Properties

Label 475.2.b.d.324.2
Level $475$
Weight $2$
Character 475.324
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(324,475)]
 
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([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("475.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.2
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 475.324
Dual form 475.2.b.d.324.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +3.35026i q^{7} -2.67513i q^{8} +2.35026 q^{9} +O(q^{10})\) \(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +3.35026i q^{7} -2.67513i q^{8} +2.35026 q^{9} +0.962389 q^{11} +0.156325i q^{12} -6.15633i q^{13} +4.96239 q^{14} -4.35026 q^{16} -6.31265i q^{17} -3.48119i q^{18} +1.00000 q^{19} +2.70052 q^{21} -1.42548i q^{22} +4.96239i q^{23} -2.15633 q^{24} -9.11871 q^{26} -4.31265i q^{27} -0.649738i q^{28} +3.61213 q^{29} -5.92478 q^{31} +1.09332i q^{32} -0.775746i q^{33} -9.35026 q^{34} -0.455802 q^{36} +10.1563i q^{37} -1.48119i q^{38} -4.96239 q^{39} +6.31265 q^{41} -4.00000i q^{42} +4.12601i q^{43} -0.186642 q^{44} +7.35026 q^{46} +3.35026i q^{47} +3.50659i q^{48} -4.22425 q^{49} -5.08840 q^{51} +1.19394i q^{52} -1.84367i q^{53} -6.38787 q^{54} +8.96239 q^{56} -0.806063i q^{57} -5.35026i q^{58} +6.38787 q^{59} -11.2750 q^{61} +8.77575i q^{62} +7.87399i q^{63} -7.08110 q^{64} -1.14903 q^{66} -6.73084i q^{67} +1.22425i q^{68} +4.00000 q^{69} -0.775746 q^{71} -6.28726i q^{72} -0.387873i q^{73} +15.0435 q^{74} -0.193937 q^{76} +3.22425i q^{77} +7.35026i q^{78} +0.836381 q^{79} +3.57452 q^{81} -9.35026i q^{82} +7.03761i q^{83} -0.523730 q^{84} +6.11142 q^{86} -2.91160i q^{87} -2.57452i q^{88} -7.08840 q^{89} +20.6253 q^{91} -0.962389i q^{92} +4.77575i q^{93} +4.96239 q^{94} +0.881286 q^{96} +10.9927i q^{97} +6.25694i q^{98} +2.26187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.48119i − 1.04736i −0.851914 0.523681i \(-0.824558\pi\)
0.851914 0.523681i \(-0.175442\pi\)
\(3\) − 0.806063i − 0.465381i −0.972551 0.232690i \(-0.925247\pi\)
0.972551 0.232690i \(-0.0747529\pi\)
\(4\) −0.193937 −0.0969683
\(5\) 0 0
\(6\) −1.19394 −0.487423
\(7\) 3.35026i 1.26628i 0.774037 + 0.633140i \(0.218234\pi\)
−0.774037 + 0.633140i \(0.781766\pi\)
\(8\) − 2.67513i − 0.945802i
\(9\) 2.35026 0.783421
\(10\) 0 0
\(11\) 0.962389 0.290171 0.145086 0.989419i \(-0.453654\pi\)
0.145086 + 0.989419i \(0.453654\pi\)
\(12\) 0.156325i 0.0451272i
\(13\) − 6.15633i − 1.70746i −0.520718 0.853729i \(-0.674336\pi\)
0.520718 0.853729i \(-0.325664\pi\)
\(14\) 4.96239 1.32625
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) − 6.31265i − 1.53104i −0.643411 0.765521i \(-0.722481\pi\)
0.643411 0.765521i \(-0.277519\pi\)
\(18\) − 3.48119i − 0.820525i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.70052 0.589303
\(22\) − 1.42548i − 0.303914i
\(23\) 4.96239i 1.03473i 0.855765 + 0.517365i \(0.173087\pi\)
−0.855765 + 0.517365i \(0.826913\pi\)
\(24\) −2.15633 −0.440158
\(25\) 0 0
\(26\) −9.11871 −1.78833
\(27\) − 4.31265i − 0.829970i
\(28\) − 0.649738i − 0.122789i
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0 0
\(31\) −5.92478 −1.06412 −0.532061 0.846706i \(-0.678582\pi\)
−0.532061 + 0.846706i \(0.678582\pi\)
\(32\) 1.09332i 0.193274i
\(33\) − 0.775746i − 0.135040i
\(34\) −9.35026 −1.60356
\(35\) 0 0
\(36\) −0.455802 −0.0759669
\(37\) 10.1563i 1.66969i 0.550485 + 0.834845i \(0.314443\pi\)
−0.550485 + 0.834845i \(0.685557\pi\)
\(38\) − 1.48119i − 0.240281i
\(39\) −4.96239 −0.794618
\(40\) 0 0
\(41\) 6.31265 0.985870 0.492935 0.870066i \(-0.335924\pi\)
0.492935 + 0.870066i \(0.335924\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 4.12601i 0.629210i 0.949223 + 0.314605i \(0.101872\pi\)
−0.949223 + 0.314605i \(0.898128\pi\)
\(44\) −0.186642 −0.0281374
\(45\) 0 0
\(46\) 7.35026 1.08374
\(47\) 3.35026i 0.488686i 0.969689 + 0.244343i \(0.0785723\pi\)
−0.969689 + 0.244343i \(0.921428\pi\)
\(48\) 3.50659i 0.506132i
\(49\) −4.22425 −0.603465
\(50\) 0 0
\(51\) −5.08840 −0.712518
\(52\) 1.19394i 0.165569i
\(53\) − 1.84367i − 0.253248i −0.991951 0.126624i \(-0.959586\pi\)
0.991951 0.126624i \(-0.0404142\pi\)
\(54\) −6.38787 −0.869279
\(55\) 0 0
\(56\) 8.96239 1.19765
\(57\) − 0.806063i − 0.106766i
\(58\) − 5.35026i − 0.702524i
\(59\) 6.38787 0.831630 0.415815 0.909449i \(-0.363496\pi\)
0.415815 + 0.909449i \(0.363496\pi\)
\(60\) 0 0
\(61\) −11.2750 −1.44362 −0.721810 0.692091i \(-0.756690\pi\)
−0.721810 + 0.692091i \(0.756690\pi\)
\(62\) 8.77575i 1.11452i
\(63\) 7.87399i 0.992030i
\(64\) −7.08110 −0.885138
\(65\) 0 0
\(66\) −1.14903 −0.141436
\(67\) − 6.73084i − 0.822303i −0.911567 0.411152i \(-0.865127\pi\)
0.911567 0.411152i \(-0.134873\pi\)
\(68\) 1.22425i 0.148463i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −0.775746 −0.0920641 −0.0460321 0.998940i \(-0.514658\pi\)
−0.0460321 + 0.998940i \(0.514658\pi\)
\(72\) − 6.28726i − 0.740960i
\(73\) − 0.387873i − 0.0453971i −0.999742 0.0226986i \(-0.992774\pi\)
0.999742 0.0226986i \(-0.00722580\pi\)
\(74\) 15.0435 1.74877
\(75\) 0 0
\(76\) −0.193937 −0.0222460
\(77\) 3.22425i 0.367438i
\(78\) 7.35026i 0.832253i
\(79\) 0.836381 0.0941002 0.0470501 0.998893i \(-0.485018\pi\)
0.0470501 + 0.998893i \(0.485018\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) − 9.35026i − 1.03256i
\(83\) 7.03761i 0.772478i 0.922399 + 0.386239i \(0.126226\pi\)
−0.922399 + 0.386239i \(0.873774\pi\)
\(84\) −0.523730 −0.0571437
\(85\) 0 0
\(86\) 6.11142 0.659011
\(87\) − 2.91160i − 0.312157i
\(88\) − 2.57452i − 0.274444i
\(89\) −7.08840 −0.751369 −0.375684 0.926748i \(-0.622592\pi\)
−0.375684 + 0.926748i \(0.622592\pi\)
\(90\) 0 0
\(91\) 20.6253 2.16212
\(92\) − 0.962389i − 0.100336i
\(93\) 4.77575i 0.495222i
\(94\) 4.96239 0.511831
\(95\) 0 0
\(96\) 0.881286 0.0899459
\(97\) 10.9927i 1.11614i 0.829794 + 0.558070i \(0.188458\pi\)
−0.829794 + 0.558070i \(0.811542\pi\)
\(98\) 6.25694i 0.632046i
\(99\) 2.26187 0.227326
\(100\) 0 0
\(101\) −2.64974 −0.263659 −0.131829 0.991272i \(-0.542085\pi\)
−0.131829 + 0.991272i \(0.542085\pi\)
\(102\) 7.53690i 0.746265i
\(103\) 10.7308i 1.05734i 0.848827 + 0.528671i \(0.177309\pi\)
−0.848827 + 0.528671i \(0.822691\pi\)
\(104\) −16.4690 −1.61492
\(105\) 0 0
\(106\) −2.73084 −0.265243
\(107\) 4.80606i 0.464620i 0.972642 + 0.232310i \(0.0746283\pi\)
−0.972642 + 0.232310i \(0.925372\pi\)
\(108\) 0.836381i 0.0804808i
\(109\) 2.77575 0.265868 0.132934 0.991125i \(-0.457560\pi\)
0.132934 + 0.991125i \(0.457560\pi\)
\(110\) 0 0
\(111\) 8.18664 0.777042
\(112\) − 14.5745i − 1.37716i
\(113\) − 6.99271i − 0.657818i −0.944362 0.328909i \(-0.893319\pi\)
0.944362 0.328909i \(-0.106681\pi\)
\(114\) −1.19394 −0.111822
\(115\) 0 0
\(116\) −0.700523 −0.0650420
\(117\) − 14.4690i − 1.33766i
\(118\) − 9.46168i − 0.871018i
\(119\) 21.1490 1.93873
\(120\) 0 0
\(121\) −10.0738 −0.915801
\(122\) 16.7005i 1.51199i
\(123\) − 5.08840i − 0.458805i
\(124\) 1.14903 0.103186
\(125\) 0 0
\(126\) 11.6629 1.03901
\(127\) 13.4314i 1.19184i 0.803043 + 0.595920i \(0.203213\pi\)
−0.803043 + 0.595920i \(0.796787\pi\)
\(128\) 12.6751i 1.12033i
\(129\) 3.32582 0.292822
\(130\) 0 0
\(131\) 20.6253 1.80204 0.901020 0.433777i \(-0.142819\pi\)
0.901020 + 0.433777i \(0.142819\pi\)
\(132\) 0.150446i 0.0130946i
\(133\) 3.35026i 0.290505i
\(134\) −9.96968 −0.861249
\(135\) 0 0
\(136\) −16.8872 −1.44806
\(137\) 20.2374i 1.72900i 0.502633 + 0.864500i \(0.332365\pi\)
−0.502633 + 0.864500i \(0.667635\pi\)
\(138\) − 5.92478i − 0.504351i
\(139\) 17.5877 1.49177 0.745884 0.666076i \(-0.232027\pi\)
0.745884 + 0.666076i \(0.232027\pi\)
\(140\) 0 0
\(141\) 2.70052 0.227425
\(142\) 1.14903i 0.0964245i
\(143\) − 5.92478i − 0.495455i
\(144\) −10.2243 −0.852021
\(145\) 0 0
\(146\) −0.574515 −0.0475472
\(147\) 3.40502i 0.280841i
\(148\) − 1.96968i − 0.161907i
\(149\) 7.42548 0.608319 0.304160 0.952621i \(-0.401624\pi\)
0.304160 + 0.952621i \(0.401624\pi\)
\(150\) 0 0
\(151\) −1.61213 −0.131193 −0.0655965 0.997846i \(-0.520895\pi\)
−0.0655965 + 0.997846i \(0.520895\pi\)
\(152\) − 2.67513i − 0.216982i
\(153\) − 14.8364i − 1.19945i
\(154\) 4.77575 0.384841
\(155\) 0 0
\(156\) 0.962389 0.0770528
\(157\) 4.38787i 0.350190i 0.984552 + 0.175095i \(0.0560233\pi\)
−0.984552 + 0.175095i \(0.943977\pi\)
\(158\) − 1.23884i − 0.0985570i
\(159\) −1.48612 −0.117857
\(160\) 0 0
\(161\) −16.6253 −1.31026
\(162\) − 5.29455i − 0.415979i
\(163\) 0.649738i 0.0508914i 0.999676 + 0.0254457i \(0.00810050\pi\)
−0.999676 + 0.0254457i \(0.991900\pi\)
\(164\) −1.22425 −0.0955982
\(165\) 0 0
\(166\) 10.4241 0.809065
\(167\) − 15.3561i − 1.18829i −0.804357 0.594147i \(-0.797490\pi\)
0.804357 0.594147i \(-0.202510\pi\)
\(168\) − 7.22425i − 0.557363i
\(169\) −24.9003 −1.91541
\(170\) 0 0
\(171\) 2.35026 0.179729
\(172\) − 0.800184i − 0.0610134i
\(173\) − 3.24472i − 0.246692i −0.992364 0.123346i \(-0.960638\pi\)
0.992364 0.123346i \(-0.0393624\pi\)
\(174\) −4.31265 −0.326941
\(175\) 0 0
\(176\) −4.18664 −0.315580
\(177\) − 5.14903i − 0.387025i
\(178\) 10.4993i 0.786955i
\(179\) −15.0132 −1.12214 −0.561069 0.827769i \(-0.689610\pi\)
−0.561069 + 0.827769i \(0.689610\pi\)
\(180\) 0 0
\(181\) 9.22425 0.685633 0.342817 0.939402i \(-0.388619\pi\)
0.342817 + 0.939402i \(0.388619\pi\)
\(182\) − 30.5501i − 2.26452i
\(183\) 9.08840i 0.671834i
\(184\) 13.2750 0.978649
\(185\) 0 0
\(186\) 7.07381 0.518677
\(187\) − 6.07522i − 0.444264i
\(188\) − 0.649738i − 0.0473870i
\(189\) 14.4485 1.05097
\(190\) 0 0
\(191\) −21.7743 −1.57554 −0.787768 0.615972i \(-0.788763\pi\)
−0.787768 + 0.615972i \(0.788763\pi\)
\(192\) 5.70782i 0.411926i
\(193\) − 12.5442i − 0.902951i −0.892283 0.451476i \(-0.850898\pi\)
0.892283 0.451476i \(-0.149102\pi\)
\(194\) 16.2823 1.16900
\(195\) 0 0
\(196\) 0.819237 0.0585169
\(197\) 24.5501i 1.74912i 0.484917 + 0.874560i \(0.338850\pi\)
−0.484917 + 0.874560i \(0.661150\pi\)
\(198\) − 3.35026i − 0.238093i
\(199\) −23.0738 −1.63566 −0.817829 0.575461i \(-0.804823\pi\)
−0.817829 + 0.575461i \(0.804823\pi\)
\(200\) 0 0
\(201\) −5.42548 −0.382684
\(202\) 3.92478i 0.276146i
\(203\) 12.1016i 0.849364i
\(204\) 0.986826 0.0690917
\(205\) 0 0
\(206\) 15.8945 1.10742
\(207\) 11.6629i 0.810628i
\(208\) 26.7816i 1.85697i
\(209\) 0.962389 0.0665698
\(210\) 0 0
\(211\) −20.9380 −1.44143 −0.720714 0.693233i \(-0.756186\pi\)
−0.720714 + 0.693233i \(0.756186\pi\)
\(212\) 0.357556i 0.0245570i
\(213\) 0.625301i 0.0428449i
\(214\) 7.11871 0.486625
\(215\) 0 0
\(216\) −11.5369 −0.784987
\(217\) − 19.8496i − 1.34748i
\(218\) − 4.11142i − 0.278460i
\(219\) −0.312650 −0.0211270
\(220\) 0 0
\(221\) −38.8627 −2.61419
\(222\) − 12.1260i − 0.813844i
\(223\) 0.0303172i 0.00203019i 0.999999 + 0.00101509i \(0.000323114\pi\)
−0.999999 + 0.00101509i \(0.999677\pi\)
\(224\) −3.66291 −0.244739
\(225\) 0 0
\(226\) −10.3576 −0.688974
\(227\) 4.80606i 0.318990i 0.987199 + 0.159495i \(0.0509865\pi\)
−0.987199 + 0.159495i \(0.949013\pi\)
\(228\) 0.156325i 0.0103529i
\(229\) −1.87399 −0.123837 −0.0619184 0.998081i \(-0.519722\pi\)
−0.0619184 + 0.998081i \(0.519722\pi\)
\(230\) 0 0
\(231\) 2.59895 0.170999
\(232\) − 9.66291i − 0.634401i
\(233\) 11.1490i 0.730397i 0.930930 + 0.365199i \(0.118999\pi\)
−0.930930 + 0.365199i \(0.881001\pi\)
\(234\) −21.4314 −1.40101
\(235\) 0 0
\(236\) −1.23884 −0.0806418
\(237\) − 0.674176i − 0.0437924i
\(238\) − 31.3258i − 2.03055i
\(239\) −9.29948 −0.601533 −0.300767 0.953698i \(-0.597243\pi\)
−0.300767 + 0.953698i \(0.597243\pi\)
\(240\) 0 0
\(241\) −2.31265 −0.148971 −0.0744855 0.997222i \(-0.523731\pi\)
−0.0744855 + 0.997222i \(0.523731\pi\)
\(242\) 14.9213i 0.959175i
\(243\) − 15.8192i − 1.01480i
\(244\) 2.18664 0.139985
\(245\) 0 0
\(246\) −7.53690 −0.480535
\(247\) − 6.15633i − 0.391718i
\(248\) 15.8496i 1.00645i
\(249\) 5.67276 0.359497
\(250\) 0 0
\(251\) 24.1016 1.52128 0.760639 0.649175i \(-0.224886\pi\)
0.760639 + 0.649175i \(0.224886\pi\)
\(252\) − 1.52705i − 0.0961954i
\(253\) 4.77575i 0.300249i
\(254\) 19.8945 1.24829
\(255\) 0 0
\(256\) 4.61213 0.288258
\(257\) − 13.3199i − 0.830875i −0.909622 0.415438i \(-0.863628\pi\)
0.909622 0.415438i \(-0.136372\pi\)
\(258\) − 4.92619i − 0.306691i
\(259\) −34.0263 −2.11429
\(260\) 0 0
\(261\) 8.48944 0.525483
\(262\) − 30.5501i − 1.88739i
\(263\) − 12.9624i − 0.799295i −0.916669 0.399648i \(-0.869133\pi\)
0.916669 0.399648i \(-0.130867\pi\)
\(264\) −2.07522 −0.127721
\(265\) 0 0
\(266\) 4.96239 0.304264
\(267\) 5.71370i 0.349673i
\(268\) 1.30536i 0.0797373i
\(269\) 11.4010 0.695134 0.347567 0.937655i \(-0.387008\pi\)
0.347567 + 0.937655i \(0.387008\pi\)
\(270\) 0 0
\(271\) 16.8119 1.02125 0.510626 0.859803i \(-0.329414\pi\)
0.510626 + 0.859803i \(0.329414\pi\)
\(272\) 27.4617i 1.66511i
\(273\) − 16.6253i − 1.00621i
\(274\) 29.9756 1.81089
\(275\) 0 0
\(276\) −0.775746 −0.0466944
\(277\) − 29.7889i − 1.78984i −0.446224 0.894921i \(-0.647231\pi\)
0.446224 0.894921i \(-0.352769\pi\)
\(278\) − 26.0508i − 1.56242i
\(279\) −13.9248 −0.833655
\(280\) 0 0
\(281\) −11.6121 −0.692721 −0.346361 0.938101i \(-0.612583\pi\)
−0.346361 + 0.938101i \(0.612583\pi\)
\(282\) − 4.00000i − 0.238197i
\(283\) − 2.26187i − 0.134454i −0.997738 0.0672270i \(-0.978585\pi\)
0.997738 0.0672270i \(-0.0214152\pi\)
\(284\) 0.150446 0.00892730
\(285\) 0 0
\(286\) −8.77575 −0.518921
\(287\) 21.1490i 1.24839i
\(288\) 2.56959i 0.151415i
\(289\) −22.8496 −1.34409
\(290\) 0 0
\(291\) 8.86082 0.519430
\(292\) 0.0752228i 0.00440208i
\(293\) − 1.84367i − 0.107709i −0.998549 0.0538543i \(-0.982849\pi\)
0.998549 0.0538543i \(-0.0171507\pi\)
\(294\) 5.04349 0.294142
\(295\) 0 0
\(296\) 27.1695 1.57920
\(297\) − 4.15045i − 0.240833i
\(298\) − 10.9986i − 0.637131i
\(299\) 30.5501 1.76676
\(300\) 0 0
\(301\) −13.8232 −0.796756
\(302\) 2.38787i 0.137407i
\(303\) 2.13586i 0.122702i
\(304\) −4.35026 −0.249505
\(305\) 0 0
\(306\) −21.9756 −1.25626
\(307\) 26.2071i 1.49572i 0.663857 + 0.747859i \(0.268918\pi\)
−0.663857 + 0.747859i \(0.731082\pi\)
\(308\) − 0.625301i − 0.0356298i
\(309\) 8.64974 0.492066
\(310\) 0 0
\(311\) 6.51388 0.369368 0.184684 0.982798i \(-0.440874\pi\)
0.184684 + 0.982798i \(0.440874\pi\)
\(312\) 13.2750i 0.751551i
\(313\) − 16.0752i − 0.908625i −0.890842 0.454313i \(-0.849885\pi\)
0.890842 0.454313i \(-0.150115\pi\)
\(314\) 6.49929 0.366776
\(315\) 0 0
\(316\) −0.162205 −0.00912473
\(317\) − 5.69323i − 0.319764i −0.987136 0.159882i \(-0.948889\pi\)
0.987136 0.159882i \(-0.0511113\pi\)
\(318\) 2.20123i 0.123439i
\(319\) 3.47627 0.194634
\(320\) 0 0
\(321\) 3.87399 0.216225
\(322\) 24.6253i 1.37231i
\(323\) − 6.31265i − 0.351245i
\(324\) −0.693229 −0.0385127
\(325\) 0 0
\(326\) 0.962389 0.0533018
\(327\) − 2.23743i − 0.123730i
\(328\) − 16.8872i − 0.932438i
\(329\) −11.2243 −0.618813
\(330\) 0 0
\(331\) −12.3127 −0.676764 −0.338382 0.941009i \(-0.609880\pi\)
−0.338382 + 0.941009i \(0.609880\pi\)
\(332\) − 1.36485i − 0.0749059i
\(333\) 23.8700i 1.30807i
\(334\) −22.7454 −1.24457
\(335\) 0 0
\(336\) −11.7480 −0.640905
\(337\) 3.76845i 0.205281i 0.994719 + 0.102640i \(0.0327291\pi\)
−0.994719 + 0.102640i \(0.967271\pi\)
\(338\) 36.8822i 2.00613i
\(339\) −5.63656 −0.306136
\(340\) 0 0
\(341\) −5.70194 −0.308777
\(342\) − 3.48119i − 0.188241i
\(343\) 9.29948i 0.502125i
\(344\) 11.0376 0.595108
\(345\) 0 0
\(346\) −4.80606 −0.258376
\(347\) − 22.3634i − 1.20053i −0.799800 0.600266i \(-0.795061\pi\)
0.799800 0.600266i \(-0.204939\pi\)
\(348\) 0.564666i 0.0302693i
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −26.5501 −1.41714
\(352\) 1.05220i 0.0560824i
\(353\) − 5.53690i − 0.294700i −0.989084 0.147350i \(-0.952926\pi\)
0.989084 0.147350i \(-0.0470743\pi\)
\(354\) −7.62672 −0.405355
\(355\) 0 0
\(356\) 1.37470 0.0728589
\(357\) − 17.0475i − 0.902247i
\(358\) 22.2374i 1.17528i
\(359\) 10.3634 0.546961 0.273481 0.961878i \(-0.411825\pi\)
0.273481 + 0.961878i \(0.411825\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 13.6629i − 0.718107i
\(363\) 8.12013i 0.426196i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 13.4617 0.703653
\(367\) 3.35026i 0.174882i 0.996170 + 0.0874411i \(0.0278690\pi\)
−0.996170 + 0.0874411i \(0.972131\pi\)
\(368\) − 21.5877i − 1.12534i
\(369\) 14.8364 0.772351
\(370\) 0 0
\(371\) 6.17679 0.320683
\(372\) − 0.926192i − 0.0480208i
\(373\) 12.6048i 0.652653i 0.945257 + 0.326327i \(0.105811\pi\)
−0.945257 + 0.326327i \(0.894189\pi\)
\(374\) −8.99859 −0.465306
\(375\) 0 0
\(376\) 8.96239 0.462200
\(377\) − 22.2374i − 1.14529i
\(378\) − 21.4010i − 1.10075i
\(379\) 37.2506 1.91343 0.956717 0.291018i \(-0.0939941\pi\)
0.956717 + 0.291018i \(0.0939941\pi\)
\(380\) 0 0
\(381\) 10.8265 0.554660
\(382\) 32.2520i 1.65016i
\(383\) − 30.8324i − 1.57546i −0.616019 0.787731i \(-0.711256\pi\)
0.616019 0.787731i \(-0.288744\pi\)
\(384\) 10.2170 0.521382
\(385\) 0 0
\(386\) −18.5804 −0.945717
\(387\) 9.69720i 0.492936i
\(388\) − 2.13189i − 0.108230i
\(389\) 1.37470 0.0697000 0.0348500 0.999393i \(-0.488905\pi\)
0.0348500 + 0.999393i \(0.488905\pi\)
\(390\) 0 0
\(391\) 31.3258 1.58422
\(392\) 11.3004i 0.570758i
\(393\) − 16.6253i − 0.838635i
\(394\) 36.3634 1.83196
\(395\) 0 0
\(396\) −0.438658 −0.0220434
\(397\) − 9.38646i − 0.471093i −0.971863 0.235546i \(-0.924312\pi\)
0.971863 0.235546i \(-0.0756880\pi\)
\(398\) 34.1768i 1.71313i
\(399\) 2.70052 0.135195
\(400\) 0 0
\(401\) 14.1016 0.704199 0.352099 0.935963i \(-0.385468\pi\)
0.352099 + 0.935963i \(0.385468\pi\)
\(402\) 8.03620i 0.400809i
\(403\) 36.4749i 1.81694i
\(404\) 0.513881 0.0255665
\(405\) 0 0
\(406\) 17.9248 0.889592
\(407\) 9.77433i 0.484496i
\(408\) 13.6121i 0.673901i
\(409\) −35.1490 −1.73801 −0.869004 0.494805i \(-0.835239\pi\)
−0.869004 + 0.494805i \(0.835239\pi\)
\(410\) 0 0
\(411\) 16.3127 0.804644
\(412\) − 2.08110i − 0.102529i
\(413\) 21.4010i 1.05308i
\(414\) 17.2750 0.849022
\(415\) 0 0
\(416\) 6.73084 0.330007
\(417\) − 14.1768i − 0.694241i
\(418\) − 1.42548i − 0.0697227i
\(419\) −9.02776 −0.441035 −0.220518 0.975383i \(-0.570775\pi\)
−0.220518 + 0.975383i \(0.570775\pi\)
\(420\) 0 0
\(421\) 15.2097 0.741274 0.370637 0.928778i \(-0.379139\pi\)
0.370637 + 0.928778i \(0.379139\pi\)
\(422\) 31.0132i 1.50970i
\(423\) 7.87399i 0.382847i
\(424\) −4.93207 −0.239523
\(425\) 0 0
\(426\) 0.926192 0.0448741
\(427\) − 37.7743i − 1.82803i
\(428\) − 0.932071i − 0.0450534i
\(429\) −4.77575 −0.230575
\(430\) 0 0
\(431\) 16.3127 0.785753 0.392876 0.919591i \(-0.371480\pi\)
0.392876 + 0.919591i \(0.371480\pi\)
\(432\) 18.7612i 0.902647i
\(433\) − 11.1432i − 0.535506i −0.963488 0.267753i \(-0.913719\pi\)
0.963488 0.267753i \(-0.0862811\pi\)
\(434\) −29.4010 −1.41130
\(435\) 0 0
\(436\) −0.538319 −0.0257808
\(437\) 4.96239i 0.237383i
\(438\) 0.463096i 0.0221276i
\(439\) −27.3865 −1.30708 −0.653542 0.756890i \(-0.726718\pi\)
−0.653542 + 0.756890i \(0.726718\pi\)
\(440\) 0 0
\(441\) −9.92810 −0.472767
\(442\) 57.5633i 2.73800i
\(443\) − 19.5125i − 0.927065i −0.886080 0.463533i \(-0.846582\pi\)
0.886080 0.463533i \(-0.153418\pi\)
\(444\) −1.58769 −0.0753484
\(445\) 0 0
\(446\) 0.0449056 0.00212634
\(447\) − 5.98541i − 0.283100i
\(448\) − 23.7235i − 1.12083i
\(449\) 22.1016 1.04304 0.521519 0.853240i \(-0.325366\pi\)
0.521519 + 0.853240i \(0.325366\pi\)
\(450\) 0 0
\(451\) 6.07522 0.286071
\(452\) 1.35614i 0.0637875i
\(453\) 1.29948i 0.0610547i
\(454\) 7.11871 0.334098
\(455\) 0 0
\(456\) −2.15633 −0.100979
\(457\) − 17.8496i − 0.834967i −0.908685 0.417483i \(-0.862912\pi\)
0.908685 0.417483i \(-0.137088\pi\)
\(458\) 2.77575i 0.129702i
\(459\) −27.2243 −1.27072
\(460\) 0 0
\(461\) −35.2506 −1.64178 −0.820892 0.571083i \(-0.806523\pi\)
−0.820892 + 0.571083i \(0.806523\pi\)
\(462\) − 3.84955i − 0.179097i
\(463\) 26.3634i 1.22521i 0.790388 + 0.612606i \(0.209879\pi\)
−0.790388 + 0.612606i \(0.790121\pi\)
\(464\) −15.7137 −0.729490
\(465\) 0 0
\(466\) 16.5139 0.764991
\(467\) − 6.78560i − 0.314000i −0.987599 0.157000i \(-0.949818\pi\)
0.987599 0.157000i \(-0.0501823\pi\)
\(468\) 2.80606i 0.129710i
\(469\) 22.5501 1.04127
\(470\) 0 0
\(471\) 3.53690 0.162972
\(472\) − 17.0884i − 0.786557i
\(473\) 3.97082i 0.182579i
\(474\) −0.998585 −0.0458665
\(475\) 0 0
\(476\) −4.10157 −0.187995
\(477\) − 4.33312i − 0.198400i
\(478\) 13.7743i 0.630023i
\(479\) −12.7104 −0.580752 −0.290376 0.956913i \(-0.593780\pi\)
−0.290376 + 0.956913i \(0.593780\pi\)
\(480\) 0 0
\(481\) 62.5256 2.85092
\(482\) 3.42548i 0.156027i
\(483\) 13.4010i 0.609769i
\(484\) 1.95368 0.0888036
\(485\) 0 0
\(486\) −23.4314 −1.06287
\(487\) − 15.7586i − 0.714090i −0.934087 0.357045i \(-0.883784\pi\)
0.934087 0.357045i \(-0.116216\pi\)
\(488\) 30.1622i 1.36538i
\(489\) 0.523730 0.0236839
\(490\) 0 0
\(491\) −14.5501 −0.656636 −0.328318 0.944567i \(-0.606482\pi\)
−0.328318 + 0.944567i \(0.606482\pi\)
\(492\) 0.986826i 0.0444896i
\(493\) − 22.8021i − 1.02695i
\(494\) −9.11871 −0.410270
\(495\) 0 0
\(496\) 25.7743 1.15730
\(497\) − 2.59895i − 0.116579i
\(498\) − 8.40246i − 0.376523i
\(499\) 5.48612 0.245592 0.122796 0.992432i \(-0.460814\pi\)
0.122796 + 0.992432i \(0.460814\pi\)
\(500\) 0 0
\(501\) −12.3780 −0.553009
\(502\) − 35.6991i − 1.59333i
\(503\) 36.6615i 1.63466i 0.576173 + 0.817328i \(0.304545\pi\)
−0.576173 + 0.817328i \(0.695455\pi\)
\(504\) 21.0640 0.938263
\(505\) 0 0
\(506\) 7.07381 0.314469
\(507\) 20.0713i 0.891396i
\(508\) − 2.60483i − 0.115571i
\(509\) −39.1900 −1.73706 −0.868532 0.495632i \(-0.834936\pi\)
−0.868532 + 0.495632i \(0.834936\pi\)
\(510\) 0 0
\(511\) 1.29948 0.0574855
\(512\) 18.5188i 0.818423i
\(513\) − 4.31265i − 0.190408i
\(514\) −19.7294 −0.870228
\(515\) 0 0
\(516\) −0.644999 −0.0283945
\(517\) 3.22425i 0.141803i
\(518\) 50.3996i 2.21443i
\(519\) −2.61545 −0.114806
\(520\) 0 0
\(521\) −17.7283 −0.776690 −0.388345 0.921514i \(-0.626953\pi\)
−0.388345 + 0.921514i \(0.626953\pi\)
\(522\) − 12.5745i − 0.550372i
\(523\) 40.7572i 1.78219i 0.453819 + 0.891094i \(0.350061\pi\)
−0.453819 + 0.891094i \(0.649939\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −19.1998 −0.837152
\(527\) 37.4010i 1.62922i
\(528\) 3.37470i 0.146865i
\(529\) −1.62530 −0.0706652
\(530\) 0 0
\(531\) 15.0132 0.651516
\(532\) − 0.649738i − 0.0281697i
\(533\) − 38.8627i − 1.68333i
\(534\) 8.46310 0.366234
\(535\) 0 0
\(536\) −18.0059 −0.777736
\(537\) 12.1016i 0.522221i
\(538\) − 16.8872i − 0.728057i
\(539\) −4.06537 −0.175108
\(540\) 0 0
\(541\) 23.9003 1.02756 0.513778 0.857923i \(-0.328246\pi\)
0.513778 + 0.857923i \(0.328246\pi\)
\(542\) − 24.9018i − 1.06962i
\(543\) − 7.43533i − 0.319081i
\(544\) 6.90175 0.295910
\(545\) 0 0
\(546\) −24.6253 −1.05387
\(547\) 8.55405i 0.365745i 0.983137 + 0.182872i \(0.0585395\pi\)
−0.983137 + 0.182872i \(0.941461\pi\)
\(548\) − 3.92478i − 0.167658i
\(549\) −26.4993 −1.13096
\(550\) 0 0
\(551\) 3.61213 0.153882
\(552\) − 10.7005i − 0.455445i
\(553\) 2.80209i 0.119157i
\(554\) −44.1232 −1.87461
\(555\) 0 0
\(556\) −3.41090 −0.144654
\(557\) − 4.23743i − 0.179546i −0.995962 0.0897728i \(-0.971386\pi\)
0.995962 0.0897728i \(-0.0286141\pi\)
\(558\) 20.6253i 0.873139i
\(559\) 25.4010 1.07435
\(560\) 0 0
\(561\) −4.89701 −0.206752
\(562\) 17.1998i 0.725530i
\(563\) − 16.4934i − 0.695114i −0.937659 0.347557i \(-0.887011\pi\)
0.937659 0.347557i \(-0.112989\pi\)
\(564\) −0.523730 −0.0220530
\(565\) 0 0
\(566\) −3.35026 −0.140822
\(567\) 11.9756i 0.502926i
\(568\) 2.07522i 0.0870744i
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −26.2619 −1.09902 −0.549512 0.835486i \(-0.685186\pi\)
−0.549512 + 0.835486i \(0.685186\pi\)
\(572\) 1.14903i 0.0480434i
\(573\) 17.5515i 0.733224i
\(574\) 31.3258 1.30751
\(575\) 0 0
\(576\) −16.6424 −0.693435
\(577\) 30.1016i 1.25314i 0.779363 + 0.626572i \(0.215543\pi\)
−0.779363 + 0.626572i \(0.784457\pi\)
\(578\) 33.8446i 1.40775i
\(579\) −10.1114 −0.420216
\(580\) 0 0
\(581\) −23.5778 −0.978174
\(582\) − 13.1246i − 0.544032i
\(583\) − 1.77433i − 0.0734853i
\(584\) −1.03761 −0.0429367
\(585\) 0 0
\(586\) −2.73084 −0.112810
\(587\) − 35.1392i − 1.45035i −0.688565 0.725175i \(-0.741759\pi\)
0.688565 0.725175i \(-0.258241\pi\)
\(588\) − 0.660357i − 0.0272327i
\(589\) −5.92478 −0.244126
\(590\) 0 0
\(591\) 19.7889 0.814007
\(592\) − 44.1827i − 1.81590i
\(593\) − 34.3244i − 1.40953i −0.709439 0.704767i \(-0.751051\pi\)
0.709439 0.704767i \(-0.248949\pi\)
\(594\) −6.14762 −0.252240
\(595\) 0 0
\(596\) −1.44007 −0.0589877
\(597\) 18.5990i 0.761204i
\(598\) − 45.2506i − 1.85043i
\(599\) −14.6107 −0.596978 −0.298489 0.954413i \(-0.596483\pi\)
−0.298489 + 0.954413i \(0.596483\pi\)
\(600\) 0 0
\(601\) 23.5633 0.961165 0.480583 0.876949i \(-0.340425\pi\)
0.480583 + 0.876949i \(0.340425\pi\)
\(602\) 20.4749i 0.834493i
\(603\) − 15.8192i − 0.644209i
\(604\) 0.312650 0.0127216
\(605\) 0 0
\(606\) 3.16362 0.128513
\(607\) − 8.80606i − 0.357427i −0.983901 0.178714i \(-0.942806\pi\)
0.983901 0.178714i \(-0.0571935\pi\)
\(608\) 1.09332i 0.0443400i
\(609\) 9.75463 0.395278
\(610\) 0 0
\(611\) 20.6253 0.834410
\(612\) 2.87732i 0.116309i
\(613\) − 10.4142i − 0.420626i −0.977634 0.210313i \(-0.932552\pi\)
0.977634 0.210313i \(-0.0674484\pi\)
\(614\) 38.8178 1.56656
\(615\) 0 0
\(616\) 8.62530 0.347523
\(617\) − 17.2849i − 0.695863i −0.937520 0.347932i \(-0.886884\pi\)
0.937520 0.347932i \(-0.113116\pi\)
\(618\) − 12.8119i − 0.515372i
\(619\) 10.6351 0.427463 0.213731 0.976892i \(-0.431438\pi\)
0.213731 + 0.976892i \(0.431438\pi\)
\(620\) 0 0
\(621\) 21.4010 0.858794
\(622\) − 9.64832i − 0.386863i
\(623\) − 23.7480i − 0.951443i
\(624\) 21.5877 0.864199
\(625\) 0 0
\(626\) −23.8105 −0.951660
\(627\) − 0.775746i − 0.0309803i
\(628\) − 0.850969i − 0.0339574i
\(629\) 64.1133 2.55637
\(630\) 0 0
\(631\) −16.5599 −0.659240 −0.329620 0.944114i \(-0.606921\pi\)
−0.329620 + 0.944114i \(0.606921\pi\)
\(632\) − 2.23743i − 0.0890001i
\(633\) 16.8773i 0.670813i
\(634\) −8.43278 −0.334908
\(635\) 0 0
\(636\) 0.288213 0.0114284
\(637\) 26.0059i 1.03039i
\(638\) − 5.14903i − 0.203852i
\(639\) −1.82321 −0.0721249
\(640\) 0 0
\(641\) −16.7612 −0.662026 −0.331013 0.943626i \(-0.607390\pi\)
−0.331013 + 0.943626i \(0.607390\pi\)
\(642\) − 5.73813i − 0.226466i
\(643\) − 5.73813i − 0.226290i −0.993578 0.113145i \(-0.963908\pi\)
0.993578 0.113145i \(-0.0360925\pi\)
\(644\) 3.22425 0.127053
\(645\) 0 0
\(646\) −9.35026 −0.367881
\(647\) − 37.2144i − 1.46305i −0.681815 0.731525i \(-0.738809\pi\)
0.681815 0.731525i \(-0.261191\pi\)
\(648\) − 9.56230i − 0.375642i
\(649\) 6.14762 0.241315
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) − 0.126008i − 0.00493485i
\(653\) − 11.7626i − 0.460305i −0.973155 0.230153i \(-0.926077\pi\)
0.973155 0.230153i \(-0.0739225\pi\)
\(654\) −3.31406 −0.129590
\(655\) 0 0
\(656\) −27.4617 −1.07220
\(657\) − 0.911603i − 0.0355650i
\(658\) 16.6253i 0.648122i
\(659\) −1.23884 −0.0482584 −0.0241292 0.999709i \(-0.507681\pi\)
−0.0241292 + 0.999709i \(0.507681\pi\)
\(660\) 0 0
\(661\) −9.53690 −0.370943 −0.185471 0.982650i \(-0.559381\pi\)
−0.185471 + 0.982650i \(0.559381\pi\)
\(662\) 18.2374i 0.708818i
\(663\) 31.3258i 1.21659i
\(664\) 18.8265 0.730611
\(665\) 0 0
\(666\) 35.3561 1.37002
\(667\) 17.9248i 0.694050i
\(668\) 2.97812i 0.115227i
\(669\) 0.0244376 0.000944811 0
\(670\) 0 0
\(671\) −10.8510 −0.418897
\(672\) 2.95254i 0.113897i
\(673\) − 39.9307i − 1.53921i −0.638518 0.769607i \(-0.720452\pi\)
0.638518 0.769607i \(-0.279548\pi\)
\(674\) 5.58181 0.215003
\(675\) 0 0
\(676\) 4.82909 0.185734
\(677\) − 3.05334i − 0.117349i −0.998277 0.0586747i \(-0.981313\pi\)
0.998277 0.0586747i \(-0.0186875\pi\)
\(678\) 8.34885i 0.320636i
\(679\) −36.8284 −1.41335
\(680\) 0 0
\(681\) 3.87399 0.148452
\(682\) 8.44568i 0.323402i
\(683\) − 29.2692i − 1.11995i −0.828508 0.559977i \(-0.810810\pi\)
0.828508 0.559977i \(-0.189190\pi\)
\(684\) −0.455802 −0.0174280
\(685\) 0 0
\(686\) 13.7743 0.525906
\(687\) 1.51056i 0.0576313i
\(688\) − 17.9492i − 0.684307i
\(689\) −11.3503 −0.432411
\(690\) 0 0
\(691\) 2.63515 0.100246 0.0501229 0.998743i \(-0.484039\pi\)
0.0501229 + 0.998743i \(0.484039\pi\)
\(692\) 0.629270i 0.0239213i
\(693\) 7.57784i 0.287858i
\(694\) −33.1246 −1.25739
\(695\) 0 0
\(696\) −7.78892 −0.295238
\(697\) − 39.8496i − 1.50941i
\(698\) − 14.8119i − 0.560640i
\(699\) 8.98683 0.339913
\(700\) 0 0
\(701\) −25.0494 −0.946102 −0.473051 0.881035i \(-0.656847\pi\)
−0.473051 + 0.881035i \(0.656847\pi\)
\(702\) 39.3258i 1.48426i
\(703\) 10.1563i 0.383053i
\(704\) −6.81477 −0.256841
\(705\) 0 0
\(706\) −8.20123 −0.308657
\(707\) − 8.87732i − 0.333866i
\(708\) 0.998585i 0.0375291i
\(709\) 41.6991 1.56604 0.783021 0.621995i \(-0.213677\pi\)
0.783021 + 0.621995i \(0.213677\pi\)
\(710\) 0 0
\(711\) 1.96571 0.0737200
\(712\) 18.9624i 0.710646i
\(713\) − 29.4010i − 1.10108i
\(714\) −25.2506 −0.944980
\(715\) 0 0
\(716\) 2.91160 0.108812
\(717\) 7.49597i 0.279942i
\(718\) − 15.3503i − 0.572867i
\(719\) −30.6351 −1.14250 −0.571249 0.820777i \(-0.693541\pi\)
−0.571249 + 0.820777i \(0.693541\pi\)
\(720\) 0 0
\(721\) −35.9511 −1.33889
\(722\) − 1.48119i − 0.0551243i
\(723\) 1.86414i 0.0693282i
\(724\) −1.78892 −0.0664847
\(725\) 0 0
\(726\) 12.0275 0.446382
\(727\) 7.50071i 0.278186i 0.990279 + 0.139093i \(0.0444187\pi\)
−0.990279 + 0.139093i \(0.955581\pi\)
\(728\) − 55.1754i − 2.04494i
\(729\) −2.02776 −0.0751023
\(730\) 0 0
\(731\) 26.0460 0.963348
\(732\) − 1.76257i − 0.0651466i
\(733\) − 9.84955i − 0.363802i −0.983317 0.181901i \(-0.941775\pi\)
0.983317 0.181901i \(-0.0582250\pi\)
\(734\) 4.96239 0.183165
\(735\) 0 0
\(736\) −5.42548 −0.199986
\(737\) − 6.47768i − 0.238609i
\(738\) − 21.9756i − 0.808932i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −4.96239 −0.182298
\(742\) − 9.14903i − 0.335871i
\(743\) 15.7177i 0.576625i 0.957536 + 0.288313i \(0.0930942\pi\)
−0.957536 + 0.288313i \(0.906906\pi\)
\(744\) 12.7757 0.468382
\(745\) 0 0
\(746\) 18.6702 0.683565
\(747\) 16.5402i 0.605175i
\(748\) 1.17821i 0.0430795i
\(749\) −16.1016 −0.588339
\(750\) 0 0
\(751\) −43.7400 −1.59610 −0.798048 0.602593i \(-0.794134\pi\)
−0.798048 + 0.602593i \(0.794134\pi\)
\(752\) − 14.5745i − 0.531478i
\(753\) − 19.4274i − 0.707974i
\(754\) −32.9380 −1.19953
\(755\) 0 0
\(756\) −2.80209 −0.101911
\(757\) − 15.7743i − 0.573328i −0.958031 0.286664i \(-0.907454\pi\)
0.958031 0.286664i \(-0.0925463\pi\)
\(758\) − 55.1754i − 2.00406i
\(759\) 3.84955 0.139730
\(760\) 0 0
\(761\) 43.2262 1.56695 0.783474 0.621425i \(-0.213446\pi\)
0.783474 + 0.621425i \(0.213446\pi\)
\(762\) − 16.0362i − 0.580930i
\(763\) 9.29948i 0.336664i
\(764\) 4.22284 0.152777
\(765\) 0 0
\(766\) −45.6688 −1.65008
\(767\) − 39.3258i − 1.41997i
\(768\) − 3.71767i − 0.134150i
\(769\) 19.1246 0.689650 0.344825 0.938667i \(-0.387938\pi\)
0.344825 + 0.938667i \(0.387938\pi\)
\(770\) 0 0
\(771\) −10.7367 −0.386674
\(772\) 2.43278i 0.0875576i
\(773\) − 25.8846i − 0.931005i −0.885047 0.465502i \(-0.845874\pi\)
0.885047 0.465502i \(-0.154126\pi\)
\(774\) 14.3634 0.516283
\(775\) 0 0
\(776\) 29.4069 1.05565
\(777\) 27.4274i 0.983952i
\(778\) − 2.03620i − 0.0730012i
\(779\) 6.31265 0.226174
\(780\) 0 0
\(781\) −0.746569 −0.0267144
\(782\) − 46.3996i − 1.65925i
\(783\) − 15.5778i − 0.556707i
\(784\) 18.3766 0.656307
\(785\) 0 0
\(786\) −24.6253 −0.878355
\(787\) 19.9814i 0.712261i 0.934436 + 0.356131i \(0.115904\pi\)
−0.934436 + 0.356131i \(0.884096\pi\)
\(788\) − 4.76116i − 0.169609i
\(789\) −10.4485 −0.371977
\(790\) 0 0
\(791\) 23.4274 0.832982
\(792\) − 6.05079i − 0.215005i
\(793\) 69.4128i 2.46492i
\(794\) −13.9032 −0.493405
\(795\) 0 0
\(796\) 4.47486 0.158607
\(797\) 28.6458i 1.01469i 0.861744 + 0.507343i \(0.169372\pi\)
−0.861744 + 0.507343i \(0.830628\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 21.1490 0.748199
\(800\) 0 0
\(801\) −16.6596 −0.588638
\(802\) − 20.8872i − 0.737551i
\(803\) − 0.373285i − 0.0131729i
\(804\) 1.05220 0.0371082
\(805\) 0 0
\(806\) 54.0263 1.90300
\(807\) − 9.18997i − 0.323502i
\(808\) 7.08840i 0.249369i
\(809\) 17.2243 0.605573 0.302786 0.953058i \(-0.402083\pi\)
0.302786 + 0.953058i \(0.402083\pi\)
\(810\) 0 0
\(811\) −15.6267 −0.548728 −0.274364 0.961626i \(-0.588467\pi\)
−0.274364 + 0.961626i \(0.588467\pi\)
\(812\) − 2.34694i − 0.0823613i
\(813\) − 13.5515i − 0.475272i
\(814\) 14.4777 0.507443
\(815\) 0 0
\(816\) 22.1359 0.774910
\(817\) 4.12601i 0.144351i
\(818\) 52.0625i 1.82032i
\(819\) 48.4749 1.69385
\(820\) 0 0
\(821\) 39.2506 1.36986 0.684928 0.728611i \(-0.259834\pi\)
0.684928 + 0.728611i \(0.259834\pi\)
\(822\) − 24.1622i − 0.842754i
\(823\) 45.5271i 1.58697i 0.608588 + 0.793487i \(0.291736\pi\)
−0.608588 + 0.793487i \(0.708264\pi\)
\(824\) 28.7064 1.00003
\(825\) 0 0
\(826\) 31.6991 1.10295
\(827\) 17.0698i 0.593576i 0.954943 + 0.296788i \(0.0959155\pi\)
−0.954943 + 0.296788i \(0.904084\pi\)
\(828\) − 2.26187i − 0.0786052i
\(829\) −1.69911 −0.0590125 −0.0295062 0.999565i \(-0.509393\pi\)
−0.0295062 + 0.999565i \(0.509393\pi\)
\(830\) 0 0
\(831\) −24.0118 −0.832959
\(832\) 43.5936i 1.51134i
\(833\) 26.6662i 0.923930i
\(834\) −20.9986 −0.727122
\(835\) 0 0
\(836\) −0.186642 −0.00645516
\(837\) 25.5515i 0.883189i
\(838\) 13.3719i 0.461924i
\(839\) 50.5910 1.74660 0.873298 0.487187i \(-0.161977\pi\)
0.873298 + 0.487187i \(0.161977\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) − 22.5285i − 0.776382i
\(843\) 9.36011i 0.322379i
\(844\) 4.06063 0.139773
\(845\) 0 0
\(846\) 11.6629 0.400979
\(847\) − 33.7499i − 1.15966i
\(848\) 8.02047i 0.275424i
\(849\) −1.82321 −0.0625723
\(850\) 0 0
\(851\) −50.3996 −1.72768
\(852\) − 0.121269i − 0.00415460i
\(853\) 22.5237i 0.771198i 0.922667 + 0.385599i \(0.126005\pi\)
−0.922667 + 0.385599i \(0.873995\pi\)
\(854\) −55.9511 −1.91461
\(855\) 0 0
\(856\) 12.8568 0.439438
\(857\) − 23.6180i − 0.806776i −0.915029 0.403388i \(-0.867833\pi\)
0.915029 0.403388i \(-0.132167\pi\)
\(858\) 7.07381i 0.241496i
\(859\) −15.1754 −0.517777 −0.258889 0.965907i \(-0.583356\pi\)
−0.258889 + 0.965907i \(0.583356\pi\)
\(860\) 0 0
\(861\) 17.0475 0.580976
\(862\) − 24.1622i − 0.822968i
\(863\) − 30.1055i − 1.02480i −0.858746 0.512402i \(-0.828756\pi\)
0.858746 0.512402i \(-0.171244\pi\)
\(864\) 4.71511 0.160411
\(865\) 0 0
\(866\) −16.5052 −0.560869
\(867\) 18.4182i 0.625515i
\(868\) 3.84955i 0.130662i
\(869\) 0.804923 0.0273051
\(870\) 0 0
\(871\) −41.4372 −1.40405
\(872\) − 7.42548i − 0.251459i
\(873\) 25.8357i 0.874407i
\(874\) 7.35026 0.248626
\(875\) 0 0
\(876\) 0.0606343 0.00204864
\(877\) − 5.53102i − 0.186769i −0.995630 0.0933847i \(-0.970231\pi\)
0.995630 0.0933847i \(-0.0297687\pi\)
\(878\) 40.5647i 1.36899i
\(879\) −1.48612 −0.0501255
\(880\) 0 0
\(881\) 20.8265 0.701664 0.350832 0.936438i \(-0.385899\pi\)
0.350832 + 0.936438i \(0.385899\pi\)
\(882\) 14.7054i 0.495158i
\(883\) − 43.1509i − 1.45214i −0.687618 0.726072i \(-0.741344\pi\)
0.687618 0.726072i \(-0.258656\pi\)
\(884\) 7.53690 0.253494
\(885\) 0 0
\(886\) −28.9018 −0.970973
\(887\) − 44.5461i − 1.49571i −0.663861 0.747856i \(-0.731083\pi\)
0.663861 0.747856i \(-0.268917\pi\)
\(888\) − 21.9003i − 0.734927i
\(889\) −44.9986 −1.50920
\(890\) 0 0
\(891\) 3.44007 0.115247
\(892\) − 0.00587961i 0 0.000196864i
\(893\) 3.35026i 0.112112i
\(894\) −8.86556 −0.296509
\(895\) 0 0
\(896\) −42.4650 −1.41866
\(897\) − 24.6253i − 0.822215i
\(898\) − 32.7367i − 1.09244i
\(899\) −21.4010 −0.713765
\(900\) 0 0
\(901\) −11.6385 −0.387734
\(902\) − 8.99859i − 0.299620i
\(903\) 11.1424i 0.370795i
\(904\) −18.7064 −0.622166
\(905\) 0 0
\(906\) 1.92478 0.0639464
\(907\) 53.7558i 1.78493i 0.451115 + 0.892466i \(0.351026\pi\)
−0.451115 + 0.892466i \(0.648974\pi\)
\(908\) − 0.932071i − 0.0309319i
\(909\) −6.22758 −0.206556
\(910\) 0 0
\(911\) −2.28630 −0.0757486 −0.0378743 0.999283i \(-0.512059\pi\)
−0.0378743 + 0.999283i \(0.512059\pi\)
\(912\) 3.50659i 0.116115i
\(913\) 6.77292i 0.224151i
\(914\) −26.4387 −0.874513
\(915\) 0 0
\(916\) 0.363436 0.0120082
\(917\) 69.1002i 2.28189i
\(918\) 40.3244i 1.33090i
\(919\) 34.8510 1.14963 0.574814 0.818284i \(-0.305075\pi\)
0.574814 + 0.818284i \(0.305075\pi\)
\(920\) 0 0
\(921\) 21.1246 0.696079
\(922\) 52.2130i 1.71954i
\(923\) 4.77575i 0.157196i
\(924\) −0.504032 −0.0165814
\(925\) 0 0
\(926\) 39.0494 1.28324
\(927\) 25.2203i 0.828343i
\(928\) 3.94921i 0.129639i
\(929\) −41.6991 −1.36810 −0.684052 0.729434i \(-0.739784\pi\)
−0.684052 + 0.729434i \(0.739784\pi\)
\(930\) 0 0
\(931\) −4.22425 −0.138444
\(932\) − 2.16220i − 0.0708254i
\(933\) − 5.25060i − 0.171897i
\(934\) −10.0508 −0.328872
\(935\) 0 0
\(936\) −38.7064 −1.26516
\(937\) 21.9102i 0.715775i 0.933765 + 0.357887i \(0.116503\pi\)
−0.933765 + 0.357887i \(0.883497\pi\)
\(938\) − 33.4010i − 1.09058i
\(939\) −12.9576 −0.422857
\(940\) 0 0
\(941\) −3.55149 −0.115775 −0.0578877 0.998323i \(-0.518437\pi\)
−0.0578877 + 0.998323i \(0.518437\pi\)
\(942\) − 5.23884i − 0.170691i
\(943\) 31.3258i 1.02011i
\(944\) −27.7889 −0.904452
\(945\) 0 0
\(946\) 5.88156 0.191226
\(947\) 38.3634i 1.24664i 0.781965 + 0.623322i \(0.214217\pi\)
−0.781965 + 0.623322i \(0.785783\pi\)
\(948\) 0.130747i 0.00424648i
\(949\) −2.38787 −0.0775136
\(950\) 0 0
\(951\) −4.58910 −0.148812
\(952\) − 56.5764i − 1.83365i
\(953\) 22.8714i 0.740879i 0.928857 + 0.370439i \(0.120793\pi\)
−0.928857 + 0.370439i \(0.879207\pi\)
\(954\) −6.41819 −0.207797
\(955\) 0 0
\(956\) 1.80351 0.0583296
\(957\) − 2.80209i − 0.0905788i
\(958\) 18.8265i 0.608258i
\(959\) −67.8007 −2.18940
\(960\) 0 0
\(961\) 4.10299 0.132354
\(962\) − 92.6126i − 2.98595i
\(963\) 11.2955i 0.363993i
\(964\) 0.448507 0.0144455
\(965\) 0 0
\(966\) 19.8496 0.638649
\(967\) 27.6629i 0.889579i 0.895635 + 0.444790i \(0.146722\pi\)
−0.895635 + 0.444790i \(0.853278\pi\)
\(968\) 26.9488i 0.866166i
\(969\) −5.08840 −0.163463
\(970\) 0 0
\(971\) −42.1768 −1.35352 −0.676759 0.736205i \(-0.736616\pi\)
−0.676759 + 0.736205i \(0.736616\pi\)
\(972\) 3.06793i 0.0984039i
\(973\) 58.9234i 1.88900i
\(974\) −23.3416 −0.747912
\(975\) 0 0
\(976\) 49.0494 1.57003
\(977\) 0.856849i 0.0274130i 0.999906 + 0.0137065i \(0.00436305\pi\)
−0.999906 + 0.0137065i \(0.995637\pi\)
\(978\) − 0.775746i − 0.0248056i
\(979\) −6.82179 −0.218025
\(980\) 0 0
\(981\) 6.52373 0.208287
\(982\) 21.5515i 0.687736i
\(983\) − 45.2809i − 1.44424i −0.691769 0.722119i \(-0.743169\pi\)
0.691769 0.722119i \(-0.256831\pi\)
\(984\) −13.6121 −0.433939
\(985\) 0 0
\(986\) −33.7743 −1.07559
\(987\) 9.04746i 0.287984i
\(988\) 1.19394i 0.0379842i
\(989\) −20.4749 −0.651063
\(990\) 0 0
\(991\) 47.0132 1.49342 0.746711 0.665148i \(-0.231632\pi\)
0.746711 + 0.665148i \(0.231632\pi\)
\(992\) − 6.47768i − 0.205667i
\(993\) 9.92478i 0.314953i
\(994\) −3.84955 −0.122100
\(995\) 0 0
\(996\) −1.10016 −0.0348598
\(997\) 13.6873i 0.433483i 0.976229 + 0.216741i \(0.0695428\pi\)
−0.976229 + 0.216741i \(0.930457\pi\)
\(998\) − 8.12601i − 0.257224i
\(999\) 43.8007 1.38579
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 475.2.b.d.324.2 6
5.2 odd 4 475.2.a.f.1.3 3
5.3 odd 4 95.2.a.a.1.1 3
5.4 even 2 inner 475.2.b.d.324.5 6
15.2 even 4 4275.2.a.bk.1.1 3
15.8 even 4 855.2.a.i.1.3 3
20.3 even 4 1520.2.a.p.1.2 3
20.7 even 4 7600.2.a.bx.1.2 3
35.13 even 4 4655.2.a.u.1.1 3
40.3 even 4 6080.2.a.by.1.2 3
40.13 odd 4 6080.2.a.bo.1.2 3
95.18 even 4 1805.2.a.f.1.3 3
95.37 even 4 9025.2.a.bb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.a.1.1 3 5.3 odd 4
475.2.a.f.1.3 3 5.2 odd 4
475.2.b.d.324.2 6 1.1 even 1 trivial
475.2.b.d.324.5 6 5.4 even 2 inner
855.2.a.i.1.3 3 15.8 even 4
1520.2.a.p.1.2 3 20.3 even 4
1805.2.a.f.1.3 3 95.18 even 4
4275.2.a.bk.1.1 3 15.2 even 4
4655.2.a.u.1.1 3 35.13 even 4
6080.2.a.bo.1.2 3 40.13 odd 4
6080.2.a.by.1.2 3 40.3 even 4
7600.2.a.bx.1.2 3 20.7 even 4
9025.2.a.bb.1.1 3 95.37 even 4