Properties

Label 475.3.c.i.151.5
Level $475$
Weight $3$
Character 475.151
Analytic conductor $12.943$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,3,Mod(151,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (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: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 151.5
Root \(-1.41833i\) of defining polynomial
Character \(\chi\) \(=\) 475.151
Dual form 475.3.c.i.151.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41833i q^{2} -1.48141i q^{3} +1.98835 q^{4} -2.10112 q^{6} -2.50491 q^{7} -8.49344i q^{8} +6.80543 q^{9} +O(q^{10})\) \(q-1.41833i q^{2} -1.48141i q^{3} +1.98835 q^{4} -2.10112 q^{6} -2.50491 q^{7} -8.49344i q^{8} +6.80543 q^{9} +16.7700 q^{11} -2.94555i q^{12} +12.8905i q^{13} +3.55278i q^{14} -4.09308 q^{16} +10.1212 q^{17} -9.65233i q^{18} +(2.53793 - 18.8297i) q^{19} +3.71079i q^{21} -23.7853i q^{22} -18.7736 q^{23} -12.5822 q^{24} +18.2829 q^{26} -23.4143i q^{27} -4.98063 q^{28} -30.4098i q^{29} +56.8943i q^{31} -28.1684i q^{32} -24.8432i q^{33} -14.3552i q^{34} +13.5316 q^{36} -19.1200i q^{37} +(-26.7067 - 3.59961i) q^{38} +19.0961 q^{39} -15.2087i q^{41} +5.26311 q^{42} +68.0964 q^{43} +33.3446 q^{44} +26.6271i q^{46} -33.9534 q^{47} +6.06352i q^{48} -42.7254 q^{49} -14.9937i q^{51} +25.6308i q^{52} +2.77237i q^{53} -33.2091 q^{54} +21.2753i q^{56} +(-27.8945 - 3.75971i) q^{57} -43.1311 q^{58} -41.0403i q^{59} -80.3350 q^{61} +80.6948 q^{62} -17.0470 q^{63} -56.3244 q^{64} -35.2358 q^{66} -73.7575i q^{67} +20.1245 q^{68} +27.8113i q^{69} +119.192i q^{71} -57.8015i q^{72} -66.3948 q^{73} -27.1185 q^{74} +(5.04629 - 37.4401i) q^{76} -42.0073 q^{77} -27.0845i q^{78} -56.0205i q^{79} +26.5628 q^{81} -21.5709 q^{82} +124.244 q^{83} +7.37834i q^{84} -96.5830i q^{86} -45.0493 q^{87} -142.435i q^{88} +9.31740i q^{89} -32.2895i q^{91} -37.3285 q^{92} +84.2836 q^{93} +48.1570i q^{94} -41.7289 q^{96} -7.61982i q^{97} +60.5986i q^{98} +114.127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41833i 0.709164i −0.935025 0.354582i \(-0.884623\pi\)
0.935025 0.354582i \(-0.115377\pi\)
\(3\) 1.48141i 0.493802i −0.969041 0.246901i \(-0.920588\pi\)
0.969041 0.246901i \(-0.0794123\pi\)
\(4\) 1.98835 0.497087
\(5\) 0 0
\(6\) −2.10112 −0.350187
\(7\) −2.50491 −0.357844 −0.178922 0.983863i \(-0.557261\pi\)
−0.178922 + 0.983863i \(0.557261\pi\)
\(8\) 8.49344i 1.06168i
\(9\) 6.80543 0.756159
\(10\) 0 0
\(11\) 16.7700 1.52454 0.762272 0.647257i \(-0.224084\pi\)
0.762272 + 0.647257i \(0.224084\pi\)
\(12\) 2.94555i 0.245463i
\(13\) 12.8905i 0.991576i 0.868444 + 0.495788i \(0.165121\pi\)
−0.868444 + 0.495788i \(0.834879\pi\)
\(14\) 3.55278i 0.253770i
\(15\) 0 0
\(16\) −4.09308 −0.255818
\(17\) 10.1212 0.595366 0.297683 0.954665i \(-0.403786\pi\)
0.297683 + 0.954665i \(0.403786\pi\)
\(18\) 9.65233i 0.536241i
\(19\) 2.53793 18.8297i 0.133575 0.991039i
\(20\) 0 0
\(21\) 3.71079i 0.176704i
\(22\) 23.7853i 1.08115i
\(23\) −18.7736 −0.816244 −0.408122 0.912927i \(-0.633816\pi\)
−0.408122 + 0.912927i \(0.633816\pi\)
\(24\) −12.5822 −0.524260
\(25\) 0 0
\(26\) 18.2829 0.703190
\(27\) 23.4143i 0.867196i
\(28\) −4.98063 −0.177880
\(29\) 30.4098i 1.04861i −0.851529 0.524307i \(-0.824324\pi\)
0.851529 0.524307i \(-0.175676\pi\)
\(30\) 0 0
\(31\) 56.8943i 1.83530i 0.397389 + 0.917650i \(0.369916\pi\)
−0.397389 + 0.917650i \(0.630084\pi\)
\(32\) 28.1684i 0.880263i
\(33\) 24.8432i 0.752824i
\(34\) 14.3552i 0.422212i
\(35\) 0 0
\(36\) 13.5316 0.375877
\(37\) 19.1200i 0.516758i −0.966044 0.258379i \(-0.916812\pi\)
0.966044 0.258379i \(-0.0831883\pi\)
\(38\) −26.7067 3.59961i −0.702809 0.0947267i
\(39\) 19.0961 0.489643
\(40\) 0 0
\(41\) 15.2087i 0.370943i −0.982650 0.185471i \(-0.940619\pi\)
0.982650 0.185471i \(-0.0593812\pi\)
\(42\) 5.26311 0.125312
\(43\) 68.0964 1.58364 0.791819 0.610756i \(-0.209135\pi\)
0.791819 + 0.610756i \(0.209135\pi\)
\(44\) 33.3446 0.757831
\(45\) 0 0
\(46\) 26.6271i 0.578850i
\(47\) −33.9534 −0.722412 −0.361206 0.932486i \(-0.617635\pi\)
−0.361206 + 0.932486i \(0.617635\pi\)
\(48\) 6.06352i 0.126323i
\(49\) −42.7254 −0.871948
\(50\) 0 0
\(51\) 14.9937i 0.293993i
\(52\) 25.6308i 0.492900i
\(53\) 2.77237i 0.0523089i 0.999658 + 0.0261544i \(0.00832617\pi\)
−0.999658 + 0.0261544i \(0.991674\pi\)
\(54\) −33.2091 −0.614984
\(55\) 0 0
\(56\) 21.2753i 0.379916i
\(57\) −27.8945 3.75971i −0.489377 0.0659598i
\(58\) −43.1311 −0.743639
\(59\) 41.0403i 0.695599i −0.937569 0.347799i \(-0.886929\pi\)
0.937569 0.347799i \(-0.113071\pi\)
\(60\) 0 0
\(61\) −80.3350 −1.31697 −0.658484 0.752595i \(-0.728802\pi\)
−0.658484 + 0.752595i \(0.728802\pi\)
\(62\) 80.6948 1.30153
\(63\) −17.0470 −0.270587
\(64\) −56.3244 −0.880068
\(65\) 0 0
\(66\) −35.2358 −0.533875
\(67\) 73.7575i 1.10086i −0.834882 0.550429i \(-0.814464\pi\)
0.834882 0.550429i \(-0.185536\pi\)
\(68\) 20.1245 0.295949
\(69\) 27.8113i 0.403063i
\(70\) 0 0
\(71\) 119.192i 1.67876i 0.543543 + 0.839381i \(0.317083\pi\)
−0.543543 + 0.839381i \(0.682917\pi\)
\(72\) 57.8015i 0.802799i
\(73\) −66.3948 −0.909517 −0.454759 0.890615i \(-0.650275\pi\)
−0.454759 + 0.890615i \(0.650275\pi\)
\(74\) −27.1185 −0.366466
\(75\) 0 0
\(76\) 5.04629 37.4401i 0.0663985 0.492632i
\(77\) −42.0073 −0.545549
\(78\) 27.0845i 0.347237i
\(79\) 56.0205i 0.709120i −0.935033 0.354560i \(-0.884631\pi\)
0.935033 0.354560i \(-0.115369\pi\)
\(80\) 0 0
\(81\) 26.5628 0.327936
\(82\) −21.5709 −0.263059
\(83\) 124.244 1.49692 0.748460 0.663180i \(-0.230793\pi\)
0.748460 + 0.663180i \(0.230793\pi\)
\(84\) 7.37834i 0.0878374i
\(85\) 0 0
\(86\) 96.5830i 1.12306i
\(87\) −45.0493 −0.517808
\(88\) 142.435i 1.61858i
\(89\) 9.31740i 0.104690i 0.998629 + 0.0523450i \(0.0166695\pi\)
−0.998629 + 0.0523450i \(0.983330\pi\)
\(90\) 0 0
\(91\) 32.2895i 0.354830i
\(92\) −37.3285 −0.405744
\(93\) 84.2836 0.906276
\(94\) 48.1570i 0.512308i
\(95\) 0 0
\(96\) −41.7289 −0.434676
\(97\) 7.61982i 0.0785548i −0.999228 0.0392774i \(-0.987494\pi\)
0.999228 0.0392774i \(-0.0125056\pi\)
\(98\) 60.5986i 0.618353i
\(99\) 114.127 1.15280
\(100\) 0 0
\(101\) −32.1518 −0.318335 −0.159168 0.987252i \(-0.550881\pi\)
−0.159168 + 0.987252i \(0.550881\pi\)
\(102\) −21.2659 −0.208489
\(103\) 43.2792i 0.420186i 0.977681 + 0.210093i \(0.0673767\pi\)
−0.977681 + 0.210093i \(0.932623\pi\)
\(104\) 109.485 1.05274
\(105\) 0 0
\(106\) 3.93213 0.0370956
\(107\) 157.913i 1.47582i 0.674899 + 0.737910i \(0.264187\pi\)
−0.674899 + 0.737910i \(0.735813\pi\)
\(108\) 46.5557i 0.431072i
\(109\) 58.7451i 0.538946i 0.963008 + 0.269473i \(0.0868494\pi\)
−0.963008 + 0.269473i \(0.913151\pi\)
\(110\) 0 0
\(111\) −28.3246 −0.255176
\(112\) 10.2528 0.0915428
\(113\) 149.248i 1.32078i 0.750924 + 0.660389i \(0.229609\pi\)
−0.750924 + 0.660389i \(0.770391\pi\)
\(114\) −5.33249 + 39.5635i −0.0467763 + 0.347049i
\(115\) 0 0
\(116\) 60.4653i 0.521253i
\(117\) 87.7254i 0.749790i
\(118\) −58.2086 −0.493293
\(119\) −25.3527 −0.213048
\(120\) 0 0
\(121\) 160.233 1.32424
\(122\) 113.941i 0.933946i
\(123\) −22.5302 −0.183172
\(124\) 113.126i 0.912304i
\(125\) 0 0
\(126\) 24.1782i 0.191891i
\(127\) 97.3189i 0.766290i 0.923688 + 0.383145i \(0.125159\pi\)
−0.923688 + 0.383145i \(0.874841\pi\)
\(128\) 32.7873i 0.256151i
\(129\) 100.878i 0.782004i
\(130\) 0 0
\(131\) 0.432007 0.00329777 0.00164888 0.999999i \(-0.499475\pi\)
0.00164888 + 0.999999i \(0.499475\pi\)
\(132\) 49.3969i 0.374219i
\(133\) −6.35728 + 47.1668i −0.0477991 + 0.354637i
\(134\) −104.612 −0.780689
\(135\) 0 0
\(136\) 85.9640i 0.632088i
\(137\) 134.070 0.978612 0.489306 0.872112i \(-0.337250\pi\)
0.489306 + 0.872112i \(0.337250\pi\)
\(138\) 39.4456 0.285838
\(139\) 16.0211 0.115260 0.0576300 0.998338i \(-0.481646\pi\)
0.0576300 + 0.998338i \(0.481646\pi\)
\(140\) 0 0
\(141\) 50.2987i 0.356729i
\(142\) 169.053 1.19052
\(143\) 216.173i 1.51170i
\(144\) −27.8552 −0.193439
\(145\) 0 0
\(146\) 94.1695i 0.644997i
\(147\) 63.2937i 0.430570i
\(148\) 38.0173i 0.256874i
\(149\) 87.2737 0.585729 0.292865 0.956154i \(-0.405391\pi\)
0.292865 + 0.956154i \(0.405391\pi\)
\(150\) 0 0
\(151\) 125.657i 0.832167i −0.909326 0.416084i \(-0.863402\pi\)
0.909326 0.416084i \(-0.136598\pi\)
\(152\) −159.929 21.5557i −1.05217 0.141814i
\(153\) 68.8793 0.450192
\(154\) 59.5801i 0.386884i
\(155\) 0 0
\(156\) 37.9696 0.243395
\(157\) 76.4512 0.486950 0.243475 0.969907i \(-0.421713\pi\)
0.243475 + 0.969907i \(0.421713\pi\)
\(158\) −79.4554 −0.502882
\(159\) 4.10701 0.0258303
\(160\) 0 0
\(161\) 47.0262 0.292088
\(162\) 37.6748i 0.232560i
\(163\) −252.353 −1.54818 −0.774088 0.633078i \(-0.781791\pi\)
−0.774088 + 0.633078i \(0.781791\pi\)
\(164\) 30.2401i 0.184391i
\(165\) 0 0
\(166\) 176.219i 1.06156i
\(167\) 305.922i 1.83187i 0.401328 + 0.915935i \(0.368549\pi\)
−0.401328 + 0.915935i \(0.631451\pi\)
\(168\) 31.5174 0.187603
\(169\) 2.83522 0.0167765
\(170\) 0 0
\(171\) 17.2717 128.145i 0.101004 0.749383i
\(172\) 135.399 0.787205
\(173\) 104.461i 0.603822i 0.953336 + 0.301911i \(0.0976245\pi\)
−0.953336 + 0.301911i \(0.902376\pi\)
\(174\) 63.8947i 0.367211i
\(175\) 0 0
\(176\) −68.6409 −0.390005
\(177\) −60.7974 −0.343488
\(178\) 13.2151 0.0742423
\(179\) 200.911i 1.12241i 0.827678 + 0.561203i \(0.189661\pi\)
−0.827678 + 0.561203i \(0.810339\pi\)
\(180\) 0 0
\(181\) 18.1515i 0.100284i 0.998742 + 0.0501421i \(0.0159674\pi\)
−0.998742 + 0.0501421i \(0.984033\pi\)
\(182\) −45.7971 −0.251632
\(183\) 119.009i 0.650322i
\(184\) 159.452i 0.866589i
\(185\) 0 0
\(186\) 119.542i 0.642698i
\(187\) 169.733 0.907662
\(188\) −67.5111 −0.359101
\(189\) 58.6506i 0.310321i
\(190\) 0 0
\(191\) 237.592 1.24394 0.621969 0.783042i \(-0.286333\pi\)
0.621969 + 0.783042i \(0.286333\pi\)
\(192\) 83.4393i 0.434580i
\(193\) 243.676i 1.26257i −0.775551 0.631285i \(-0.782528\pi\)
0.775551 0.631285i \(-0.217472\pi\)
\(194\) −10.8074 −0.0557082
\(195\) 0 0
\(196\) −84.9530 −0.433434
\(197\) 102.247 0.519020 0.259510 0.965740i \(-0.416439\pi\)
0.259510 + 0.965740i \(0.416439\pi\)
\(198\) 161.869i 0.817523i
\(199\) −93.7720 −0.471216 −0.235608 0.971848i \(-0.575708\pi\)
−0.235608 + 0.971848i \(0.575708\pi\)
\(200\) 0 0
\(201\) −109.265 −0.543607
\(202\) 45.6018i 0.225752i
\(203\) 76.1739i 0.375241i
\(204\) 29.8126i 0.146140i
\(205\) 0 0
\(206\) 61.3840 0.297981
\(207\) −127.763 −0.617210
\(208\) 52.7618i 0.253663i
\(209\) 42.5610 315.774i 0.203641 1.51088i
\(210\) 0 0
\(211\) 66.2032i 0.313759i −0.987618 0.156880i \(-0.949856\pi\)
0.987618 0.156880i \(-0.0501435\pi\)
\(212\) 5.51244i 0.0260021i
\(213\) 176.572 0.828977
\(214\) 223.972 1.04660
\(215\) 0 0
\(216\) −198.868 −0.920684
\(217\) 142.515i 0.656752i
\(218\) 83.3197 0.382201
\(219\) 98.3577i 0.449122i
\(220\) 0 0
\(221\) 130.468i 0.590351i
\(222\) 40.1735i 0.180962i
\(223\) 226.345i 1.01500i 0.861652 + 0.507500i \(0.169430\pi\)
−0.861652 + 0.507500i \(0.830570\pi\)
\(224\) 70.5593i 0.314997i
\(225\) 0 0
\(226\) 211.682 0.936648
\(227\) 88.6093i 0.390349i −0.980768 0.195175i \(-0.937473\pi\)
0.980768 0.195175i \(-0.0625274\pi\)
\(228\) −55.4640 7.47560i −0.243263 0.0327877i
\(229\) −359.123 −1.56822 −0.784112 0.620619i \(-0.786881\pi\)
−0.784112 + 0.620619i \(0.786881\pi\)
\(230\) 0 0
\(231\) 62.2299i 0.269394i
\(232\) −258.284 −1.11329
\(233\) −242.528 −1.04089 −0.520446 0.853895i \(-0.674234\pi\)
−0.520446 + 0.853895i \(0.674234\pi\)
\(234\) 124.423 0.531724
\(235\) 0 0
\(236\) 81.6024i 0.345773i
\(237\) −82.9892 −0.350165
\(238\) 35.9585i 0.151086i
\(239\) 143.682 0.601182 0.300591 0.953753i \(-0.402816\pi\)
0.300591 + 0.953753i \(0.402816\pi\)
\(240\) 0 0
\(241\) 401.898i 1.66763i 0.552046 + 0.833813i \(0.313847\pi\)
−0.552046 + 0.833813i \(0.686153\pi\)
\(242\) 227.262i 0.939100i
\(243\) 250.079i 1.02913i
\(244\) −159.734 −0.654647
\(245\) 0 0
\(246\) 31.9552i 0.129899i
\(247\) 242.725 + 32.7152i 0.982690 + 0.132450i
\(248\) 483.228 1.94850
\(249\) 184.057i 0.739183i
\(250\) 0 0
\(251\) −45.3071 −0.180506 −0.0902531 0.995919i \(-0.528768\pi\)
−0.0902531 + 0.995919i \(0.528768\pi\)
\(252\) −33.8954 −0.134505
\(253\) −314.833 −1.24440
\(254\) 138.030 0.543425
\(255\) 0 0
\(256\) −271.801 −1.06172
\(257\) 47.2850i 0.183988i −0.995760 0.0919941i \(-0.970676\pi\)
0.995760 0.0919941i \(-0.0293241\pi\)
\(258\) −143.079 −0.554569
\(259\) 47.8940i 0.184919i
\(260\) 0 0
\(261\) 206.952i 0.792920i
\(262\) 0.612728i 0.00233866i
\(263\) 326.709 1.24224 0.621119 0.783716i \(-0.286678\pi\)
0.621119 + 0.783716i \(0.286678\pi\)
\(264\) −211.004 −0.799257
\(265\) 0 0
\(266\) 66.8979 + 9.01671i 0.251496 + 0.0338974i
\(267\) 13.8029 0.0516961
\(268\) 146.656i 0.547223i
\(269\) 415.031i 1.54287i −0.636311 0.771433i \(-0.719540\pi\)
0.636311 0.771433i \(-0.280460\pi\)
\(270\) 0 0
\(271\) −363.836 −1.34257 −0.671285 0.741200i \(-0.734257\pi\)
−0.671285 + 0.741200i \(0.734257\pi\)
\(272\) −41.4270 −0.152305
\(273\) −47.8339 −0.175216
\(274\) 190.155i 0.693996i
\(275\) 0 0
\(276\) 55.2986i 0.200357i
\(277\) −332.903 −1.20182 −0.600909 0.799318i \(-0.705194\pi\)
−0.600909 + 0.799318i \(0.705194\pi\)
\(278\) 22.7232i 0.0817382i
\(279\) 387.190i 1.38778i
\(280\) 0 0
\(281\) 258.538i 0.920063i 0.887903 + 0.460032i \(0.152162\pi\)
−0.887903 + 0.460032i \(0.847838\pi\)
\(282\) 71.3401 0.252979
\(283\) 191.340 0.676114 0.338057 0.941126i \(-0.390230\pi\)
0.338057 + 0.941126i \(0.390230\pi\)
\(284\) 236.995i 0.834491i
\(285\) 0 0
\(286\) 306.605 1.07204
\(287\) 38.0963i 0.132740i
\(288\) 191.698i 0.665619i
\(289\) −186.561 −0.645539
\(290\) 0 0
\(291\) −11.2881 −0.0387906
\(292\) −132.016 −0.452109
\(293\) 479.052i 1.63499i 0.575935 + 0.817495i \(0.304638\pi\)
−0.575935 + 0.817495i \(0.695362\pi\)
\(294\) 89.7712 0.305344
\(295\) 0 0
\(296\) −162.395 −0.548631
\(297\) 392.657i 1.32208i
\(298\) 123.783i 0.415378i
\(299\) 242.001i 0.809368i
\(300\) 0 0
\(301\) −170.575 −0.566695
\(302\) −178.223 −0.590143
\(303\) 47.6300i 0.157195i
\(304\) −10.3880 + 77.0716i −0.0341709 + 0.253525i
\(305\) 0 0
\(306\) 97.6934i 0.319260i
\(307\) 270.388i 0.880742i 0.897816 + 0.440371i \(0.145153\pi\)
−0.897816 + 0.440371i \(0.854847\pi\)
\(308\) −83.5251 −0.271185
\(309\) 64.1140 0.207489
\(310\) 0 0
\(311\) 20.7303 0.0666569 0.0333285 0.999444i \(-0.489389\pi\)
0.0333285 + 0.999444i \(0.489389\pi\)
\(312\) 162.191i 0.519844i
\(313\) 337.610 1.07863 0.539313 0.842106i \(-0.318684\pi\)
0.539313 + 0.842106i \(0.318684\pi\)
\(314\) 108.433i 0.345327i
\(315\) 0 0
\(316\) 111.388i 0.352494i
\(317\) 392.652i 1.23865i −0.785135 0.619324i \(-0.787407\pi\)
0.785135 0.619324i \(-0.212593\pi\)
\(318\) 5.82508i 0.0183179i
\(319\) 509.972i 1.59866i
\(320\) 0 0
\(321\) 233.933 0.728763
\(322\) 66.6985i 0.207138i
\(323\) 25.6870 190.580i 0.0795262 0.590031i
\(324\) 52.8161 0.163013
\(325\) 0 0
\(326\) 357.919i 1.09791i
\(327\) 87.0254 0.266133
\(328\) −129.174 −0.393822
\(329\) 85.0501 0.258511
\(330\) 0 0
\(331\) 101.930i 0.307946i −0.988075 0.153973i \(-0.950793\pi\)
0.988075 0.153973i \(-0.0492069\pi\)
\(332\) 247.041 0.744100
\(333\) 130.120i 0.390751i
\(334\) 433.898 1.29909
\(335\) 0 0
\(336\) 15.1886i 0.0452041i
\(337\) 356.299i 1.05727i −0.848850 0.528633i \(-0.822705\pi\)
0.848850 0.528633i \(-0.177295\pi\)
\(338\) 4.02127i 0.0118973i
\(339\) 221.097 0.652203
\(340\) 0 0
\(341\) 954.117i 2.79800i
\(342\) −181.751 24.4969i −0.531435 0.0716285i
\(343\) 229.764 0.669866
\(344\) 578.372i 1.68132i
\(345\) 0 0
\(346\) 148.160 0.428208
\(347\) −457.590 −1.31870 −0.659352 0.751835i \(-0.729169\pi\)
−0.659352 + 0.751835i \(0.729169\pi\)
\(348\) −89.5737 −0.257396
\(349\) −292.657 −0.838559 −0.419280 0.907857i \(-0.637717\pi\)
−0.419280 + 0.907857i \(0.637717\pi\)
\(350\) 0 0
\(351\) 301.822 0.859891
\(352\) 472.384i 1.34200i
\(353\) −643.379 −1.82260 −0.911302 0.411739i \(-0.864922\pi\)
−0.911302 + 0.411739i \(0.864922\pi\)
\(354\) 86.2306i 0.243589i
\(355\) 0 0
\(356\) 18.5262i 0.0520400i
\(357\) 37.5577i 0.105204i
\(358\) 284.957 0.795970
\(359\) −512.837 −1.42851 −0.714257 0.699883i \(-0.753235\pi\)
−0.714257 + 0.699883i \(0.753235\pi\)
\(360\) 0 0
\(361\) −348.118 95.5771i −0.964315 0.264756i
\(362\) 25.7447 0.0711180
\(363\) 237.370i 0.653911i
\(364\) 64.2028i 0.176381i
\(365\) 0 0
\(366\) 168.794 0.461185
\(367\) −374.320 −1.01994 −0.509972 0.860191i \(-0.670344\pi\)
−0.509972 + 0.860191i \(0.670344\pi\)
\(368\) 76.8419 0.208809
\(369\) 103.501i 0.280492i
\(370\) 0 0
\(371\) 6.94454i 0.0187184i
\(372\) 167.585 0.450498
\(373\) 338.585i 0.907736i −0.891069 0.453868i \(-0.850044\pi\)
0.891069 0.453868i \(-0.149956\pi\)
\(374\) 240.737i 0.643681i
\(375\) 0 0
\(376\) 288.381i 0.766970i
\(377\) 391.998 1.03978
\(378\) 83.1858 0.220068
\(379\) 511.061i 1.34845i 0.738527 + 0.674224i \(0.235522\pi\)
−0.738527 + 0.674224i \(0.764478\pi\)
\(380\) 0 0
\(381\) 144.169 0.378396
\(382\) 336.984i 0.882156i
\(383\) 196.052i 0.511886i 0.966692 + 0.255943i \(0.0823859\pi\)
−0.966692 + 0.255943i \(0.917614\pi\)
\(384\) −48.5713 −0.126488
\(385\) 0 0
\(386\) −345.612 −0.895369
\(387\) 463.426 1.19748
\(388\) 15.1509i 0.0390486i
\(389\) −271.187 −0.697140 −0.348570 0.937283i \(-0.613333\pi\)
−0.348570 + 0.937283i \(0.613333\pi\)
\(390\) 0 0
\(391\) −190.012 −0.485964
\(392\) 362.886i 0.925729i
\(393\) 0.639979i 0.00162844i
\(394\) 145.020i 0.368070i
\(395\) 0 0
\(396\) 226.924 0.573041
\(397\) 538.992 1.35766 0.678831 0.734294i \(-0.262487\pi\)
0.678831 + 0.734294i \(0.262487\pi\)
\(398\) 132.999i 0.334169i
\(399\) 69.8732 + 9.41772i 0.175121 + 0.0236033i
\(400\) 0 0
\(401\) 247.355i 0.616846i −0.951249 0.308423i \(-0.900199\pi\)
0.951249 0.308423i \(-0.0998013\pi\)
\(402\) 154.973i 0.385506i
\(403\) −733.396 −1.81984
\(404\) −63.9290 −0.158240
\(405\) 0 0
\(406\) 108.039 0.266107
\(407\) 320.643i 0.787820i
\(408\) −127.348 −0.312127
\(409\) 705.490i 1.72491i 0.506130 + 0.862457i \(0.331075\pi\)
−0.506130 + 0.862457i \(0.668925\pi\)
\(410\) 0 0
\(411\) 198.612i 0.483241i
\(412\) 86.0540i 0.208869i
\(413\) 102.802i 0.248916i
\(414\) 181.209i 0.437703i
\(415\) 0 0
\(416\) 363.105 0.872848
\(417\) 23.7338i 0.0569157i
\(418\) −447.871 60.3655i −1.07146 0.144415i
\(419\) −53.2615 −0.127116 −0.0635578 0.997978i \(-0.520245\pi\)
−0.0635578 + 0.997978i \(0.520245\pi\)
\(420\) 0 0
\(421\) 632.760i 1.50299i −0.659738 0.751496i \(-0.729333\pi\)
0.659738 0.751496i \(-0.270667\pi\)
\(422\) −93.8979 −0.222507
\(423\) −231.067 −0.546258
\(424\) 23.5470 0.0555353
\(425\) 0 0
\(426\) 250.437i 0.587880i
\(427\) 201.232 0.471269
\(428\) 313.985i 0.733611i
\(429\) 320.241 0.746482
\(430\) 0 0
\(431\) 114.829i 0.266425i −0.991088 0.133212i \(-0.957471\pi\)
0.991088 0.133212i \(-0.0425293\pi\)
\(432\) 95.8366i 0.221844i
\(433\) 757.439i 1.74928i −0.484772 0.874641i \(-0.661097\pi\)
0.484772 0.874641i \(-0.338903\pi\)
\(434\) −202.133 −0.465744
\(435\) 0 0
\(436\) 116.806i 0.267903i
\(437\) −47.6461 + 353.502i −0.109030 + 0.808929i
\(438\) 139.503 0.318501
\(439\) 290.888i 0.662614i 0.943523 + 0.331307i \(0.107490\pi\)
−0.943523 + 0.331307i \(0.892510\pi\)
\(440\) 0 0
\(441\) −290.765 −0.659331
\(442\) 185.046 0.418655
\(443\) −251.148 −0.566926 −0.283463 0.958983i \(-0.591483\pi\)
−0.283463 + 0.958983i \(0.591483\pi\)
\(444\) −56.3191 −0.126845
\(445\) 0 0
\(446\) 321.031 0.719801
\(447\) 129.288i 0.289235i
\(448\) 141.087 0.314927
\(449\) 569.985i 1.26945i −0.772736 0.634727i \(-0.781112\pi\)
0.772736 0.634727i \(-0.218888\pi\)
\(450\) 0 0
\(451\) 255.049i 0.565519i
\(452\) 296.757i 0.656541i
\(453\) −186.149 −0.410926
\(454\) −125.677 −0.276822
\(455\) 0 0
\(456\) −31.9328 + 236.920i −0.0700281 + 0.519562i
\(457\) −97.0937 −0.212459 −0.106229 0.994342i \(-0.533878\pi\)
−0.106229 + 0.994342i \(0.533878\pi\)
\(458\) 509.355i 1.11213i
\(459\) 236.981i 0.516299i
\(460\) 0 0
\(461\) 242.929 0.526961 0.263480 0.964665i \(-0.415130\pi\)
0.263480 + 0.964665i \(0.415130\pi\)
\(462\) 88.2624 0.191044
\(463\) 666.435 1.43938 0.719692 0.694293i \(-0.244283\pi\)
0.719692 + 0.694293i \(0.244283\pi\)
\(464\) 124.470i 0.268254i
\(465\) 0 0
\(466\) 343.984i 0.738162i
\(467\) −453.552 −0.971204 −0.485602 0.874180i \(-0.661399\pi\)
−0.485602 + 0.874180i \(0.661399\pi\)
\(468\) 174.429i 0.372711i
\(469\) 184.756i 0.393936i
\(470\) 0 0
\(471\) 113.255i 0.240457i
\(472\) −348.573 −0.738503
\(473\) 1141.98 2.41433
\(474\) 117.706i 0.248324i
\(475\) 0 0
\(476\) −50.4101 −0.105904
\(477\) 18.8672i 0.0395539i
\(478\) 203.789i 0.426336i
\(479\) 922.912 1.92675 0.963373 0.268163i \(-0.0864167\pi\)
0.963373 + 0.268163i \(0.0864167\pi\)
\(480\) 0 0
\(481\) 246.467 0.512405
\(482\) 570.023 1.18262
\(483\) 69.6649i 0.144234i
\(484\) 318.598 0.658260
\(485\) 0 0
\(486\) −354.694 −0.729822
\(487\) 345.986i 0.710444i 0.934782 + 0.355222i \(0.115595\pi\)
−0.934782 + 0.355222i \(0.884405\pi\)
\(488\) 682.321i 1.39820i
\(489\) 373.837i 0.764493i
\(490\) 0 0
\(491\) 237.336 0.483372 0.241686 0.970354i \(-0.422300\pi\)
0.241686 + 0.970354i \(0.422300\pi\)
\(492\) −44.7979 −0.0910526
\(493\) 307.785i 0.624310i
\(494\) 46.4008 344.263i 0.0939287 0.696888i
\(495\) 0 0
\(496\) 232.873i 0.469502i
\(497\) 298.566i 0.600736i
\(498\) −261.052 −0.524202
\(499\) 352.497 0.706407 0.353204 0.935547i \(-0.385092\pi\)
0.353204 + 0.935547i \(0.385092\pi\)
\(500\) 0 0
\(501\) 453.195 0.904581
\(502\) 64.2602i 0.128008i
\(503\) −191.562 −0.380839 −0.190420 0.981703i \(-0.560985\pi\)
−0.190420 + 0.981703i \(0.560985\pi\)
\(504\) 144.788i 0.287277i
\(505\) 0 0
\(506\) 446.536i 0.882483i
\(507\) 4.20012i 0.00828425i
\(508\) 193.504i 0.380913i
\(509\) 372.483i 0.731794i −0.930655 0.365897i \(-0.880762\pi\)
0.930655 0.365897i \(-0.119238\pi\)
\(510\) 0 0
\(511\) 166.313 0.325466
\(512\) 254.353i 0.496783i
\(513\) −440.885 59.4238i −0.859424 0.115836i
\(514\) −67.0656 −0.130478
\(515\) 0 0
\(516\) 200.581i 0.388724i
\(517\) −569.397 −1.10135
\(518\) 67.9293 0.131138
\(519\) 154.749 0.298169
\(520\) 0 0
\(521\) 304.679i 0.584797i 0.956297 + 0.292399i \(0.0944534\pi\)
−0.956297 + 0.292399i \(0.905547\pi\)
\(522\) −293.526 −0.562310
\(523\) 721.855i 1.38022i 0.723705 + 0.690110i \(0.242438\pi\)
−0.723705 + 0.690110i \(0.757562\pi\)
\(524\) 0.858981 0.00163928
\(525\) 0 0
\(526\) 463.380i 0.880950i
\(527\) 575.840i 1.09268i
\(528\) 101.685i 0.192586i
\(529\) −176.552 −0.333746
\(530\) 0 0
\(531\) 279.297i 0.525983i
\(532\) −12.6405 + 93.7840i −0.0237603 + 0.176286i
\(533\) 196.047 0.367818
\(534\) 19.5770i 0.0366610i
\(535\) 0 0
\(536\) −626.455 −1.16876
\(537\) 297.631 0.554247
\(538\) −588.650 −1.09414
\(539\) −716.505 −1.32932
\(540\) 0 0
\(541\) −652.655 −1.20639 −0.603193 0.797595i \(-0.706105\pi\)
−0.603193 + 0.797595i \(0.706105\pi\)
\(542\) 516.039i 0.952101i
\(543\) 26.8897 0.0495206
\(544\) 285.099i 0.524079i
\(545\) 0 0
\(546\) 67.8441i 0.124257i
\(547\) 1019.58i 1.86394i 0.362531 + 0.931972i \(0.381913\pi\)
−0.362531 + 0.931972i \(0.618087\pi\)
\(548\) 266.578 0.486455
\(549\) −546.715 −0.995837
\(550\) 0 0
\(551\) −572.609 77.1780i −1.03922 0.140069i
\(552\) 236.214 0.427924
\(553\) 140.326i 0.253755i
\(554\) 472.166i 0.852285i
\(555\) 0 0
\(556\) 31.8556 0.0572943
\(557\) 209.280 0.375727 0.187863 0.982195i \(-0.439844\pi\)
0.187863 + 0.982195i \(0.439844\pi\)
\(558\) 549.163 0.984163
\(559\) 877.796i 1.57030i
\(560\) 0 0
\(561\) 251.443i 0.448206i
\(562\) 366.691 0.652475
\(563\) 9.61456i 0.0170774i 0.999964 + 0.00853868i \(0.00271798\pi\)
−0.999964 + 0.00853868i \(0.997282\pi\)
\(564\) 100.011i 0.177325i
\(565\) 0 0
\(566\) 271.383i 0.479476i
\(567\) −66.5375 −0.117350
\(568\) 1012.35 1.78231
\(569\) 734.454i 1.29078i 0.763853 + 0.645390i \(0.223305\pi\)
−0.763853 + 0.645390i \(0.776695\pi\)
\(570\) 0 0
\(571\) 26.9034 0.0471163 0.0235581 0.999722i \(-0.492501\pi\)
0.0235581 + 0.999722i \(0.492501\pi\)
\(572\) 429.828i 0.751447i
\(573\) 351.971i 0.614260i
\(574\) 54.0330 0.0941342
\(575\) 0 0
\(576\) −383.312 −0.665472
\(577\) −722.739 −1.25258 −0.626290 0.779590i \(-0.715428\pi\)
−0.626290 + 0.779590i \(0.715428\pi\)
\(578\) 264.604i 0.457793i
\(579\) −360.983 −0.623460
\(580\) 0 0
\(581\) −311.221 −0.535664
\(582\) 16.0102i 0.0275089i
\(583\) 46.4926i 0.0797472i
\(584\) 563.920i 0.965616i
\(585\) 0 0
\(586\) 679.453 1.15948
\(587\) 539.053 0.918318 0.459159 0.888354i \(-0.348151\pi\)
0.459159 + 0.888354i \(0.348151\pi\)
\(588\) 125.850i 0.214031i
\(589\) 1071.30 + 144.394i 1.81885 + 0.245151i
\(590\) 0 0
\(591\) 151.469i 0.256293i
\(592\) 78.2599i 0.132196i
\(593\) −349.866 −0.589994 −0.294997 0.955498i \(-0.595319\pi\)
−0.294997 + 0.955498i \(0.595319\pi\)
\(594\) −556.916 −0.937570
\(595\) 0 0
\(596\) 173.530 0.291158
\(597\) 138.914i 0.232688i
\(598\) −343.237 −0.573974
\(599\) 432.796i 0.722530i −0.932463 0.361265i \(-0.882345\pi\)
0.932463 0.361265i \(-0.117655\pi\)
\(600\) 0 0
\(601\) 548.198i 0.912143i −0.889943 0.456072i \(-0.849256\pi\)
0.889943 0.456072i \(-0.150744\pi\)
\(602\) 241.932i 0.401880i
\(603\) 501.952i 0.832425i
\(604\) 249.850i 0.413659i
\(605\) 0 0
\(606\) 67.5549 0.111477
\(607\) 145.705i 0.240040i −0.992771 0.120020i \(-0.961704\pi\)
0.992771 0.120020i \(-0.0382959\pi\)
\(608\) −530.404 71.4895i −0.872375 0.117581i
\(609\) 112.844 0.185295
\(610\) 0 0
\(611\) 437.675i 0.716326i
\(612\) 136.956 0.223784
\(613\) 572.788 0.934401 0.467201 0.884151i \(-0.345263\pi\)
0.467201 + 0.884151i \(0.345263\pi\)
\(614\) 383.498 0.624590
\(615\) 0 0
\(616\) 356.786i 0.579199i
\(617\) −548.520 −0.889012 −0.444506 0.895776i \(-0.646621\pi\)
−0.444506 + 0.895776i \(0.646621\pi\)
\(618\) 90.9347i 0.147144i
\(619\) 317.700 0.513248 0.256624 0.966511i \(-0.417390\pi\)
0.256624 + 0.966511i \(0.417390\pi\)
\(620\) 0 0
\(621\) 439.570i 0.707843i
\(622\) 29.4024i 0.0472707i
\(623\) 23.3392i 0.0374627i
\(624\) −78.1618 −0.125259
\(625\) 0 0
\(626\) 478.841i 0.764922i
\(627\) −467.790 63.0502i −0.746077 0.100559i
\(628\) 152.011 0.242057
\(629\) 193.518i 0.307660i
\(630\) 0 0
\(631\) −320.013 −0.507152 −0.253576 0.967315i \(-0.581607\pi\)
−0.253576 + 0.967315i \(0.581607\pi\)
\(632\) −475.807 −0.752859
\(633\) −98.0739 −0.154935
\(634\) −556.908 −0.878405
\(635\) 0 0
\(636\) 8.16616 0.0128399
\(637\) 550.752i 0.864603i
\(638\) −723.308 −1.13371
\(639\) 811.154i 1.26941i
\(640\) 0 0
\(641\) 529.676i 0.826328i 0.910657 + 0.413164i \(0.135576\pi\)
−0.910657 + 0.413164i \(0.864424\pi\)
\(642\) 331.794i 0.516812i
\(643\) 876.680 1.36342 0.681710 0.731622i \(-0.261236\pi\)
0.681710 + 0.731622i \(0.261236\pi\)
\(644\) 93.5044 0.145193
\(645\) 0 0
\(646\) −270.305 36.4325i −0.418428 0.0563971i
\(647\) 664.468 1.02700 0.513499 0.858090i \(-0.328349\pi\)
0.513499 + 0.858090i \(0.328349\pi\)
\(648\) 225.610i 0.348163i
\(649\) 688.246i 1.06047i
\(650\) 0 0
\(651\) −211.123 −0.324305
\(652\) −501.765 −0.769578
\(653\) 502.217 0.769091 0.384546 0.923106i \(-0.374358\pi\)
0.384546 + 0.923106i \(0.374358\pi\)
\(654\) 123.430i 0.188732i
\(655\) 0 0
\(656\) 62.2503i 0.0948937i
\(657\) −451.845 −0.687740
\(658\) 120.629i 0.183327i
\(659\) 760.007i 1.15327i 0.817001 + 0.576636i \(0.195635\pi\)
−0.817001 + 0.576636i \(0.804365\pi\)
\(660\) 0 0
\(661\) 855.497i 1.29425i −0.762385 0.647123i \(-0.775972\pi\)
0.762385 0.647123i \(-0.224028\pi\)
\(662\) −144.570 −0.218384
\(663\) 193.276 0.291517
\(664\) 1055.26i 1.58925i
\(665\) 0 0
\(666\) −184.553 −0.277107
\(667\) 570.902i 0.855925i
\(668\) 608.280i 0.910598i
\(669\) 335.309 0.501209
\(670\) 0 0
\(671\) −1347.22 −2.00778
\(672\) 104.527 0.155546
\(673\) 413.053i 0.613749i −0.951750 0.306875i \(-0.900717\pi\)
0.951750 0.306875i \(-0.0992832\pi\)
\(674\) −505.348 −0.749775
\(675\) 0 0
\(676\) 5.63740 0.00833936
\(677\) 184.789i 0.272952i −0.990643 0.136476i \(-0.956422\pi\)
0.990643 0.136476i \(-0.0435777\pi\)
\(678\) 313.588i 0.462519i
\(679\) 19.0870i 0.0281104i
\(680\) 0 0
\(681\) −131.266 −0.192755
\(682\) 1353.25 1.98424
\(683\) 1166.19i 1.70745i 0.520723 + 0.853726i \(0.325663\pi\)
−0.520723 + 0.853726i \(0.674337\pi\)
\(684\) 34.3422 254.796i 0.0502078 0.372509i
\(685\) 0 0
\(686\) 325.880i 0.475044i
\(687\) 532.008i 0.774393i
\(688\) −278.724 −0.405122
\(689\) −35.7372 −0.0518683
\(690\) 0 0
\(691\) 1210.49 1.75180 0.875898 0.482496i \(-0.160270\pi\)
0.875898 + 0.482496i \(0.160270\pi\)
\(692\) 207.705i 0.300152i
\(693\) −285.878 −0.412522
\(694\) 649.012i 0.935176i
\(695\) 0 0
\(696\) 382.624i 0.549747i
\(697\) 153.930i 0.220847i
\(698\) 415.084i 0.594676i
\(699\) 359.282i 0.513995i
\(700\) 0 0
\(701\) 799.506 1.14052 0.570261 0.821464i \(-0.306842\pi\)
0.570261 + 0.821464i \(0.306842\pi\)
\(702\) 428.082i 0.609803i
\(703\) −360.025 48.5253i −0.512127 0.0690261i
\(704\) −944.559 −1.34170
\(705\) 0 0
\(706\) 912.522i 1.29252i
\(707\) 80.5375 0.113914
\(708\) −120.886 −0.170743
\(709\) 688.559 0.971169 0.485584 0.874190i \(-0.338607\pi\)
0.485584 + 0.874190i \(0.338607\pi\)
\(710\) 0 0
\(711\) 381.244i 0.536208i
\(712\) 79.1368 0.111147
\(713\) 1068.11i 1.49805i
\(714\) 53.2692 0.0746067
\(715\) 0 0
\(716\) 399.480i 0.557934i
\(717\) 212.852i 0.296865i
\(718\) 727.370i 1.01305i
\(719\) −828.034 −1.15165 −0.575823 0.817574i \(-0.695318\pi\)
−0.575823 + 0.817574i \(0.695318\pi\)
\(720\) 0 0
\(721\) 108.410i 0.150361i
\(722\) −135.560 + 493.745i −0.187756 + 0.683857i
\(723\) 595.375 0.823478
\(724\) 36.0914i 0.0498500i
\(725\) 0 0
\(726\) −336.668 −0.463730
\(727\) 11.4411 0.0157374 0.00786868 0.999969i \(-0.497495\pi\)
0.00786868 + 0.999969i \(0.497495\pi\)
\(728\) −274.249 −0.376716
\(729\) −131.403 −0.180251
\(730\) 0 0
\(731\) 689.219 0.942844
\(732\) 236.631i 0.323266i
\(733\) −1402.01 −1.91270 −0.956352 0.292216i \(-0.905607\pi\)
−0.956352 + 0.292216i \(0.905607\pi\)
\(734\) 530.908i 0.723307i
\(735\) 0 0
\(736\) 528.823i 0.718509i
\(737\) 1236.91i 1.67831i
\(738\) −146.799 −0.198915
\(739\) 57.5237 0.0778400 0.0389200 0.999242i \(-0.487608\pi\)
0.0389200 + 0.999242i \(0.487608\pi\)
\(740\) 0 0
\(741\) 48.4645 359.574i 0.0654041 0.485255i
\(742\) −9.84963 −0.0132744
\(743\) 1315.28i 1.77023i −0.465373 0.885115i \(-0.654080\pi\)
0.465373 0.885115i \(-0.345920\pi\)
\(744\) 715.858i 0.962174i
\(745\) 0 0
\(746\) −480.225 −0.643733
\(747\) 845.537 1.13191
\(748\) 337.488 0.451187
\(749\) 395.557i 0.528114i
\(750\) 0 0
\(751\) 494.439i 0.658374i −0.944265 0.329187i \(-0.893225\pi\)
0.944265 0.329187i \(-0.106775\pi\)
\(752\) 138.974 0.184806
\(753\) 67.1182i 0.0891344i
\(754\) 555.981i 0.737375i
\(755\) 0 0
\(756\) 116.618i 0.154256i
\(757\) −582.272 −0.769183 −0.384591 0.923087i \(-0.625658\pi\)
−0.384591 + 0.923087i \(0.625658\pi\)
\(758\) 724.852 0.956270
\(759\) 466.396i 0.614487i
\(760\) 0 0
\(761\) 879.724 1.15601 0.578006 0.816033i \(-0.303831\pi\)
0.578006 + 0.816033i \(0.303831\pi\)
\(762\) 204.479i 0.268345i
\(763\) 147.151i 0.192859i
\(764\) 472.416 0.618346
\(765\) 0 0
\(766\) 278.066 0.363011
\(767\) 529.030 0.689739
\(768\) 402.647i 0.524280i
\(769\) 664.542 0.864164 0.432082 0.901834i \(-0.357779\pi\)
0.432082 + 0.901834i \(0.357779\pi\)
\(770\) 0 0
\(771\) −70.0483 −0.0908538
\(772\) 484.513i 0.627607i
\(773\) 246.948i 0.319467i −0.987160 0.159733i \(-0.948937\pi\)
0.987160 0.159733i \(-0.0510635\pi\)
\(774\) 657.289i 0.849211i
\(775\) 0 0
\(776\) −64.7185 −0.0834001
\(777\) 70.9505 0.0913133
\(778\) 384.633i 0.494386i
\(779\) −286.375 38.5985i −0.367619 0.0495488i
\(780\) 0 0
\(781\) 1998.85i 2.55935i
\(782\) 269.499i 0.344628i
\(783\) −712.024 −0.909354
\(784\) 174.879 0.223060
\(785\) 0 0
\(786\) −0.907699 −0.00115483
\(787\) 423.704i 0.538379i 0.963087 + 0.269189i \(0.0867558\pi\)
−0.963087 + 0.269189i \(0.913244\pi\)
\(788\) 203.303 0.257998
\(789\) 483.988i 0.613420i
\(790\) 0 0
\(791\) 373.852i 0.472633i
\(792\) 969.331i 1.22390i
\(793\) 1035.56i 1.30587i
\(794\) 764.467i 0.962805i
\(795\) 0 0
\(796\) −186.451 −0.234235
\(797\) 1487.62i 1.86653i −0.359190 0.933264i \(-0.616947\pi\)
0.359190 0.933264i \(-0.383053\pi\)
\(798\) 13.3574 99.1031i 0.0167386 0.124189i
\(799\) −343.650 −0.430100
\(800\) 0 0
\(801\) 63.4090i 0.0791622i
\(802\) −350.831 −0.437445
\(803\) −1113.44 −1.38660
\(804\) −217.257 −0.270220
\(805\) 0 0
\(806\) 1040.20i 1.29056i
\(807\) −614.830 −0.761871
\(808\) 273.080i 0.337970i
\(809\) 775.380 0.958443 0.479221 0.877694i \(-0.340919\pi\)
0.479221 + 0.877694i \(0.340919\pi\)
\(810\) 0 0
\(811\) 315.542i 0.389078i −0.980895 0.194539i \(-0.937679\pi\)
0.980895 0.194539i \(-0.0623211\pi\)
\(812\) 151.460i 0.186527i
\(813\) 538.990i 0.662964i
\(814\) −454.777 −0.558694
\(815\) 0 0
\(816\) 61.3702i 0.0752086i
\(817\) 172.824 1282.24i 0.211535 1.56945i
\(818\) 1000.62 1.22325
\(819\) 219.744i 0.268308i
\(820\) 0 0
\(821\) 878.348 1.06985 0.534926 0.844899i \(-0.320340\pi\)
0.534926 + 0.844899i \(0.320340\pi\)
\(822\) −281.697 −0.342697
\(823\) −1178.55 −1.43202 −0.716009 0.698091i \(-0.754033\pi\)
−0.716009 + 0.698091i \(0.754033\pi\)
\(824\) 367.589 0.446103
\(825\) 0 0
\(826\) 145.807 0.176522
\(827\) 14.2277i 0.0172040i 0.999963 + 0.00860201i \(0.00273814\pi\)
−0.999963 + 0.00860201i \(0.997262\pi\)
\(828\) −254.036 −0.306807
\(829\) 839.046i 1.01212i −0.862499 0.506059i \(-0.831102\pi\)
0.862499 0.506059i \(-0.168898\pi\)
\(830\) 0 0
\(831\) 493.165i 0.593460i
\(832\) 726.049i 0.872655i
\(833\) −432.434 −0.519128
\(834\) −33.6624 −0.0403625
\(835\) 0 0
\(836\) 84.6262 627.869i 0.101227 0.751040i
\(837\) 1332.14 1.59156
\(838\) 75.5422i 0.0901458i
\(839\) 809.554i 0.964903i 0.875923 + 0.482452i \(0.160254\pi\)
−0.875923 + 0.482452i \(0.839746\pi\)
\(840\) 0 0
\(841\) −83.7574 −0.0995927
\(842\) −897.460 −1.06587
\(843\) 383.000 0.454329
\(844\) 131.635i 0.155966i
\(845\) 0 0
\(846\) 327.729i 0.387387i
\(847\) −401.368 −0.473870
\(848\) 11.3475i 0.0133815i
\(849\) 283.453i 0.333867i
\(850\) 0 0
\(851\) 358.952i 0.421800i
\(852\) 351.087 0.412074
\(853\) 1263.87 1.48168 0.740840 0.671681i \(-0.234428\pi\)
0.740840 + 0.671681i \(0.234428\pi\)
\(854\) 285.413i 0.334207i
\(855\) 0 0
\(856\) 1341.22 1.56685
\(857\) 226.840i 0.264691i −0.991204 0.132346i \(-0.957749\pi\)
0.991204 0.132346i \(-0.0422508\pi\)
\(858\) 454.206i 0.529378i
\(859\) −263.283 −0.306499 −0.153249 0.988188i \(-0.548974\pi\)
−0.153249 + 0.988188i \(0.548974\pi\)
\(860\) 0 0
\(861\) 56.4361 0.0655472
\(862\) −162.865 −0.188939
\(863\) 430.419i 0.498747i −0.968407 0.249373i \(-0.919775\pi\)
0.968407 0.249373i \(-0.0802247\pi\)
\(864\) −659.543 −0.763360
\(865\) 0 0
\(866\) −1074.30 −1.24053
\(867\) 276.373i 0.318769i
\(868\) 283.370i 0.326463i
\(869\) 939.463i 1.08109i
\(870\) 0 0
\(871\) 950.771 1.09159
\(872\) 498.948 0.572188
\(873\) 51.8562i 0.0594000i
\(874\) 501.381 + 67.5777i 0.573663 + 0.0773201i
\(875\) 0 0
\(876\) 195.569i 0.223253i
\(877\) 825.399i 0.941162i −0.882357 0.470581i \(-0.844044\pi\)
0.882357 0.470581i \(-0.155956\pi\)
\(878\) 412.574 0.469902
\(879\) 709.671 0.807362
\(880\) 0 0
\(881\) −340.053 −0.385985 −0.192992 0.981200i \(-0.561819\pi\)
−0.192992 + 0.981200i \(0.561819\pi\)
\(882\) 412.400i 0.467574i
\(883\) 1129.39 1.27904 0.639518 0.768776i \(-0.279134\pi\)
0.639518 + 0.768776i \(0.279134\pi\)
\(884\) 259.415i 0.293456i
\(885\) 0 0
\(886\) 356.211i 0.402044i
\(887\) 240.033i 0.270612i 0.990804 + 0.135306i \(0.0432018\pi\)
−0.990804 + 0.135306i \(0.956798\pi\)
\(888\) 240.573i 0.270915i
\(889\) 243.775i 0.274213i
\(890\) 0 0
\(891\) 445.458 0.499953
\(892\) 450.052i 0.504543i
\(893\) −86.1712 + 639.333i −0.0964963 + 0.715938i
\(894\) −183.372 −0.205115
\(895\) 0 0
\(896\) 82.1292i 0.0916621i
\(897\) −358.502 −0.399668
\(898\) −808.425 −0.900251
\(899\) 1730.15 1.92452
\(900\) 0 0
\(901\) 28.0598i 0.0311429i
\(902\) −361.743 −0.401045
\(903\) 252.691i 0.279835i
\(904\) 1267.63 1.40224
\(905\) 0 0
\(906\) 264.021i 0.291414i
\(907\) 162.059i 0.178676i 0.996001 + 0.0893379i \(0.0284751\pi\)
−0.996001 + 0.0893379i \(0.971525\pi\)
\(908\) 176.186i 0.194038i
\(909\) −218.807 −0.240712
\(910\) 0 0
\(911\) 1545.15i 1.69611i −0.529911 0.848054i \(-0.677775\pi\)
0.529911 0.848054i \(-0.322225\pi\)
\(912\) 114.174 + 15.3888i 0.125191 + 0.0168737i
\(913\) 2083.58 2.28212
\(914\) 137.711i 0.150668i
\(915\) 0 0
\(916\) −714.062 −0.779544
\(917\) −1.08214 −0.00118009
\(918\) −336.117 −0.366140
\(919\) −607.021 −0.660524 −0.330262 0.943889i \(-0.607137\pi\)
−0.330262 + 0.943889i \(0.607137\pi\)
\(920\) 0 0
\(921\) 400.554 0.434912
\(922\) 344.553i 0.373702i
\(923\) −1536.45 −1.66462
\(924\) 123.735i 0.133912i
\(925\) 0 0
\(926\) 945.223i 1.02076i
\(927\) 294.533i 0.317728i
\(928\) −856.597 −0.923057
\(929\) −696.933 −0.750197 −0.375099 0.926985i \(-0.622391\pi\)
−0.375099 + 0.926985i \(0.622391\pi\)
\(930\) 0 0
\(931\) −108.434 + 804.508i −0.116471 + 0.864134i
\(932\) −482.229 −0.517414
\(933\) 30.7100i 0.0329153i
\(934\) 643.285i 0.688742i
\(935\) 0 0
\(936\) 745.090 0.796036
\(937\) −896.631 −0.956916 −0.478458 0.878110i \(-0.658804\pi\)
−0.478458 + 0.878110i \(0.658804\pi\)
\(938\) 262.044 0.279365
\(939\) 500.137i 0.532628i
\(940\) 0 0
\(941\) 1627.92i 1.72999i 0.501782 + 0.864994i \(0.332678\pi\)
−0.501782 + 0.864994i \(0.667322\pi\)
\(942\) −160.633 −0.170523
\(943\) 285.521i 0.302780i
\(944\) 167.981i 0.177946i
\(945\) 0 0
\(946\) 1619.70i 1.71215i
\(947\) 373.464 0.394365 0.197182 0.980367i \(-0.436821\pi\)
0.197182 + 0.980367i \(0.436821\pi\)
\(948\) −165.011 −0.174063
\(949\) 855.861i 0.901856i
\(950\) 0 0
\(951\) −581.677 −0.611648
\(952\) 215.332i 0.226189i
\(953\) 17.8029i 0.0186809i −0.999956 0.00934043i \(-0.997027\pi\)
0.999956 0.00934043i \(-0.00297319\pi\)
\(954\) 26.7598 0.0280502
\(955\) 0 0
\(956\) 285.691 0.298840
\(957\) −755.477 −0.789422
\(958\) 1308.99i 1.36638i
\(959\) −335.833 −0.350191
\(960\) 0 0
\(961\) −2275.96 −2.36833
\(962\) 349.571i 0.363379i
\(963\) 1074.66i 1.11596i
\(964\) 799.113i 0.828955i
\(965\) 0 0
\(966\) −98.8076 −0.102285
\(967\) −446.545 −0.461784 −0.230892 0.972979i \(-0.574164\pi\)
−0.230892 + 0.972979i \(0.574164\pi\)
\(968\) 1360.92i 1.40591i
\(969\) −282.326 38.0528i −0.291359 0.0392702i
\(970\) 0 0
\(971\) 928.748i 0.956486i 0.878228 + 0.478243i \(0.158726\pi\)
−0.878228 + 0.478243i \(0.841274\pi\)
\(972\) 497.244i 0.511568i
\(973\) −40.1315 −0.0412451
\(974\) 490.722 0.503821
\(975\) 0 0
\(976\) 328.818 0.336904
\(977\) 1334.75i 1.36617i −0.730338 0.683086i \(-0.760637\pi\)
0.730338 0.683086i \(-0.239363\pi\)
\(978\) 530.223 0.542151
\(979\) 156.253i 0.159604i
\(980\) 0 0
\(981\) 399.786i 0.407529i
\(982\) 336.620i 0.342790i
\(983\) 715.994i 0.728376i −0.931325 0.364188i \(-0.881346\pi\)
0.931325 0.364188i \(-0.118654\pi\)
\(984\) 191.359i 0.194470i
\(985\) 0 0
\(986\) −436.539 −0.442738
\(987\) 125.994i 0.127653i
\(988\) 482.621 + 65.0491i 0.488483 + 0.0658392i
\(989\) −1278.41 −1.29263
\(990\) 0 0
\(991\) 246.568i 0.248807i −0.992232 0.124404i \(-0.960298\pi\)
0.992232 0.124404i \(-0.0397018\pi\)
\(992\) 1602.62 1.61555
\(993\) −151.000 −0.152064
\(994\) −423.464 −0.426020
\(995\) 0 0
\(996\) 365.968i 0.367438i
\(997\) 1358.93 1.36301 0.681507 0.731812i \(-0.261325\pi\)
0.681507 + 0.731812i \(0.261325\pi\)
\(998\) 499.956i 0.500958i
\(999\) −447.682 −0.448130
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 475.3.c.i.151.5 yes 14
5.2 odd 4 475.3.d.d.474.19 28
5.3 odd 4 475.3.d.d.474.10 28
5.4 even 2 475.3.c.h.151.10 yes 14
19.18 odd 2 inner 475.3.c.i.151.10 yes 14
95.18 even 4 475.3.d.d.474.20 28
95.37 even 4 475.3.d.d.474.9 28
95.94 odd 2 475.3.c.h.151.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.3.c.h.151.5 14 95.94 odd 2
475.3.c.h.151.10 yes 14 5.4 even 2
475.3.c.i.151.5 yes 14 1.1 even 1 trivial
475.3.c.i.151.10 yes 14 19.18 odd 2 inner
475.3.d.d.474.9 28 95.37 even 4
475.3.d.d.474.10 28 5.3 odd 4
475.3.d.d.474.19 28 5.2 odd 4
475.3.d.d.474.20 28 95.18 even 4