Properties

Label 1512.2.j.d.323.4
Level $1512$
Weight $2$
Character 1512.323
Analytic conductor $12.073$
Analytic rank $0$
Dimension $48$
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] = [1512,2,Mod(323,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("1512.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.j (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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.4
Character \(\chi\) \(=\) 1512.323
Dual form 1512.2.j.d.323.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.37397 + 0.334962i) q^{2} +(1.77560 - 0.920458i) q^{4} +4.18263 q^{5} +1.00000i q^{7} +(-2.13131 + 1.85944i) q^{8} +O(q^{10})\) \(q+(-1.37397 + 0.334962i) q^{2} +(1.77560 - 0.920458i) q^{4} +4.18263 q^{5} +1.00000i q^{7} +(-2.13131 + 1.85944i) q^{8} +(-5.74682 + 1.40102i) q^{10} -1.86730i q^{11} +3.07604i q^{13} +(-0.334962 - 1.37397i) q^{14} +(2.30551 - 3.26873i) q^{16} +0.504885i q^{17} +3.05596 q^{19} +(7.42668 - 3.84993i) q^{20} +(0.625474 + 2.56561i) q^{22} -5.97229 q^{23} +12.4944 q^{25} +(-1.03036 - 4.22639i) q^{26} +(0.920458 + 1.77560i) q^{28} +3.52884 q^{29} +10.8980i q^{31} +(-2.07281 + 5.26341i) q^{32} +(-0.169117 - 0.693698i) q^{34} +4.18263i q^{35} +11.3660i q^{37} +(-4.19880 + 1.02363i) q^{38} +(-8.91447 + 7.77736i) q^{40} -7.26960i q^{41} +9.02120 q^{43} +(-1.71877 - 3.31557i) q^{44} +(8.20576 - 2.00049i) q^{46} +0.327367 q^{47} -1.00000 q^{49} +(-17.1669 + 4.18514i) q^{50} +(2.83136 + 5.46181i) q^{52} +8.32940 q^{53} -7.81021i q^{55} +(-1.85944 - 2.13131i) q^{56} +(-4.84853 + 1.18203i) q^{58} -5.98008i q^{59} -13.2783i q^{61} +(-3.65043 - 14.9736i) q^{62} +(1.08494 - 7.92609i) q^{64} +12.8659i q^{65} -9.21499 q^{67} +(0.464725 + 0.896474i) q^{68} +(-1.40102 - 5.74682i) q^{70} -9.02460 q^{71} +0.416491 q^{73} +(-3.80718 - 15.6166i) q^{74} +(5.42616 - 2.81288i) q^{76} +1.86730 q^{77} -6.00459i q^{79} +(9.64311 - 13.6719i) q^{80} +(2.43504 + 9.98822i) q^{82} +2.62368i q^{83} +2.11175i q^{85} +(-12.3949 + 3.02176i) q^{86} +(3.47213 + 3.97978i) q^{88} +5.69486i q^{89} -3.07604 q^{91} +(-10.6044 + 5.49724i) q^{92} +(-0.449793 + 0.109655i) q^{94} +12.7819 q^{95} +6.21289 q^{97} +(1.37397 - 0.334962i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{4} - 12 q^{10} - 12 q^{16} + 16 q^{19} - 24 q^{22} + 48 q^{25} + 8 q^{28} + 12 q^{34} + 32 q^{40} + 64 q^{43} + 60 q^{46} - 48 q^{49} + 16 q^{52} + 36 q^{58} - 4 q^{64} + 32 q^{67} + 12 q^{70} - 32 q^{76} + 36 q^{82} + 24 q^{88} - 60 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37397 + 0.334962i −0.971545 + 0.236854i
\(3\) 0 0
\(4\) 1.77560 0.920458i 0.887800 0.460229i
\(5\) 4.18263 1.87053 0.935264 0.353951i \(-0.115162\pi\)
0.935264 + 0.353951i \(0.115162\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) −2.13131 + 1.85944i −0.753531 + 0.657412i
\(9\) 0 0
\(10\) −5.74682 + 1.40102i −1.81730 + 0.443042i
\(11\) 1.86730i 0.563011i −0.959560 0.281506i \(-0.909166\pi\)
0.959560 0.281506i \(-0.0908338\pi\)
\(12\) 0 0
\(13\) 3.07604i 0.853139i 0.904455 + 0.426569i \(0.140278\pi\)
−0.904455 + 0.426569i \(0.859722\pi\)
\(14\) −0.334962 1.37397i −0.0895224 0.367210i
\(15\) 0 0
\(16\) 2.30551 3.26873i 0.576379 0.817183i
\(17\) 0.504885i 0.122453i 0.998124 + 0.0612263i \(0.0195011\pi\)
−0.998124 + 0.0612263i \(0.980499\pi\)
\(18\) 0 0
\(19\) 3.05596 0.701085 0.350543 0.936547i \(-0.385997\pi\)
0.350543 + 0.936547i \(0.385997\pi\)
\(20\) 7.42668 3.84993i 1.66066 0.860871i
\(21\) 0 0
\(22\) 0.625474 + 2.56561i 0.133352 + 0.546991i
\(23\) −5.97229 −1.24531 −0.622654 0.782497i \(-0.713946\pi\)
−0.622654 + 0.782497i \(0.713946\pi\)
\(24\) 0 0
\(25\) 12.4944 2.49887
\(26\) −1.03036 4.22639i −0.202069 0.828863i
\(27\) 0 0
\(28\) 0.920458 + 1.77560i 0.173950 + 0.335557i
\(29\) 3.52884 0.655289 0.327644 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(30\) 0 0
\(31\) 10.8980i 1.95734i 0.205435 + 0.978671i \(0.434139\pi\)
−0.205435 + 0.978671i \(0.565861\pi\)
\(32\) −2.07281 + 5.26341i −0.366425 + 0.930448i
\(33\) 0 0
\(34\) −0.169117 0.693698i −0.0290034 0.118968i
\(35\) 4.18263i 0.706993i
\(36\) 0 0
\(37\) 11.3660i 1.86856i 0.356542 + 0.934279i \(0.383956\pi\)
−0.356542 + 0.934279i \(0.616044\pi\)
\(38\) −4.19880 + 1.02363i −0.681136 + 0.166055i
\(39\) 0 0
\(40\) −8.91447 + 7.77736i −1.40950 + 1.22971i
\(41\) 7.26960i 1.13532i −0.823263 0.567660i \(-0.807849\pi\)
0.823263 0.567660i \(-0.192151\pi\)
\(42\) 0 0
\(43\) 9.02120 1.37572 0.687860 0.725844i \(-0.258550\pi\)
0.687860 + 0.725844i \(0.258550\pi\)
\(44\) −1.71877 3.31557i −0.259114 0.499841i
\(45\) 0 0
\(46\) 8.20576 2.00049i 1.20987 0.294957i
\(47\) 0.327367 0.0477513 0.0238757 0.999715i \(-0.492399\pi\)
0.0238757 + 0.999715i \(0.492399\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) −17.1669 + 4.18514i −2.42777 + 0.591869i
\(51\) 0 0
\(52\) 2.83136 + 5.46181i 0.392639 + 0.757417i
\(53\) 8.32940 1.14413 0.572066 0.820208i \(-0.306142\pi\)
0.572066 + 0.820208i \(0.306142\pi\)
\(54\) 0 0
\(55\) 7.81021i 1.05313i
\(56\) −1.85944 2.13131i −0.248479 0.284808i
\(57\) 0 0
\(58\) −4.84853 + 1.18203i −0.636643 + 0.155208i
\(59\) 5.98008i 0.778540i −0.921124 0.389270i \(-0.872727\pi\)
0.921124 0.389270i \(-0.127273\pi\)
\(60\) 0 0
\(61\) 13.2783i 1.70011i −0.526691 0.850057i \(-0.676568\pi\)
0.526691 0.850057i \(-0.323432\pi\)
\(62\) −3.65043 14.9736i −0.463604 1.90165i
\(63\) 0 0
\(64\) 1.08494 7.92609i 0.135618 0.990761i
\(65\) 12.8659i 1.59582i
\(66\) 0 0
\(67\) −9.21499 −1.12579 −0.562895 0.826528i \(-0.690312\pi\)
−0.562895 + 0.826528i \(0.690312\pi\)
\(68\) 0.464725 + 0.896474i 0.0563562 + 0.108713i
\(69\) 0 0
\(70\) −1.40102 5.74682i −0.167454 0.686876i
\(71\) −9.02460 −1.07102 −0.535512 0.844528i \(-0.679881\pi\)
−0.535512 + 0.844528i \(0.679881\pi\)
\(72\) 0 0
\(73\) 0.416491 0.0487466 0.0243733 0.999703i \(-0.492241\pi\)
0.0243733 + 0.999703i \(0.492241\pi\)
\(74\) −3.80718 15.6166i −0.442576 1.81539i
\(75\) 0 0
\(76\) 5.42616 2.81288i 0.622424 0.322660i
\(77\) 1.86730 0.212798
\(78\) 0 0
\(79\) 6.00459i 0.675569i −0.941224 0.337784i \(-0.890323\pi\)
0.941224 0.337784i \(-0.109677\pi\)
\(80\) 9.64311 13.6719i 1.07813 1.52856i
\(81\) 0 0
\(82\) 2.43504 + 9.98822i 0.268905 + 1.10301i
\(83\) 2.62368i 0.287987i 0.989579 + 0.143993i \(0.0459944\pi\)
−0.989579 + 0.143993i \(0.954006\pi\)
\(84\) 0 0
\(85\) 2.11175i 0.229051i
\(86\) −12.3949 + 3.02176i −1.33657 + 0.325845i
\(87\) 0 0
\(88\) 3.47213 + 3.97978i 0.370131 + 0.424246i
\(89\) 5.69486i 0.603654i 0.953363 + 0.301827i \(0.0975966\pi\)
−0.953363 + 0.301827i \(0.902403\pi\)
\(90\) 0 0
\(91\) −3.07604 −0.322456
\(92\) −10.6044 + 5.49724i −1.10559 + 0.573127i
\(93\) 0 0
\(94\) −0.449793 + 0.109655i −0.0463926 + 0.0113101i
\(95\) 12.7819 1.31140
\(96\) 0 0
\(97\) 6.21289 0.630824 0.315412 0.948955i \(-0.397857\pi\)
0.315412 + 0.948955i \(0.397857\pi\)
\(98\) 1.37397 0.334962i 0.138792 0.0338363i
\(99\) 0 0
\(100\) 22.1850 11.5005i 2.21850 1.15005i
\(101\) −1.85216 −0.184296 −0.0921482 0.995745i \(-0.529373\pi\)
−0.0921482 + 0.995745i \(0.529373\pi\)
\(102\) 0 0
\(103\) 2.74001i 0.269982i −0.990847 0.134991i \(-0.956900\pi\)
0.990847 0.134991i \(-0.0431005\pi\)
\(104\) −5.71971 6.55598i −0.560864 0.642867i
\(105\) 0 0
\(106\) −11.4444 + 2.79004i −1.11158 + 0.270992i
\(107\) 4.22506i 0.408452i −0.978924 0.204226i \(-0.934532\pi\)
0.978924 0.204226i \(-0.0654677\pi\)
\(108\) 0 0
\(109\) 13.9420i 1.33540i 0.744432 + 0.667698i \(0.232720\pi\)
−0.744432 + 0.667698i \(0.767280\pi\)
\(110\) 2.61613 + 10.7310i 0.249438 + 1.02316i
\(111\) 0 0
\(112\) 3.26873 + 2.30551i 0.308866 + 0.217851i
\(113\) 7.49447i 0.705020i 0.935808 + 0.352510i \(0.114672\pi\)
−0.935808 + 0.352510i \(0.885328\pi\)
\(114\) 0 0
\(115\) −24.9799 −2.32938
\(116\) 6.26581 3.24815i 0.581766 0.301583i
\(117\) 0 0
\(118\) 2.00310 + 8.21646i 0.184400 + 0.756387i
\(119\) −0.504885 −0.0462827
\(120\) 0 0
\(121\) 7.51320 0.683018
\(122\) 4.44773 + 18.2440i 0.402679 + 1.65174i
\(123\) 0 0
\(124\) 10.0312 + 19.3505i 0.900825 + 1.73773i
\(125\) 31.3462 2.80369
\(126\) 0 0
\(127\) 18.0939i 1.60557i −0.596267 0.802787i \(-0.703350\pi\)
0.596267 0.802787i \(-0.296650\pi\)
\(128\) 1.16426 + 11.2536i 0.102907 + 0.994691i
\(129\) 0 0
\(130\) −4.30960 17.6774i −0.377977 1.55041i
\(131\) 9.32625i 0.814839i −0.913241 0.407419i \(-0.866429\pi\)
0.913241 0.407419i \(-0.133571\pi\)
\(132\) 0 0
\(133\) 3.05596i 0.264985i
\(134\) 12.6611 3.08667i 1.09376 0.266648i
\(135\) 0 0
\(136\) −0.938805 1.07607i −0.0805019 0.0922718i
\(137\) 6.53069i 0.557955i −0.960298 0.278977i \(-0.910005\pi\)
0.960298 0.278977i \(-0.0899954\pi\)
\(138\) 0 0
\(139\) −10.1589 −0.861664 −0.430832 0.902432i \(-0.641780\pi\)
−0.430832 + 0.902432i \(0.641780\pi\)
\(140\) 3.84993 + 7.42668i 0.325379 + 0.627669i
\(141\) 0 0
\(142\) 12.3996 3.02290i 1.04055 0.253676i
\(143\) 5.74387 0.480327
\(144\) 0 0
\(145\) 14.7598 1.22574
\(146\) −0.572247 + 0.139509i −0.0473595 + 0.0115458i
\(147\) 0 0
\(148\) 10.4619 + 20.1815i 0.859965 + 1.65891i
\(149\) 9.72516 0.796716 0.398358 0.917230i \(-0.369580\pi\)
0.398358 + 0.917230i \(0.369580\pi\)
\(150\) 0 0
\(151\) 12.4664i 1.01450i 0.861800 + 0.507249i \(0.169338\pi\)
−0.861800 + 0.507249i \(0.830662\pi\)
\(152\) −6.51319 + 5.68238i −0.528289 + 0.460902i
\(153\) 0 0
\(154\) −2.56561 + 0.625474i −0.206743 + 0.0504021i
\(155\) 45.5823i 3.66126i
\(156\) 0 0
\(157\) 11.3627i 0.906838i 0.891297 + 0.453419i \(0.149796\pi\)
−0.891297 + 0.453419i \(0.850204\pi\)
\(158\) 2.01131 + 8.25014i 0.160011 + 0.656346i
\(159\) 0 0
\(160\) −8.66980 + 22.0149i −0.685408 + 1.74043i
\(161\) 5.97229i 0.470682i
\(162\) 0 0
\(163\) −3.59449 −0.281542 −0.140771 0.990042i \(-0.544958\pi\)
−0.140771 + 0.990042i \(0.544958\pi\)
\(164\) −6.69136 12.9079i −0.522507 1.00794i
\(165\) 0 0
\(166\) −0.878835 3.60487i −0.0682108 0.279792i
\(167\) 20.3801 1.57706 0.788531 0.614995i \(-0.210842\pi\)
0.788531 + 0.614995i \(0.210842\pi\)
\(168\) 0 0
\(169\) 3.53800 0.272154
\(170\) −0.707355 2.90148i −0.0542517 0.222533i
\(171\) 0 0
\(172\) 16.0180 8.30363i 1.22136 0.633146i
\(173\) −4.49761 −0.341947 −0.170974 0.985276i \(-0.554691\pi\)
−0.170974 + 0.985276i \(0.554691\pi\)
\(174\) 0 0
\(175\) 12.4944i 0.944486i
\(176\) −6.10369 4.30508i −0.460083 0.324508i
\(177\) 0 0
\(178\) −1.90756 7.82459i −0.142978 0.586477i
\(179\) 2.63430i 0.196897i 0.995142 + 0.0984485i \(0.0313880\pi\)
−0.995142 + 0.0984485i \(0.968612\pi\)
\(180\) 0 0
\(181\) 10.3748i 0.771151i −0.922676 0.385576i \(-0.874003\pi\)
0.922676 0.385576i \(-0.125997\pi\)
\(182\) 4.22639 1.03036i 0.313281 0.0763751i
\(183\) 0 0
\(184\) 12.7288 11.1051i 0.938379 0.818681i
\(185\) 47.5397i 3.49519i
\(186\) 0 0
\(187\) 0.942770 0.0689422
\(188\) 0.581272 0.301327i 0.0423936 0.0219765i
\(189\) 0 0
\(190\) −17.5620 + 4.28147i −1.27408 + 0.310610i
\(191\) 18.1707 1.31479 0.657395 0.753547i \(-0.271659\pi\)
0.657395 + 0.753547i \(0.271659\pi\)
\(192\) 0 0
\(193\) −16.7704 −1.20716 −0.603578 0.797304i \(-0.706259\pi\)
−0.603578 + 0.797304i \(0.706259\pi\)
\(194\) −8.53634 + 2.08109i −0.612874 + 0.149413i
\(195\) 0 0
\(196\) −1.77560 + 0.920458i −0.126829 + 0.0657470i
\(197\) −19.2240 −1.36965 −0.684826 0.728706i \(-0.740122\pi\)
−0.684826 + 0.728706i \(0.740122\pi\)
\(198\) 0 0
\(199\) 1.68931i 0.119752i 0.998206 + 0.0598761i \(0.0190706\pi\)
−0.998206 + 0.0598761i \(0.980929\pi\)
\(200\) −26.6294 + 23.2326i −1.88298 + 1.64279i
\(201\) 0 0
\(202\) 2.54481 0.620402i 0.179052 0.0436514i
\(203\) 3.52884i 0.247676i
\(204\) 0 0
\(205\) 30.4060i 2.12365i
\(206\) 0.917801 + 3.76470i 0.0639463 + 0.262299i
\(207\) 0 0
\(208\) 10.0547 + 7.09184i 0.697170 + 0.491731i
\(209\) 5.70638i 0.394719i
\(210\) 0 0
\(211\) −14.1544 −0.974427 −0.487213 0.873283i \(-0.661987\pi\)
−0.487213 + 0.873283i \(0.661987\pi\)
\(212\) 14.7897 7.66687i 1.01576 0.526563i
\(213\) 0 0
\(214\) 1.41523 + 5.80511i 0.0967434 + 0.396829i
\(215\) 37.7323 2.57332
\(216\) 0 0
\(217\) −10.8980 −0.739806
\(218\) −4.67003 19.1559i −0.316294 1.29740i
\(219\) 0 0
\(220\) −7.18897 13.8678i −0.484680 0.934967i
\(221\) −1.55304 −0.104469
\(222\) 0 0
\(223\) 1.33032i 0.0890851i 0.999007 + 0.0445425i \(0.0141830\pi\)
−0.999007 + 0.0445425i \(0.985817\pi\)
\(224\) −5.26341 2.07281i −0.351676 0.138496i
\(225\) 0 0
\(226\) −2.51036 10.2972i −0.166987 0.684959i
\(227\) 6.34646i 0.421229i −0.977569 0.210615i \(-0.932453\pi\)
0.977569 0.210615i \(-0.0675465\pi\)
\(228\) 0 0
\(229\) 13.9920i 0.924619i −0.886719 0.462309i \(-0.847021\pi\)
0.886719 0.462309i \(-0.152979\pi\)
\(230\) 34.3217 8.36732i 2.26310 0.551724i
\(231\) 0 0
\(232\) −7.52104 + 6.56168i −0.493781 + 0.430795i
\(233\) 14.0797i 0.922392i −0.887298 0.461196i \(-0.847420\pi\)
0.887298 0.461196i \(-0.152580\pi\)
\(234\) 0 0
\(235\) 1.36925 0.0893202
\(236\) −5.50441 10.6182i −0.358307 0.691188i
\(237\) 0 0
\(238\) 0.693698 0.169117i 0.0449658 0.0109623i
\(239\) −13.1737 −0.852134 −0.426067 0.904691i \(-0.640101\pi\)
−0.426067 + 0.904691i \(0.640101\pi\)
\(240\) 0 0
\(241\) 7.46846 0.481086 0.240543 0.970639i \(-0.422675\pi\)
0.240543 + 0.970639i \(0.422675\pi\)
\(242\) −10.3229 + 2.51664i −0.663583 + 0.161776i
\(243\) 0 0
\(244\) −12.2221 23.5770i −0.782442 1.50936i
\(245\) −4.18263 −0.267218
\(246\) 0 0
\(247\) 9.40024i 0.598123i
\(248\) −20.2642 23.2270i −1.28678 1.47492i
\(249\) 0 0
\(250\) −43.0688 + 10.4998i −2.72391 + 0.664065i
\(251\) 16.3203i 1.03013i 0.857151 + 0.515065i \(0.172232\pi\)
−0.857151 + 0.515065i \(0.827768\pi\)
\(252\) 0 0
\(253\) 11.1520i 0.701123i
\(254\) 6.06077 + 24.8605i 0.380287 + 1.55989i
\(255\) 0 0
\(256\) −5.36921 15.0722i −0.335576 0.942013i
\(257\) 13.6939i 0.854205i −0.904203 0.427102i \(-0.859534\pi\)
0.904203 0.427102i \(-0.140466\pi\)
\(258\) 0 0
\(259\) −11.3660 −0.706249
\(260\) 11.8425 + 22.8447i 0.734443 + 1.41677i
\(261\) 0 0
\(262\) 3.12394 + 12.8140i 0.192998 + 0.791653i
\(263\) −4.63480 −0.285794 −0.142897 0.989738i \(-0.545642\pi\)
−0.142897 + 0.989738i \(0.545642\pi\)
\(264\) 0 0
\(265\) 34.8388 2.14013
\(266\) −1.02363 4.19880i −0.0627629 0.257445i
\(267\) 0 0
\(268\) −16.3621 + 8.48201i −0.999477 + 0.518121i
\(269\) −28.6544 −1.74709 −0.873545 0.486743i \(-0.838185\pi\)
−0.873545 + 0.486743i \(0.838185\pi\)
\(270\) 0 0
\(271\) 8.27458i 0.502645i −0.967903 0.251322i \(-0.919135\pi\)
0.967903 0.251322i \(-0.0808654\pi\)
\(272\) 1.65033 + 1.16402i 0.100066 + 0.0705791i
\(273\) 0 0
\(274\) 2.18754 + 8.97299i 0.132154 + 0.542078i
\(275\) 23.3307i 1.40689i
\(276\) 0 0
\(277\) 12.5278i 0.752724i 0.926473 + 0.376362i \(0.122825\pi\)
−0.926473 + 0.376362i \(0.877175\pi\)
\(278\) 13.9580 3.40284i 0.837146 0.204089i
\(279\) 0 0
\(280\) −7.77736 8.91447i −0.464786 0.532741i
\(281\) 8.63112i 0.514889i −0.966293 0.257445i \(-0.917119\pi\)
0.966293 0.257445i \(-0.0828805\pi\)
\(282\) 0 0
\(283\) 8.73498 0.519241 0.259620 0.965711i \(-0.416403\pi\)
0.259620 + 0.965711i \(0.416403\pi\)
\(284\) −16.0241 + 8.30677i −0.950855 + 0.492916i
\(285\) 0 0
\(286\) −7.89192 + 1.92398i −0.466659 + 0.113767i
\(287\) 7.26960 0.429111
\(288\) 0 0
\(289\) 16.7451 0.985005
\(290\) −20.2796 + 4.94398i −1.19086 + 0.290321i
\(291\) 0 0
\(292\) 0.739522 0.383362i 0.0432772 0.0224346i
\(293\) −30.3572 −1.77348 −0.886742 0.462265i \(-0.847037\pi\)
−0.886742 + 0.462265i \(0.847037\pi\)
\(294\) 0 0
\(295\) 25.0124i 1.45628i
\(296\) −21.1344 24.2244i −1.22841 1.40802i
\(297\) 0 0
\(298\) −13.3621 + 3.25756i −0.774046 + 0.188706i
\(299\) 18.3710i 1.06242i
\(300\) 0 0
\(301\) 9.02120i 0.519973i
\(302\) −4.17576 17.1284i −0.240288 0.985630i
\(303\) 0 0
\(304\) 7.04556 9.98911i 0.404091 0.572915i
\(305\) 55.5382i 3.18011i
\(306\) 0 0
\(307\) −29.1705 −1.66485 −0.832425 0.554138i \(-0.813048\pi\)
−0.832425 + 0.554138i \(0.813048\pi\)
\(308\) 3.31557 1.71877i 0.188922 0.0979359i
\(309\) 0 0
\(310\) −15.2684 62.6289i −0.867185 3.55708i
\(311\) 13.4778 0.764253 0.382127 0.924110i \(-0.375192\pi\)
0.382127 + 0.924110i \(0.375192\pi\)
\(312\) 0 0
\(313\) 20.7291 1.17168 0.585838 0.810428i \(-0.300765\pi\)
0.585838 + 0.810428i \(0.300765\pi\)
\(314\) −3.80606 15.6120i −0.214788 0.881034i
\(315\) 0 0
\(316\) −5.52697 10.6617i −0.310916 0.599770i
\(317\) −0.453146 −0.0254512 −0.0127256 0.999919i \(-0.504051\pi\)
−0.0127256 + 0.999919i \(0.504051\pi\)
\(318\) 0 0
\(319\) 6.58939i 0.368935i
\(320\) 4.53791 33.1519i 0.253677 1.85325i
\(321\) 0 0
\(322\) 2.00049 + 8.20576i 0.111483 + 0.457289i
\(323\) 1.54291i 0.0858497i
\(324\) 0 0
\(325\) 38.4331i 2.13189i
\(326\) 4.93873 1.20402i 0.273531 0.0666844i
\(327\) 0 0
\(328\) 13.5174 + 15.4937i 0.746374 + 0.855499i
\(329\) 0.327367i 0.0180483i
\(330\) 0 0
\(331\) 24.8798 1.36752 0.683759 0.729708i \(-0.260344\pi\)
0.683759 + 0.729708i \(0.260344\pi\)
\(332\) 2.41499 + 4.65861i 0.132540 + 0.255675i
\(333\) 0 0
\(334\) −28.0018 + 6.82658i −1.53219 + 0.373534i
\(335\) −38.5429 −2.10582
\(336\) 0 0
\(337\) 21.8036 1.18772 0.593860 0.804568i \(-0.297603\pi\)
0.593860 + 0.804568i \(0.297603\pi\)
\(338\) −4.86112 + 1.18510i −0.264410 + 0.0644608i
\(339\) 0 0
\(340\) 1.94377 + 3.74962i 0.105416 + 0.203352i
\(341\) 20.3498 1.10201
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −19.2269 + 16.7744i −1.03665 + 0.904415i
\(345\) 0 0
\(346\) 6.17960 1.50653i 0.332217 0.0809916i
\(347\) 20.7794i 1.11550i 0.830010 + 0.557748i \(0.188335\pi\)
−0.830010 + 0.557748i \(0.811665\pi\)
\(348\) 0 0
\(349\) 30.5928i 1.63760i −0.574081 0.818799i \(-0.694640\pi\)
0.574081 0.818799i \(-0.305360\pi\)
\(350\) −4.18514 17.1669i −0.223705 0.917611i
\(351\) 0 0
\(352\) 9.82834 + 3.87055i 0.523852 + 0.206301i
\(353\) 19.5174i 1.03881i 0.854529 + 0.519403i \(0.173846\pi\)
−0.854529 + 0.519403i \(0.826154\pi\)
\(354\) 0 0
\(355\) −37.7466 −2.00338
\(356\) 5.24188 + 10.1118i 0.277819 + 0.535924i
\(357\) 0 0
\(358\) −0.882392 3.61946i −0.0466358 0.191294i
\(359\) 20.3969 1.07651 0.538254 0.842783i \(-0.319084\pi\)
0.538254 + 0.842783i \(0.319084\pi\)
\(360\) 0 0
\(361\) −9.66111 −0.508479
\(362\) 3.47516 + 14.2547i 0.182650 + 0.749208i
\(363\) 0 0
\(364\) −5.46181 + 2.83136i −0.286277 + 0.148404i
\(365\) 1.74203 0.0911818
\(366\) 0 0
\(367\) 25.6785i 1.34041i −0.742177 0.670203i \(-0.766207\pi\)
0.742177 0.670203i \(-0.233793\pi\)
\(368\) −13.7692 + 19.5218i −0.717769 + 1.01764i
\(369\) 0 0
\(370\) −15.9240 65.3183i −0.827850 3.39574i
\(371\) 8.32940i 0.432441i
\(372\) 0 0
\(373\) 12.4313i 0.643669i −0.946796 0.321835i \(-0.895700\pi\)
0.946796 0.321835i \(-0.104300\pi\)
\(374\) −1.29534 + 0.315792i −0.0669804 + 0.0163292i
\(375\) 0 0
\(376\) −0.697719 + 0.608720i −0.0359821 + 0.0313923i
\(377\) 10.8548i 0.559053i
\(378\) 0 0
\(379\) −9.90048 −0.508553 −0.254277 0.967132i \(-0.581837\pi\)
−0.254277 + 0.967132i \(0.581837\pi\)
\(380\) 22.6956 11.7652i 1.16426 0.603544i
\(381\) 0 0
\(382\) −24.9661 + 6.08652i −1.27738 + 0.311413i
\(383\) −23.6536 −1.20864 −0.604320 0.796741i \(-0.706555\pi\)
−0.604320 + 0.796741i \(0.706555\pi\)
\(384\) 0 0
\(385\) 7.81021 0.398045
\(386\) 23.0420 5.61744i 1.17281 0.285920i
\(387\) 0 0
\(388\) 11.0316 5.71871i 0.560045 0.290323i
\(389\) 3.14513 0.159465 0.0797323 0.996816i \(-0.474593\pi\)
0.0797323 + 0.996816i \(0.474593\pi\)
\(390\) 0 0
\(391\) 3.01532i 0.152491i
\(392\) 2.13131 1.85944i 0.107647 0.0939161i
\(393\) 0 0
\(394\) 26.4132 6.43931i 1.33068 0.324408i
\(395\) 25.1150i 1.26367i
\(396\) 0 0
\(397\) 12.2176i 0.613183i 0.951841 + 0.306592i \(0.0991886\pi\)
−0.951841 + 0.306592i \(0.900811\pi\)
\(398\) −0.565856 2.32107i −0.0283638 0.116345i
\(399\) 0 0
\(400\) 28.8060 40.8408i 1.44030 2.04204i
\(401\) 35.0797i 1.75180i 0.482494 + 0.875899i \(0.339731\pi\)
−0.482494 + 0.875899i \(0.660269\pi\)
\(402\) 0 0
\(403\) −33.5227 −1.66988
\(404\) −3.28869 + 1.70483i −0.163618 + 0.0848186i
\(405\) 0 0
\(406\) −1.18203 4.84853i −0.0586631 0.240628i
\(407\) 21.2237 1.05202
\(408\) 0 0
\(409\) −27.3489 −1.35231 −0.676157 0.736757i \(-0.736356\pi\)
−0.676157 + 0.736757i \(0.736356\pi\)
\(410\) 10.1849 + 41.7770i 0.502995 + 2.06322i
\(411\) 0 0
\(412\) −2.52207 4.86517i −0.124253 0.239690i
\(413\) 5.98008 0.294260
\(414\) 0 0
\(415\) 10.9739i 0.538687i
\(416\) −16.1904 6.37604i −0.793801 0.312611i
\(417\) 0 0
\(418\) 1.91142 + 7.84041i 0.0934908 + 0.383487i
\(419\) 14.6442i 0.715417i 0.933833 + 0.357709i \(0.116442\pi\)
−0.933833 + 0.357709i \(0.883558\pi\)
\(420\) 0 0
\(421\) 20.2168i 0.985305i −0.870226 0.492653i \(-0.836027\pi\)
0.870226 0.492653i \(-0.163973\pi\)
\(422\) 19.4477 4.74118i 0.946700 0.230797i
\(423\) 0 0
\(424\) −17.7525 + 15.4881i −0.862139 + 0.752166i
\(425\) 6.30822i 0.305994i
\(426\) 0 0
\(427\) 13.2783 0.642583
\(428\) −3.88899 7.50201i −0.187981 0.362623i
\(429\) 0 0
\(430\) −51.8431 + 12.6389i −2.50010 + 0.609502i
\(431\) −19.6656 −0.947260 −0.473630 0.880724i \(-0.657057\pi\)
−0.473630 + 0.880724i \(0.657057\pi\)
\(432\) 0 0
\(433\) 11.9996 0.576666 0.288333 0.957530i \(-0.406899\pi\)
0.288333 + 0.957530i \(0.406899\pi\)
\(434\) 14.9736 3.65043i 0.718755 0.175226i
\(435\) 0 0
\(436\) 12.8330 + 24.7553i 0.614588 + 1.18557i
\(437\) −18.2511 −0.873068
\(438\) 0 0
\(439\) 7.82113i 0.373282i −0.982428 0.186641i \(-0.940240\pi\)
0.982428 0.186641i \(-0.0597601\pi\)
\(440\) 14.5226 + 16.6460i 0.692340 + 0.793565i
\(441\) 0 0
\(442\) 2.13384 0.520211i 0.101496 0.0247439i
\(443\) 29.9854i 1.42465i −0.701850 0.712325i \(-0.747642\pi\)
0.701850 0.712325i \(-0.252358\pi\)
\(444\) 0 0
\(445\) 23.8195i 1.12915i
\(446\) −0.445608 1.82783i −0.0211002 0.0865502i
\(447\) 0 0
\(448\) 7.92609 + 1.08494i 0.374473 + 0.0512587i
\(449\) 42.0892i 1.98631i −0.116797 0.993156i \(-0.537263\pi\)
0.116797 0.993156i \(-0.462737\pi\)
\(450\) 0 0
\(451\) −13.5745 −0.639198
\(452\) 6.89834 + 13.3072i 0.324471 + 0.625917i
\(453\) 0 0
\(454\) 2.12582 + 8.71986i 0.0997699 + 0.409243i
\(455\) −12.8659 −0.603163
\(456\) 0 0
\(457\) −12.1655 −0.569079 −0.284540 0.958664i \(-0.591841\pi\)
−0.284540 + 0.958664i \(0.591841\pi\)
\(458\) 4.68680 + 19.2247i 0.219000 + 0.898309i
\(459\) 0 0
\(460\) −44.3543 + 22.9929i −2.06803 + 1.07205i
\(461\) −29.0054 −1.35091 −0.675457 0.737399i \(-0.736054\pi\)
−0.675457 + 0.737399i \(0.736054\pi\)
\(462\) 0 0
\(463\) 5.21317i 0.242277i −0.992636 0.121138i \(-0.961346\pi\)
0.992636 0.121138i \(-0.0386545\pi\)
\(464\) 8.13579 11.5348i 0.377695 0.535491i
\(465\) 0 0
\(466\) 4.71617 + 19.3451i 0.218472 + 0.896146i
\(467\) 12.3061i 0.569460i 0.958608 + 0.284730i \(0.0919039\pi\)
−0.958608 + 0.284730i \(0.908096\pi\)
\(468\) 0 0
\(469\) 9.21499i 0.425509i
\(470\) −1.88132 + 0.458648i −0.0867786 + 0.0211559i
\(471\) 0 0
\(472\) 11.1196 + 12.7454i 0.511822 + 0.586654i
\(473\) 16.8452i 0.774545i
\(474\) 0 0
\(475\) 38.1823 1.75192
\(476\) −0.896474 + 0.464725i −0.0410898 + 0.0213007i
\(477\) 0 0
\(478\) 18.1003 4.41269i 0.827887 0.201832i
\(479\) −19.9609 −0.912038 −0.456019 0.889970i \(-0.650725\pi\)
−0.456019 + 0.889970i \(0.650725\pi\)
\(480\) 0 0
\(481\) −34.9622 −1.59414
\(482\) −10.2615 + 2.50165i −0.467396 + 0.113947i
\(483\) 0 0
\(484\) 13.3404 6.91559i 0.606384 0.314345i
\(485\) 25.9862 1.17997
\(486\) 0 0
\(487\) 5.61420i 0.254404i 0.991877 + 0.127202i \(0.0405996\pi\)
−0.991877 + 0.127202i \(0.959400\pi\)
\(488\) 24.6903 + 28.3002i 1.11768 + 1.28109i
\(489\) 0 0
\(490\) 5.74682 1.40102i 0.259615 0.0632918i
\(491\) 9.07131i 0.409382i −0.978827 0.204691i \(-0.934381\pi\)
0.978827 0.204691i \(-0.0656190\pi\)
\(492\) 0 0
\(493\) 1.78166i 0.0802418i
\(494\) −3.14873 12.9157i −0.141668 0.581104i
\(495\) 0 0
\(496\) 35.6227 + 25.1255i 1.59951 + 1.12817i
\(497\) 9.02460i 0.404809i
\(498\) 0 0
\(499\) −36.4346 −1.63103 −0.815517 0.578732i \(-0.803548\pi\)
−0.815517 + 0.578732i \(0.803548\pi\)
\(500\) 55.6583 28.8528i 2.48911 1.29034i
\(501\) 0 0
\(502\) −5.46669 22.4237i −0.243990 1.00082i
\(503\) −4.10995 −0.183253 −0.0916267 0.995793i \(-0.529207\pi\)
−0.0916267 + 0.995793i \(0.529207\pi\)
\(504\) 0 0
\(505\) −7.74688 −0.344732
\(506\) −3.73551 15.3226i −0.166064 0.681172i
\(507\) 0 0
\(508\) −16.6547 32.1275i −0.738931 1.42543i
\(509\) −17.5175 −0.776450 −0.388225 0.921565i \(-0.626912\pi\)
−0.388225 + 0.921565i \(0.626912\pi\)
\(510\) 0 0
\(511\) 0.416491i 0.0184245i
\(512\) 12.4258 + 18.9103i 0.549147 + 0.835726i
\(513\) 0 0
\(514\) 4.58696 + 18.8151i 0.202322 + 0.829899i
\(515\) 11.4605i 0.505008i
\(516\) 0 0
\(517\) 0.611291i 0.0268845i
\(518\) 15.6166 3.80718i 0.686153 0.167278i
\(519\) 0 0
\(520\) −23.9234 27.4212i −1.04911 1.20250i
\(521\) 40.7567i 1.78558i 0.450472 + 0.892791i \(0.351256\pi\)
−0.450472 + 0.892791i \(0.648744\pi\)
\(522\) 0 0
\(523\) 7.03412 0.307581 0.153790 0.988103i \(-0.450852\pi\)
0.153790 + 0.988103i \(0.450852\pi\)
\(524\) −8.58442 16.5597i −0.375012 0.723414i
\(525\) 0 0
\(526\) 6.36808 1.55248i 0.277662 0.0676915i
\(527\) −5.50225 −0.239682
\(528\) 0 0
\(529\) 12.6683 0.550794
\(530\) −47.8675 + 11.6697i −2.07923 + 0.506899i
\(531\) 0 0
\(532\) 2.81288 + 5.42616i 0.121954 + 0.235254i
\(533\) 22.3615 0.968586
\(534\) 0 0
\(535\) 17.6718i 0.764020i
\(536\) 19.6400 17.1347i 0.848318 0.740108i
\(537\) 0 0
\(538\) 39.3704 9.59815i 1.69738 0.413806i
\(539\) 1.86730i 0.0804302i
\(540\) 0 0
\(541\) 19.6764i 0.845953i −0.906141 0.422976i \(-0.860985\pi\)
0.906141 0.422976i \(-0.139015\pi\)
\(542\) 2.77167 + 11.3690i 0.119053 + 0.488342i
\(543\) 0 0
\(544\) −2.65742 1.04653i −0.113936 0.0448697i
\(545\) 58.3140i 2.49790i
\(546\) 0 0
\(547\) 17.8078 0.761408 0.380704 0.924697i \(-0.375682\pi\)
0.380704 + 0.924697i \(0.375682\pi\)
\(548\) −6.01123 11.5959i −0.256787 0.495352i
\(549\) 0 0
\(550\) 7.81491 + 32.0557i 0.333229 + 1.36686i
\(551\) 10.7840 0.459413
\(552\) 0 0
\(553\) 6.00459 0.255341
\(554\) −4.19635 17.2129i −0.178286 0.731306i
\(555\) 0 0
\(556\) −18.0381 + 9.35081i −0.764986 + 0.396563i
\(557\) 5.99646 0.254078 0.127039 0.991898i \(-0.459453\pi\)
0.127039 + 0.991898i \(0.459453\pi\)
\(558\) 0 0
\(559\) 27.7495i 1.17368i
\(560\) 13.6719 + 9.64311i 0.577743 + 0.407496i
\(561\) 0 0
\(562\) 2.89110 + 11.8589i 0.121954 + 0.500238i
\(563\) 20.4848i 0.863333i −0.902033 0.431667i \(-0.857926\pi\)
0.902033 0.431667i \(-0.142074\pi\)
\(564\) 0 0
\(565\) 31.3466i 1.31876i
\(566\) −12.0016 + 2.92589i −0.504466 + 0.122984i
\(567\) 0 0
\(568\) 19.2342 16.7807i 0.807049 0.704104i
\(569\) 38.1889i 1.60096i −0.599359 0.800480i \(-0.704578\pi\)
0.599359 0.800480i \(-0.295422\pi\)
\(570\) 0 0
\(571\) 11.3551 0.475198 0.237599 0.971363i \(-0.423640\pi\)
0.237599 + 0.971363i \(0.423640\pi\)
\(572\) 10.1988 5.28699i 0.426434 0.221060i
\(573\) 0 0
\(574\) −9.98822 + 2.43504i −0.416900 + 0.101637i
\(575\) −74.6200 −3.11187
\(576\) 0 0
\(577\) −20.7869 −0.865371 −0.432685 0.901545i \(-0.642434\pi\)
−0.432685 + 0.901545i \(0.642434\pi\)
\(578\) −23.0073 + 5.60897i −0.956977 + 0.233303i
\(579\) 0 0
\(580\) 26.2075 13.5858i 1.08821 0.564119i
\(581\) −2.62368 −0.108849
\(582\) 0 0
\(583\) 15.5535i 0.644159i
\(584\) −0.887670 + 0.774441i −0.0367321 + 0.0320466i
\(585\) 0 0
\(586\) 41.7099 10.1685i 1.72302 0.420057i
\(587\) 24.3849i 1.00647i 0.864148 + 0.503237i \(0.167858\pi\)
−0.864148 + 0.503237i \(0.832142\pi\)
\(588\) 0 0
\(589\) 33.3039i 1.37226i
\(590\) 8.37823 + 34.3664i 0.344926 + 1.41484i
\(591\) 0 0
\(592\) 37.1524 + 26.2045i 1.52695 + 1.07700i
\(593\) 32.5053i 1.33483i −0.744684 0.667417i \(-0.767400\pi\)
0.744684 0.667417i \(-0.232600\pi\)
\(594\) 0 0
\(595\) −2.11175 −0.0865731
\(596\) 17.2680 8.95160i 0.707325 0.366672i
\(597\) 0 0
\(598\) 6.15359 + 25.2412i 0.251639 + 1.03219i
\(599\) 10.4800 0.428200 0.214100 0.976812i \(-0.431318\pi\)
0.214100 + 0.976812i \(0.431318\pi\)
\(600\) 0 0
\(601\) −22.4946 −0.917575 −0.458787 0.888546i \(-0.651716\pi\)
−0.458787 + 0.888546i \(0.651716\pi\)
\(602\) −3.02176 12.3949i −0.123158 0.505177i
\(603\) 0 0
\(604\) 11.4748 + 22.1353i 0.466901 + 0.900671i
\(605\) 31.4249 1.27761
\(606\) 0 0
\(607\) 32.3672i 1.31374i 0.754002 + 0.656872i \(0.228121\pi\)
−0.754002 + 0.656872i \(0.771879\pi\)
\(608\) −6.33443 + 16.0848i −0.256895 + 0.652323i
\(609\) 0 0
\(610\) 18.6032 + 76.3080i 0.753222 + 3.08962i
\(611\) 1.00699i 0.0407385i
\(612\) 0 0
\(613\) 34.8714i 1.40844i −0.709981 0.704220i \(-0.751297\pi\)
0.709981 0.704220i \(-0.248703\pi\)
\(614\) 40.0795 9.77102i 1.61748 0.394326i
\(615\) 0 0
\(616\) −3.97978 + 3.47213i −0.160350 + 0.139896i
\(617\) 2.47324i 0.0995688i 0.998760 + 0.0497844i \(0.0158534\pi\)
−0.998760 + 0.0497844i \(0.984147\pi\)
\(618\) 0 0
\(619\) −15.2375 −0.612447 −0.306223 0.951960i \(-0.599065\pi\)
−0.306223 + 0.951960i \(0.599065\pi\)
\(620\) 41.9566 + 80.9360i 1.68502 + 3.25047i
\(621\) 0 0
\(622\) −18.5181 + 4.51454i −0.742507 + 0.181017i
\(623\) −5.69486 −0.228160
\(624\) 0 0
\(625\) 68.6375 2.74550
\(626\) −28.4812 + 6.94346i −1.13834 + 0.277516i
\(627\) 0 0
\(628\) 10.4588 + 20.1755i 0.417353 + 0.805091i
\(629\) −5.73852 −0.228810
\(630\) 0 0
\(631\) 23.2064i 0.923832i −0.886924 0.461916i \(-0.847162\pi\)
0.886924 0.461916i \(-0.152838\pi\)
\(632\) 11.1652 + 12.7976i 0.444127 + 0.509062i
\(633\) 0 0
\(634\) 0.622610 0.151787i 0.0247270 0.00602823i
\(635\) 75.6800i 3.00327i
\(636\) 0 0
\(637\) 3.07604i 0.121877i
\(638\) 2.20720 + 9.05364i 0.0873838 + 0.358437i
\(639\) 0 0
\(640\) 4.86967 + 47.0698i 0.192491 + 1.86060i
\(641\) 29.5988i 1.16908i −0.811364 0.584541i \(-0.801275\pi\)
0.811364 0.584541i \(-0.198725\pi\)
\(642\) 0 0
\(643\) 28.8060 1.13600 0.567999 0.823029i \(-0.307718\pi\)
0.567999 + 0.823029i \(0.307718\pi\)
\(644\) −5.49724 10.6044i −0.216622 0.417872i
\(645\) 0 0
\(646\) −0.516816 2.11991i −0.0203339 0.0834069i
\(647\) −27.3622 −1.07572 −0.537860 0.843034i \(-0.680767\pi\)
−0.537860 + 0.843034i \(0.680767\pi\)
\(648\) 0 0
\(649\) −11.1666 −0.438327
\(650\) −12.8737 52.8061i −0.504946 2.07123i
\(651\) 0 0
\(652\) −6.38238 + 3.30858i −0.249953 + 0.129574i
\(653\) −39.9292 −1.56255 −0.781275 0.624187i \(-0.785430\pi\)
−0.781275 + 0.624187i \(0.785430\pi\)
\(654\) 0 0
\(655\) 39.0082i 1.52418i
\(656\) −23.7624 16.7602i −0.927764 0.654374i
\(657\) 0 0
\(658\) −0.109655 0.449793i −0.00427482 0.0175347i
\(659\) 14.9809i 0.583572i 0.956484 + 0.291786i \(0.0942495\pi\)
−0.956484 + 0.291786i \(0.905751\pi\)
\(660\) 0 0
\(661\) 33.0963i 1.28730i 0.765322 + 0.643648i \(0.222580\pi\)
−0.765322 + 0.643648i \(0.777420\pi\)
\(662\) −34.1842 + 8.33380i −1.32861 + 0.323902i
\(663\) 0 0
\(664\) −4.87859 5.59187i −0.189326 0.217007i
\(665\) 12.7819i 0.495662i
\(666\) 0 0
\(667\) −21.0753 −0.816037
\(668\) 36.1870 18.7591i 1.40012 0.725810i
\(669\) 0 0
\(670\) 52.9568 12.9104i 2.04590 0.498773i
\(671\) −24.7945 −0.957183
\(672\) 0 0
\(673\) −12.1189 −0.467150 −0.233575 0.972339i \(-0.575043\pi\)
−0.233575 + 0.972339i \(0.575043\pi\)
\(674\) −29.9576 + 7.30340i −1.15392 + 0.281316i
\(675\) 0 0
\(676\) 6.28208 3.25658i 0.241618 0.125253i
\(677\) 27.7231 1.06549 0.532743 0.846277i \(-0.321161\pi\)
0.532743 + 0.846277i \(0.321161\pi\)
\(678\) 0 0
\(679\) 6.21289i 0.238429i
\(680\) −3.92667 4.50078i −0.150581 0.172597i
\(681\) 0 0
\(682\) −27.9601 + 6.81643i −1.07065 + 0.261014i
\(683\) 9.79813i 0.374915i −0.982273 0.187457i \(-0.939975\pi\)
0.982273 0.187457i \(-0.0600247\pi\)
\(684\) 0 0
\(685\) 27.3155i 1.04367i
\(686\) 0.334962 + 1.37397i 0.0127889 + 0.0524585i
\(687\) 0 0
\(688\) 20.7985 29.4879i 0.792935 1.12421i
\(689\) 25.6215i 0.976103i
\(690\) 0 0
\(691\) −37.4765 −1.42567 −0.712837 0.701330i \(-0.752590\pi\)
−0.712837 + 0.701330i \(0.752590\pi\)
\(692\) −7.98597 + 4.13987i −0.303581 + 0.157374i
\(693\) 0 0
\(694\) −6.96032 28.5503i −0.264210 1.08376i
\(695\) −42.4908 −1.61177
\(696\) 0 0
\(697\) 3.67031 0.139023
\(698\) 10.2475 + 42.0337i 0.387872 + 1.59100i
\(699\) 0 0
\(700\) 11.5005 + 22.1850i 0.434680 + 0.838515i
\(701\) −22.6525 −0.855575 −0.427787 0.903879i \(-0.640707\pi\)
−0.427787 + 0.903879i \(0.640707\pi\)
\(702\) 0 0
\(703\) 34.7340i 1.31002i
\(704\) −14.8004 2.02591i −0.557810 0.0763543i
\(705\) 0 0
\(706\) −6.53759 26.8164i −0.246045 1.00925i
\(707\) 1.85216i 0.0696575i
\(708\) 0 0
\(709\) 1.58535i 0.0595393i −0.999557 0.0297696i \(-0.990523\pi\)
0.999557 0.0297696i \(-0.00947737\pi\)
\(710\) 51.8627 12.6437i 1.94637 0.474509i
\(711\) 0 0
\(712\) −10.5893 12.1375i −0.396850 0.454872i
\(713\) 65.0861i 2.43749i
\(714\) 0 0
\(715\) 24.0245 0.898465
\(716\) 2.42476 + 4.67747i 0.0906177 + 0.174805i
\(717\) 0 0
\(718\) −28.0248 + 6.83220i −1.04588 + 0.254975i
\(719\) −17.7228 −0.660948 −0.330474 0.943815i \(-0.607209\pi\)
−0.330474 + 0.943815i \(0.607209\pi\)
\(720\) 0 0
\(721\) 2.74001 0.102043
\(722\) 13.2741 3.23611i 0.494011 0.120435i
\(723\) 0 0
\(724\) −9.54955 18.4215i −0.354906 0.684628i
\(725\) 44.0906 1.63749
\(726\) 0 0
\(727\) 3.31812i 0.123062i 0.998105 + 0.0615312i \(0.0195984\pi\)
−0.998105 + 0.0615312i \(0.980402\pi\)
\(728\) 6.55598 5.71971i 0.242981 0.211987i
\(729\) 0 0
\(730\) −2.39350 + 0.583513i −0.0885873 + 0.0215968i
\(731\) 4.55467i 0.168460i
\(732\) 0 0
\(733\) 18.8547i 0.696414i −0.937418 0.348207i \(-0.886791\pi\)
0.937418 0.348207i \(-0.113209\pi\)
\(734\) 8.60133 + 35.2816i 0.317481 + 1.30227i
\(735\) 0 0
\(736\) 12.3794 31.4346i 0.456312 1.15869i
\(737\) 17.2071i 0.633832i
\(738\) 0 0
\(739\) −33.3560 −1.22702 −0.613510 0.789687i \(-0.710243\pi\)
−0.613510 + 0.789687i \(0.710243\pi\)
\(740\) 43.7583 + 84.4116i 1.60859 + 3.10303i
\(741\) 0 0
\(742\) −2.79004 11.4444i −0.102425 0.420136i
\(743\) 27.0826 0.993565 0.496783 0.867875i \(-0.334515\pi\)
0.496783 + 0.867875i \(0.334515\pi\)
\(744\) 0 0
\(745\) 40.6767 1.49028
\(746\) 4.16402 + 17.0803i 0.152456 + 0.625354i
\(747\) 0 0
\(748\) 1.67398 0.867780i 0.0612069 0.0317292i
\(749\) 4.22506 0.154380
\(750\) 0 0
\(751\) 44.3284i 1.61757i −0.588107 0.808783i \(-0.700126\pi\)
0.588107 0.808783i \(-0.299874\pi\)
\(752\) 0.754748 1.07007i 0.0275228 0.0390216i
\(753\) 0 0
\(754\) −3.63596 14.9142i −0.132414 0.543145i
\(755\) 52.1421i 1.89765i
\(756\) 0 0
\(757\) 12.3093i 0.447391i 0.974659 + 0.223695i \(0.0718121\pi\)
−0.974659 + 0.223695i \(0.928188\pi\)
\(758\) 13.6030 3.31629i 0.494083 0.120453i
\(759\) 0 0
\(760\) −27.2422 + 23.7673i −0.988180 + 0.862130i
\(761\) 1.85054i 0.0670820i 0.999437 + 0.0335410i \(0.0106784\pi\)
−0.999437 + 0.0335410i \(0.989322\pi\)
\(762\) 0 0
\(763\) −13.9420 −0.504733
\(764\) 32.2640 16.7254i 1.16727 0.605104i
\(765\) 0 0
\(766\) 32.4993 7.92305i 1.17425 0.286272i
\(767\) 18.3949 0.664203
\(768\) 0 0
\(769\) −31.2463 −1.12677 −0.563386 0.826194i \(-0.690502\pi\)
−0.563386 + 0.826194i \(0.690502\pi\)
\(770\) −10.7310 + 2.61613i −0.386719 + 0.0942786i
\(771\) 0 0
\(772\) −29.7775 + 15.4364i −1.07171 + 0.555569i
\(773\) −44.4521 −1.59883 −0.799415 0.600779i \(-0.794857\pi\)
−0.799415 + 0.600779i \(0.794857\pi\)
\(774\) 0 0
\(775\) 136.164i 4.89115i
\(776\) −13.2416 + 11.5525i −0.475345 + 0.414711i
\(777\) 0 0
\(778\) −4.32133 + 1.05350i −0.154927 + 0.0377698i
\(779\) 22.2156i 0.795956i
\(780\) 0 0
\(781\) 16.8516i 0.602998i
\(782\) 1.01002 + 4.14297i 0.0361182 + 0.148152i
\(783\) 0 0
\(784\) −2.30551 + 3.26873i −0.0823398 + 0.116740i
\(785\) 47.5257i 1.69627i
\(786\) 0 0
\(787\) 28.0995 1.00164 0.500820 0.865551i \(-0.333032\pi\)
0.500820 + 0.865551i \(0.333032\pi\)
\(788\) −34.1341 + 17.6949i −1.21598 + 0.630354i
\(789\) 0 0
\(790\) 8.41256 + 34.5073i 0.299306 + 1.22771i
\(791\) −7.49447 −0.266473
\(792\) 0 0
\(793\) 40.8446 1.45043
\(794\) −4.09243 16.7866i −0.145235 0.595735i
\(795\) 0 0
\(796\) 1.55494 + 2.99955i 0.0551135 + 0.106316i
\(797\) 42.9492 1.52134 0.760670 0.649139i \(-0.224871\pi\)
0.760670 + 0.649139i \(0.224871\pi\)
\(798\) 0 0
\(799\) 0.165282i 0.00584727i
\(800\) −25.8985 + 65.7630i −0.915649 + 2.32507i
\(801\) 0 0
\(802\) −11.7504 48.1986i −0.414921 1.70195i
\(803\) 0.777712i 0.0274449i
\(804\) 0 0
\(805\) 24.9799i 0.880425i
\(806\) 46.0593 11.2288i 1.62237 0.395519i
\(807\) 0 0
\(808\) 3.94751 3.44398i 0.138873 0.121159i
\(809\) 8.54301i 0.300356i 0.988659 + 0.150178i \(0.0479847\pi\)
−0.988659 + 0.150178i \(0.952015\pi\)
\(810\) 0 0
\(811\) 34.6880 1.21806 0.609030 0.793147i \(-0.291559\pi\)
0.609030 + 0.793147i \(0.291559\pi\)
\(812\) 3.24815 + 6.26581i 0.113988 + 0.219887i
\(813\) 0 0
\(814\) −29.1608 + 7.10914i −1.02208 + 0.249175i
\(815\) −15.0344 −0.526632
\(816\) 0 0
\(817\) 27.5684 0.964497
\(818\) 37.5766 9.16084i 1.31383 0.320301i
\(819\) 0 0
\(820\) −27.9875 53.9889i −0.977364 1.88538i
\(821\) −8.46951 −0.295588 −0.147794 0.989018i \(-0.547217\pi\)
−0.147794 + 0.989018i \(0.547217\pi\)
\(822\) 0 0
\(823\) 44.5109i 1.55155i 0.631008 + 0.775777i \(0.282642\pi\)
−0.631008 + 0.775777i \(0.717358\pi\)
\(824\) 5.09490 + 5.83981i 0.177489 + 0.203439i
\(825\) 0 0
\(826\) −8.21646 + 2.00310i −0.285887 + 0.0696968i
\(827\) 36.3895i 1.26539i 0.774403 + 0.632693i \(0.218051\pi\)
−0.774403 + 0.632693i \(0.781949\pi\)
\(828\) 0 0
\(829\) 39.2607i 1.36358i −0.731548 0.681790i \(-0.761202\pi\)
0.731548 0.681790i \(-0.238798\pi\)
\(830\) −3.67584 15.0778i −0.127590 0.523359i
\(831\) 0 0
\(832\) 24.3809 + 3.33732i 0.845257 + 0.115701i
\(833\) 0.504885i 0.0174932i
\(834\) 0 0
\(835\) 85.2425 2.94994
\(836\) −5.25249 10.1323i −0.181661 0.350431i
\(837\) 0 0
\(838\) −4.90527 20.1208i −0.169450 0.695060i
\(839\) −15.1101 −0.521659 −0.260830 0.965385i \(-0.583996\pi\)
−0.260830 + 0.965385i \(0.583996\pi\)
\(840\) 0 0
\(841\) −16.5473 −0.570596
\(842\) 6.77186 + 27.7773i 0.233374 + 0.957268i
\(843\) 0 0
\(844\) −25.1325 + 13.0285i −0.865096 + 0.448459i
\(845\) 14.7981 0.509072
\(846\) 0 0
\(847\) 7.51320i 0.258157i
\(848\) 19.2036 27.2266i 0.659453 0.934965i
\(849\) 0 0
\(850\) −2.11302 8.66732i −0.0724759 0.297287i
\(851\) 67.8810i 2.32693i
\(852\) 0 0
\(853\) 23.4153i 0.801724i −0.916138 0.400862i \(-0.868711\pi\)
0.916138 0.400862i \(-0.131289\pi\)
\(854\) −18.2440 + 4.44773i −0.624298 + 0.152198i
\(855\) 0 0
\(856\) 7.85625 + 9.00489i 0.268521 + 0.307781i
\(857\) 20.7800i 0.709833i 0.934898 + 0.354916i \(0.115491\pi\)
−0.934898 + 0.354916i \(0.884509\pi\)
\(858\) 0 0
\(859\) 12.0088 0.409735 0.204868 0.978790i \(-0.434324\pi\)
0.204868 + 0.978790i \(0.434324\pi\)
\(860\) 66.9975 34.7310i 2.28460 1.18432i
\(861\) 0 0
\(862\) 27.0200 6.58725i 0.920306 0.224363i
\(863\) −11.5254 −0.392328 −0.196164 0.980571i \(-0.562849\pi\)
−0.196164 + 0.980571i \(0.562849\pi\)
\(864\) 0 0
\(865\) −18.8118 −0.639622
\(866\) −16.4872 + 4.01943i −0.560257 + 0.136586i
\(867\) 0 0
\(868\) −19.3505 + 10.0312i −0.656800 + 0.340480i
\(869\) −11.2123 −0.380353
\(870\) 0 0
\(871\) 28.3456i 0.960455i
\(872\) −25.9243 29.7146i −0.877906 1.00626i
\(873\) 0 0
\(874\) 25.0765 6.11342i 0.848225 0.206790i
\(875\) 31.3462i 1.05969i
\(876\) 0 0
\(877\) 7.25986i 0.245148i −0.992459 0.122574i \(-0.960885\pi\)
0.992459 0.122574i \(-0.0391149\pi\)
\(878\) 2.61978 + 10.7460i 0.0884134 + 0.362660i
\(879\) 0 0
\(880\) −25.5295 18.0065i −0.860598 0.607000i
\(881\) 10.7411i 0.361876i 0.983495 + 0.180938i \(0.0579133\pi\)
−0.983495 + 0.180938i \(0.942087\pi\)
\(882\) 0 0
\(883\) 9.09009 0.305906 0.152953 0.988233i \(-0.451122\pi\)
0.152953 + 0.988233i \(0.451122\pi\)
\(884\) −2.75759 + 1.42951i −0.0927477 + 0.0480797i
\(885\) 0 0
\(886\) 10.0440 + 41.1991i 0.337434 + 1.38411i
\(887\) 29.0988 0.977041 0.488521 0.872552i \(-0.337537\pi\)
0.488521 + 0.872552i \(0.337537\pi\)
\(888\) 0 0
\(889\) 18.0939 0.606850
\(890\) −7.97863 32.7273i −0.267444 1.09702i
\(891\) 0 0
\(892\) 1.22451 + 2.36212i 0.0409995 + 0.0790898i
\(893\) 1.00042 0.0334778
\(894\) 0 0
\(895\) 11.0183i 0.368301i
\(896\) −11.2536 + 1.16426i −0.375958 + 0.0388952i
\(897\) 0 0
\(898\) 14.0983 + 57.8294i 0.470466 + 1.92979i
\(899\) 38.4574i 1.28262i
\(900\) 0 0
\(901\) 4.20539i 0.140102i
\(902\) 18.6510 4.54694i 0.621010 0.151397i
\(903\) 0 0
\(904\) −13.9355 15.9730i −0.463489 0.531254i
\(905\) 43.3938i 1.44246i
\(906\) 0 0
\(907\) 29.9962 0.996008 0.498004 0.867175i \(-0.334067\pi\)
0.498004 + 0.867175i \(0.334067\pi\)
\(908\) −5.84165 11.2688i −0.193862 0.373967i
\(909\) 0 0
\(910\) 17.6774 4.30960i 0.586000 0.142862i
\(911\) 43.3253 1.43543 0.717716 0.696336i \(-0.245188\pi\)
0.717716 + 0.696336i \(0.245188\pi\)
\(912\) 0 0
\(913\) 4.89919 0.162140
\(914\) 16.7151 4.07499i 0.552886 0.134789i
\(915\) 0 0
\(916\) −12.8791 24.8442i −0.425536 0.820877i
\(917\) 9.32625 0.307980
\(918\) 0 0
\(919\) 44.6079i 1.47148i −0.677264 0.735740i \(-0.736835\pi\)
0.677264 0.735740i \(-0.263165\pi\)
\(920\) 53.2398 46.4486i 1.75526 1.53137i
\(921\) 0 0
\(922\) 39.8526 9.71570i 1.31247 0.319970i
\(923\) 27.7600i 0.913732i
\(924\) 0 0
\(925\) 142.011i 4.66929i
\(926\) 1.74622 + 7.16275i 0.0573842 + 0.235383i
\(927\) 0 0
\(928\) −7.31462 + 18.5737i −0.240114 + 0.609712i
\(929\) 7.73251i 0.253695i −0.991922 0.126848i \(-0.959514\pi\)
0.991922 0.126848i \(-0.0404860\pi\)
\(930\) 0 0
\(931\) −3.05596 −0.100155
\(932\) −12.9598 24.9999i −0.424512 0.818900i
\(933\) 0 0
\(934\) −4.12209 16.9083i −0.134879 0.553256i
\(935\) 3.94326 0.128958
\(936\) 0 0
\(937\) 25.6210 0.837002 0.418501 0.908216i \(-0.362556\pi\)
0.418501 + 0.908216i \(0.362556\pi\)
\(938\) 3.08667 + 12.6611i 0.100783 + 0.413401i
\(939\) 0 0
\(940\) 2.43125 1.26034i 0.0792985 0.0411077i
\(941\) 30.8529 1.00578 0.502888 0.864352i \(-0.332271\pi\)
0.502888 + 0.864352i \(0.332271\pi\)
\(942\) 0 0
\(943\) 43.4161i 1.41382i
\(944\) −19.5473 13.7872i −0.636209 0.448734i
\(945\) 0 0
\(946\) 5.64252 + 23.1449i 0.183454 + 0.752506i
\(947\) 39.5682i 1.28579i 0.765953 + 0.642897i \(0.222268\pi\)
−0.765953 + 0.642897i \(0.777732\pi\)
\(948\) 0 0
\(949\) 1.28114i 0.0415876i
\(950\) −52.4614 + 12.7896i −1.70207 + 0.414951i
\(951\) 0 0
\(952\) 1.07607 0.938805i 0.0348755 0.0304268i
\(953\) 4.36408i 0.141366i 0.997499 + 0.0706832i \(0.0225179\pi\)
−0.997499 + 0.0706832i \(0.977482\pi\)
\(954\) 0 0
\(955\) 76.0015 2.45935
\(956\) −23.3912 + 12.1258i −0.756525 + 0.392177i
\(957\) 0 0
\(958\) 27.4258 6.68616i 0.886086 0.216020i
\(959\) 6.53069 0.210887
\(960\) 0 0
\(961\) −87.7668 −2.83119
\(962\) 48.0371 11.7110i 1.54878 0.377579i
\(963\) 0 0
\(964\) 13.2610 6.87440i 0.427108 0.221410i
\(965\) −70.1442 −2.25802
\(966\) 0 0
\(967\) 51.1938i 1.64628i −0.567837 0.823141i \(-0.692220\pi\)
0.567837 0.823141i \(-0.307780\pi\)
\(968\) −16.0129 + 13.9704i −0.514676 + 0.449025i
\(969\) 0 0
\(970\) −35.7044 + 8.70440i −1.14640 + 0.279482i
\(971\) 47.6773i 1.53004i 0.644008 + 0.765019i \(0.277270\pi\)
−0.644008 + 0.765019i \(0.722730\pi\)
\(972\) 0 0
\(973\) 10.1589i 0.325678i
\(974\) −1.88055 7.71376i −0.0602566 0.247165i
\(975\) 0 0
\(976\) −43.4032 30.6133i −1.38930 0.979909i
\(977\) 10.1656i 0.325225i −0.986690 0.162613i \(-0.948008\pi\)
0.986690 0.162613i \(-0.0519921\pi\)
\(978\) 0 0
\(979\) 10.6340 0.339864
\(980\) −7.42668 + 3.84993i −0.237236 + 0.122982i
\(981\) 0 0
\(982\) 3.03855 + 12.4637i 0.0969639 + 0.397734i
\(983\) 7.35716 0.234657 0.117328 0.993093i \(-0.462567\pi\)
0.117328 + 0.993093i \(0.462567\pi\)
\(984\) 0 0
\(985\) −80.4068 −2.56197
\(986\) −0.596788 2.44795i −0.0190056 0.0779586i
\(987\) 0 0
\(988\) 8.65253 + 16.6911i 0.275274 + 0.531014i
\(989\) −53.8772 −1.71320
\(990\) 0 0
\(991\) 18.5907i 0.590552i −0.955412 0.295276i \(-0.904588\pi\)
0.955412 0.295276i \(-0.0954116\pi\)
\(992\) −57.3607 22.5895i −1.82120 0.717218i
\(993\) 0 0
\(994\) 3.02290 + 12.3996i 0.0958806 + 0.393290i
\(995\) 7.06577i 0.224000i
\(996\) 0 0
\(997\) 4.19288i 0.132790i 0.997793 + 0.0663950i \(0.0211497\pi\)
−0.997793 + 0.0663950i \(0.978850\pi\)
\(998\) 50.0601 12.2042i 1.58462 0.386317i
\(999\) 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 1512.2.j.d.323.4 yes 48
3.2 odd 2 inner 1512.2.j.d.323.45 yes 48
4.3 odd 2 6048.2.j.d.5615.47 48
8.3 odd 2 inner 1512.2.j.d.323.46 yes 48
8.5 even 2 6048.2.j.d.5615.1 48
12.11 even 2 6048.2.j.d.5615.2 48
24.5 odd 2 6048.2.j.d.5615.48 48
24.11 even 2 inner 1512.2.j.d.323.3 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.3 48 24.11 even 2 inner
1512.2.j.d.323.4 yes 48 1.1 even 1 trivial
1512.2.j.d.323.45 yes 48 3.2 odd 2 inner
1512.2.j.d.323.46 yes 48 8.3 odd 2 inner
6048.2.j.d.5615.1 48 8.5 even 2
6048.2.j.d.5615.2 48 12.11 even 2
6048.2.j.d.5615.47 48 4.3 odd 2
6048.2.j.d.5615.48 48 24.5 odd 2