562\) 2.15864 0.0910566
\(563\) −32.1608 −1.35542 −0.677709 0.735330i \(-0.737027\pi\)
−0.677709 + 0.735330i \(0.737027\pi\)
\(564\) 0 0
\(565\) 26.3286i 1.10765i
\(566\) −6.34613 −0.266748
\(567\) 0 0
\(568\) 15.6668 0.657365
\(569\) 28.3101i 1.18682i 0.804901 + 0.593410i \(0.202219\pi\)
−0.804901 + 0.593410i \(0.797781\pi\)
\(570\) 0 0
\(571\) 13.6999 0.573324 0.286662 0.958032i \(-0.407454\pi\)
0.286662 + 0.958032i \(0.407454\pi\)
\(572\) −8.25124 −0.345002
\(573\) 0 0
\(574\) 4.57018 + 22.0665i 0.190756 + 0.921039i
\(575\) 24.7727i 1.03309i
\(576\) 0 0
\(577\) 18.1791i 0.756804i 0.925641 + 0.378402i \(0.123526\pi\)
−0.925641 + 0.378402i \(0.876474\pi\)
\(578\) 11.5619i 0.480911i
\(579\) 0 0
\(580\) 15.2472i 0.633104i
\(581\) 7.82789 1.62123i 0.324756 0.0672598i
\(582\) 0 0
\(583\) 3.54462 0.146803
\(584\) 7.65133 0.316614
\(585\) 0 0
\(586\) 27.7945i 1.14818i
\(587\) 11.0814 0.457378 0.228689 0.973500i \(-0.426556\pi\)
0.228689 + 0.973500i \(0.426556\pi\)
\(588\) 0 0
\(589\) −33.9214 −1.39771
\(590\) 5.56753i 0.229211i
\(591\) 0 0
\(592\) −5.74907 −0.236285
\(593\) 32.4304 1.33175 0.665877 0.746061i \(-0.268057\pi\)
0.665877 + 0.746061i \(0.268057\pi\)
\(594\) 0 0
\(595\) −22.9456 + 4.75223i −0.940676 + 0.194823i
\(596\) 19.7434i 0.808722i
\(597\) 0 0
\(598\) 13.4382i 0.549528i
\(599\) 12.7938i 0.522740i −0.965239 0.261370i \(-0.915826\pi\)
0.965239 0.261370i \(-0.0841743\pi\)
\(600\) 0 0
\(601\) 31.2244i 1.27367i −0.771000 0.636836i \(-0.780243\pi\)
0.771000 0.636836i \(-0.219757\pi\)
\(602\) 4.96718 + 23.9834i 0.202447 + 0.977491i
\(603\) 0 0
\(604\) −12.1750 −0.495394
\(605\) −31.8834 −1.29624
\(606\) 0 0
\(607\) 24.1607i 0.980653i 0.871539 + 0.490327i \(0.163123\pi\)
−0.871539 + 0.490327i \(0.836877\pi\)
\(608\) −4.24621 −0.172207
\(609\) 0 0
\(610\) 54.7478 2.21667
\(611\) 49.5671i 2.00527i
\(612\) 0 0
\(613\) −4.99979 −0.201939 −0.100970 0.994889i \(-0.532195\pi\)
−0.100970 + 0.994889i \(0.532195\pi\)
\(614\) −7.30829 −0.294939
\(615\) 0 0
\(616\) 4.18154 0.866035i 0.168479 0.0348935i
\(617\) 18.0949i 0.728473i 0.931307 + 0.364236i \(0.118670\pi\)
−0.931307 + 0.364236i \(0.881330\pi\)
\(618\) 0 0
\(619\) 8.97221i 0.360624i −0.983610 0.180312i \(-0.942289\pi\)
0.983610 0.180312i \(-0.0577107\pi\)
\(620\) 30.3402i 1.21849i
\(621\) 0 0
\(622\) 3.65543i 0.146570i
\(623\) 45.3997 9.40270i 1.81890 0.376711i
\(624\) 0 0
\(625\) 16.6943 0.667771
\(626\) 20.1737 0.806304
\(627\) 0 0
\(628\) 10.0794i 0.402213i
\(629\) −13.4067 −0.534560
\(630\) 0 0
\(631\) −35.6267 −1.41827 −0.709137 0.705070i \(-0.750915\pi\)
−0.709137 + 0.705070i \(0.750915\pi\)
\(632\) 1.64324i 0.0653645i
\(633\) 0 0
\(634\) −12.3330 −0.489805
\(635\) −56.1053 −2.22647
\(636\) 0 0
\(637\) −14.2134 32.8420i −0.563156 1.30125i
\(638\) 6.47964i 0.256531i
\(639\) 0 0
\(640\) 3.79792i 0.150126i
\(641\) 16.7338i 0.660944i 0.943816 + 0.330472i \(0.107208\pi\)
−0.943816 + 0.330472i \(0.892792\pi\)
\(642\) 0 0
\(643\) 17.4707i 0.688978i −0.938790 0.344489i \(-0.888052\pi\)
0.938790 0.344489i \(-0.111948\pi\)
\(644\) −1.41045 6.81017i −0.0555794 0.268358i
\(645\) 0 0
\(646\) −9.90206 −0.389591
\(647\) −21.0554 −0.827775 −0.413887 0.910328i \(-0.635829\pi\)
−0.413887 + 0.910328i \(0.635829\pi\)
\(648\) 0 0
\(649\) 2.36605i 0.0928756i
\(650\) 48.1787 1.88973
\(651\) 0 0
\(652\) 5.05934 0.198139
\(653\) 1.19637i 0.0468174i 0.999726 + 0.0234087i \(0.00745190\pi\)
−0.999726 + 0.0234087i \(0.992548\pi\)
\(654\) 0 0
\(655\) 29.2699 1.14367
\(656\) 8.51737 0.332547
\(657\) 0 0
\(658\) −5.20247 25.1195i −0.202813 0.979259i
\(659\) 32.1986i 1.25428i 0.778907 + 0.627140i \(0.215774\pi\)
−0.778907 + 0.627140i \(0.784226\pi\)
\(660\) 0 0
\(661\) 23.2109i 0.902798i −0.892322 0.451399i \(-0.850925\pi\)
0.892322 0.451399i \(-0.149075\pi\)
\(662\) 22.5152i 0.875077i
\(663\) 0 0
\(664\) 3.02145i 0.117255i
\(665\) −8.65317 41.7807i −0.335555 1.62019i
\(666\) 0 0
\(667\) −10.5529 −0.408610
\(668\) 2.38747 0.0923740
\(669\) 0 0
\(670\) 21.9555i 0.848217i
\(671\) −23.2664 −0.898188
\(672\) 0 0
\(673\) 24.0153 0.925721 0.462861 0.886431i \(-0.346823\pi\)
0.462861 + 0.886431i \(0.346823\pi\)
\(674\) 9.78726i 0.376991i
\(675\) 0 0
\(676\) 13.1351 0.505194
\(677\) −3.45488 −0.132782 −0.0663908 0.997794i \(-0.521148\pi\)
−0.0663908 + 0.997794i \(0.521148\pi\)
\(678\) 0 0
\(679\) −5.56753 26.8821i −0.213662 1.03164i
\(680\) 8.85666i 0.339637i
\(681\) 0 0
\(682\) 12.8938i 0.493728i
\(683\) 20.4853i 0.783848i −0.919998 0.391924i \(-0.871810\pi\)
0.919998 0.391924i \(-0.128190\pi\)
\(684\) 0 0
\(685\) 1.80898i 0.0691177i
\(686\) 10.6501 + 15.1518i 0.406622 + 0.578497i
\(687\) 0 0
\(688\) 9.25725 0.352929
\(689\) −11.2273 −0.427725
\(690\) 0 0
\(691\) 40.4968i 1.54057i −0.637698 0.770286i \(-0.720113\pi\)
0.637698 0.770286i \(-0.279887\pi\)
\(692\) 0.232053 0.00882134
\(693\) 0 0
\(694\) −10.8185 −0.410663
\(695\) 29.7644i 1.12903i
\(696\) 0 0
\(697\) 19.8623 0.752338
\(698\) −9.87806 −0.373890
\(699\) 0 0
\(700\) −24.4159 + 5.05675i −0.922834 + 0.191127i
\(701\) 17.2261i 0.650619i 0.945608 + 0.325310i \(0.105469\pi\)
−0.945608 + 0.325310i \(0.894531\pi\)
\(702\) 0 0
\(703\) 24.4117i 0.920707i
\(704\) 1.61401i 0.0608305i
\(705\) 0 0
\(706\) 10.4167i 0.392039i
\(707\) 19.9666 4.13527i 0.750922 0.155523i
\(708\) 0 0
\(709\) 27.2472 1.02329 0.511645 0.859197i \(-0.329036\pi\)
0.511645 + 0.859197i \(0.329036\pi\)
\(710\) 59.5013 2.23304
\(711\) 0 0
\(712\) 17.5236i 0.656726i
\(713\) −20.9991 −0.786424
\(714\) 0 0
\(715\) −31.3375 −1.17196
\(716\) 24.4451i 0.913557i
\(717\) 0 0
\(718\) 9.97078 0.372106
\(719\) 33.6709 1.25571 0.627857 0.778329i \(-0.283932\pi\)
0.627857 + 0.778329i \(0.283932\pi\)
\(720\) 0 0
\(721\) 1.57317 + 7.59584i 0.0585878 + 0.282884i
\(722\) 0.969696i 0.0360883i
\(723\) 0 0
\(724\) 10.5527i 0.392187i
\(725\) 37.8344i 1.40514i
\(726\) 0 0
\(727\) 44.4862i 1.64990i −0.565205 0.824951i \(-0.691203\pi\)
0.565205 0.824951i \(-0.308797\pi\)
\(728\) −13.2447 + 2.74309i −0.490880 + 0.101666i
\(729\) 0 0
\(730\) 29.0591 1.07553
\(731\) 21.5877 0.798450
\(732\) 0 0
\(733\) 15.7418i 0.581438i 0.956809 + 0.290719i \(0.0938944\pi\)
−0.956809 + 0.290719i \(0.906106\pi\)
\(734\) −24.2534 −0.895211
\(735\) 0 0
\(736\) −2.62863 −0.0968925
\(737\) 9.33053i 0.343694i
\(738\) 0 0
\(739\) 2.27818 0.0838043 0.0419022 0.999122i \(-0.486658\pi\)
0.0419022 + 0.999122i \(0.486658\pi\)
\(740\) −21.8345 −0.802652
\(741\) 0 0
\(742\) 5.68973 1.17839i 0.208876 0.0432602i
\(743\) 34.8955i 1.28019i −0.768295 0.640096i \(-0.778894\pi\)
0.768295 0.640096i \(-0.221106\pi\)
\(744\) 0 0
\(745\) 74.9839i 2.74720i
\(746\) 2.65153i 0.0970794i
\(747\) 0 0
\(748\) 3.76384i 0.137620i
\(749\) 6.60874 + 31.9095i 0.241478 + 1.16595i
\(750\) 0 0
\(751\) 12.2409 0.446676 0.223338 0.974741i \(-0.428305\pi\)
0.223338 + 0.974741i \(0.428305\pi\)
\(752\) −9.69576 −0.353568
\(753\) 0 0
\(754\) 20.5237i 0.747428i
\(755\) −46.2397 −1.68283
\(756\) 0 0
\(757\) −15.4361 −0.561036 −0.280518 0.959849i \(-0.590506\pi\)
−0.280518 + 0.959849i \(0.590506\pi\)
\(758\) 19.4305i 0.705748i
\(759\) 0 0
\(760\) −16.1268 −0.584979
\(761\) 10.4962 0.380488 0.190244 0.981737i \(-0.439072\pi\)
0.190244 + 0.981737i \(0.439072\pi\)
\(762\) 0 0
\(763\) 3.82709 0.792625i 0.138550 0.0286950i
\(764\) 12.1732i 0.440413i
\(765\) 0 0
\(766\) 21.2320i 0.767143i
\(767\) 7.49425i 0.270602i
\(768\) 0 0
\(769\) 42.4382i 1.53036i −0.643815 0.765181i \(-0.722649\pi\)
0.643815 0.765181i \(-0.277351\pi\)
\(770\) 15.8811 3.28913i 0.572317 0.118532i
\(771\) 0 0
\(772\) 24.3784 0.877397
\(773\) 27.7852 0.999366 0.499683 0.866208i \(-0.333450\pi\)
0.499683 + 0.866208i \(0.333450\pi\)
\(774\) 0 0
\(775\) 75.2863i 2.70436i
\(776\) −10.3761 −0.372480
\(777\) 0 0
\(778\) −4.47156 −0.160313
\(779\) 36.1665i 1.29580i
\(780\) 0 0
\(781\) −25.2865 −0.904821
\(782\) −6.12989 −0.219205
\(783\) 0 0
\(784\) 6.42418 2.78027i 0.229435 0.0992953i
\(785\) 38.2808i 1.36630i
\(786\) 0 0
\(787\) 33.1387i 1.18127i 0.806940 + 0.590633i \(0.201122\pi\)
−0.806940 + 0.590633i \(0.798878\pi\)
\(788\) 2.10759i 0.0750797i
\(789\) 0 0
\(790\) 6.24088i 0.222041i
\(791\) −3.71971 17.9602i −0.132258 0.638590i
\(792\) 0 0
\(793\) 73.6941 2.61695
\(794\) 35.0315 1.24322
\(795\) 0 0
\(796\) 19.1753i 0.679650i
\(797\) 0.232053 0.00821975 0.00410987 0.999992i \(-0.498692\pi\)
0.00410987 + 0.999992i \(0.498692\pi\)
\(798\) 0 0
\(799\) −22.6103 −0.799894
\(800\) 9.42418i 0.333195i
\(801\) 0 0
\(802\) −21.3229 −0.752938
\(803\) −12.3494 −0.435799
\(804\) 0 0
\(805\) −5.35676 25.8645i −0.188801 0.911602i
\(806\) 40.8399i 1.43852i
\(807\) 0 0
\(808\) 7.70683i 0.271125i
\(809\) 31.8244i 1.11889i −0.828869 0.559443i \(-0.811015\pi\)
0.828869 0.559443i \(-0.188985\pi\)
\(810\) 0 0
\(811\) 28.9878i 1.01790i −0.860797 0.508949i \(-0.830034\pi\)
0.860797 0.508949i \(-0.169966\pi\)
\(812\) −2.15413 10.4009i −0.0755951 0.365001i
\(813\) 0 0
\(814\) 9.27908 0.325231
\(815\) 19.2150 0.673071
\(816\) 0 0
\(817\) 39.3082i 1.37522i
\(818\) −24.0927 −0.842382
\(819\) 0 0
\(820\) 32.3483 1.12965
\(821\) 11.9562i 0.417273i −0.977993 0.208637i \(-0.933097\pi\)
0.977993 0.208637i \(-0.0669026\pi\)
\(822\) 0 0
\(823\) −15.7729 −0.549807 −0.274904 0.961472i \(-0.588646\pi\)
−0.274904 + 0.961472i \(0.588646\pi\)
\(824\) 2.93188 0.102137
\(825\) 0 0
\(826\) −0.786583 3.79792i −0.0273687 0.132146i
\(827\) 13.4682i 0.468336i 0.972196 + 0.234168i \(0.0752366\pi\)
−0.972196 + 0.234168i \(0.924763\pi\)
\(828\) 0 0
\(829\) 14.7775i 0.513242i 0.966512 + 0.256621i \(0.0826092\pi\)
−0.966512 + 0.256621i \(0.917391\pi\)
\(830\) 11.4752i 0.398311i
\(831\) 0 0
\(832\) 5.11225i 0.177235i
\(833\) 14.9810 6.48352i 0.519062 0.224641i
\(834\) 0 0
\(835\) 9.06742 0.313791
\(836\) 6.85345 0.237031
\(837\) 0 0
\(838\) 25.8304i 0.892296i
\(839\) −41.3777 −1.42852 −0.714259 0.699881i \(-0.753236\pi\)
−0.714259 + 0.699881i \(0.753236\pi\)
\(840\) 0 0
\(841\) 12.8829 0.444238
\(842\) 17.0648i 0.588091i
\(843\) 0 0
\(844\) −8.67511 −0.298610
\(845\) 49.8859 1.71613
\(846\) 0 0
\(847\) 21.7494 4.50450i 0.747318 0.154776i
\(848\) 2.19615i 0.0754162i
\(849\) 0 0
\(850\) 21.9770i 0.753804i
\(851\) 15.1121i 0.518038i
\(852\) 0 0
\(853\) 9.08578i 0.311091i −0.987829 0.155546i \(-0.950286\pi\)
0.987829 0.155546i \(-0.0497135\pi\)
\(854\) −37.3465 + 7.73479i −1.27797 + 0.264679i
\(855\) 0 0
\(856\) 12.3166 0.420972
\(857\) 0.0314947 0.00107584 0.000537919 1.00000i \(-0.499829\pi\)
0.000537919 1.00000i \(0.499829\pi\)
\(858\) 0 0
\(859\) 24.8682i 0.848493i 0.905547 + 0.424246i \(0.139461\pi\)
−0.905547 + 0.424246i \(0.860539\pi\)
\(860\) 35.1583 1.19889
\(861\) 0 0
\(862\) −27.4224 −0.934011
\(863\) 13.7637i 0.468521i −0.972174 0.234261i \(-0.924733\pi\)
0.972174 0.234261i \(-0.0752669\pi\)
\(864\) 0 0
\(865\) 0.881319 0.0299658
\(866\) −21.1385 −0.718314
\(867\) 0 0
\(868\) −4.28648 20.6967i −0.145492 0.702492i
\(869\) 2.65221i 0.0899701i
\(870\) 0 0
\(871\) 29.5536i 1.00139i
\(872\) 1.47720i 0.0500244i
\(873\) 0 0
\(874\) 11.1617i 0.377550i
\(875\) −43.5319 + 9.01584i −1.47165 + 0.304791i
\(876\) 0 0
\(877\) 14.5812 0.492374 0.246187 0.969222i \(-0.420822\pi\)
0.246187 + 0.969222i \(0.420822\pi\)
\(878\) 25.5428 0.862029
\(879\) 0 0
\(880\) 6.12989i 0.206639i
\(881\) −24.9089 −0.839201 −0.419600 0.907709i \(-0.637830\pi\)
−0.419600 + 0.907709i \(0.637830\pi\)
\(882\) 0 0
\(883\) 53.0681 1.78588 0.892942 0.450172i \(-0.148637\pi\)
0.892942 + 0.450172i \(0.148637\pi\)
\(884\) 11.9216i 0.400968i
\(885\) 0 0
\(886\) 30.9184 1.03873
\(887\) −39.1668 −1.31509 −0.657547 0.753414i \(-0.728406\pi\)
−0.657547 + 0.753414i \(0.728406\pi\)
\(888\) 0 0
\(889\) 38.2725 7.92659i 1.28362 0.265849i
\(890\) 66.5534i 2.23087i
\(891\) 0 0
\(892\) 3.71748i 0.124470i
\(893\) 41.1702i 1.37771i
\(894\) 0 0
\(895\) 92.8405i 3.10332i
\(896\) −0.536572 2.59077i −0.0179256 0.0865516i
\(897\) 0 0
\(898\) 1.16428 0.0388524
\(899\) −32.0713 −1.06964
\(900\) 0 0
\(901\) 5.12138i 0.170618i
\(902\) −13.7472 −0.457730
\(903\) 0 0
\(904\) −6.93237 −0.230567
\(905\) 40.0782i 1.33224i
\(906\) 0 0
\(907\) 27.8605 0.925094 0.462547 0.886595i \(-0.346936\pi\)
0.462547 + 0.886595i \(0.346936\pi\)
\(908\) 3.21943 0.106841
\(909\) 0 0
\(910\) −50.3021 + 10.4180i −1.66750 + 0.345354i
\(911\) 37.5327i 1.24351i −0.783211 0.621756i \(-0.786419\pi\)
0.783211 0.621756i \(-0.213581\pi\)
\(912\) 0 0
\(913\) 4.87667i 0.161394i
\(914\) 7.21166i 0.238540i
\(915\) 0 0
\(916\) 8.99060i 0.297058i
\(917\) −19.9666 + 4.13527i −0.659356 + 0.136559i
\(918\) 0 0
\(919\) −14.4325 −0.476083 −0.238042 0.971255i \(-0.576505\pi\)
−0.238042 + 0.971255i \(0.576505\pi\)
\(920\) −9.98331 −0.329140
\(921\) 0 0
\(922\) 28.6513i 0.943579i
\(923\) 80.0926 2.63628
\(924\) 0 0
\(925\) −54.1802 −1.78144
\(926\) 28.3402i 0.931316i
\(927\) 0 0
\(928\) −4.01461 −0.131786
\(929\) 6.00625 0.197059 0.0985293 0.995134i \(-0.468586\pi\)
0.0985293 + 0.995134i \(0.468586\pi\)
\(930\) 0 0
\(931\) 11.8056 + 27.2784i 0.386913 + 0.894014i
\(932\) 10.1604i 0.332815i
\(933\) 0 0
\(934\) 10.2188i 0.334368i
\(935\) 14.2948i 0.467489i
\(936\) 0 0
\(937\) 7.11686i 0.232498i 0.993220 + 0.116249i \(0.0370870\pi\)
−0.993220 + 0.116249i \(0.962913\pi\)
\(938\) −3.10189 14.9771i −0.101280 0.489020i
\(939\) 0 0
\(940\) −36.8237 −1.20106
\(941\) 1.74214 0.0567923 0.0283961 0.999597i \(-0.490960\pi\)
0.0283961 + 0.999597i \(0.490960\pi\)
\(942\) 0 0
\(943\) 22.3890i 0.729085i
\(944\) −1.46594 −0.0477123
\(945\) 0 0
\(946\) −14.9413 −0.485785
\(947\) 40.4694i 1.31508i −0.753421 0.657539i \(-0.771597\pi\)
0.753421 0.657539i \(-0.228403\pi\)
\(948\) 0 0
\(949\) 39.1155 1.26974
\(950\) −40.0171 −1.29832
\(951\) 0 0
\(952\) −1.25127 6.04162i −0.0405540 0.195810i
\(953\) 14.0692i 0.455745i −0.973691 0.227873i \(-0.926823\pi\)
0.973691 0.227873i \(-0.0731770\pi\)
\(954\) 0 0
\(955\) 46.2330i 1.49606i
\(956\) 3.59851i 0.116384i
\(957\) 0 0
\(958\) 12.6276i 0.407981i
\(959\) −0.255574 1.23401i −0.00825292 0.0398482i
\(960\) 0 0
\(961\) −32.8183 −1.05865
\(962\) −29.3906 −0.947592
\(963\) 0 0
\(964\) 23.2144i 0.747685i
\(965\) 92.5871 2.98048
\(966\) 0 0
\(967\) 32.6269 1.04921 0.524605 0.851346i \(-0.324213\pi\)
0.524605 + 0.851346i \(0.324213\pi\)
\(968\) 8.39496i 0.269824i
\(969\) 0 0
\(970\) −39.4076 −1.26530
\(971\) 35.0687 1.12541 0.562704 0.826658i \(-0.309761\pi\)
0.562704 + 0.826658i \(0.309761\pi\)
\(972\) 0 0
\(973\) 4.20512 + 20.3039i 0.134810 + 0.650914i
\(974\) 21.8841i 0.701212i
\(975\) 0 0
\(976\) 14.4152i 0.461420i
\(977\) 26.7096i 0.854515i 0.904130 + 0.427258i \(0.140520\pi\)
−0.904130 + 0.427258i \(0.859480\pi\)
\(978\) 0 0
\(979\) 28.2834i 0.903942i
\(980\) 24.3985 10.5592i 0.779382 0.337302i
\(981\) 0 0
\(982\) −7.60770 −0.242771
\(983\) 17.9470 0.572420 0.286210 0.958167i \(-0.407604\pi\)
0.286210 + 0.958167i \(0.407604\pi\)
\(984\) 0 0
\(985\) 8.00445i 0.255043i
\(986\) −9.36198 −0.298146
\(987\) 0 0
\(988\) −21.7077 −0.690613
\(989\) 24.3339i 0.773772i
\(990\) 0 0
\(991\) 25.9725 0.825043 0.412521 0.910948i \(-0.364648\pi\)
0.412521 + 0.910948i \(0.364648\pi\)
\(992\) −7.98863 −0.253639
\(993\) 0 0
\(994\) −40.5891 + 8.40638i −1.28741 + 0.266634i
\(995\) 72.8261i 2.30874i
\(996\) 0 0
\(997\) 22.7106i 0.719252i −0.933097 0.359626i \(-0.882904\pi\)
0.933097 0.359626i \(-0.117096\pi\)
\(998\) 3.85666i 0.122080i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1134.2.d.b.1133.9 yes 16
3.2 odd 2 inner 1134.2.d.b.1133.8 yes 16
7.6 odd 2 inner 1134.2.d.b.1133.16 yes 16
9.2 odd 6 1134.2.m.i.377.1 16
9.4 even 3 1134.2.m.i.755.4 16
9.5 odd 6 1134.2.m.h.755.5 16
9.7 even 3 1134.2.m.h.377.8 16
21.20 even 2 inner 1134.2.d.b.1133.1 16
63.13 odd 6 1134.2.m.i.755.1 16
63.20 even 6 1134.2.m.i.377.4 16
63.34 odd 6 1134.2.m.h.377.5 16
63.41 even 6 1134.2.m.h.755.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.d.b.1133.1 16 21.20 even 2 inner
1134.2.d.b.1133.8 yes 16 3.2 odd 2 inner
1134.2.d.b.1133.9 yes 16 1.1 even 1 trivial
1134.2.d.b.1133.16 yes 16 7.6 odd 2 inner
1134.2.m.h.377.5 16 63.34 odd 6
1134.2.m.h.377.8 16 9.7 even 3
1134.2.m.h.755.5 16 9.5 odd 6
1134.2.m.h.755.8 16 63.41 even 6
1134.2.m.i.377.1 16 9.2 odd 6
1134.2.m.i.377.4 16 63.20 even 6
1134.2.m.i.755.1 16 63.13 odd 6
1134.2.m.i.755.4 16 9.4 even 3