Properties

Label 1600.2.o.l.607.2
Level $1600$
Weight $2$
Character 1600.607
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(543,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.543");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 607.2
Root \(0.228425 - 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 1600.607
Dual form 1600.2.o.l.543.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.456850 + 0.456850i) q^{3} +(2.79129 - 2.79129i) q^{7} +2.58258i q^{9} +O(q^{10})\) \(q+(-0.456850 + 0.456850i) q^{3} +(2.79129 - 2.79129i) q^{7} +2.58258i q^{9} -4.37780 q^{11} +(-1.73205 - 1.73205i) q^{13} +(-3.00000 - 3.00000i) q^{17} -3.46410i q^{19} +2.55040i q^{21} +(0.791288 + 0.791288i) q^{23} +(-2.55040 - 2.55040i) q^{27} -5.29150 q^{29} +1.58258i q^{31} +(2.00000 - 2.00000i) q^{33} +(-5.19615 + 5.19615i) q^{37} +1.58258 q^{39} -7.58258 q^{41} +(8.29875 - 8.29875i) q^{43} +(-0.791288 + 0.791288i) q^{47} -8.58258i q^{49} +2.74110 q^{51} +(-2.64575 - 2.64575i) q^{53} +(1.58258 + 1.58258i) q^{57} +5.29150i q^{59} +9.66930i q^{61} +(7.20871 + 7.20871i) q^{63} +(-8.29875 - 8.29875i) q^{67} -0.723000 q^{69} -13.5826i q^{71} +(-0.582576 + 0.582576i) q^{73} +(-12.2197 + 12.2197i) q^{77} +12.0000 q^{79} -5.41742 q^{81} +(-4.83465 + 4.83465i) q^{83} +(2.41742 - 2.41742i) q^{87} -3.16515i q^{89} -9.66930 q^{91} +(-0.723000 - 0.723000i) q^{93} +(-0.582576 - 0.582576i) q^{97} -11.3060i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} - 24 q^{17} - 12 q^{23} + 16 q^{33} - 24 q^{39} - 24 q^{41} + 12 q^{47} - 24 q^{57} + 76 q^{63} + 32 q^{73} + 96 q^{79} - 80 q^{81} + 56 q^{87} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.456850 + 0.456850i −0.263763 + 0.263763i −0.826581 0.562818i \(-0.809717\pi\)
0.562818 + 0.826581i \(0.309717\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.79129 2.79129i 1.05501 1.05501i 0.0566113 0.998396i \(-0.481970\pi\)
0.998396 0.0566113i \(-0.0180296\pi\)
\(8\) 0 0
\(9\) 2.58258i 0.860859i
\(10\) 0 0
\(11\) −4.37780 −1.31996 −0.659979 0.751284i \(-0.729435\pi\)
−0.659979 + 0.751284i \(0.729435\pi\)
\(12\) 0 0
\(13\) −1.73205 1.73205i −0.480384 0.480384i 0.424870 0.905254i \(-0.360320\pi\)
−0.905254 + 0.424870i \(0.860320\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 3.00000i −0.727607 0.727607i 0.242536 0.970143i \(-0.422021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 2.55040i 0.556543i
\(22\) 0 0
\(23\) 0.791288 + 0.791288i 0.164995 + 0.164995i 0.784775 0.619780i \(-0.212778\pi\)
−0.619780 + 0.784775i \(0.712778\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.55040 2.55040i −0.490825 0.490825i
\(28\) 0 0
\(29\) −5.29150 −0.982607 −0.491304 0.870988i \(-0.663479\pi\)
−0.491304 + 0.870988i \(0.663479\pi\)
\(30\) 0 0
\(31\) 1.58258i 0.284239i 0.989850 + 0.142119i \(0.0453917\pi\)
−0.989850 + 0.142119i \(0.954608\pi\)
\(32\) 0 0
\(33\) 2.00000 2.00000i 0.348155 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.19615 + 5.19615i −0.854242 + 0.854242i −0.990652 0.136410i \(-0.956443\pi\)
0.136410 + 0.990652i \(0.456443\pi\)
\(38\) 0 0
\(39\) 1.58258 0.253415
\(40\) 0 0
\(41\) −7.58258 −1.18420 −0.592100 0.805865i \(-0.701701\pi\)
−0.592100 + 0.805865i \(0.701701\pi\)
\(42\) 0 0
\(43\) 8.29875 8.29875i 1.26555 1.26555i 0.317184 0.948364i \(-0.397263\pi\)
0.948364 0.317184i \(-0.102737\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.791288 + 0.791288i −0.115421 + 0.115421i −0.762458 0.647037i \(-0.776008\pi\)
0.647037 + 0.762458i \(0.276008\pi\)
\(48\) 0 0
\(49\) 8.58258i 1.22608i
\(50\) 0 0
\(51\) 2.74110 0.383831
\(52\) 0 0
\(53\) −2.64575 2.64575i −0.363422 0.363422i 0.501649 0.865071i \(-0.332727\pi\)
−0.865071 + 0.501649i \(0.832727\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.58258 + 1.58258i 0.209617 + 0.209617i
\(58\) 0 0
\(59\) 5.29150i 0.688895i 0.938806 + 0.344447i \(0.111934\pi\)
−0.938806 + 0.344447i \(0.888066\pi\)
\(60\) 0 0
\(61\) 9.66930i 1.23803i 0.785380 + 0.619014i \(0.212468\pi\)
−0.785380 + 0.619014i \(0.787532\pi\)
\(62\) 0 0
\(63\) 7.20871 + 7.20871i 0.908212 + 0.908212i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.29875 8.29875i −1.01385 1.01385i −0.999903 0.0139515i \(-0.995559\pi\)
−0.0139515 0.999903i \(-0.504441\pi\)
\(68\) 0 0
\(69\) −0.723000 −0.0870390
\(70\) 0 0
\(71\) 13.5826i 1.61196i −0.591946 0.805978i \(-0.701640\pi\)
0.591946 0.805978i \(-0.298360\pi\)
\(72\) 0 0
\(73\) −0.582576 + 0.582576i −0.0681853 + 0.0681853i −0.740377 0.672192i \(-0.765353\pi\)
0.672192 + 0.740377i \(0.265353\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.2197 + 12.2197i −1.39256 + 1.39256i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −5.41742 −0.601936
\(82\) 0 0
\(83\) −4.83465 + 4.83465i −0.530672 + 0.530672i −0.920772 0.390100i \(-0.872440\pi\)
0.390100 + 0.920772i \(0.372440\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.41742 2.41742i 0.259175 0.259175i
\(88\) 0 0
\(89\) 3.16515i 0.335505i −0.985829 0.167753i \(-0.946349\pi\)
0.985829 0.167753i \(-0.0536510\pi\)
\(90\) 0 0
\(91\) −9.66930 −1.01362
\(92\) 0 0
\(93\) −0.723000 0.723000i −0.0749716 0.0749716i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.582576 0.582576i −0.0591516 0.0591516i 0.676912 0.736064i \(-0.263318\pi\)
−0.736064 + 0.676912i \(0.763318\pi\)
\(98\) 0 0
\(99\) 11.3060i 1.13630i
\(100\) 0 0
\(101\) 3.65480i 0.363666i −0.983329 0.181833i \(-0.941797\pi\)
0.983329 0.181833i \(-0.0582031\pi\)
\(102\) 0 0
\(103\) −10.3739 10.3739i −1.02217 1.02217i −0.999749 0.0224185i \(-0.992863\pi\)
−0.0224185 0.999749i \(-0.507137\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.29875 8.29875i −0.802271 0.802271i 0.181179 0.983450i \(-0.442009\pi\)
−0.983450 + 0.181179i \(0.942009\pi\)
\(108\) 0 0
\(109\) 6.20520 0.594351 0.297175 0.954823i \(-0.403955\pi\)
0.297175 + 0.954823i \(0.403955\pi\)
\(110\) 0 0
\(111\) 4.74773i 0.450634i
\(112\) 0 0
\(113\) 4.58258 4.58258i 0.431092 0.431092i −0.457907 0.889000i \(-0.651401\pi\)
0.889000 + 0.457907i \(0.151401\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.47315 4.47315i 0.413543 0.413543i
\(118\) 0 0
\(119\) −16.7477 −1.53526
\(120\) 0 0
\(121\) 8.16515 0.742286
\(122\) 0 0
\(123\) 3.46410 3.46410i 0.312348 0.312348i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.20871 + 1.20871i −0.107256 + 0.107256i −0.758698 0.651442i \(-0.774164\pi\)
0.651442 + 0.758698i \(0.274164\pi\)
\(128\) 0 0
\(129\) 7.58258i 0.667609i
\(130\) 0 0
\(131\) −11.3060 −0.987810 −0.493905 0.869516i \(-0.664431\pi\)
−0.493905 + 0.869516i \(0.664431\pi\)
\(132\) 0 0
\(133\) −9.66930 9.66930i −0.838435 0.838435i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.1652 12.1652i −1.03934 1.03934i −0.999194 0.0401452i \(-0.987218\pi\)
−0.0401452 0.999194i \(-0.512782\pi\)
\(138\) 0 0
\(139\) 8.94630i 0.758816i −0.925230 0.379408i \(-0.876128\pi\)
0.925230 0.379408i \(-0.123872\pi\)
\(140\) 0 0
\(141\) 0.723000i 0.0608876i
\(142\) 0 0
\(143\) 7.58258 + 7.58258i 0.634087 + 0.634087i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.92095 + 3.92095i 0.323395 + 0.323395i
\(148\) 0 0
\(149\) −4.37780 −0.358644 −0.179322 0.983790i \(-0.557390\pi\)
−0.179322 + 0.983790i \(0.557390\pi\)
\(150\) 0 0
\(151\) 12.7477i 1.03740i 0.854958 + 0.518698i \(0.173583\pi\)
−0.854958 + 0.518698i \(0.826417\pi\)
\(152\) 0 0
\(153\) 7.74773 7.74773i 0.626367 0.626367i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.4014 11.4014i 0.909927 0.909927i −0.0863386 0.996266i \(-0.527517\pi\)
0.996266 + 0.0863386i \(0.0275167\pi\)
\(158\) 0 0
\(159\) 2.41742 0.191714
\(160\) 0 0
\(161\) 4.41742 0.348142
\(162\) 0 0
\(163\) 8.29875 8.29875i 0.650009 0.650009i −0.302986 0.952995i \(-0.597984\pi\)
0.952995 + 0.302986i \(0.0979837\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.2087 11.2087i 0.867356 0.867356i −0.124823 0.992179i \(-0.539836\pi\)
0.992179 + 0.124823i \(0.0398363\pi\)
\(168\) 0 0
\(169\) 7.00000i 0.538462i
\(170\) 0 0
\(171\) 8.94630 0.684141
\(172\) 0 0
\(173\) 8.66025 + 8.66025i 0.658427 + 0.658427i 0.955008 0.296581i \(-0.0958464\pi\)
−0.296581 + 0.955008i \(0.595846\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.41742 2.41742i −0.181705 0.181705i
\(178\) 0 0
\(179\) 8.56490i 0.640171i 0.947389 + 0.320085i \(0.103712\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) −4.41742 4.41742i −0.326545 0.326545i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 13.1334 + 13.1334i 0.960410 + 0.960410i
\(188\) 0 0
\(189\) −14.2378 −1.03565
\(190\) 0 0
\(191\) 13.5826i 0.982801i 0.870934 + 0.491400i \(0.163515\pi\)
−0.870934 + 0.491400i \(0.836485\pi\)
\(192\) 0 0
\(193\) −14.1652 + 14.1652i −1.01963 + 1.01963i −0.0198265 + 0.999803i \(0.506311\pi\)
−0.999803 + 0.0198265i \(0.993689\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.64575 + 2.64575i −0.188502 + 0.188502i −0.795048 0.606546i \(-0.792554\pi\)
0.606546 + 0.795048i \(0.292554\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 7.58258 0.534834
\(202\) 0 0
\(203\) −14.7701 + 14.7701i −1.03666 + 1.03666i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.04356 + 2.04356i −0.142037 + 0.142037i
\(208\) 0 0
\(209\) 15.1652i 1.04900i
\(210\) 0 0
\(211\) 7.65120 0.526731 0.263365 0.964696i \(-0.415168\pi\)
0.263365 + 0.964696i \(0.415168\pi\)
\(212\) 0 0
\(213\) 6.20520 + 6.20520i 0.425174 + 0.425174i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.41742 + 4.41742i 0.299874 + 0.299874i
\(218\) 0 0
\(219\) 0.532300i 0.0359695i
\(220\) 0 0
\(221\) 10.3923i 0.699062i
\(222\) 0 0
\(223\) 12.3739 + 12.3739i 0.828615 + 0.828615i 0.987325 0.158710i \(-0.0507335\pi\)
−0.158710 + 0.987325i \(0.550734\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.66205 + 6.66205i 0.442176 + 0.442176i 0.892743 0.450567i \(-0.148778\pi\)
−0.450567 + 0.892743i \(0.648778\pi\)
\(228\) 0 0
\(229\) 3.46410 0.228914 0.114457 0.993428i \(-0.463487\pi\)
0.114457 + 0.993428i \(0.463487\pi\)
\(230\) 0 0
\(231\) 11.1652i 0.734613i
\(232\) 0 0
\(233\) −6.16515 + 6.16515i −0.403892 + 0.403892i −0.879602 0.475710i \(-0.842191\pi\)
0.475710 + 0.879602i \(0.342191\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.48220 + 5.48220i −0.356107 + 0.356107i
\(238\) 0 0
\(239\) 3.16515 0.204737 0.102368 0.994747i \(-0.467358\pi\)
0.102368 + 0.994747i \(0.467358\pi\)
\(240\) 0 0
\(241\) −10.7477 −0.692322 −0.346161 0.938175i \(-0.612515\pi\)
−0.346161 + 0.938175i \(0.612515\pi\)
\(242\) 0 0
\(243\) 10.1262 10.1262i 0.649593 0.649593i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 + 6.00000i −0.381771 + 0.381771i
\(248\) 0 0
\(249\) 4.41742i 0.279943i
\(250\) 0 0
\(251\) 18.2342 1.15093 0.575467 0.817825i \(-0.304821\pi\)
0.575467 + 0.817825i \(0.304821\pi\)
\(252\) 0 0
\(253\) −3.46410 3.46410i −0.217786 0.217786i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.7477 + 19.7477i 1.23183 + 1.23183i 0.963260 + 0.268569i \(0.0865507\pi\)
0.268569 + 0.963260i \(0.413449\pi\)
\(258\) 0 0
\(259\) 29.0079i 1.80246i
\(260\) 0 0
\(261\) 13.6657i 0.845886i
\(262\) 0 0
\(263\) −18.7913 18.7913i −1.15872 1.15872i −0.984751 0.173969i \(-0.944341\pi\)
−0.173969 0.984751i \(-0.555659\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.44600 + 1.44600i 0.0884938 + 0.0884938i
\(268\) 0 0
\(269\) 23.7164 1.44602 0.723008 0.690840i \(-0.242759\pi\)
0.723008 + 0.690840i \(0.242759\pi\)
\(270\) 0 0
\(271\) 8.74773i 0.531387i 0.964058 + 0.265693i \(0.0856008\pi\)
−0.964058 + 0.265693i \(0.914399\pi\)
\(272\) 0 0
\(273\) 4.41742 4.41742i 0.267355 0.267355i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.91915 5.91915i 0.355647 0.355647i −0.506558 0.862206i \(-0.669083\pi\)
0.862206 + 0.506558i \(0.169083\pi\)
\(278\) 0 0
\(279\) −4.08712 −0.244690
\(280\) 0 0
\(281\) −10.7477 −0.641156 −0.320578 0.947222i \(-0.603877\pi\)
−0.320578 + 0.947222i \(0.603877\pi\)
\(282\) 0 0
\(283\) −17.2451 + 17.2451i −1.02511 + 1.02511i −0.0254359 + 0.999676i \(0.508097\pi\)
−0.999676 + 0.0254359i \(0.991903\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −21.1652 + 21.1652i −1.24934 + 1.24934i
\(288\) 0 0
\(289\) 1.00000i 0.0588235i
\(290\) 0 0
\(291\) 0.532300 0.0312040
\(292\) 0 0
\(293\) 23.4304 + 23.4304i 1.36882 + 1.36882i 0.862131 + 0.506685i \(0.169129\pi\)
0.506685 + 0.862131i \(0.330871\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 11.1652 + 11.1652i 0.647868 + 0.647868i
\(298\) 0 0
\(299\) 2.74110i 0.158522i
\(300\) 0 0
\(301\) 46.3284i 2.67033i
\(302\) 0 0
\(303\) 1.66970 + 1.66970i 0.0959216 + 0.0959216i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.37055 1.37055i −0.0782215 0.0782215i 0.666914 0.745135i \(-0.267615\pi\)
−0.745135 + 0.666914i \(0.767615\pi\)
\(308\) 0 0
\(309\) 9.47860 0.539219
\(310\) 0 0
\(311\) 10.4174i 0.590718i −0.955386 0.295359i \(-0.904561\pi\)
0.955386 0.295359i \(-0.0954392\pi\)
\(312\) 0 0
\(313\) −2.58258 + 2.58258i −0.145976 + 0.145976i −0.776318 0.630342i \(-0.782915\pi\)
0.630342 + 0.776318i \(0.282915\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.6066 17.6066i 0.988883 0.988883i −0.0110560 0.999939i \(-0.503519\pi\)
0.999939 + 0.0110560i \(0.00351931\pi\)
\(318\) 0 0
\(319\) 23.1652 1.29700
\(320\) 0 0
\(321\) 7.58258 0.423218
\(322\) 0 0
\(323\) −10.3923 + 10.3923i −0.578243 + 0.578243i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.83485 + 2.83485i −0.156767 + 0.156767i
\(328\) 0 0
\(329\) 4.41742i 0.243540i
\(330\) 0 0
\(331\) 6.20520 0.341069 0.170534 0.985352i \(-0.445451\pi\)
0.170534 + 0.985352i \(0.445451\pi\)
\(332\) 0 0
\(333\) −13.4195 13.4195i −0.735382 0.735382i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 1.00000i −0.0544735 0.0544735i 0.679345 0.733819i \(-0.262264\pi\)
−0.733819 + 0.679345i \(0.762264\pi\)
\(338\) 0 0
\(339\) 4.18710i 0.227412i
\(340\) 0 0
\(341\) 6.92820i 0.375183i
\(342\) 0 0
\(343\) −4.41742 4.41742i −0.238518 0.238518i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.92095 3.92095i −0.210488 0.210488i 0.593987 0.804475i \(-0.297553\pi\)
−0.804475 + 0.593987i \(0.797553\pi\)
\(348\) 0 0
\(349\) −10.3923 −0.556287 −0.278144 0.960539i \(-0.589719\pi\)
−0.278144 + 0.960539i \(0.589719\pi\)
\(350\) 0 0
\(351\) 8.83485i 0.471569i
\(352\) 0 0
\(353\) −5.83485 + 5.83485i −0.310558 + 0.310558i −0.845126 0.534568i \(-0.820474\pi\)
0.534568 + 0.845126i \(0.320474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.65120 7.65120i 0.404945 0.404945i
\(358\) 0 0
\(359\) −8.83485 −0.466285 −0.233143 0.972443i \(-0.574901\pi\)
−0.233143 + 0.972443i \(0.574901\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) −3.73025 + 3.73025i −0.195787 + 0.195787i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −15.5390 + 15.5390i −0.811130 + 0.811130i −0.984803 0.173673i \(-0.944436\pi\)
0.173673 + 0.984803i \(0.444436\pi\)
\(368\) 0 0
\(369\) 19.5826i 1.01943i
\(370\) 0 0
\(371\) −14.7701 −0.766826
\(372\) 0 0
\(373\) −5.19615 5.19615i −0.269047 0.269047i 0.559669 0.828716i \(-0.310928\pi\)
−0.828716 + 0.559669i \(0.810928\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.16515 + 9.16515i 0.472029 + 0.472029i
\(378\) 0 0
\(379\) 10.3923i 0.533817i −0.963722 0.266908i \(-0.913998\pi\)
0.963722 0.266908i \(-0.0860021\pi\)
\(380\) 0 0
\(381\) 1.10440i 0.0565802i
\(382\) 0 0
\(383\) −3.62614 3.62614i −0.185287 0.185287i 0.608368 0.793655i \(-0.291824\pi\)
−0.793655 + 0.608368i \(0.791824\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 21.4322 + 21.4322i 1.08946 + 1.08946i
\(388\) 0 0
\(389\) 30.2632 1.53441 0.767203 0.641404i \(-0.221648\pi\)
0.767203 + 0.641404i \(0.221648\pi\)
\(390\) 0 0
\(391\) 4.74773i 0.240103i
\(392\) 0 0
\(393\) 5.16515 5.16515i 0.260547 0.260547i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.93725 7.93725i 0.398359 0.398359i −0.479295 0.877654i \(-0.659107\pi\)
0.877654 + 0.479295i \(0.159107\pi\)
\(398\) 0 0
\(399\) 8.83485 0.442296
\(400\) 0 0
\(401\) 12.3303 0.615746 0.307873 0.951427i \(-0.400383\pi\)
0.307873 + 0.951427i \(0.400383\pi\)
\(402\) 0 0
\(403\) 2.74110 2.74110i 0.136544 0.136544i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.7477 22.7477i 1.12756 1.12756i
\(408\) 0 0
\(409\) 25.5826i 1.26498i 0.774570 + 0.632488i \(0.217966\pi\)
−0.774570 + 0.632488i \(0.782034\pi\)
\(410\) 0 0
\(411\) 11.1153 0.548278
\(412\) 0 0
\(413\) 14.7701 + 14.7701i 0.726789 + 0.726789i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.08712 + 4.08712i 0.200147 + 0.200147i
\(418\) 0 0
\(419\) 0.190700i 0.00931632i −0.999989 0.00465816i \(-0.998517\pi\)
0.999989 0.00465816i \(-0.00148274\pi\)
\(420\) 0 0
\(421\) 2.74110i 0.133593i 0.997767 + 0.0667966i \(0.0212778\pi\)
−0.997767 + 0.0667966i \(0.978722\pi\)
\(422\) 0 0
\(423\) −2.04356 2.04356i −0.0993613 0.0993613i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 26.9898 + 26.9898i 1.30613 + 1.30613i
\(428\) 0 0
\(429\) −6.92820 −0.334497
\(430\) 0 0
\(431\) 16.7477i 0.806710i 0.915044 + 0.403355i \(0.132156\pi\)
−0.915044 + 0.403355i \(0.867844\pi\)
\(432\) 0 0
\(433\) 14.5826 14.5826i 0.700794 0.700794i −0.263787 0.964581i \(-0.584972\pi\)
0.964581 + 0.263787i \(0.0849716\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.74110 2.74110i 0.131125 0.131125i
\(438\) 0 0
\(439\) −29.4955 −1.40774 −0.703871 0.710328i \(-0.748547\pi\)
−0.703871 + 0.710328i \(0.748547\pi\)
\(440\) 0 0
\(441\) 22.1652 1.05548
\(442\) 0 0
\(443\) −1.17985 + 1.17985i −0.0560564 + 0.0560564i −0.734579 0.678523i \(-0.762620\pi\)
0.678523 + 0.734579i \(0.262620\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.00000 2.00000i 0.0945968 0.0945968i
\(448\) 0 0
\(449\) 10.4174i 0.491629i 0.969317 + 0.245814i \(0.0790553\pi\)
−0.969317 + 0.245814i \(0.920945\pi\)
\(450\) 0 0
\(451\) 33.1950 1.56309
\(452\) 0 0
\(453\) −5.82380 5.82380i −0.273626 0.273626i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.41742 3.41742i −0.159860 0.159860i 0.622644 0.782505i \(-0.286058\pi\)
−0.782505 + 0.622644i \(0.786058\pi\)
\(458\) 0 0
\(459\) 15.3024i 0.714255i
\(460\) 0 0
\(461\) 17.5112i 0.815578i 0.913076 + 0.407789i \(0.133700\pi\)
−0.913076 + 0.407789i \(0.866300\pi\)
\(462\) 0 0
\(463\) 16.7913 + 16.7913i 0.780357 + 0.780357i 0.979891 0.199534i \(-0.0639427\pi\)
−0.199534 + 0.979891i \(0.563943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.57575 7.57575i −0.350564 0.350564i 0.509755 0.860319i \(-0.329736\pi\)
−0.860319 + 0.509755i \(0.829736\pi\)
\(468\) 0 0
\(469\) −46.3284 −2.13925
\(470\) 0 0
\(471\) 10.4174i 0.480010i
\(472\) 0 0
\(473\) −36.3303 + 36.3303i −1.67047 + 1.67047i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.83285 6.83285i 0.312855 0.312855i
\(478\) 0 0
\(479\) −21.4955 −0.982152 −0.491076 0.871117i \(-0.663396\pi\)
−0.491076 + 0.871117i \(0.663396\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) −2.01810 + 2.01810i −0.0918268 + 0.0918268i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.373864 + 0.373864i −0.0169414 + 0.0169414i −0.715527 0.698585i \(-0.753813\pi\)
0.698585 + 0.715527i \(0.253813\pi\)
\(488\) 0 0
\(489\) 7.58258i 0.342896i
\(490\) 0 0
\(491\) 9.86001 0.444976 0.222488 0.974935i \(-0.428582\pi\)
0.222488 + 0.974935i \(0.428582\pi\)
\(492\) 0 0
\(493\) 15.8745 + 15.8745i 0.714952 + 0.714952i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −37.9129 37.9129i −1.70063 1.70063i
\(498\) 0 0
\(499\) 24.2487i 1.08552i −0.839887 0.542761i \(-0.817379\pi\)
0.839887 0.542761i \(-0.182621\pi\)
\(500\) 0 0
\(501\) 10.2414i 0.457552i
\(502\) 0 0
\(503\) −17.5390 17.5390i −0.782026 0.782026i 0.198146 0.980172i \(-0.436508\pi\)
−0.980172 + 0.198146i \(0.936508\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.19795 + 3.19795i 0.142026 + 0.142026i
\(508\) 0 0
\(509\) 38.4865 1.70588 0.852942 0.522005i \(-0.174816\pi\)
0.852942 + 0.522005i \(0.174816\pi\)
\(510\) 0 0
\(511\) 3.25227i 0.143872i
\(512\) 0 0
\(513\) −8.83485 + 8.83485i −0.390068 + 0.390068i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.46410 3.46410i 0.152351 0.152351i
\(518\) 0 0
\(519\) −7.91288 −0.347337
\(520\) 0 0
\(521\) −33.1652 −1.45299 −0.726496 0.687171i \(-0.758852\pi\)
−0.726496 + 0.687171i \(0.758852\pi\)
\(522\) 0 0
\(523\) 15.9500 15.9500i 0.697443 0.697443i −0.266415 0.963858i \(-0.585839\pi\)
0.963858 + 0.266415i \(0.0858393\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.74773 4.74773i 0.206814 0.206814i
\(528\) 0 0
\(529\) 21.7477i 0.945553i
\(530\) 0 0
\(531\) −13.6657 −0.593041
\(532\) 0 0
\(533\) 13.1334 + 13.1334i 0.568871 + 0.568871i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.91288 3.91288i −0.168853 0.168853i
\(538\) 0 0
\(539\) 37.5728i 1.61838i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −3.16515 3.16515i −0.135830 0.135830i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.647551 0.647551i −0.0276873 0.0276873i 0.693128 0.720815i \(-0.256232\pi\)
−0.720815 + 0.693128i \(0.756232\pi\)
\(548\) 0 0
\(549\) −24.9717 −1.06577
\(550\) 0 0
\(551\) 18.3303i 0.780897i
\(552\) 0 0
\(553\) 33.4955 33.4955i 1.42437 1.42437i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.2970 + 10.2970i −0.436296 + 0.436296i −0.890763 0.454467i \(-0.849830\pi\)
0.454467 + 0.890763i \(0.349830\pi\)
\(558\) 0 0
\(559\) −28.7477 −1.21590
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) −14.3133 + 14.3133i −0.603232 + 0.603232i −0.941169 0.337937i \(-0.890271\pi\)
0.337937 + 0.941169i \(0.390271\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −15.1216 + 15.1216i −0.635047 + 0.635047i
\(568\) 0 0
\(569\) 4.74773i 0.199035i 0.995036 + 0.0995175i \(0.0317299\pi\)
−0.995036 + 0.0995175i \(0.968270\pi\)
\(570\) 0 0
\(571\) −39.4002 −1.64885 −0.824424 0.565973i \(-0.808501\pi\)
−0.824424 + 0.565973i \(0.808501\pi\)
\(572\) 0 0
\(573\) −6.20520 6.20520i −0.259226 0.259226i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.7477 + 11.7477i 0.489064 + 0.489064i 0.908011 0.418947i \(-0.137601\pi\)
−0.418947 + 0.908011i \(0.637601\pi\)
\(578\) 0 0
\(579\) 12.9427i 0.537881i
\(580\) 0 0
\(581\) 26.9898i 1.11973i
\(582\) 0 0
\(583\) 11.5826 + 11.5826i 0.479701 + 0.479701i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.3604 28.3604i −1.17056 1.17056i −0.982077 0.188481i \(-0.939644\pi\)
−0.188481 0.982077i \(-0.560356\pi\)
\(588\) 0 0
\(589\) 5.48220 0.225890
\(590\) 0 0
\(591\) 2.41742i 0.0994395i
\(592\) 0 0
\(593\) −0.165151 + 0.165151i −0.00678195 + 0.00678195i −0.710490 0.703708i \(-0.751526\pi\)
0.703708 + 0.710490i \(0.251526\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.82740 1.82740i 0.0747905 0.0747905i
\(598\) 0 0
\(599\) 42.3303 1.72957 0.864785 0.502143i \(-0.167455\pi\)
0.864785 + 0.502143i \(0.167455\pi\)
\(600\) 0 0
\(601\) −2.74773 −0.112082 −0.0560411 0.998428i \(-0.517848\pi\)
−0.0560411 + 0.998428i \(0.517848\pi\)
\(602\) 0 0
\(603\) 21.4322 21.4322i 0.872785 0.872785i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 6.37386 6.37386i 0.258707 0.258707i −0.565821 0.824528i \(-0.691441\pi\)
0.824528 + 0.565821i \(0.191441\pi\)
\(608\) 0 0
\(609\) 13.4955i 0.546863i
\(610\) 0 0
\(611\) 2.74110 0.110893
\(612\) 0 0
\(613\) −8.66025 8.66025i −0.349784 0.349784i 0.510245 0.860029i \(-0.329555\pi\)
−0.860029 + 0.510245i \(0.829555\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.00000 3.00000i −0.120775 0.120775i 0.644136 0.764911i \(-0.277217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 29.7309i 1.19499i −0.801874 0.597493i \(-0.796164\pi\)
0.801874 0.597493i \(-0.203836\pi\)
\(620\) 0 0
\(621\) 4.03620i 0.161967i
\(622\) 0 0
\(623\) −8.83485 8.83485i −0.353961 0.353961i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −6.92820 6.92820i −0.276686 0.276686i
\(628\) 0 0
\(629\) 31.1769 1.24310
\(630\) 0 0
\(631\) 47.0780i 1.87415i −0.349132 0.937073i \(-0.613524\pi\)
0.349132 0.937073i \(-0.386476\pi\)
\(632\) 0 0
\(633\) −3.49545 + 3.49545i −0.138932 + 0.138932i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −14.8655 + 14.8655i −0.588991 + 0.588991i
\(638\) 0 0
\(639\) 35.0780 1.38767
\(640\) 0 0
\(641\) 25.9129 1.02350 0.511749 0.859135i \(-0.328998\pi\)
0.511749 + 0.859135i \(0.328998\pi\)
\(642\) 0 0
\(643\) −18.6911 + 18.6911i −0.737103 + 0.737103i −0.972016 0.234913i \(-0.924519\pi\)
0.234913 + 0.972016i \(0.424519\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.460985 0.460985i 0.0181232 0.0181232i −0.697987 0.716110i \(-0.745921\pi\)
0.716110 + 0.697987i \(0.245921\pi\)
\(648\) 0 0
\(649\) 23.1652i 0.909312i
\(650\) 0 0
\(651\) −4.03620 −0.158191
\(652\) 0 0
\(653\) −11.4014 11.4014i −0.446170 0.446170i 0.447909 0.894079i \(-0.352169\pi\)
−0.894079 + 0.447909i \(0.852169\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.50455 1.50455i −0.0586979 0.0586979i
\(658\) 0 0
\(659\) 3.84550i 0.149800i −0.997191 0.0748998i \(-0.976136\pi\)
0.997191 0.0748998i \(-0.0238637\pi\)
\(660\) 0 0
\(661\) 41.4183i 1.61099i −0.592605 0.805493i \(-0.701901\pi\)
0.592605 0.805493i \(-0.298099\pi\)
\(662\) 0 0
\(663\) −4.74773 4.74773i −0.184386 0.184386i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4.18710 4.18710i −0.162125 0.162125i
\(668\) 0 0
\(669\) −11.3060 −0.437115
\(670\) 0 0
\(671\) 42.3303i 1.63414i
\(672\) 0 0
\(673\) 18.5826 18.5826i 0.716306 0.716306i −0.251541 0.967847i \(-0.580937\pi\)
0.967847 + 0.251541i \(0.0809373\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.3296 + 18.3296i −0.704462 + 0.704462i −0.965365 0.260903i \(-0.915980\pi\)
0.260903 + 0.965365i \(0.415980\pi\)
\(678\) 0 0
\(679\) −3.25227 −0.124811
\(680\) 0 0
\(681\) −6.08712 −0.233259
\(682\) 0 0
\(683\) 6.85275 6.85275i 0.262213 0.262213i −0.563739 0.825953i \(-0.690638\pi\)
0.825953 + 0.563739i \(0.190638\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.58258 + 1.58258i −0.0603790 + 0.0603790i
\(688\) 0 0
\(689\) 9.16515i 0.349164i
\(690\) 0 0
\(691\) 32.4720 1.23529 0.617647 0.786456i \(-0.288086\pi\)
0.617647 + 0.786456i \(0.288086\pi\)
\(692\) 0 0
\(693\) −31.5583 31.5583i −1.19880 1.19880i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 22.7477 + 22.7477i 0.861632 + 0.861632i
\(698\) 0 0
\(699\) 5.63310i 0.213063i
\(700\) 0 0
\(701\) 19.8709i 0.750514i −0.926921 0.375257i \(-0.877554\pi\)
0.926921 0.375257i \(-0.122446\pi\)
\(702\) 0 0
\(703\) 18.0000 + 18.0000i 0.678883 + 0.678883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.2016 10.2016i −0.383671 0.383671i
\(708\) 0 0
\(709\) 4.91010 0.184403 0.0922014 0.995740i \(-0.470610\pi\)
0.0922014 + 0.995740i \(0.470610\pi\)
\(710\) 0 0
\(711\) 30.9909i 1.16225i
\(712\) 0 0
\(713\) −1.25227 + 1.25227i −0.0468980 + 0.0468980i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.44600 + 1.44600i −0.0540019 + 0.0540019i
\(718\) 0 0
\(719\) 39.8258 1.48525 0.742625 0.669707i \(-0.233580\pi\)
0.742625 + 0.669707i \(0.233580\pi\)
\(720\) 0 0
\(721\) −57.9129 −2.15679
\(722\) 0 0
\(723\) 4.91010 4.91010i 0.182609 0.182609i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 5.95644 5.95644i 0.220912 0.220912i −0.587970 0.808882i \(-0.700073\pi\)
0.808882 + 0.587970i \(0.200073\pi\)
\(728\) 0 0
\(729\) 7.00000i 0.259259i
\(730\) 0 0
\(731\) −49.7925 −1.84164
\(732\) 0 0
\(733\) −17.6066 17.6066i −0.650313 0.650313i 0.302755 0.953068i \(-0.402094\pi\)
−0.953068 + 0.302755i \(0.902094\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.3303 + 36.3303i 1.33824 + 1.33824i
\(738\) 0 0
\(739\) 2.01810i 0.0742371i 0.999311 + 0.0371185i \(0.0118179\pi\)
−0.999311 + 0.0371185i \(0.988182\pi\)
\(740\) 0 0
\(741\) 5.48220i 0.201394i
\(742\) 0 0
\(743\) 17.2087 + 17.2087i 0.631326 + 0.631326i 0.948401 0.317074i \(-0.102700\pi\)
−0.317074 + 0.948401i \(0.602700\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.4859 12.4859i −0.456834 0.456834i
\(748\) 0 0
\(749\) −46.3284 −1.69280
\(750\) 0 0
\(751\) 23.0780i 0.842129i −0.907031 0.421065i \(-0.861657\pi\)
0.907031 0.421065i \(-0.138343\pi\)
\(752\) 0 0
\(753\) −8.33030 + 8.33030i −0.303573 + 0.303573i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.3477 + 20.3477i −0.739548 + 0.739548i −0.972491 0.232942i \(-0.925165\pi\)
0.232942 + 0.972491i \(0.425165\pi\)
\(758\) 0 0
\(759\) 3.16515 0.114888
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 17.3205 17.3205i 0.627044 0.627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.16515 9.16515i 0.330934 0.330934i
\(768\) 0 0
\(769\) 37.4955i 1.35212i 0.736846 + 0.676060i \(0.236314\pi\)
−0.736846 + 0.676060i \(0.763686\pi\)
\(770\) 0 0
\(771\) −18.0435 −0.649821
\(772\) 0 0
\(773\) −0.286051 0.286051i −0.0102885 0.0102885i 0.701944 0.712232i \(-0.252316\pi\)
−0.712232 + 0.701944i \(0.752316\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −13.2523 13.2523i −0.475423 0.475423i
\(778\) 0 0
\(779\) 26.2668i 0.941106i
\(780\) 0 0
\(781\) 59.4618i 2.12771i
\(782\) 0 0
\(783\) 13.4955 + 13.4955i 0.482288 + 0.482288i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −20.7092 20.7092i −0.738202 0.738202i 0.234028 0.972230i \(-0.424809\pi\)
−0.972230 + 0.234028i \(0.924809\pi\)
\(788\) 0 0
\(789\) 17.1696 0.611254
\(790\) 0 0
\(791\) 25.5826i 0.909612i
\(792\) 0 0
\(793\) 16.7477 16.7477i 0.594729 0.594729i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.9271 + 34.9271i −1.23718 + 1.23718i −0.276032 + 0.961149i \(0.589019\pi\)
−0.961149 + 0.276032i \(0.910981\pi\)
\(798\) 0 0
\(799\) 4.74773 0.167963
\(800\) 0 0
\(801\) 8.17424 0.288823
\(802\) 0 0
\(803\) 2.55040 2.55040i 0.0900017 0.0900017i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.8348 + 10.8348i −0.381405 + 0.381405i
\(808\) 0 0
\(809\) 30.3303i 1.06636i −0.846003 0.533178i \(-0.820997\pi\)
0.846003 0.533178i \(-0.179003\pi\)
\(810\) 0 0
\(811\) −2.16900 −0.0761639 −0.0380820 0.999275i \(-0.512125\pi\)
−0.0380820 + 0.999275i \(0.512125\pi\)
\(812\) 0 0
\(813\) −3.99640 3.99640i −0.140160 0.140160i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −28.7477 28.7477i −1.00576 1.00576i
\(818\) 0 0
\(819\) 24.9717i 0.872582i
\(820\) 0 0
\(821\) 54.8933i 1.91579i 0.287118 + 0.957895i \(0.407303\pi\)
−0.287118 + 0.957895i \(0.592697\pi\)
\(822\) 0 0
\(823\) −3.20871 3.20871i −0.111849 0.111849i 0.648967 0.760816i \(-0.275201\pi\)
−0.760816 + 0.648967i \(0.775201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.1226 14.1226i −0.491089 0.491089i 0.417560 0.908649i \(-0.362885\pi\)
−0.908649 + 0.417560i \(0.862885\pi\)
\(828\) 0 0
\(829\) −7.65120 −0.265737 −0.132869 0.991134i \(-0.542419\pi\)
−0.132869 + 0.991134i \(0.542419\pi\)
\(830\) 0 0
\(831\) 5.40833i 0.187613i
\(832\) 0 0
\(833\) −25.7477 + 25.7477i −0.892106 + 0.892106i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.03620 4.03620i 0.139512 0.139512i
\(838\) 0 0
\(839\) −30.3303 −1.04712 −0.523559 0.851989i \(-0.675396\pi\)
−0.523559 + 0.851989i \(0.675396\pi\)
\(840\) 0 0
\(841\) −1.00000 −0.0344828
\(842\) 0 0
\(843\) 4.91010 4.91010i 0.169113 0.169113i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 22.7913 22.7913i 0.783118 0.783118i
\(848\) 0 0
\(849\) 15.7568i 0.540773i
\(850\) 0 0
\(851\) −8.22330 −0.281891
\(852\) 0 0
\(853\) 36.9452 + 36.9452i 1.26498 + 1.26498i 0.948650 + 0.316329i \(0.102450\pi\)
0.316329 + 0.948650i \(0.397550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.3303 39.3303i −1.34350 1.34350i −0.892551 0.450947i \(-0.851086\pi\)
−0.450947 0.892551i \(-0.648914\pi\)
\(858\) 0 0
\(859\) 2.01810i 0.0688567i −0.999407 0.0344284i \(-0.989039\pi\)
0.999407 0.0344284i \(-0.0109611\pi\)
\(860\) 0 0
\(861\) 19.3386i 0.659058i
\(862\) 0 0
\(863\) −32.7042 32.7042i −1.11326 1.11326i −0.992706 0.120556i \(-0.961532\pi\)
−0.120556 0.992706i \(-0.538468\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.456850 0.456850i −0.0155154 0.0155154i
\(868\) 0 0
\(869\) −52.5336 −1.78208
\(870\) 0 0
\(871\) 28.7477i 0.974080i
\(872\) 0 0
\(873\) 1.50455 1.50455i 0.0509212 0.0509212i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.7756 19.7756i 0.667773 0.667773i −0.289427 0.957200i \(-0.593465\pi\)
0.957200 + 0.289427i \(0.0934647\pi\)
\(878\) 0 0
\(879\) −21.4083 −0.722085
\(880\) 0 0
\(881\) −55.5826 −1.87262 −0.936312 0.351169i \(-0.885784\pi\)
−0.936312 + 0.351169i \(0.885784\pi\)
\(882\) 0 0
\(883\) 22.1552 22.1552i 0.745581 0.745581i −0.228065 0.973646i \(-0.573240\pi\)
0.973646 + 0.228065i \(0.0732400\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.9564 21.9564i 0.737225 0.737225i −0.234815 0.972040i \(-0.575448\pi\)
0.972040 + 0.234815i \(0.0754484\pi\)
\(888\) 0 0
\(889\) 6.74773i 0.226312i
\(890\) 0 0
\(891\) 23.7164 0.794530
\(892\) 0 0
\(893\) 2.74110 + 2.74110i 0.0917275 + 0.0917275i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.25227 + 1.25227i 0.0418122 + 0.0418122i
\(898\) 0 0
\(899\) 8.37420i 0.279295i
\(900\) 0 0
\(901\) 15.8745i 0.528857i
\(902\) 0 0
\(903\) 21.1652 + 21.1652i 0.704332 + 0.704332i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11.0399 + 11.0399i 0.366572 + 0.366572i 0.866226 0.499653i \(-0.166539\pi\)
−0.499653 + 0.866226i \(0.666539\pi\)
\(908\) 0 0
\(909\) 9.43880 0.313065
\(910\) 0 0
\(911\) 49.5826i 1.64274i 0.570393 + 0.821372i \(0.306791\pi\)
−0.570393 + 0.821372i \(0.693209\pi\)
\(912\) 0 0
\(913\) 21.1652 21.1652i 0.700464 0.700464i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −31.5583 + 31.5583i −1.04215 + 1.04215i
\(918\) 0 0
\(919\) 42.3303 1.39635 0.698174 0.715928i \(-0.253996\pi\)
0.698174 + 0.715928i \(0.253996\pi\)
\(920\) 0 0
\(921\) 1.25227 0.0412638
\(922\) 0 0
\(923\) −23.5257 + 23.5257i −0.774358 + 0.774358i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 26.7913 26.7913i 0.879941 0.879941i
\(928\) 0 0
\(929\) 19.9129i 0.653320i 0.945142 + 0.326660i \(0.105923\pi\)
−0.945142 + 0.326660i \(0.894077\pi\)
\(930\) 0 0
\(931\) −29.7309 −0.974391
\(932\) 0 0
\(933\) 4.75920 + 4.75920i 0.155809 + 0.155809i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −34.1652 34.1652i −1.11613 1.11613i −0.992304 0.123822i \(-0.960485\pi\)
−0.123822 0.992304i \(-0.539515\pi\)
\(938\) 0 0
\(939\) 2.35970i 0.0770059i
\(940\) 0 0
\(941\) 24.4394i 0.796702i 0.917233 + 0.398351i \(0.130417\pi\)
−0.917233 + 0.398351i \(0.869583\pi\)
\(942\) 0 0
\(943\) −6.00000 6.00000i −0.195387 0.195387i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.6947 + 14.6947i 0.477512 + 0.477512i 0.904335 0.426823i \(-0.140367\pi\)
−0.426823 + 0.904335i \(0.640367\pi\)
\(948\) 0 0
\(949\) 2.01810 0.0655103
\(950\) 0 0
\(951\) 16.0871i 0.521661i
\(952\) 0 0
\(953\) 10.9129 10.9129i 0.353503 0.353503i −0.507908 0.861411i \(-0.669581\pi\)
0.861411 + 0.507908i \(0.169581\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −10.5830 + 10.5830i −0.342100 + 0.342100i
\(958\) 0 0
\(959\) −67.9129 −2.19302
\(960\) 0 0
\(961\) 28.4955 0.919208
\(962\) 0 0
\(963\) 21.4322 21.4322i 0.690642 0.690642i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.79129 + 8.79129i −0.282709 + 0.282709i −0.834188 0.551480i \(-0.814063\pi\)
0.551480 + 0.834188i \(0.314063\pi\)
\(968\) 0 0
\(969\) 9.49545i 0.305038i
\(970\) 0 0
\(971\) −37.5728 −1.20577 −0.602885 0.797828i \(-0.705982\pi\)
−0.602885 + 0.797828i \(0.705982\pi\)
\(972\) 0 0
\(973\) −24.9717 24.9717i −0.800556 0.800556i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.5826 28.5826i −0.914438 0.914438i 0.0821799 0.996618i \(-0.473812\pi\)
−0.996618 + 0.0821799i \(0.973812\pi\)
\(978\) 0 0
\(979\) 13.8564i 0.442853i
\(980\) 0 0
\(981\) 16.0254i 0.511652i
\(982\) 0 0
\(983\) 26.7042 + 26.7042i 0.851731 + 0.851731i 0.990346 0.138616i \(-0.0442653\pi\)
−0.138616 + 0.990346i \(0.544265\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.01810 2.01810i −0.0642369 0.0642369i
\(988\) 0 0
\(989\) 13.1334 0.417618
\(990\) 0 0
\(991\) 3.25227i 0.103312i −0.998665 0.0516559i \(-0.983550\pi\)
0.998665 0.0516559i \(-0.0164499\pi\)
\(992\) 0 0
\(993\) −2.83485 + 2.83485i −0.0899612 + 0.0899612i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 17.6066 17.6066i 0.557605 0.557605i −0.371020 0.928625i \(-0.620992\pi\)
0.928625 + 0.371020i \(0.120992\pi\)
\(998\) 0 0
\(999\) 26.5045 0.838567
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 1600.2.o.l.607.2 8
4.3 odd 2 1600.2.o.e.607.3 8
5.2 odd 4 320.2.o.f.223.3 yes 8
5.3 odd 4 1600.2.o.e.543.2 8
5.4 even 2 320.2.o.e.287.3 yes 8
8.3 odd 2 1600.2.o.e.607.2 8
8.5 even 2 inner 1600.2.o.l.607.3 8
20.3 even 4 inner 1600.2.o.l.543.3 8
20.7 even 4 320.2.o.e.223.2 8
20.19 odd 2 320.2.o.f.287.2 yes 8
40.3 even 4 inner 1600.2.o.l.543.2 8
40.13 odd 4 1600.2.o.e.543.3 8
40.19 odd 2 320.2.o.f.287.3 yes 8
40.27 even 4 320.2.o.e.223.3 yes 8
40.29 even 2 320.2.o.e.287.2 yes 8
40.37 odd 4 320.2.o.f.223.2 yes 8
80.19 odd 4 1280.2.n.n.767.2 8
80.27 even 4 1280.2.n.p.1023.2 8
80.29 even 4 1280.2.n.p.767.3 8
80.37 odd 4 1280.2.n.n.1023.3 8
80.59 odd 4 1280.2.n.n.767.3 8
80.67 even 4 1280.2.n.p.1023.3 8
80.69 even 4 1280.2.n.p.767.2 8
80.77 odd 4 1280.2.n.n.1023.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.o.e.223.2 8 20.7 even 4
320.2.o.e.223.3 yes 8 40.27 even 4
320.2.o.e.287.2 yes 8 40.29 even 2
320.2.o.e.287.3 yes 8 5.4 even 2
320.2.o.f.223.2 yes 8 40.37 odd 4
320.2.o.f.223.3 yes 8 5.2 odd 4
320.2.o.f.287.2 yes 8 20.19 odd 2
320.2.o.f.287.3 yes 8 40.19 odd 2
1280.2.n.n.767.2 8 80.19 odd 4
1280.2.n.n.767.3 8 80.59 odd 4
1280.2.n.n.1023.2 8 80.77 odd 4
1280.2.n.n.1023.3 8 80.37 odd 4
1280.2.n.p.767.2 8 80.69 even 4
1280.2.n.p.767.3 8 80.29 even 4
1280.2.n.p.1023.2 8 80.27 even 4
1280.2.n.p.1023.3 8 80.67 even 4
1600.2.o.e.543.2 8 5.3 odd 4
1600.2.o.e.543.3 8 40.13 odd 4
1600.2.o.e.607.2 8 8.3 odd 2
1600.2.o.e.607.3 8 4.3 odd 2
1600.2.o.l.543.2 8 40.3 even 4 inner
1600.2.o.l.543.3 8 20.3 even 4 inner
1600.2.o.l.607.2 8 1.1 even 1 trivial
1600.2.o.l.607.3 8 8.5 even 2 inner