Properties

Label 1425.2.c.k.799.1
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.k.799.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.73205i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.73205i q^{7} -1.73205i q^{8} -1.00000 q^{9} +4.73205 q^{11} +1.00000i q^{12} -0.732051i q^{13} -4.73205 q^{14} -5.00000 q^{16} +1.73205i q^{18} -1.00000 q^{19} -2.73205 q^{21} -8.19615i q^{22} -3.46410i q^{23} -1.73205 q^{24} -1.26795 q^{26} +1.00000i q^{27} +2.73205i q^{28} -8.19615 q^{29} +8.92820 q^{31} +5.19615i q^{32} -4.73205i q^{33} +1.00000 q^{36} -6.19615i q^{37} +1.73205i q^{38} -0.732051 q^{39} +1.26795 q^{41} +4.73205i q^{42} -4.19615i q^{43} -4.73205 q^{44} -6.00000 q^{46} -3.46410i q^{47} +5.00000i q^{48} -0.464102 q^{49} +0.732051i q^{52} +9.46410i q^{53} +1.73205 q^{54} -4.73205 q^{56} +1.00000i q^{57} +14.1962i q^{58} -2.53590 q^{59} -6.53590 q^{61} -15.4641i q^{62} +2.73205i q^{63} -1.00000 q^{64} -8.19615 q^{66} +8.00000i q^{67} -3.46410 q^{69} -4.39230 q^{71} +1.73205i q^{72} +16.9282i q^{73} -10.7321 q^{74} +1.00000 q^{76} -12.9282i q^{77} +1.26795i q^{78} +10.9282 q^{79} +1.00000 q^{81} -2.19615i q^{82} +12.9282i q^{83} +2.73205 q^{84} -7.26795 q^{86} +8.19615i q^{87} -8.19615i q^{88} -10.7321 q^{89} -2.00000 q^{91} +3.46410i q^{92} -8.92820i q^{93} -6.00000 q^{94} +5.19615 q^{96} -6.19615i q^{97} +0.803848i q^{98} -4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 4 q^{19} - 4 q^{21} - 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{36} + 4 q^{39} + 12 q^{41} - 12 q^{44} - 24 q^{46} + 12 q^{49} - 12 q^{56} - 24 q^{59} - 40 q^{61} - 4 q^{64} - 12 q^{66} + 24 q^{71} - 36 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{81} + 4 q^{84} - 36 q^{86} - 36 q^{89} - 8 q^{91} - 24 q^{94} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.73205 −0.707107
\(7\) − 2.73205i − 1.03262i −0.856402 0.516309i \(-0.827306\pi\)
0.856402 0.516309i \(-0.172694\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 0.732051i − 0.203034i −0.994834 0.101517i \(-0.967630\pi\)
0.994834 0.101517i \(-0.0323697\pi\)
\(14\) −4.73205 −1.26469
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.73205i 0.408248i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.73205 −0.596182
\(22\) − 8.19615i − 1.74743i
\(23\) − 3.46410i − 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) −1.26795 −0.248665
\(27\) 1.00000i 0.192450i
\(28\) 2.73205i 0.516309i
\(29\) −8.19615 −1.52199 −0.760994 0.648759i \(-0.775288\pi\)
−0.760994 + 0.648759i \(0.775288\pi\)
\(30\) 0 0
\(31\) 8.92820 1.60355 0.801776 0.597624i \(-0.203889\pi\)
0.801776 + 0.597624i \(0.203889\pi\)
\(32\) 5.19615i 0.918559i
\(33\) − 4.73205i − 0.823744i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 6.19615i − 1.01864i −0.860577 0.509321i \(-0.829897\pi\)
0.860577 0.509321i \(-0.170103\pi\)
\(38\) 1.73205i 0.280976i
\(39\) −0.732051 −0.117222
\(40\) 0 0
\(41\) 1.26795 0.198020 0.0990102 0.995086i \(-0.468432\pi\)
0.0990102 + 0.995086i \(0.468432\pi\)
\(42\) 4.73205i 0.730171i
\(43\) − 4.19615i − 0.639907i −0.947433 0.319954i \(-0.896333\pi\)
0.947433 0.319954i \(-0.103667\pi\)
\(44\) −4.73205 −0.713384
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) − 3.46410i − 0.505291i −0.967559 0.252646i \(-0.918699\pi\)
0.967559 0.252646i \(-0.0813007\pi\)
\(48\) 5.00000i 0.721688i
\(49\) −0.464102 −0.0663002
\(50\) 0 0
\(51\) 0 0
\(52\) 0.732051i 0.101517i
\(53\) 9.46410i 1.29999i 0.759937 + 0.649997i \(0.225230\pi\)
−0.759937 + 0.649997i \(0.774770\pi\)
\(54\) 1.73205 0.235702
\(55\) 0 0
\(56\) −4.73205 −0.632347
\(57\) 1.00000i 0.132453i
\(58\) 14.1962i 1.86405i
\(59\) −2.53590 −0.330146 −0.165073 0.986281i \(-0.552786\pi\)
−0.165073 + 0.986281i \(0.552786\pi\)
\(60\) 0 0
\(61\) −6.53590 −0.836836 −0.418418 0.908255i \(-0.637415\pi\)
−0.418418 + 0.908255i \(0.637415\pi\)
\(62\) − 15.4641i − 1.96394i
\(63\) 2.73205i 0.344206i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −8.19615 −1.00888
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −3.46410 −0.417029
\(70\) 0 0
\(71\) −4.39230 −0.521271 −0.260635 0.965437i \(-0.583932\pi\)
−0.260635 + 0.965437i \(0.583932\pi\)
\(72\) 1.73205i 0.204124i
\(73\) 16.9282i 1.98130i 0.136441 + 0.990648i \(0.456434\pi\)
−0.136441 + 0.990648i \(0.543566\pi\)
\(74\) −10.7321 −1.24758
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) − 12.9282i − 1.47331i
\(78\) 1.26795i 0.143567i
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.19615i − 0.242524i
\(83\) 12.9282i 1.41905i 0.704678 + 0.709527i \(0.251092\pi\)
−0.704678 + 0.709527i \(0.748908\pi\)
\(84\) 2.73205 0.298091
\(85\) 0 0
\(86\) −7.26795 −0.783723
\(87\) 8.19615i 0.878720i
\(88\) − 8.19615i − 0.873713i
\(89\) −10.7321 −1.13760 −0.568798 0.822478i \(-0.692591\pi\)
−0.568798 + 0.822478i \(0.692591\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 3.46410i 0.361158i
\(93\) − 8.92820i − 0.925812i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 5.19615 0.530330
\(97\) − 6.19615i − 0.629124i −0.949237 0.314562i \(-0.898142\pi\)
0.949237 0.314562i \(-0.101858\pi\)
\(98\) 0.803848i 0.0812009i
\(99\) −4.73205 −0.475589
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) − 9.85641i − 0.971181i −0.874187 0.485590i \(-0.838605\pi\)
0.874187 0.485590i \(-0.161395\pi\)
\(104\) −1.26795 −0.124333
\(105\) 0 0
\(106\) 16.3923 1.59216
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 14.3923 1.37853 0.689266 0.724508i \(-0.257933\pi\)
0.689266 + 0.724508i \(0.257933\pi\)
\(110\) 0 0
\(111\) −6.19615 −0.588113
\(112\) 13.6603i 1.29077i
\(113\) − 18.9282i − 1.78062i −0.455359 0.890308i \(-0.650489\pi\)
0.455359 0.890308i \(-0.349511\pi\)
\(114\) 1.73205 0.162221
\(115\) 0 0
\(116\) 8.19615 0.760994
\(117\) 0.732051i 0.0676781i
\(118\) 4.39230i 0.404344i
\(119\) 0 0
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) 11.3205i 1.02491i
\(123\) − 1.26795i − 0.114327i
\(124\) −8.92820 −0.801776
\(125\) 0 0
\(126\) 4.73205 0.421565
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 12.1244i 1.07165i
\(129\) −4.19615 −0.369451
\(130\) 0 0
\(131\) −9.12436 −0.797199 −0.398599 0.917125i \(-0.630504\pi\)
−0.398599 + 0.917125i \(0.630504\pi\)
\(132\) 4.73205i 0.411872i
\(133\) 2.73205i 0.236899i
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 19.8564i 1.69645i 0.529638 + 0.848224i \(0.322328\pi\)
−0.529638 + 0.848224i \(0.677672\pi\)
\(138\) 6.00000i 0.510754i
\(139\) 8.39230 0.711826 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(140\) 0 0
\(141\) −3.46410 −0.291730
\(142\) 7.60770i 0.638424i
\(143\) − 3.46410i − 0.289683i
\(144\) 5.00000 0.416667
\(145\) 0 0
\(146\) 29.3205 2.42658
\(147\) 0.464102i 0.0382785i
\(148\) 6.19615i 0.509321i
\(149\) 19.8564 1.62670 0.813350 0.581775i \(-0.197641\pi\)
0.813350 + 0.581775i \(0.197641\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 1.73205i 0.140488i
\(153\) 0 0
\(154\) −22.3923 −1.80442
\(155\) 0 0
\(156\) 0.732051 0.0586110
\(157\) 6.39230i 0.510161i 0.966920 + 0.255081i \(0.0821021\pi\)
−0.966920 + 0.255081i \(0.917898\pi\)
\(158\) − 18.9282i − 1.50585i
\(159\) 9.46410 0.750552
\(160\) 0 0
\(161\) −9.46410 −0.745876
\(162\) − 1.73205i − 0.136083i
\(163\) − 9.26795i − 0.725922i −0.931804 0.362961i \(-0.881766\pi\)
0.931804 0.362961i \(-0.118234\pi\)
\(164\) −1.26795 −0.0990102
\(165\) 0 0
\(166\) 22.3923 1.73798
\(167\) 3.46410i 0.268060i 0.990977 + 0.134030i \(0.0427919\pi\)
−0.990977 + 0.134030i \(0.957208\pi\)
\(168\) 4.73205i 0.365086i
\(169\) 12.4641 0.958777
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 4.19615i 0.319954i
\(173\) − 6.92820i − 0.526742i −0.964695 0.263371i \(-0.915166\pi\)
0.964695 0.263371i \(-0.0848343\pi\)
\(174\) 14.1962 1.07621
\(175\) 0 0
\(176\) −23.6603 −1.78346
\(177\) 2.53590i 0.190610i
\(178\) 18.5885i 1.39326i
\(179\) −23.3205 −1.74306 −0.871528 0.490345i \(-0.836871\pi\)
−0.871528 + 0.490345i \(0.836871\pi\)
\(180\) 0 0
\(181\) −2.39230 −0.177819 −0.0889093 0.996040i \(-0.528338\pi\)
−0.0889093 + 0.996040i \(0.528338\pi\)
\(182\) 3.46410i 0.256776i
\(183\) 6.53590i 0.483148i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) −15.4641 −1.13388
\(187\) 0 0
\(188\) 3.46410i 0.252646i
\(189\) 2.73205 0.198727
\(190\) 0 0
\(191\) 0.339746 0.0245832 0.0122916 0.999924i \(-0.496087\pi\)
0.0122916 + 0.999924i \(0.496087\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 17.1244i − 1.23264i −0.787497 0.616319i \(-0.788623\pi\)
0.787497 0.616319i \(-0.211377\pi\)
\(194\) −10.7321 −0.770516
\(195\) 0 0
\(196\) 0.464102 0.0331501
\(197\) − 24.0000i − 1.70993i −0.518686 0.854965i \(-0.673579\pi\)
0.518686 0.854965i \(-0.326421\pi\)
\(198\) 8.19615i 0.582475i
\(199\) 15.3205 1.08604 0.543021 0.839719i \(-0.317280\pi\)
0.543021 + 0.839719i \(0.317280\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 18.0000i 1.26648i
\(203\) 22.3923i 1.57163i
\(204\) 0 0
\(205\) 0 0
\(206\) −17.0718 −1.18945
\(207\) 3.46410i 0.240772i
\(208\) 3.66025i 0.253793i
\(209\) −4.73205 −0.327323
\(210\) 0 0
\(211\) 1.07180 0.0737855 0.0368928 0.999319i \(-0.488254\pi\)
0.0368928 + 0.999319i \(0.488254\pi\)
\(212\) − 9.46410i − 0.649997i
\(213\) 4.39230i 0.300956i
\(214\) 0 0
\(215\) 0 0
\(216\) 1.73205 0.117851
\(217\) − 24.3923i − 1.65586i
\(218\) − 24.9282i − 1.68835i
\(219\) 16.9282 1.14390
\(220\) 0 0
\(221\) 0 0
\(222\) 10.7321i 0.720288i
\(223\) 17.8564i 1.19575i 0.801588 + 0.597877i \(0.203989\pi\)
−0.801588 + 0.597877i \(0.796011\pi\)
\(224\) 14.1962 0.948520
\(225\) 0 0
\(226\) −32.7846 −2.18080
\(227\) 10.3923i 0.689761i 0.938647 + 0.344881i \(0.112081\pi\)
−0.938647 + 0.344881i \(0.887919\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 18.5359 1.22489 0.612443 0.790515i \(-0.290187\pi\)
0.612443 + 0.790515i \(0.290187\pi\)
\(230\) 0 0
\(231\) −12.9282 −0.850613
\(232\) 14.1962i 0.932023i
\(233\) 7.85641i 0.514690i 0.966320 + 0.257345i \(0.0828477\pi\)
−0.966320 + 0.257345i \(0.917152\pi\)
\(234\) 1.26795 0.0828884
\(235\) 0 0
\(236\) 2.53590 0.165073
\(237\) − 10.9282i − 0.709863i
\(238\) 0 0
\(239\) 9.80385 0.634158 0.317079 0.948399i \(-0.397298\pi\)
0.317079 + 0.948399i \(0.397298\pi\)
\(240\) 0 0
\(241\) −3.07180 −0.197872 −0.0989359 0.995094i \(-0.531544\pi\)
−0.0989359 + 0.995094i \(0.531544\pi\)
\(242\) − 19.7321i − 1.26842i
\(243\) − 1.00000i − 0.0641500i
\(244\) 6.53590 0.418418
\(245\) 0 0
\(246\) −2.19615 −0.140022
\(247\) 0.732051i 0.0465793i
\(248\) − 15.4641i − 0.981971i
\(249\) 12.9282 0.819292
\(250\) 0 0
\(251\) 28.0526 1.77066 0.885331 0.464961i \(-0.153932\pi\)
0.885331 + 0.464961i \(0.153932\pi\)
\(252\) − 2.73205i − 0.172103i
\(253\) − 16.3923i − 1.03058i
\(254\) −6.92820 −0.434714
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 24.0000i − 1.49708i −0.663090 0.748539i \(-0.730755\pi\)
0.663090 0.748539i \(-0.269245\pi\)
\(258\) 7.26795i 0.452483i
\(259\) −16.9282 −1.05187
\(260\) 0 0
\(261\) 8.19615 0.507329
\(262\) 15.8038i 0.976365i
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) −8.19615 −0.504438
\(265\) 0 0
\(266\) 4.73205 0.290141
\(267\) 10.7321i 0.656791i
\(268\) − 8.00000i − 0.488678i
\(269\) −0.588457 −0.0358789 −0.0179394 0.999839i \(-0.505711\pi\)
−0.0179394 + 0.999839i \(0.505711\pi\)
\(270\) 0 0
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 34.3923 2.07772
\(275\) 0 0
\(276\) 3.46410 0.208514
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 14.5359i − 0.871805i
\(279\) −8.92820 −0.534518
\(280\) 0 0
\(281\) 1.26795 0.0756395 0.0378198 0.999285i \(-0.487959\pi\)
0.0378198 + 0.999285i \(0.487959\pi\)
\(282\) 6.00000i 0.357295i
\(283\) − 24.9808i − 1.48495i −0.669873 0.742476i \(-0.733651\pi\)
0.669873 0.742476i \(-0.266349\pi\)
\(284\) 4.39230 0.260635
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) − 3.46410i − 0.204479i
\(288\) − 5.19615i − 0.306186i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −6.19615 −0.363225
\(292\) − 16.9282i − 0.990648i
\(293\) − 27.7128i − 1.61900i −0.587120 0.809500i \(-0.699738\pi\)
0.587120 0.809500i \(-0.300262\pi\)
\(294\) 0.803848 0.0468813
\(295\) 0 0
\(296\) −10.7321 −0.623788
\(297\) 4.73205i 0.274581i
\(298\) − 34.3923i − 1.99229i
\(299\) −2.53590 −0.146655
\(300\) 0 0
\(301\) −11.4641 −0.660780
\(302\) − 24.2487i − 1.39536i
\(303\) 10.3923i 0.597022i
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) − 32.3923i − 1.84873i −0.381514 0.924363i \(-0.624597\pi\)
0.381514 0.924363i \(-0.375403\pi\)
\(308\) 12.9282i 0.736653i
\(309\) −9.85641 −0.560711
\(310\) 0 0
\(311\) 32.4449 1.83978 0.919890 0.392177i \(-0.128278\pi\)
0.919890 + 0.392177i \(0.128278\pi\)
\(312\) 1.26795i 0.0717835i
\(313\) − 6.39230i − 0.361314i −0.983546 0.180657i \(-0.942178\pi\)
0.983546 0.180657i \(-0.0578225\pi\)
\(314\) 11.0718 0.624818
\(315\) 0 0
\(316\) −10.9282 −0.614759
\(317\) − 11.3205i − 0.635823i −0.948120 0.317912i \(-0.897018\pi\)
0.948120 0.317912i \(-0.102982\pi\)
\(318\) − 16.3923i − 0.919235i
\(319\) −38.7846 −2.17152
\(320\) 0 0
\(321\) 0 0
\(322\) 16.3923i 0.913507i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0526 −0.889069
\(327\) − 14.3923i − 0.795896i
\(328\) − 2.19615i − 0.121262i
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) −25.7128 −1.41330 −0.706652 0.707561i \(-0.749795\pi\)
−0.706652 + 0.707561i \(0.749795\pi\)
\(332\) − 12.9282i − 0.709527i
\(333\) 6.19615i 0.339547i
\(334\) 6.00000 0.328305
\(335\) 0 0
\(336\) 13.6603 0.745228
\(337\) 5.12436i 0.279141i 0.990212 + 0.139571i \(0.0445723\pi\)
−0.990212 + 0.139571i \(0.955428\pi\)
\(338\) − 21.5885i − 1.17426i
\(339\) −18.9282 −1.02804
\(340\) 0 0
\(341\) 42.2487 2.28790
\(342\) − 1.73205i − 0.0936586i
\(343\) − 17.8564i − 0.964155i
\(344\) −7.26795 −0.391862
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) − 0.928203i − 0.0498286i −0.999690 0.0249143i \(-0.992069\pi\)
0.999690 0.0249143i \(-0.00793128\pi\)
\(348\) − 8.19615i − 0.439360i
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0.732051 0.0390740
\(352\) 24.5885i 1.31057i
\(353\) 14.7846i 0.786905i 0.919345 + 0.393453i \(0.128719\pi\)
−0.919345 + 0.393453i \(0.871281\pi\)
\(354\) 4.39230 0.233448
\(355\) 0 0
\(356\) 10.7321 0.568798
\(357\) 0 0
\(358\) 40.3923i 2.13480i
\(359\) −0.339746 −0.0179311 −0.00896555 0.999960i \(-0.502854\pi\)
−0.00896555 + 0.999960i \(0.502854\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.14359i 0.217782i
\(363\) − 11.3923i − 0.597941i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 11.3205 0.591732
\(367\) 16.1962i 0.845432i 0.906262 + 0.422716i \(0.138923\pi\)
−0.906262 + 0.422716i \(0.861077\pi\)
\(368\) 17.3205i 0.902894i
\(369\) −1.26795 −0.0660068
\(370\) 0 0
\(371\) 25.8564 1.34240
\(372\) 8.92820i 0.462906i
\(373\) 6.19615i 0.320825i 0.987050 + 0.160412i \(0.0512824\pi\)
−0.987050 + 0.160412i \(0.948718\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 6.00000i 0.309016i
\(378\) − 4.73205i − 0.243390i
\(379\) −20.9282 −1.07501 −0.537505 0.843261i \(-0.680633\pi\)
−0.537505 + 0.843261i \(0.680633\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) − 0.588457i − 0.0301081i
\(383\) 17.0718i 0.872328i 0.899867 + 0.436164i \(0.143663\pi\)
−0.899867 + 0.436164i \(0.856337\pi\)
\(384\) 12.1244 0.618718
\(385\) 0 0
\(386\) −29.6603 −1.50967
\(387\) 4.19615i 0.213302i
\(388\) 6.19615i 0.314562i
\(389\) −7.85641 −0.398336 −0.199168 0.979965i \(-0.563824\pi\)
−0.199168 + 0.979965i \(0.563824\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.803848i 0.0406004i
\(393\) 9.12436i 0.460263i
\(394\) −41.5692 −2.09423
\(395\) 0 0
\(396\) 4.73205 0.237795
\(397\) 8.92820i 0.448094i 0.974578 + 0.224047i \(0.0719269\pi\)
−0.974578 + 0.224047i \(0.928073\pi\)
\(398\) − 26.5359i − 1.33012i
\(399\) 2.73205 0.136774
\(400\) 0 0
\(401\) −34.0526 −1.70050 −0.850252 0.526376i \(-0.823550\pi\)
−0.850252 + 0.526376i \(0.823550\pi\)
\(402\) − 13.8564i − 0.691095i
\(403\) − 6.53590i − 0.325576i
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) 38.7846 1.92485
\(407\) − 29.3205i − 1.45336i
\(408\) 0 0
\(409\) 26.3923 1.30502 0.652508 0.757782i \(-0.273717\pi\)
0.652508 + 0.757782i \(0.273717\pi\)
\(410\) 0 0
\(411\) 19.8564 0.979444
\(412\) 9.85641i 0.485590i
\(413\) 6.92820i 0.340915i
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 3.80385 0.186499
\(417\) − 8.39230i − 0.410973i
\(418\) 8.19615i 0.400887i
\(419\) 28.0526 1.37046 0.685229 0.728328i \(-0.259702\pi\)
0.685229 + 0.728328i \(0.259702\pi\)
\(420\) 0 0
\(421\) −18.7846 −0.915506 −0.457753 0.889079i \(-0.651346\pi\)
−0.457753 + 0.889079i \(0.651346\pi\)
\(422\) − 1.85641i − 0.0903685i
\(423\) 3.46410i 0.168430i
\(424\) 16.3923 0.796081
\(425\) 0 0
\(426\) 7.60770 0.368594
\(427\) 17.8564i 0.864132i
\(428\) 0 0
\(429\) −3.46410 −0.167248
\(430\) 0 0
\(431\) −11.3205 −0.545290 −0.272645 0.962115i \(-0.587898\pi\)
−0.272645 + 0.962115i \(0.587898\pi\)
\(432\) − 5.00000i − 0.240563i
\(433\) 10.5885i 0.508849i 0.967093 + 0.254424i \(0.0818860\pi\)
−0.967093 + 0.254424i \(0.918114\pi\)
\(434\) −42.2487 −2.02800
\(435\) 0 0
\(436\) −14.3923 −0.689266
\(437\) 3.46410i 0.165710i
\(438\) − 29.3205i − 1.40099i
\(439\) −26.9282 −1.28521 −0.642607 0.766196i \(-0.722147\pi\)
−0.642607 + 0.766196i \(0.722147\pi\)
\(440\) 0 0
\(441\) 0.464102 0.0221001
\(442\) 0 0
\(443\) 5.32051i 0.252785i 0.991980 + 0.126392i \(0.0403399\pi\)
−0.991980 + 0.126392i \(0.959660\pi\)
\(444\) 6.19615 0.294056
\(445\) 0 0
\(446\) 30.9282 1.46449
\(447\) − 19.8564i − 0.939176i
\(448\) 2.73205i 0.129077i
\(449\) 5.66025 0.267124 0.133562 0.991040i \(-0.457358\pi\)
0.133562 + 0.991040i \(0.457358\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 18.9282i 0.890308i
\(453\) − 14.0000i − 0.657777i
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 1.73205 0.0811107
\(457\) 4.53590i 0.212180i 0.994357 + 0.106090i \(0.0338332\pi\)
−0.994357 + 0.106090i \(0.966167\pi\)
\(458\) − 32.1051i − 1.50017i
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 22.3923i 1.04178i
\(463\) 35.5167i 1.65060i 0.564695 + 0.825300i \(0.308994\pi\)
−0.564695 + 0.825300i \(0.691006\pi\)
\(464\) 40.9808 1.90248
\(465\) 0 0
\(466\) 13.6077 0.630364
\(467\) 20.5359i 0.950288i 0.879908 + 0.475144i \(0.157604\pi\)
−0.879908 + 0.475144i \(0.842396\pi\)
\(468\) − 0.732051i − 0.0338391i
\(469\) 21.8564 1.00924
\(470\) 0 0
\(471\) 6.39230 0.294542
\(472\) 4.39230i 0.202172i
\(473\) − 19.8564i − 0.912999i
\(474\) −18.9282 −0.869401
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.46410i − 0.433331i
\(478\) − 16.9808i − 0.776682i
\(479\) −25.5167 −1.16589 −0.582943 0.812513i \(-0.698099\pi\)
−0.582943 + 0.812513i \(0.698099\pi\)
\(480\) 0 0
\(481\) −4.53590 −0.206819
\(482\) 5.32051i 0.242343i
\(483\) 9.46410i 0.430632i
\(484\) −11.3923 −0.517832
\(485\) 0 0
\(486\) −1.73205 −0.0785674
\(487\) − 32.3923i − 1.46784i −0.679238 0.733918i \(-0.737690\pi\)
0.679238 0.733918i \(-0.262310\pi\)
\(488\) 11.3205i 0.512455i
\(489\) −9.26795 −0.419111
\(490\) 0 0
\(491\) 16.0526 0.724442 0.362221 0.932092i \(-0.382019\pi\)
0.362221 + 0.932092i \(0.382019\pi\)
\(492\) 1.26795i 0.0571636i
\(493\) 0 0
\(494\) 1.26795 0.0570477
\(495\) 0 0
\(496\) −44.6410 −2.00444
\(497\) 12.0000i 0.538274i
\(498\) − 22.3923i − 1.00342i
\(499\) −10.5359 −0.471652 −0.235826 0.971795i \(-0.575779\pi\)
−0.235826 + 0.971795i \(0.575779\pi\)
\(500\) 0 0
\(501\) 3.46410 0.154765
\(502\) − 48.5885i − 2.16861i
\(503\) 23.0718i 1.02872i 0.857574 + 0.514360i \(0.171971\pi\)
−0.857574 + 0.514360i \(0.828029\pi\)
\(504\) 4.73205 0.210782
\(505\) 0 0
\(506\) −28.3923 −1.26219
\(507\) − 12.4641i − 0.553550i
\(508\) 4.00000i 0.177471i
\(509\) 10.0526 0.445572 0.222786 0.974867i \(-0.428485\pi\)
0.222786 + 0.974867i \(0.428485\pi\)
\(510\) 0 0
\(511\) 46.2487 2.04592
\(512\) − 8.66025i − 0.382733i
\(513\) − 1.00000i − 0.0441511i
\(514\) −41.5692 −1.83354
\(515\) 0 0
\(516\) 4.19615 0.184725
\(517\) − 16.3923i − 0.720933i
\(518\) 29.3205i 1.28827i
\(519\) −6.92820 −0.304114
\(520\) 0 0
\(521\) 37.2679 1.63274 0.816369 0.577530i \(-0.195983\pi\)
0.816369 + 0.577530i \(0.195983\pi\)
\(522\) − 14.1962i − 0.621349i
\(523\) − 8.67949i − 0.379528i −0.981830 0.189764i \(-0.939228\pi\)
0.981830 0.189764i \(-0.0607722\pi\)
\(524\) 9.12436 0.398599
\(525\) 0 0
\(526\) −10.3923 −0.453126
\(527\) 0 0
\(528\) 23.6603i 1.02968i
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 2.53590 0.110049
\(532\) − 2.73205i − 0.118449i
\(533\) − 0.928203i − 0.0402049i
\(534\) 18.5885 0.804401
\(535\) 0 0
\(536\) 13.8564 0.598506
\(537\) 23.3205i 1.00635i
\(538\) 1.01924i 0.0439425i
\(539\) −2.19615 −0.0945950
\(540\) 0 0
\(541\) 41.7128 1.79337 0.896687 0.442665i \(-0.145967\pi\)
0.896687 + 0.442665i \(0.145967\pi\)
\(542\) − 0.679492i − 0.0291867i
\(543\) 2.39230i 0.102664i
\(544\) 0 0
\(545\) 0 0
\(546\) 3.46410 0.148250
\(547\) 43.3205i 1.85225i 0.377215 + 0.926126i \(0.376882\pi\)
−0.377215 + 0.926126i \(0.623118\pi\)
\(548\) − 19.8564i − 0.848224i
\(549\) 6.53590 0.278945
\(550\) 0 0
\(551\) 8.19615 0.349168
\(552\) 6.00000i 0.255377i
\(553\) − 29.8564i − 1.26962i
\(554\) 3.46410 0.147176
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) 0.928203i 0.0393292i 0.999807 + 0.0196646i \(0.00625985\pi\)
−0.999807 + 0.0196646i \(0.993740\pi\)
\(558\) 15.4641i 0.654648i
\(559\) −3.07180 −0.129923
\(560\) 0 0
\(561\) 0 0
\(562\) − 2.19615i − 0.0926391i
\(563\) 27.4641i 1.15747i 0.815514 + 0.578737i \(0.196454\pi\)
−0.815514 + 0.578737i \(0.803546\pi\)
\(564\) 3.46410 0.145865
\(565\) 0 0
\(566\) −43.2679 −1.81869
\(567\) − 2.73205i − 0.114735i
\(568\) 7.60770i 0.319212i
\(569\) −22.0526 −0.924491 −0.462246 0.886752i \(-0.652956\pi\)
−0.462246 + 0.886752i \(0.652956\pi\)
\(570\) 0 0
\(571\) −34.2487 −1.43326 −0.716632 0.697452i \(-0.754317\pi\)
−0.716632 + 0.697452i \(0.754317\pi\)
\(572\) 3.46410i 0.144841i
\(573\) − 0.339746i − 0.0141931i
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 15.1769i 0.631823i 0.948789 + 0.315912i \(0.102310\pi\)
−0.948789 + 0.315912i \(0.897690\pi\)
\(578\) − 29.4449i − 1.22474i
\(579\) −17.1244 −0.711664
\(580\) 0 0
\(581\) 35.3205 1.46534
\(582\) 10.7321i 0.444858i
\(583\) 44.7846i 1.85479i
\(584\) 29.3205 1.21329
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) − 3.46410i − 0.142979i −0.997441 0.0714894i \(-0.977225\pi\)
0.997441 0.0714894i \(-0.0227752\pi\)
\(588\) − 0.464102i − 0.0191392i
\(589\) −8.92820 −0.367880
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 30.9808i 1.27330i
\(593\) 38.7846i 1.59269i 0.604841 + 0.796347i \(0.293237\pi\)
−0.604841 + 0.796347i \(0.706763\pi\)
\(594\) 8.19615 0.336292
\(595\) 0 0
\(596\) −19.8564 −0.813350
\(597\) − 15.3205i − 0.627027i
\(598\) 4.39230i 0.179615i
\(599\) 13.8564 0.566157 0.283079 0.959097i \(-0.408644\pi\)
0.283079 + 0.959097i \(0.408644\pi\)
\(600\) 0 0
\(601\) −47.1769 −1.92439 −0.962193 0.272368i \(-0.912193\pi\)
−0.962193 + 0.272368i \(0.912193\pi\)
\(602\) 19.8564i 0.809287i
\(603\) − 8.00000i − 0.325785i
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) − 11.6077i − 0.471142i −0.971857 0.235571i \(-0.924304\pi\)
0.971857 0.235571i \(-0.0756960\pi\)
\(608\) − 5.19615i − 0.210732i
\(609\) 22.3923 0.907382
\(610\) 0 0
\(611\) −2.53590 −0.102591
\(612\) 0 0
\(613\) − 42.3923i − 1.71221i −0.516803 0.856105i \(-0.672878\pi\)
0.516803 0.856105i \(-0.327122\pi\)
\(614\) −56.1051 −2.26422
\(615\) 0 0
\(616\) −22.3923 −0.902212
\(617\) 27.7128i 1.11568i 0.829950 + 0.557838i \(0.188369\pi\)
−0.829950 + 0.557838i \(0.811631\pi\)
\(618\) 17.0718i 0.686728i
\(619\) 15.3205 0.615783 0.307892 0.951421i \(-0.400377\pi\)
0.307892 + 0.951421i \(0.400377\pi\)
\(620\) 0 0
\(621\) 3.46410 0.139010
\(622\) − 56.1962i − 2.25326i
\(623\) 29.3205i 1.17470i
\(624\) 3.66025 0.146527
\(625\) 0 0
\(626\) −11.0718 −0.442518
\(627\) 4.73205i 0.188980i
\(628\) − 6.39230i − 0.255081i
\(629\) 0 0
\(630\) 0 0
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) − 18.9282i − 0.752923i
\(633\) − 1.07180i − 0.0426001i
\(634\) −19.6077 −0.778721
\(635\) 0 0
\(636\) −9.46410 −0.375276
\(637\) 0.339746i 0.0134612i
\(638\) 67.1769i 2.65956i
\(639\) 4.39230 0.173757
\(640\) 0 0
\(641\) 48.5885 1.91913 0.959564 0.281489i \(-0.0908284\pi\)
0.959564 + 0.281489i \(0.0908284\pi\)
\(642\) 0 0
\(643\) 12.1962i 0.480969i 0.970653 + 0.240485i \(0.0773064\pi\)
−0.970653 + 0.240485i \(0.922694\pi\)
\(644\) 9.46410 0.372938
\(645\) 0 0
\(646\) 0 0
\(647\) − 4.14359i − 0.162901i −0.996677 0.0814507i \(-0.974045\pi\)
0.996677 0.0814507i \(-0.0259553\pi\)
\(648\) − 1.73205i − 0.0680414i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −24.3923 −0.956010
\(652\) 9.26795i 0.362961i
\(653\) − 17.0718i − 0.668071i −0.942560 0.334036i \(-0.891589\pi\)
0.942560 0.334036i \(-0.108411\pi\)
\(654\) −24.9282 −0.974770
\(655\) 0 0
\(656\) −6.33975 −0.247525
\(657\) − 16.9282i − 0.660432i
\(658\) 16.3923i 0.639039i
\(659\) −5.07180 −0.197569 −0.0987846 0.995109i \(-0.531495\pi\)
−0.0987846 + 0.995109i \(0.531495\pi\)
\(660\) 0 0
\(661\) 39.1769 1.52381 0.761903 0.647692i \(-0.224265\pi\)
0.761903 + 0.647692i \(0.224265\pi\)
\(662\) 44.5359i 1.73094i
\(663\) 0 0
\(664\) 22.3923 0.868990
\(665\) 0 0
\(666\) 10.7321 0.415859
\(667\) 28.3923i 1.09935i
\(668\) − 3.46410i − 0.134030i
\(669\) 17.8564 0.690369
\(670\) 0 0
\(671\) −30.9282 −1.19397
\(672\) − 14.1962i − 0.547628i
\(673\) − 17.1244i − 0.660095i −0.943964 0.330048i \(-0.892935\pi\)
0.943964 0.330048i \(-0.107065\pi\)
\(674\) 8.87564 0.341877
\(675\) 0 0
\(676\) −12.4641 −0.479389
\(677\) 0.679492i 0.0261150i 0.999915 + 0.0130575i \(0.00415645\pi\)
−0.999915 + 0.0130575i \(0.995844\pi\)
\(678\) 32.7846i 1.25909i
\(679\) −16.9282 −0.649645
\(680\) 0 0
\(681\) 10.3923 0.398234
\(682\) − 73.1769i − 2.80209i
\(683\) 5.07180i 0.194067i 0.995281 + 0.0970335i \(0.0309354\pi\)
−0.995281 + 0.0970335i \(0.969065\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −30.9282 −1.18084
\(687\) − 18.5359i − 0.707189i
\(688\) 20.9808i 0.799884i
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) 12.3923 0.471425 0.235713 0.971823i \(-0.424258\pi\)
0.235713 + 0.971823i \(0.424258\pi\)
\(692\) 6.92820i 0.263371i
\(693\) 12.9282i 0.491102i
\(694\) −1.60770 −0.0610273
\(695\) 0 0
\(696\) 14.1962 0.538104
\(697\) 0 0
\(698\) − 38.1051i − 1.44230i
\(699\) 7.85641 0.297157
\(700\) 0 0
\(701\) −33.7128 −1.27332 −0.636658 0.771147i \(-0.719684\pi\)
−0.636658 + 0.771147i \(0.719684\pi\)
\(702\) − 1.26795i − 0.0478557i
\(703\) 6.19615i 0.233692i
\(704\) −4.73205 −0.178346
\(705\) 0 0
\(706\) 25.6077 0.963758
\(707\) 28.3923i 1.06780i
\(708\) − 2.53590i − 0.0953049i
\(709\) 29.1769 1.09576 0.547881 0.836556i \(-0.315435\pi\)
0.547881 + 0.836556i \(0.315435\pi\)
\(710\) 0 0
\(711\) −10.9282 −0.409840
\(712\) 18.5885i 0.696632i
\(713\) − 30.9282i − 1.15827i
\(714\) 0 0
\(715\) 0 0
\(716\) 23.3205 0.871528
\(717\) − 9.80385i − 0.366131i
\(718\) 0.588457i 0.0219610i
\(719\) 11.6603 0.434854 0.217427 0.976077i \(-0.430234\pi\)
0.217427 + 0.976077i \(0.430234\pi\)
\(720\) 0 0
\(721\) −26.9282 −1.00286
\(722\) − 1.73205i − 0.0644603i
\(723\) 3.07180i 0.114241i
\(724\) 2.39230 0.0889093
\(725\) 0 0
\(726\) −19.7321 −0.732325
\(727\) 25.6603i 0.951686i 0.879530 + 0.475843i \(0.157857\pi\)
−0.879530 + 0.475843i \(0.842143\pi\)
\(728\) 3.46410i 0.128388i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 6.53590i − 0.241574i
\(733\) 18.7846i 0.693825i 0.937897 + 0.346913i \(0.112770\pi\)
−0.937897 + 0.346913i \(0.887230\pi\)
\(734\) 28.0526 1.03544
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) 37.8564i 1.39446i
\(738\) 2.19615i 0.0808415i
\(739\) −6.14359 −0.225996 −0.112998 0.993595i \(-0.536045\pi\)
−0.112998 + 0.993595i \(0.536045\pi\)
\(740\) 0 0
\(741\) 0.732051 0.0268926
\(742\) − 44.7846i − 1.64409i
\(743\) 3.21539i 0.117961i 0.998259 + 0.0589806i \(0.0187850\pi\)
−0.998259 + 0.0589806i \(0.981215\pi\)
\(744\) −15.4641 −0.566941
\(745\) 0 0
\(746\) 10.7321 0.392928
\(747\) − 12.9282i − 0.473018i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 17.3205i 0.631614i
\(753\) − 28.0526i − 1.02229i
\(754\) 10.3923 0.378465
\(755\) 0 0
\(756\) −2.73205 −0.0993637
\(757\) 32.2487i 1.17210i 0.810275 + 0.586050i \(0.199318\pi\)
−0.810275 + 0.586050i \(0.800682\pi\)
\(758\) 36.2487i 1.31661i
\(759\) −16.3923 −0.595003
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 6.92820i 0.250982i
\(763\) − 39.3205i − 1.42350i
\(764\) −0.339746 −0.0122916
\(765\) 0 0
\(766\) 29.5692 1.06838
\(767\) 1.85641i 0.0670310i
\(768\) − 19.0000i − 0.685603i
\(769\) 20.6410 0.744334 0.372167 0.928166i \(-0.378615\pi\)
0.372167 + 0.928166i \(0.378615\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 17.1244i 0.616319i
\(773\) 25.1769i 0.905551i 0.891625 + 0.452775i \(0.149566\pi\)
−0.891625 + 0.452775i \(0.850434\pi\)
\(774\) 7.26795 0.261241
\(775\) 0 0
\(776\) −10.7321 −0.385258
\(777\) 16.9282i 0.607296i
\(778\) 13.6077i 0.487860i
\(779\) −1.26795 −0.0454290
\(780\) 0 0
\(781\) −20.7846 −0.743732
\(782\) 0 0
\(783\) − 8.19615i − 0.292907i
\(784\) 2.32051 0.0828753
\(785\) 0 0
\(786\) 15.8038 0.563705
\(787\) 8.67949i 0.309390i 0.987962 + 0.154695i \(0.0494396\pi\)
−0.987962 + 0.154695i \(0.950560\pi\)
\(788\) 24.0000i 0.854965i
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −51.7128 −1.83870
\(792\) 8.19615i 0.291238i
\(793\) 4.78461i 0.169906i
\(794\) 15.4641 0.548800
\(795\) 0 0
\(796\) −15.3205 −0.543021
\(797\) 44.7846i 1.58635i 0.608992 + 0.793176i \(0.291574\pi\)
−0.608992 + 0.793176i \(0.708426\pi\)
\(798\) − 4.73205i − 0.167513i
\(799\) 0 0
\(800\) 0 0
\(801\) 10.7321 0.379198
\(802\) 58.9808i 2.08268i
\(803\) 80.1051i 2.82685i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −11.3205 −0.398748
\(807\) 0.588457i 0.0207147i
\(808\) 18.0000i 0.633238i
\(809\) −14.7846 −0.519799 −0.259900 0.965636i \(-0.583689\pi\)
−0.259900 + 0.965636i \(0.583689\pi\)
\(810\) 0 0
\(811\) 37.5692 1.31923 0.659617 0.751602i \(-0.270719\pi\)
0.659617 + 0.751602i \(0.270719\pi\)
\(812\) − 22.3923i − 0.785816i
\(813\) − 0.392305i − 0.0137587i
\(814\) −50.7846 −1.78000
\(815\) 0 0
\(816\) 0 0
\(817\) 4.19615i 0.146805i
\(818\) − 45.7128i − 1.59831i
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −32.5359 −1.13551 −0.567755 0.823197i \(-0.692188\pi\)
−0.567755 + 0.823197i \(0.692188\pi\)
\(822\) − 34.3923i − 1.19957i
\(823\) − 12.9808i − 0.452481i −0.974071 0.226240i \(-0.927356\pi\)
0.974071 0.226240i \(-0.0726435\pi\)
\(824\) −17.0718 −0.594724
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) − 5.32051i − 0.185012i −0.995712 0.0925061i \(-0.970512\pi\)
0.995712 0.0925061i \(-0.0294878\pi\)
\(828\) − 3.46410i − 0.120386i
\(829\) −34.1051 −1.18452 −0.592260 0.805747i \(-0.701764\pi\)
−0.592260 + 0.805747i \(0.701764\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0.732051i 0.0253793i
\(833\) 0 0
\(834\) −14.5359 −0.503337
\(835\) 0 0
\(836\) 4.73205 0.163661
\(837\) 8.92820i 0.308604i
\(838\) − 48.5885i − 1.67846i
\(839\) 19.6077 0.676933 0.338466 0.940978i \(-0.390092\pi\)
0.338466 + 0.940978i \(0.390092\pi\)
\(840\) 0 0
\(841\) 38.1769 1.31645
\(842\) 32.5359i 1.12126i
\(843\) − 1.26795i − 0.0436705i
\(844\) −1.07180 −0.0368928
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 31.1244i − 1.06945i
\(848\) − 47.3205i − 1.62499i
\(849\) −24.9808 −0.857338
\(850\) 0 0
\(851\) −21.4641 −0.735780
\(852\) − 4.39230i − 0.150478i
\(853\) − 27.1769i − 0.930520i −0.885174 0.465260i \(-0.845961\pi\)
0.885174 0.465260i \(-0.154039\pi\)
\(854\) 30.9282 1.05834
\(855\) 0 0
\(856\) 0 0
\(857\) − 42.2487i − 1.44319i −0.692316 0.721594i \(-0.743410\pi\)
0.692316 0.721594i \(-0.256590\pi\)
\(858\) 6.00000i 0.204837i
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) −3.46410 −0.118056
\(862\) 19.6077i 0.667841i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) −5.19615 −0.176777
\(865\) 0 0
\(866\) 18.3397 0.623210
\(867\) − 17.0000i − 0.577350i
\(868\) 24.3923i 0.827929i
\(869\) 51.7128 1.75424
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) − 24.9282i − 0.844175i
\(873\) 6.19615i 0.209708i
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) −16.9282 −0.571951
\(877\) 53.1244i 1.79388i 0.442150 + 0.896941i \(0.354216\pi\)
−0.442150 + 0.896941i \(0.645784\pi\)
\(878\) 46.6410i 1.57406i
\(879\) −27.7128 −0.934730
\(880\) 0 0
\(881\) 8.53590 0.287582 0.143791 0.989608i \(-0.454071\pi\)
0.143791 + 0.989608i \(0.454071\pi\)
\(882\) − 0.803848i − 0.0270670i
\(883\) − 36.9808i − 1.24450i −0.782818 0.622251i \(-0.786218\pi\)
0.782818 0.622251i \(-0.213782\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 9.21539 0.309597
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) 10.7321i 0.360144i
\(889\) −10.9282 −0.366520
\(890\) 0 0
\(891\) 4.73205 0.158530
\(892\) − 17.8564i − 0.597877i
\(893\) 3.46410i 0.115922i
\(894\) −34.3923 −1.15025
\(895\) 0 0
\(896\) 33.1244 1.10661
\(897\) 2.53590i 0.0846712i
\(898\) − 9.80385i − 0.327159i
\(899\) −73.1769 −2.44059
\(900\) 0 0
\(901\) 0 0
\(902\) − 10.3923i − 0.346026i
\(903\) 11.4641i 0.381501i
\(904\) −32.7846 −1.09040
\(905\) 0 0
\(906\) −24.2487 −0.805609
\(907\) − 32.3923i − 1.07557i −0.843082 0.537784i \(-0.819261\pi\)
0.843082 0.537784i \(-0.180739\pi\)
\(908\) − 10.3923i − 0.344881i
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) −54.9282 −1.81985 −0.909926 0.414770i \(-0.863862\pi\)
−0.909926 + 0.414770i \(0.863862\pi\)
\(912\) − 5.00000i − 0.165567i
\(913\) 61.1769i 2.02466i
\(914\) 7.85641 0.259867
\(915\) 0 0
\(916\) −18.5359 −0.612443
\(917\) 24.9282i 0.823202i
\(918\) 0 0
\(919\) −51.4256 −1.69637 −0.848187 0.529696i \(-0.822306\pi\)
−0.848187 + 0.529696i \(0.822306\pi\)
\(920\) 0 0
\(921\) −32.3923 −1.06736
\(922\) − 10.3923i − 0.342252i
\(923\) 3.21539i 0.105836i
\(924\) 12.9282 0.425307
\(925\) 0 0
\(926\) 61.5167 2.02156
\(927\) 9.85641i 0.323727i
\(928\) − 42.5885i − 1.39803i
\(929\) 1.60770 0.0527468 0.0263734 0.999652i \(-0.491604\pi\)
0.0263734 + 0.999652i \(0.491604\pi\)
\(930\) 0 0
\(931\) 0.464102 0.0152103
\(932\) − 7.85641i − 0.257345i
\(933\) − 32.4449i − 1.06220i
\(934\) 35.5692 1.16386
\(935\) 0 0
\(936\) 1.26795 0.0414442
\(937\) − 16.2487i − 0.530822i −0.964135 0.265411i \(-0.914492\pi\)
0.964135 0.265411i \(-0.0855077\pi\)
\(938\) − 37.8564i − 1.23606i
\(939\) −6.39230 −0.208605
\(940\) 0 0
\(941\) 0.588457 0.0191832 0.00959158 0.999954i \(-0.496947\pi\)
0.00959158 + 0.999954i \(0.496947\pi\)
\(942\) − 11.0718i − 0.360739i
\(943\) − 4.39230i − 0.143033i
\(944\) 12.6795 0.412682
\(945\) 0 0
\(946\) −34.3923 −1.11819
\(947\) 28.1436i 0.914544i 0.889327 + 0.457272i \(0.151173\pi\)
−0.889327 + 0.457272i \(0.848827\pi\)
\(948\) 10.9282i 0.354932i
\(949\) 12.3923 0.402271
\(950\) 0 0
\(951\) −11.3205 −0.367093
\(952\) 0 0
\(953\) − 37.8564i − 1.22629i −0.789971 0.613145i \(-0.789904\pi\)
0.789971 0.613145i \(-0.210096\pi\)
\(954\) −16.3923 −0.530720
\(955\) 0 0
\(956\) −9.80385 −0.317079
\(957\) 38.7846i 1.25373i
\(958\) 44.1962i 1.42791i
\(959\) 54.2487 1.75178
\(960\) 0 0
\(961\) 48.7128 1.57138
\(962\) 7.85641i 0.253301i
\(963\) 0 0
\(964\) 3.07180 0.0989359
\(965\) 0 0
\(966\) 16.3923 0.527414
\(967\) 4.87564i 0.156790i 0.996922 + 0.0783951i \(0.0249796\pi\)
−0.996922 + 0.0783951i \(0.975020\pi\)
\(968\) − 19.7321i − 0.634212i
\(969\) 0 0
\(970\) 0 0
\(971\) −27.7128 −0.889346 −0.444673 0.895693i \(-0.646680\pi\)
−0.444673 + 0.895693i \(0.646680\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 22.9282i − 0.735044i
\(974\) −56.1051 −1.79772
\(975\) 0 0
\(976\) 32.6795 1.04605
\(977\) − 39.0333i − 1.24879i −0.781110 0.624393i \(-0.785346\pi\)
0.781110 0.624393i \(-0.214654\pi\)
\(978\) 16.0526i 0.513304i
\(979\) −50.7846 −1.62308
\(980\) 0 0
\(981\) −14.3923 −0.459511
\(982\) − 27.8038i − 0.887256i
\(983\) 41.3205i 1.31792i 0.752178 + 0.658960i \(0.229003\pi\)
−0.752178 + 0.658960i \(0.770997\pi\)
\(984\) −2.19615 −0.0700108
\(985\) 0 0
\(986\) 0 0
\(987\) 9.46410i 0.301246i
\(988\) − 0.732051i − 0.0232896i
\(989\) −14.5359 −0.462215
\(990\) 0 0
\(991\) 13.0718 0.415239 0.207620 0.978210i \(-0.433428\pi\)
0.207620 + 0.978210i \(0.433428\pi\)
\(992\) 46.3923i 1.47296i
\(993\) 25.7128i 0.815971i
\(994\) 20.7846 0.659248
\(995\) 0 0
\(996\) −12.9282 −0.409646
\(997\) − 17.6077i − 0.557641i −0.960343 0.278821i \(-0.910056\pi\)
0.960343 0.278821i \(-0.0899435\pi\)
\(998\) 18.2487i 0.577653i
\(999\) 6.19615 0.196038
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 1425.2.c.k.799.1 4
5.2 odd 4 1425.2.a.o.1.2 2
5.3 odd 4 285.2.a.e.1.1 2
5.4 even 2 inner 1425.2.c.k.799.4 4
15.2 even 4 4275.2.a.t.1.1 2
15.8 even 4 855.2.a.f.1.2 2
20.3 even 4 4560.2.a.bh.1.2 2
95.18 even 4 5415.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.1 2 5.3 odd 4
855.2.a.f.1.2 2 15.8 even 4
1425.2.a.o.1.2 2 5.2 odd 4
1425.2.c.k.799.1 4 1.1 even 1 trivial
1425.2.c.k.799.4 4 5.4 even 2 inner
4275.2.a.t.1.1 2 15.2 even 4
4560.2.a.bh.1.2 2 20.3 even 4
5415.2.a.r.1.2 2 95.18 even 4