Properties

Label 192.2.k.a.47.5
Level $192$
Weight $2$
Character 192.47
Analytic conductor $1.533$
Analytic rank $0$
Dimension $12$
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] = [192,2,Mod(47,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 192.k (of order \(4\), degree \(2\), 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: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.163368480538624.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.5
Root \(1.35164 - 0.416001i\) of defining polynomial
Character \(\chi\) \(=\) 192.47
Dual form 192.2.k.a.143.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.43726 + 0.966579i) q^{3} +(1.57184 + 1.57184i) q^{5} -2.24914 q^{7} +(1.13145 + 2.77846i) q^{9} +O(q^{10})\) \(q+(1.43726 + 0.966579i) q^{3} +(1.57184 + 1.57184i) q^{5} -2.24914 q^{7} +(1.13145 + 2.77846i) q^{9} +(1.13145 - 1.13145i) q^{11} +(-3.24914 - 3.24914i) q^{13} +(0.739839 + 3.77846i) q^{15} +1.66400i q^{17} +(3.77846 - 3.77846i) q^{19} +(-3.23261 - 2.17397i) q^{21} +2.26290i q^{23} -0.0586332i q^{25} +(-1.05941 + 5.08701i) q^{27} +(3.23584 - 3.23584i) q^{29} -1.30777i q^{31} +(2.71982 - 0.532554i) q^{33} +(-3.53529 - 3.53529i) q^{35} +(2.30777 - 2.30777i) q^{37} +(-1.52932 - 7.81042i) q^{39} -10.2143 q^{41} +(-3.77846 - 3.77846i) q^{43} +(-2.58884 + 6.14575i) q^{45} -3.74258 q^{47} -1.94137 q^{49} +(-1.60839 + 2.39161i) q^{51} +(-0.972946 - 0.972946i) q^{53} +3.55691 q^{55} +(9.08281 - 1.77846i) q^{57} +(3.88352 - 3.88352i) q^{59} +(4.19051 + 4.19051i) q^{61} +(-2.54479 - 6.24914i) q^{63} -10.2143i q^{65} +(-8.02760 + 8.02760i) q^{67} +(-2.18727 + 3.25238i) q^{69} +11.0950i q^{71} -6.38101i q^{73} +(0.0566736 - 0.0842713i) q^{75} +(-2.54479 + 2.54479i) q^{77} -2.69223i q^{79} +(-6.43965 + 6.28736i) q^{81} +(2.61113 + 2.61113i) q^{83} +(-2.61555 + 2.61555i) q^{85} +(7.77846 - 1.52306i) q^{87} +7.35247 q^{89} +(7.30777 + 7.30777i) q^{91} +(1.26407 - 1.87961i) q^{93} +11.8783 q^{95} -5.67418 q^{97} +(4.42386 + 1.86351i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} + 8 q^{7} - 4 q^{13} + 12 q^{19} - 8 q^{21} - 10 q^{27} - 4 q^{33} - 4 q^{37} - 20 q^{39} - 12 q^{43} - 12 q^{45} - 20 q^{49} - 24 q^{51} - 24 q^{55} + 12 q^{61} - 28 q^{67} + 4 q^{69} + 34 q^{75} - 4 q^{81} + 32 q^{85} + 60 q^{87} + 56 q^{91} + 28 q^{93} - 8 q^{97} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 1.43726 + 0.966579i 0.829804 + 0.558055i
\(4\) 0 0
\(5\) 1.57184 + 1.57184i 0.702949 + 0.702949i 0.965042 0.262094i \(-0.0844130\pi\)
−0.262094 + 0.965042i \(0.584413\pi\)
\(6\) 0 0
\(7\) −2.24914 −0.850095 −0.425048 0.905171i \(-0.639743\pi\)
−0.425048 + 0.905171i \(0.639743\pi\)
\(8\) 0 0
\(9\) 1.13145 + 2.77846i 0.377150 + 0.926152i
\(10\) 0 0
\(11\) 1.13145 1.13145i 0.341145 0.341145i −0.515653 0.856798i \(-0.672451\pi\)
0.856798 + 0.515653i \(0.172451\pi\)
\(12\) 0 0
\(13\) −3.24914 3.24914i −0.901149 0.901149i 0.0943862 0.995536i \(-0.469911\pi\)
−0.995536 + 0.0943862i \(0.969911\pi\)
\(14\) 0 0
\(15\) 0.739839 + 3.77846i 0.191026 + 0.975593i
\(16\) 0 0
\(17\) 1.66400i 0.403580i 0.979429 + 0.201790i \(0.0646758\pi\)
−0.979429 + 0.201790i \(0.935324\pi\)
\(18\) 0 0
\(19\) 3.77846 3.77846i 0.866838 0.866838i −0.125283 0.992121i \(-0.539984\pi\)
0.992121 + 0.125283i \(0.0399840\pi\)
\(20\) 0 0
\(21\) −3.23261 2.17397i −0.705412 0.474400i
\(22\) 0 0
\(23\) 2.26290i 0.471847i 0.971772 + 0.235923i \(0.0758114\pi\)
−0.971772 + 0.235923i \(0.924189\pi\)
\(24\) 0 0
\(25\) 0.0586332i 0.0117266i
\(26\) 0 0
\(27\) −1.05941 + 5.08701i −0.203884 + 0.978995i
\(28\) 0 0
\(29\) 3.23584 3.23584i 0.600881 0.600881i −0.339665 0.940546i \(-0.610314\pi\)
0.940546 + 0.339665i \(0.110314\pi\)
\(30\) 0 0
\(31\) 1.30777i 0.234883i −0.993080 0.117442i \(-0.962531\pi\)
0.993080 0.117442i \(-0.0374693\pi\)
\(32\) 0 0
\(33\) 2.71982 0.532554i 0.473461 0.0927058i
\(34\) 0 0
\(35\) −3.53529 3.53529i −0.597573 0.597573i
\(36\) 0 0
\(37\) 2.30777 2.30777i 0.379396 0.379396i −0.491488 0.870884i \(-0.663547\pi\)
0.870884 + 0.491488i \(0.163547\pi\)
\(38\) 0 0
\(39\) −1.52932 7.81042i −0.244887 1.25067i
\(40\) 0 0
\(41\) −10.2143 −1.59520 −0.797600 0.603187i \(-0.793897\pi\)
−0.797600 + 0.603187i \(0.793897\pi\)
\(42\) 0 0
\(43\) −3.77846 3.77846i −0.576209 0.576209i 0.357647 0.933857i \(-0.383579\pi\)
−0.933857 + 0.357647i \(0.883579\pi\)
\(44\) 0 0
\(45\) −2.58884 + 6.14575i −0.385921 + 0.916154i
\(46\) 0 0
\(47\) −3.74258 −0.545911 −0.272955 0.962027i \(-0.588001\pi\)
−0.272955 + 0.962027i \(0.588001\pi\)
\(48\) 0 0
\(49\) −1.94137 −0.277338
\(50\) 0 0
\(51\) −1.60839 + 2.39161i −0.225220 + 0.334892i
\(52\) 0 0
\(53\) −0.972946 0.972946i −0.133644 0.133644i 0.637120 0.770765i \(-0.280126\pi\)
−0.770765 + 0.637120i \(0.780126\pi\)
\(54\) 0 0
\(55\) 3.55691 0.479614
\(56\) 0 0
\(57\) 9.08281 1.77846i 1.20305 0.235562i
\(58\) 0 0
\(59\) 3.88352 3.88352i 0.505591 0.505591i −0.407579 0.913170i \(-0.633627\pi\)
0.913170 + 0.407579i \(0.133627\pi\)
\(60\) 0 0
\(61\) 4.19051 + 4.19051i 0.536539 + 0.536539i 0.922511 0.385971i \(-0.126134\pi\)
−0.385971 + 0.922511i \(0.626134\pi\)
\(62\) 0 0
\(63\) −2.54479 6.24914i −0.320613 0.787318i
\(64\) 0 0
\(65\) 10.2143i 1.26692i
\(66\) 0 0
\(67\) −8.02760 + 8.02760i −0.980727 + 0.980727i −0.999818 0.0190906i \(-0.993923\pi\)
0.0190906 + 0.999818i \(0.493923\pi\)
\(68\) 0 0
\(69\) −2.18727 + 3.25238i −0.263316 + 0.391540i
\(70\) 0 0
\(71\) 11.0950i 1.31674i 0.752695 + 0.658370i \(0.228754\pi\)
−0.752695 + 0.658370i \(0.771246\pi\)
\(72\) 0 0
\(73\) 6.38101i 0.746841i −0.927662 0.373421i \(-0.878185\pi\)
0.927662 0.373421i \(-0.121815\pi\)
\(74\) 0 0
\(75\) 0.0566736 0.0842713i 0.00654410 0.00973081i
\(76\) 0 0
\(77\) −2.54479 + 2.54479i −0.290005 + 0.290005i
\(78\) 0 0
\(79\) 2.69223i 0.302899i −0.988465 0.151450i \(-0.951606\pi\)
0.988465 0.151450i \(-0.0483941\pi\)
\(80\) 0 0
\(81\) −6.43965 + 6.28736i −0.715516 + 0.698596i
\(82\) 0 0
\(83\) 2.61113 + 2.61113i 0.286608 + 0.286608i 0.835738 0.549129i \(-0.185040\pi\)
−0.549129 + 0.835738i \(0.685040\pi\)
\(84\) 0 0
\(85\) −2.61555 + 2.61555i −0.283696 + 0.283696i
\(86\) 0 0
\(87\) 7.77846 1.52306i 0.833938 0.163289i
\(88\) 0 0
\(89\) 7.35247 0.779360 0.389680 0.920950i \(-0.372586\pi\)
0.389680 + 0.920950i \(0.372586\pi\)
\(90\) 0 0
\(91\) 7.30777 + 7.30777i 0.766063 + 0.766063i
\(92\) 0 0
\(93\) 1.26407 1.87961i 0.131078 0.194907i
\(94\) 0 0
\(95\) 11.8783 1.21868
\(96\) 0 0
\(97\) −5.67418 −0.576126 −0.288063 0.957611i \(-0.593011\pi\)
−0.288063 + 0.957611i \(0.593011\pi\)
\(98\) 0 0
\(99\) 4.42386 + 1.86351i 0.444614 + 0.187289i
\(100\) 0 0
\(101\) 10.3064 + 10.3064i 1.02553 + 1.02553i 0.999666 + 0.0258621i \(0.00823308\pi\)
0.0258621 + 0.999666i \(0.491767\pi\)
\(102\) 0 0
\(103\) 8.13187 0.801257 0.400629 0.916241i \(-0.368792\pi\)
0.400629 + 0.916241i \(0.368792\pi\)
\(104\) 0 0
\(105\) −1.66400 8.49828i −0.162390 0.829347i
\(106\) 0 0
\(107\) −2.40384 + 2.40384i −0.232388 + 0.232388i −0.813689 0.581301i \(-0.802544\pi\)
0.581301 + 0.813689i \(0.302544\pi\)
\(108\) 0 0
\(109\) 8.92332 + 8.92332i 0.854699 + 0.854699i 0.990708 0.136009i \(-0.0434275\pi\)
−0.136009 + 0.990708i \(0.543427\pi\)
\(110\) 0 0
\(111\) 5.54752 1.08623i 0.526548 0.103100i
\(112\) 0 0
\(113\) 15.9027i 1.49600i 0.663697 + 0.748002i \(0.268986\pi\)
−0.663697 + 0.748002i \(0.731014\pi\)
\(114\) 0 0
\(115\) −3.55691 + 3.55691i −0.331684 + 0.331684i
\(116\) 0 0
\(117\) 5.35136 12.7038i 0.494734 1.17447i
\(118\) 0 0
\(119\) 3.74258i 0.343081i
\(120\) 0 0
\(121\) 8.43965i 0.767241i
\(122\) 0 0
\(123\) −14.6806 9.87290i −1.32370 0.890209i
\(124\) 0 0
\(125\) 7.95137 7.95137i 0.711192 0.711192i
\(126\) 0 0
\(127\) 7.42504i 0.658866i 0.944179 + 0.329433i \(0.106858\pi\)
−0.944179 + 0.329433i \(0.893142\pi\)
\(128\) 0 0
\(129\) −1.77846 9.08281i −0.156584 0.799697i
\(130\) 0 0
\(131\) −3.88352 3.88352i −0.339305 0.339305i 0.516801 0.856106i \(-0.327123\pi\)
−0.856106 + 0.516801i \(0.827123\pi\)
\(132\) 0 0
\(133\) −8.49828 + 8.49828i −0.736894 + 0.736894i
\(134\) 0 0
\(135\) −9.66119 + 6.33074i −0.831503 + 0.544864i
\(136\) 0 0
\(137\) −2.72911 −0.233164 −0.116582 0.993181i \(-0.537194\pi\)
−0.116582 + 0.993181i \(0.537194\pi\)
\(138\) 0 0
\(139\) 0.0275977 + 0.0275977i 0.00234080 + 0.00234080i 0.708276 0.705935i \(-0.249473\pi\)
−0.705935 + 0.708276i \(0.749473\pi\)
\(140\) 0 0
\(141\) −5.37907 3.61750i −0.452999 0.304648i
\(142\) 0 0
\(143\) −7.35247 −0.614845
\(144\) 0 0
\(145\) 10.1725 0.844777
\(146\) 0 0
\(147\) −2.79025 1.87649i −0.230136 0.154770i
\(148\) 0 0
\(149\) −12.5693 12.5693i −1.02972 1.02972i −0.999545 0.0301744i \(-0.990394\pi\)
−0.0301744 0.999545i \(-0.509606\pi\)
\(150\) 0 0
\(151\) −16.8647 −1.37243 −0.686214 0.727399i \(-0.740729\pi\)
−0.686214 + 0.727399i \(0.740729\pi\)
\(152\) 0 0
\(153\) −4.62336 + 1.88273i −0.373777 + 0.152210i
\(154\) 0 0
\(155\) 2.05561 2.05561i 0.165111 0.165111i
\(156\) 0 0
\(157\) −5.36641 5.36641i −0.428286 0.428286i 0.459758 0.888044i \(-0.347936\pi\)
−0.888044 + 0.459758i \(0.847936\pi\)
\(158\) 0 0
\(159\) −0.457950 2.33881i −0.0363178 0.185480i
\(160\) 0 0
\(161\) 5.08957i 0.401115i
\(162\) 0 0
\(163\) 8.77502 8.77502i 0.687313 0.687313i −0.274325 0.961637i \(-0.588454\pi\)
0.961637 + 0.274325i \(0.0884543\pi\)
\(164\) 0 0
\(165\) 5.11222 + 3.43804i 0.397986 + 0.267651i
\(166\) 0 0
\(167\) 16.9678i 1.31301i −0.754321 0.656505i \(-0.772034\pi\)
0.754321 0.656505i \(-0.227966\pi\)
\(168\) 0 0
\(169\) 8.11383i 0.624141i
\(170\) 0 0
\(171\) 14.7734 + 6.22315i 1.12975 + 0.475896i
\(172\) 0 0
\(173\) −16.3119 + 16.3119i −1.24017 + 1.24017i −0.280241 + 0.959930i \(0.590414\pi\)
−0.959930 + 0.280241i \(0.909586\pi\)
\(174\) 0 0
\(175\) 0.131874i 0.00996875i
\(176\) 0 0
\(177\) 9.33537 1.82791i 0.701689 0.137394i
\(178\) 0 0
\(179\) −1.33873 1.33873i −0.100062 0.100062i 0.655304 0.755365i \(-0.272541\pi\)
−0.755365 + 0.655304i \(0.772541\pi\)
\(180\) 0 0
\(181\) 10.2457 10.2457i 0.761557 0.761557i −0.215047 0.976604i \(-0.568990\pi\)
0.976604 + 0.215047i \(0.0689904\pi\)
\(182\) 0 0
\(183\) 1.97240 + 10.0733i 0.145804 + 0.744641i
\(184\) 0 0
\(185\) 7.25491 0.533391
\(186\) 0 0
\(187\) 1.88273 + 1.88273i 0.137679 + 0.137679i
\(188\) 0 0
\(189\) 2.38276 11.4414i 0.173320 0.832239i
\(190\) 0 0
\(191\) −24.5398 −1.77563 −0.887817 0.460197i \(-0.847779\pi\)
−0.887817 + 0.460197i \(0.847779\pi\)
\(192\) 0 0
\(193\) 8.38101 0.603279 0.301639 0.953422i \(-0.402466\pi\)
0.301639 + 0.953422i \(0.402466\pi\)
\(194\) 0 0
\(195\) 9.87290 14.6806i 0.707013 1.05130i
\(196\) 0 0
\(197\) −1.28995 1.28995i −0.0919052 0.0919052i 0.659659 0.751565i \(-0.270701\pi\)
−0.751565 + 0.659659i \(0.770701\pi\)
\(198\) 0 0
\(199\) 13.0992 0.928579 0.464290 0.885683i \(-0.346310\pi\)
0.464290 + 0.885683i \(0.346310\pi\)
\(200\) 0 0
\(201\) −19.2971 + 3.77846i −1.36111 + 0.266512i
\(202\) 0 0
\(203\) −7.27787 + 7.27787i −0.510806 + 0.510806i
\(204\) 0 0
\(205\) −16.0552 16.0552i −1.12134 1.12134i
\(206\) 0 0
\(207\) −6.28736 + 2.56035i −0.437002 + 0.177957i
\(208\) 0 0
\(209\) 8.55026i 0.591434i
\(210\) 0 0
\(211\) 8.47068 8.47068i 0.583146 0.583146i −0.352621 0.935766i \(-0.614709\pi\)
0.935766 + 0.352621i \(0.114709\pi\)
\(212\) 0 0
\(213\) −10.7242 + 15.9465i −0.734813 + 1.09264i
\(214\) 0 0
\(215\) 11.8783i 0.810091i
\(216\) 0 0
\(217\) 2.94137i 0.199673i
\(218\) 0 0
\(219\) 6.16776 9.17120i 0.416778 0.619732i
\(220\) 0 0
\(221\) 5.40658 5.40658i 0.363686 0.363686i
\(222\) 0 0
\(223\) 21.5715i 1.44454i 0.691613 + 0.722268i \(0.256900\pi\)
−0.691613 + 0.722268i \(0.743100\pi\)
\(224\) 0 0
\(225\) 0.162910 0.0663404i 0.0108606 0.00442269i
\(226\) 0 0
\(227\) 16.7523 + 16.7523i 1.11189 + 1.11189i 0.992895 + 0.118994i \(0.0379668\pi\)
0.118994 + 0.992895i \(0.462033\pi\)
\(228\) 0 0
\(229\) 3.36641 3.36641i 0.222458 0.222458i −0.587074 0.809533i \(-0.699720\pi\)
0.809533 + 0.587074i \(0.199720\pi\)
\(230\) 0 0
\(231\) −6.11727 + 1.19779i −0.402487 + 0.0788087i
\(232\) 0 0
\(233\) 0.501329 0.0328431 0.0164216 0.999865i \(-0.494773\pi\)
0.0164216 + 0.999865i \(0.494773\pi\)
\(234\) 0 0
\(235\) −5.88273 5.88273i −0.383747 0.383747i
\(236\) 0 0
\(237\) 2.60225 3.86944i 0.169034 0.251347i
\(238\) 0 0
\(239\) 30.4585 1.97019 0.985097 0.171999i \(-0.0550225\pi\)
0.985097 + 0.171999i \(0.0550225\pi\)
\(240\) 0 0
\(241\) −15.4948 −0.998111 −0.499055 0.866570i \(-0.666320\pi\)
−0.499055 + 0.866570i \(0.666320\pi\)
\(242\) 0 0
\(243\) −15.3327 + 2.81216i −0.983593 + 0.180400i
\(244\) 0 0
\(245\) −3.05152 3.05152i −0.194954 0.194954i
\(246\) 0 0
\(247\) −24.5535 −1.56230
\(248\) 0 0
\(249\) 1.22901 + 6.27674i 0.0778856 + 0.397772i
\(250\) 0 0
\(251\) −12.2265 + 12.2265i −0.771730 + 0.771730i −0.978409 0.206679i \(-0.933734\pi\)
0.206679 + 0.978409i \(0.433734\pi\)
\(252\) 0 0
\(253\) 2.56035 + 2.56035i 0.160968 + 0.160968i
\(254\) 0 0
\(255\) −6.28736 + 1.23109i −0.393730 + 0.0770941i
\(256\) 0 0
\(257\) 7.48515i 0.466911i 0.972367 + 0.233455i \(0.0750033\pi\)
−0.972367 + 0.233455i \(0.924997\pi\)
\(258\) 0 0
\(259\) −5.19051 + 5.19051i −0.322522 + 0.322522i
\(260\) 0 0
\(261\) 12.6518 + 5.32946i 0.783129 + 0.329885i
\(262\) 0 0
\(263\) 10.2659i 0.633023i 0.948589 + 0.316511i \(0.102512\pi\)
−0.948589 + 0.316511i \(0.897488\pi\)
\(264\) 0 0
\(265\) 3.05863i 0.187890i
\(266\) 0 0
\(267\) 10.5674 + 7.10675i 0.646716 + 0.434926i
\(268\) 0 0
\(269\) 2.76963 2.76963i 0.168867 0.168867i −0.617614 0.786481i \(-0.711901\pi\)
0.786481 + 0.617614i \(0.211901\pi\)
\(270\) 0 0
\(271\) 28.6854i 1.74251i −0.490830 0.871255i \(-0.663306\pi\)
0.490830 0.871255i \(-0.336694\pi\)
\(272\) 0 0
\(273\) 3.43965 + 17.5667i 0.208177 + 1.06319i
\(274\) 0 0
\(275\) −0.0663404 0.0663404i −0.00400048 0.00400048i
\(276\) 0 0
\(277\) −0.806055 + 0.806055i −0.0484311 + 0.0484311i −0.730908 0.682476i \(-0.760903\pi\)
0.682476 + 0.730908i \(0.260903\pi\)
\(278\) 0 0
\(279\) 3.63359 1.47968i 0.217538 0.0885861i
\(280\) 0 0
\(281\) 22.3228 1.33167 0.665833 0.746101i \(-0.268076\pi\)
0.665833 + 0.746101i \(0.268076\pi\)
\(282\) 0 0
\(283\) −8.08279 8.08279i −0.480472 0.480472i 0.424810 0.905282i \(-0.360341\pi\)
−0.905282 + 0.424810i \(0.860341\pi\)
\(284\) 0 0
\(285\) 17.0722 + 11.4813i 1.01127 + 0.680093i
\(286\) 0 0
\(287\) 22.9733 1.35607
\(288\) 0 0
\(289\) 14.2311 0.837123
\(290\) 0 0
\(291\) −8.15529 5.48455i −0.478071 0.321510i
\(292\) 0 0
\(293\) −6.37953 6.37953i −0.372696 0.372696i 0.495762 0.868458i \(-0.334889\pi\)
−0.868458 + 0.495762i \(0.834889\pi\)
\(294\) 0 0
\(295\) 12.2086 0.710809
\(296\) 0 0
\(297\) 4.55702 + 6.95436i 0.264425 + 0.403533i
\(298\) 0 0
\(299\) 7.35247 7.35247i 0.425204 0.425204i
\(300\) 0 0
\(301\) 8.49828 + 8.49828i 0.489833 + 0.489833i
\(302\) 0 0
\(303\) 4.85106 + 24.7750i 0.278686 + 1.42329i
\(304\) 0 0
\(305\) 13.1736i 0.754319i
\(306\) 0 0
\(307\) 6.58795 6.58795i 0.375994 0.375994i −0.493661 0.869655i \(-0.664341\pi\)
0.869655 + 0.493661i \(0.164341\pi\)
\(308\) 0 0
\(309\) 11.6876 + 7.86010i 0.664887 + 0.447146i
\(310\) 0 0
\(311\) 9.52861i 0.540318i 0.962816 + 0.270159i \(0.0870763\pi\)
−0.962816 + 0.270159i \(0.912924\pi\)
\(312\) 0 0
\(313\) 25.1690i 1.42264i −0.702870 0.711319i \(-0.748098\pi\)
0.702870 0.711319i \(-0.251902\pi\)
\(314\) 0 0
\(315\) 5.82265 13.8227i 0.328069 0.778818i
\(316\) 0 0
\(317\) −15.5287 + 15.5287i −0.872178 + 0.872178i −0.992709 0.120532i \(-0.961540\pi\)
0.120532 + 0.992709i \(0.461540\pi\)
\(318\) 0 0
\(319\) 7.32238i 0.409975i
\(320\) 0 0
\(321\) −5.77846 + 1.13145i −0.322522 + 0.0631513i
\(322\) 0 0
\(323\) 6.28736 + 6.28736i 0.349838 + 0.349838i
\(324\) 0 0
\(325\) −0.190507 + 0.190507i −0.0105674 + 0.0105674i
\(326\) 0 0
\(327\) 4.20006 + 21.4503i 0.232264 + 1.18620i
\(328\) 0 0
\(329\) 8.41758 0.464076
\(330\) 0 0
\(331\) −2.58795 2.58795i −0.142247 0.142247i 0.632397 0.774644i \(-0.282071\pi\)
−0.774644 + 0.632397i \(0.782071\pi\)
\(332\) 0 0
\(333\) 9.02318 + 3.80092i 0.494467 + 0.208289i
\(334\) 0 0
\(335\) −25.2362 −1.37880
\(336\) 0 0
\(337\) −23.1690 −1.26210 −0.631049 0.775743i \(-0.717375\pi\)
−0.631049 + 0.775743i \(0.717375\pi\)
\(338\) 0 0
\(339\) −15.3713 + 22.8564i −0.834852 + 1.24139i
\(340\) 0 0
\(341\) −1.47968 1.47968i −0.0801291 0.0801291i
\(342\) 0 0
\(343\) 20.1104 1.08586
\(344\) 0 0
\(345\) −8.55026 + 1.67418i −0.460331 + 0.0901349i
\(346\) 0 0
\(347\) −6.72235 + 6.72235i −0.360875 + 0.360875i −0.864135 0.503260i \(-0.832134\pi\)
0.503260 + 0.864135i \(0.332134\pi\)
\(348\) 0 0
\(349\) −2.75086 2.75086i −0.147250 0.147250i 0.629638 0.776888i \(-0.283203\pi\)
−0.776888 + 0.629638i \(0.783203\pi\)
\(350\) 0 0
\(351\) 19.9706 13.0862i 1.06595 0.698491i
\(352\) 0 0
\(353\) 23.1928i 1.23443i −0.786796 0.617213i \(-0.788262\pi\)
0.786796 0.617213i \(-0.211738\pi\)
\(354\) 0 0
\(355\) −17.4396 + 17.4396i −0.925600 + 0.925600i
\(356\) 0 0
\(357\) 3.61750 5.37907i 0.191458 0.284690i
\(358\) 0 0
\(359\) 27.3664i 1.44434i −0.691713 0.722172i \(-0.743144\pi\)
0.691713 0.722172i \(-0.256856\pi\)
\(360\) 0 0
\(361\) 9.55348i 0.502815i
\(362\) 0 0
\(363\) −8.15759 + 12.1300i −0.428162 + 0.636659i
\(364\) 0 0
\(365\) 10.0299 10.0299i 0.524991 0.524991i
\(366\) 0 0
\(367\) 26.0406i 1.35931i −0.733533 0.679654i \(-0.762130\pi\)
0.733533 0.679654i \(-0.237870\pi\)
\(368\) 0 0
\(369\) −11.5569 28.3799i −0.601629 1.47740i
\(370\) 0 0
\(371\) 2.18829 + 2.18829i 0.113611 + 0.113611i
\(372\) 0 0
\(373\) −13.1319 + 13.1319i −0.679943 + 0.679943i −0.959987 0.280044i \(-0.909651\pi\)
0.280044 + 0.959987i \(0.409651\pi\)
\(374\) 0 0
\(375\) 19.1138 3.74258i 0.987034 0.193266i
\(376\) 0 0
\(377\) −21.0274 −1.08297
\(378\) 0 0
\(379\) 17.4526 + 17.4526i 0.896482 + 0.896482i 0.995123 0.0986413i \(-0.0314496\pi\)
−0.0986413 + 0.995123i \(0.531450\pi\)
\(380\) 0 0
\(381\) −7.17689 + 10.6717i −0.367683 + 0.546729i
\(382\) 0 0
\(383\) −26.4965 −1.35391 −0.676953 0.736027i \(-0.736700\pi\)
−0.676953 + 0.736027i \(0.736700\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) 6.22315 14.7734i 0.316341 0.750975i
\(388\) 0 0
\(389\) 2.35506 + 2.35506i 0.119406 + 0.119406i 0.764285 0.644879i \(-0.223092\pi\)
−0.644879 + 0.764285i \(0.723092\pi\)
\(390\) 0 0
\(391\) −3.76547 −0.190428
\(392\) 0 0
\(393\) −1.82791 9.33537i −0.0922058 0.470907i
\(394\) 0 0
\(395\) 4.23175 4.23175i 0.212923 0.212923i
\(396\) 0 0
\(397\) 4.68879 + 4.68879i 0.235324 + 0.235324i 0.814910 0.579587i \(-0.196786\pi\)
−0.579587 + 0.814910i \(0.696786\pi\)
\(398\) 0 0
\(399\) −20.4285 + 4.00000i −1.02271 + 0.200250i
\(400\) 0 0
\(401\) 5.18714i 0.259033i −0.991577 0.129517i \(-0.958657\pi\)
0.991577 0.129517i \(-0.0413426\pi\)
\(402\) 0 0
\(403\) −4.24914 + 4.24914i −0.211665 + 0.211665i
\(404\) 0 0
\(405\) −20.0048 0.239367i −0.994048 0.0118943i
\(406\) 0 0
\(407\) 5.22225i 0.258858i
\(408\) 0 0
\(409\) 14.8793i 0.735734i −0.929878 0.367867i \(-0.880088\pi\)
0.929878 0.367867i \(-0.119912\pi\)
\(410\) 0 0
\(411\) −3.92245 2.63790i −0.193480 0.130118i
\(412\) 0 0
\(413\) −8.73458 + 8.73458i −0.429801 + 0.429801i
\(414\) 0 0
\(415\) 8.20855i 0.402942i
\(416\) 0 0
\(417\) 0.0129898 + 0.0663404i 0.000636111 + 0.00324870i
\(418\) 0 0
\(419\) −25.3026 25.3026i −1.23611 1.23611i −0.961578 0.274533i \(-0.911477\pi\)
−0.274533 0.961578i \(-0.588523\pi\)
\(420\) 0 0
\(421\) 7.13187 7.13187i 0.347586 0.347586i −0.511623 0.859210i \(-0.670956\pi\)
0.859210 + 0.511623i \(0.170956\pi\)
\(422\) 0 0
\(423\) −4.23453 10.3986i −0.205890 0.505597i
\(424\) 0 0
\(425\) 0.0975657 0.00473263
\(426\) 0 0
\(427\) −9.42504 9.42504i −0.456110 0.456110i
\(428\) 0 0
\(429\) −10.5674 7.10675i −0.510200 0.343117i
\(430\) 0 0
\(431\) 15.4882 0.746038 0.373019 0.927824i \(-0.378323\pi\)
0.373019 + 0.927824i \(0.378323\pi\)
\(432\) 0 0
\(433\) 25.5500 1.22786 0.613928 0.789362i \(-0.289588\pi\)
0.613928 + 0.789362i \(0.289588\pi\)
\(434\) 0 0
\(435\) 14.6205 + 9.83249i 0.700999 + 0.471432i
\(436\) 0 0
\(437\) 8.55026 + 8.55026i 0.409014 + 0.409014i
\(438\) 0 0
\(439\) −2.63703 −0.125859 −0.0629293 0.998018i \(-0.520044\pi\)
−0.0629293 + 0.998018i \(0.520044\pi\)
\(440\) 0 0
\(441\) −2.19656 5.39400i −0.104598 0.256857i
\(442\) 0 0
\(443\) 14.8580 14.8580i 0.705927 0.705927i −0.259749 0.965676i \(-0.583640\pi\)
0.965676 + 0.259749i \(0.0836399\pi\)
\(444\) 0 0
\(445\) 11.5569 + 11.5569i 0.547850 + 0.547850i
\(446\) 0 0
\(447\) −5.91617 30.2147i −0.279825 1.42910i
\(448\) 0 0
\(449\) 31.7079i 1.49639i 0.663480 + 0.748194i \(0.269079\pi\)
−0.663480 + 0.748194i \(0.730921\pi\)
\(450\) 0 0
\(451\) −11.5569 + 11.5569i −0.544194 + 0.544194i
\(452\) 0 0
\(453\) −24.2390 16.3011i −1.13885 0.765891i
\(454\) 0 0
\(455\) 22.9733i 1.07701i
\(456\) 0 0
\(457\) 23.8759i 1.11687i 0.829550 + 0.558433i \(0.188597\pi\)
−0.829550 + 0.558433i \(0.811403\pi\)
\(458\) 0 0
\(459\) −8.46480 1.76286i −0.395103 0.0822833i
\(460\) 0 0
\(461\) −0.921303 + 0.921303i −0.0429094 + 0.0429094i −0.728236 0.685327i \(-0.759659\pi\)
0.685327 + 0.728236i \(0.259659\pi\)
\(462\) 0 0
\(463\) 26.1510i 1.21534i 0.794190 + 0.607670i \(0.207895\pi\)
−0.794190 + 0.607670i \(0.792105\pi\)
\(464\) 0 0
\(465\) 4.94137 0.967542i 0.229150 0.0448687i
\(466\) 0 0
\(467\) −16.2510 16.2510i −0.752005 0.752005i 0.222848 0.974853i \(-0.428465\pi\)
−0.974853 + 0.222848i \(0.928465\pi\)
\(468\) 0 0
\(469\) 18.0552 18.0552i 0.833711 0.833711i
\(470\) 0 0
\(471\) −2.52588 12.9000i −0.116386 0.594400i
\(472\) 0 0
\(473\) −8.55026 −0.393141
\(474\) 0 0
\(475\) −0.221543 0.221543i −0.0101651 0.0101651i
\(476\) 0 0
\(477\) 1.60245 3.80413i 0.0733712 0.174179i
\(478\) 0 0
\(479\) 11.7456 0.536669 0.268335 0.963326i \(-0.413527\pi\)
0.268335 + 0.963326i \(0.413527\pi\)
\(480\) 0 0
\(481\) −14.9966 −0.683784
\(482\) 0 0
\(483\) 4.91948 7.31506i 0.223844 0.332847i
\(484\) 0 0
\(485\) −8.91891 8.91891i −0.404987 0.404987i
\(486\) 0 0
\(487\) 0.783513 0.0355044 0.0177522 0.999842i \(-0.494349\pi\)
0.0177522 + 0.999842i \(0.494349\pi\)
\(488\) 0 0
\(489\) 21.0938 4.13026i 0.953893 0.186777i
\(490\) 0 0
\(491\) −10.0382 + 10.0382i −0.453018 + 0.453018i −0.896355 0.443337i \(-0.853794\pi\)
0.443337 + 0.896355i \(0.353794\pi\)
\(492\) 0 0
\(493\) 5.38445 + 5.38445i 0.242504 + 0.242504i
\(494\) 0 0
\(495\) 4.02447 + 9.88273i 0.180886 + 0.444196i
\(496\) 0 0
\(497\) 24.9543i 1.11935i
\(498\) 0 0
\(499\) 29.9655 29.9655i 1.34144 1.34144i 0.446815 0.894627i \(-0.352558\pi\)
0.894627 0.446815i \(-0.147442\pi\)
\(500\) 0 0
\(501\) 16.4008 24.3872i 0.732732 1.08954i
\(502\) 0 0
\(503\) 21.7131i 0.968138i 0.875030 + 0.484069i \(0.160842\pi\)
−0.875030 + 0.484069i \(0.839158\pi\)
\(504\) 0 0
\(505\) 32.4001i 1.44179i
\(506\) 0 0
\(507\) −7.84266 + 11.6617i −0.348305 + 0.517914i
\(508\) 0 0
\(509\) 16.1276 16.1276i 0.714842 0.714842i −0.252702 0.967544i \(-0.581319\pi\)
0.967544 + 0.252702i \(0.0813192\pi\)
\(510\) 0 0
\(511\) 14.3518i 0.634886i
\(512\) 0 0
\(513\) 15.2181 + 23.2240i 0.671896 + 1.02536i
\(514\) 0 0
\(515\) 12.7820 + 12.7820i 0.563243 + 0.563243i
\(516\) 0 0
\(517\) −4.23453 + 4.23453i −0.186235 + 0.186235i
\(518\) 0 0
\(519\) −39.2112 + 7.67774i −1.72118 + 0.337015i
\(520\) 0 0
\(521\) 5.68847 0.249216 0.124608 0.992206i \(-0.460233\pi\)
0.124608 + 0.992206i \(0.460233\pi\)
\(522\) 0 0
\(523\) −13.6612 13.6612i −0.597362 0.597362i 0.342248 0.939610i \(-0.388812\pi\)
−0.939610 + 0.342248i \(0.888812\pi\)
\(524\) 0 0
\(525\) −0.127467 + 0.189538i −0.00556311 + 0.00827211i
\(526\) 0 0
\(527\) 2.17614 0.0947941
\(528\) 0 0
\(529\) 17.8793 0.777361
\(530\) 0 0
\(531\) 15.1842 + 6.39619i 0.658938 + 0.277571i
\(532\) 0 0
\(533\) 33.1876 + 33.1876i 1.43751 + 1.43751i
\(534\) 0 0
\(535\) −7.55691 −0.326714
\(536\) 0 0
\(537\) −0.630120 3.21811i −0.0271917 0.138871i
\(538\) 0 0
\(539\) −2.19656 + 2.19656i −0.0946124 + 0.0946124i
\(540\) 0 0
\(541\) −24.5715 24.5715i −1.05641 1.05641i −0.998311 0.0581016i \(-0.981495\pi\)
−0.0581016 0.998311i \(-0.518505\pi\)
\(542\) 0 0
\(543\) 24.6291 4.82248i 1.05693 0.206953i
\(544\) 0 0
\(545\) 28.0521i 1.20162i
\(546\) 0 0
\(547\) −3.56990 + 3.56990i −0.152638 + 0.152638i −0.779295 0.626657i \(-0.784423\pi\)
0.626657 + 0.779295i \(0.284423\pi\)
\(548\) 0 0
\(549\) −6.90180 + 16.3845i −0.294562 + 0.699273i
\(550\) 0 0
\(551\) 24.4530i 1.04173i
\(552\) 0 0
\(553\) 6.05520i 0.257493i
\(554\) 0 0
\(555\) 10.4272 + 7.01244i 0.442610 + 0.297662i
\(556\) 0 0
\(557\) 18.1602 18.1602i 0.769473 0.769473i −0.208540 0.978014i \(-0.566871\pi\)
0.978014 + 0.208540i \(0.0668713\pi\)
\(558\) 0 0
\(559\) 24.5535i 1.03850i
\(560\) 0 0
\(561\) 0.886172 + 4.52579i 0.0374142 + 0.191079i
\(562\) 0 0
\(563\) 6.91748 + 6.91748i 0.291537 + 0.291537i 0.837687 0.546150i \(-0.183907\pi\)
−0.546150 + 0.837687i \(0.683907\pi\)
\(564\) 0 0
\(565\) −24.9966 + 24.9966i −1.05161 + 1.05161i
\(566\) 0 0
\(567\) 14.4837 14.1412i 0.608257 0.593873i
\(568\) 0 0
\(569\) −36.2961 −1.52161 −0.760807 0.648979i \(-0.775196\pi\)
−0.760807 + 0.648979i \(0.775196\pi\)
\(570\) 0 0
\(571\) 33.5224 + 33.5224i 1.40287 + 1.40287i 0.790814 + 0.612056i \(0.209657\pi\)
0.612056 + 0.790814i \(0.290343\pi\)
\(572\) 0 0
\(573\) −35.2701 23.7196i −1.47343 0.990901i
\(574\) 0 0
\(575\) 0.132681 0.00553317
\(576\) 0 0
\(577\) −18.9345 −0.788253 −0.394127 0.919056i \(-0.628953\pi\)
−0.394127 + 0.919056i \(0.628953\pi\)
\(578\) 0 0
\(579\) 12.0457 + 8.10092i 0.500603 + 0.336663i
\(580\) 0 0
\(581\) −5.87279 5.87279i −0.243644 0.243644i
\(582\) 0 0
\(583\) −2.20168 −0.0911842
\(584\) 0 0
\(585\) 28.3799 11.5569i 1.17336 0.477820i
\(586\) 0 0
\(587\) 29.6211 29.6211i 1.22259 1.22259i 0.255885 0.966707i \(-0.417633\pi\)
0.966707 0.255885i \(-0.0823667\pi\)
\(588\) 0 0
\(589\) −4.94137 4.94137i −0.203605 0.203605i
\(590\) 0 0
\(591\) −0.607159 3.10084i −0.0249752 0.127551i
\(592\) 0 0
\(593\) 21.6263i 0.888086i −0.896005 0.444043i \(-0.853544\pi\)
0.896005 0.444043i \(-0.146456\pi\)
\(594\) 0 0
\(595\) 5.88273 5.88273i 0.241169 0.241169i
\(596\) 0 0
\(597\) 18.8270 + 12.6614i 0.770539 + 0.518198i
\(598\) 0 0
\(599\) 29.8079i 1.21792i 0.793201 + 0.608959i \(0.208413\pi\)
−0.793201 + 0.608959i \(0.791587\pi\)
\(600\) 0 0
\(601\) 32.8432i 1.33970i 0.742495 + 0.669851i \(0.233642\pi\)
−0.742495 + 0.669851i \(0.766358\pi\)
\(602\) 0 0
\(603\) −31.3872 13.2215i −1.27818 0.538422i
\(604\) 0 0
\(605\) −13.2658 + 13.2658i −0.539331 + 0.539331i
\(606\) 0 0
\(607\) 6.95597i 0.282334i 0.989986 + 0.141167i \(0.0450855\pi\)
−0.989986 + 0.141167i \(0.954915\pi\)
\(608\) 0 0
\(609\) −17.4948 + 3.42557i −0.708927 + 0.138811i
\(610\) 0 0
\(611\) 12.1602 + 12.1602i 0.491947 + 0.491947i
\(612\) 0 0
\(613\) −13.5389 + 13.5389i −0.546830 + 0.546830i −0.925522 0.378693i \(-0.876374\pi\)
0.378693 + 0.925522i \(0.376374\pi\)
\(614\) 0 0
\(615\) −7.55691 38.5942i −0.304724 1.55627i
\(616\) 0 0
\(617\) 8.91891 0.359062 0.179531 0.983752i \(-0.442542\pi\)
0.179531 + 0.983752i \(0.442542\pi\)
\(618\) 0 0
\(619\) 1.64658 + 1.64658i 0.0661818 + 0.0661818i 0.739423 0.673241i \(-0.235098\pi\)
−0.673241 + 0.739423i \(0.735098\pi\)
\(620\) 0 0
\(621\) −11.5114 2.39734i −0.461936 0.0962018i
\(622\) 0 0
\(623\) −16.5367 −0.662531
\(624\) 0 0
\(625\) 24.7034 0.988136
\(626\) 0 0
\(627\) 8.26451 12.2890i 0.330053 0.490774i
\(628\) 0 0
\(629\) 3.84014 + 3.84014i 0.153116 + 0.153116i
\(630\) 0 0
\(631\) −19.2457 −0.766159 −0.383080 0.923715i \(-0.625137\pi\)
−0.383080 + 0.923715i \(0.625137\pi\)
\(632\) 0 0
\(633\) 20.3622 3.98701i 0.809324 0.158469i
\(634\) 0 0
\(635\) −11.6710 + 11.6710i −0.463149 + 0.463149i
\(636\) 0 0
\(637\) 6.30777 + 6.30777i 0.249923 + 0.249923i
\(638\) 0 0
\(639\) −30.8271 + 12.5535i −1.21950 + 0.496608i
\(640\) 0 0
\(641\) 16.6343i 0.657016i −0.944501 0.328508i \(-0.893454\pi\)
0.944501 0.328508i \(-0.106546\pi\)
\(642\) 0 0
\(643\) 4.77502 4.77502i 0.188308 0.188308i −0.606656 0.794964i \(-0.707489\pi\)
0.794964 + 0.606656i \(0.207489\pi\)
\(644\) 0 0
\(645\) 11.4813 17.0722i 0.452075 0.672217i
\(646\) 0 0
\(647\) 48.2095i 1.89531i 0.319293 + 0.947656i \(0.396555\pi\)
−0.319293 + 0.947656i \(0.603445\pi\)
\(648\) 0 0
\(649\) 8.78801i 0.344960i
\(650\) 0 0
\(651\) −2.84306 + 4.22752i −0.111428 + 0.165689i
\(652\) 0 0
\(653\) 24.2281 24.2281i 0.948121 0.948121i −0.0505983 0.998719i \(-0.516113\pi\)
0.998719 + 0.0505983i \(0.0161128\pi\)
\(654\) 0 0
\(655\) 12.2086i 0.477028i
\(656\) 0 0
\(657\) 17.7294 7.21979i 0.691689 0.281671i
\(658\) 0 0
\(659\) 9.47442 + 9.47442i 0.369071 + 0.369071i 0.867138 0.498067i \(-0.165957\pi\)
−0.498067 + 0.867138i \(0.665957\pi\)
\(660\) 0 0
\(661\) 23.0406 23.0406i 0.896175 0.896175i −0.0989204 0.995095i \(-0.531539\pi\)
0.995095 + 0.0989204i \(0.0315389\pi\)
\(662\) 0 0
\(663\) 12.9966 2.54479i 0.504745 0.0988313i
\(664\) 0 0
\(665\) −26.7159 −1.03600
\(666\) 0 0
\(667\) 7.32238 + 7.32238i 0.283524 + 0.283524i
\(668\) 0 0
\(669\) −20.8506 + 31.0039i −0.806130 + 1.19868i
\(670\) 0 0
\(671\) 9.48269 0.366075
\(672\) 0 0
\(673\) 29.7846 1.14811 0.574055 0.818816i \(-0.305369\pi\)
0.574055 + 0.818816i \(0.305369\pi\)
\(674\) 0 0
\(675\) 0.298267 + 0.0621166i 0.0114803 + 0.00239087i
\(676\) 0 0
\(677\) −5.59631 5.59631i −0.215084 0.215084i 0.591339 0.806423i \(-0.298599\pi\)
−0.806423 + 0.591339i \(0.798599\pi\)
\(678\) 0 0
\(679\) 12.7620 0.489762
\(680\) 0 0
\(681\) 7.88503 + 40.2699i 0.302155 + 1.54314i
\(682\) 0 0
\(683\) 19.5790 19.5790i 0.749168 0.749168i −0.225155 0.974323i \(-0.572289\pi\)
0.974323 + 0.225155i \(0.0722887\pi\)
\(684\) 0 0
\(685\) −4.28973 4.28973i −0.163902 0.163902i
\(686\) 0 0
\(687\) 8.09231 1.58451i 0.308741 0.0604529i
\(688\) 0 0
\(689\) 6.32248i 0.240867i
\(690\) 0 0
\(691\) −3.98701 + 3.98701i −0.151673 + 0.151673i −0.778865 0.627192i \(-0.784204\pi\)
0.627192 + 0.778865i \(0.284204\pi\)
\(692\) 0 0
\(693\) −9.94988 4.19129i −0.377965 0.159214i
\(694\) 0 0
\(695\) 0.0867582i 0.00329093i
\(696\) 0 0
\(697\) 16.9966i 0.643791i
\(698\) 0 0
\(699\) 0.720541 + 0.484574i 0.0272534 + 0.0183283i
\(700\) 0 0
\(701\) −15.2117 + 15.2117i −0.574537 + 0.574537i −0.933393 0.358856i \(-0.883167\pi\)
0.358856 + 0.933393i \(0.383167\pi\)
\(702\) 0 0
\(703\) 17.4396i 0.657749i
\(704\) 0 0
\(705\) −2.76891 14.1412i −0.104283 0.532587i
\(706\) 0 0
\(707\) −23.1806 23.1806i −0.871796 0.871796i
\(708\) 0 0
\(709\) 20.3009 20.3009i 0.762416 0.762416i −0.214342 0.976759i \(-0.568761\pi\)
0.976759 + 0.214342i \(0.0687608\pi\)
\(710\) 0 0
\(711\) 7.48024 3.04612i 0.280531 0.114238i
\(712\) 0 0
\(713\) 2.95936 0.110829
\(714\) 0 0
\(715\) −11.5569 11.5569i −0.432204 0.432204i
\(716\) 0 0
\(717\) 43.7768 + 29.4405i 1.63488 + 1.09948i
\(718\) 0 0
\(719\) −3.52314 −0.131391 −0.0656954 0.997840i \(-0.520927\pi\)
−0.0656954 + 0.997840i \(0.520927\pi\)
\(720\) 0 0
\(721\) −18.2897 −0.681145
\(722\) 0 0
\(723\) −22.2702 14.9770i −0.828236 0.557000i
\(724\) 0 0
\(725\) −0.189728 0.189728i −0.00704631 0.00704631i
\(726\) 0 0
\(727\) −20.3664 −0.755348 −0.377674 0.925939i \(-0.623276\pi\)
−0.377674 + 0.925939i \(0.623276\pi\)
\(728\) 0 0
\(729\) −24.7553 10.7785i −0.916863 0.399202i
\(730\) 0 0
\(731\) 6.28736 6.28736i 0.232547 0.232547i
\(732\) 0 0
\(733\) 2.48024 + 2.48024i 0.0916096 + 0.0916096i 0.751426 0.659817i \(-0.229366\pi\)
−0.659817 + 0.751426i \(0.729366\pi\)
\(734\) 0 0
\(735\) −1.43630 7.33537i −0.0529787 0.270569i
\(736\) 0 0
\(737\) 18.1656i 0.669140i
\(738\) 0 0
\(739\) −15.7931 + 15.7931i −0.580957 + 0.580957i −0.935166 0.354209i \(-0.884750\pi\)
0.354209 + 0.935166i \(0.384750\pi\)
\(740\) 0 0
\(741\) −35.2898 23.7329i −1.29640 0.871849i
\(742\) 0 0
\(743\) 38.5942i 1.41588i −0.706271 0.707941i \(-0.749624\pi\)
0.706271 0.707941i \(-0.250376\pi\)
\(744\) 0 0
\(745\) 39.5139i 1.44768i
\(746\) 0 0
\(747\) −4.30055 + 10.2093i −0.157349 + 0.373537i
\(748\) 0 0
\(749\) 5.40658 5.40658i 0.197552 0.197552i
\(750\) 0 0
\(751\) 17.6527i 0.644156i −0.946713 0.322078i \(-0.895619\pi\)
0.946713 0.322078i \(-0.104381\pi\)
\(752\) 0 0
\(753\) −29.3906 + 5.75481i −1.07105 + 0.209717i
\(754\) 0 0
\(755\) −26.5086 26.5086i −0.964747 0.964747i
\(756\) 0 0
\(757\) −32.7440 + 32.7440i −1.19010 + 1.19010i −0.213062 + 0.977039i \(0.568343\pi\)
−0.977039 + 0.213062i \(0.931657\pi\)
\(758\) 0 0
\(759\) 1.20512 + 6.15468i 0.0437429 + 0.223401i
\(760\) 0 0
\(761\) −6.69113 −0.242553 −0.121277 0.992619i \(-0.538699\pi\)
−0.121277 + 0.992619i \(0.538699\pi\)
\(762\) 0 0
\(763\) −20.0698 20.0698i −0.726576 0.726576i
\(764\) 0 0
\(765\) −10.2265 4.30783i −0.369741 0.155750i
\(766\) 0 0
\(767\) −25.2362 −0.911227
\(768\) 0 0
\(769\) −5.03265 −0.181482 −0.0907411 0.995875i \(-0.528924\pi\)
−0.0907411 + 0.995875i \(0.528924\pi\)
\(770\) 0 0
\(771\) −7.23499 + 10.7581i −0.260562 + 0.387445i
\(772\) 0 0
\(773\) −10.6859 10.6859i −0.384344 0.384344i 0.488320 0.872665i \(-0.337610\pi\)
−0.872665 + 0.488320i \(0.837610\pi\)
\(774\) 0 0
\(775\) −0.0766789 −0.00275439
\(776\) 0 0
\(777\) −12.4772 + 2.44309i −0.447616 + 0.0876452i
\(778\) 0 0
\(779\) −38.5942 + 38.5942i −1.38278 + 1.38278i
\(780\) 0 0
\(781\) 12.5535 + 12.5535i 0.449199 + 0.449199i
\(782\) 0 0
\(783\) 13.0327 + 19.8888i 0.465750 + 0.710769i
\(784\) 0 0
\(785\) 16.8703i 0.602126i
\(786\) 0 0
\(787\) −16.0974 + 16.0974i −0.573810 + 0.573810i −0.933191 0.359381i \(-0.882988\pi\)
0.359381 + 0.933191i \(0.382988\pi\)
\(788\) 0 0
\(789\) −9.92281 + 14.7548i −0.353262 + 0.525285i
\(790\) 0 0
\(791\) 35.7675i 1.27175i
\(792\) 0 0
\(793\) 27.2311i 0.967005i
\(794\) 0 0
\(795\) 2.95641 4.39606i 0.104853 0.155912i
\(796\) 0 0
\(797\) −2.63695 + 2.63695i −0.0934055 + 0.0934055i −0.752266 0.658860i \(-0.771039\pi\)
0.658860 + 0.752266i \(0.271039\pi\)
\(798\) 0 0
\(799\) 6.22766i 0.220319i
\(800\) 0 0
\(801\) 8.31894 + 20.4285i 0.293935 + 0.721806i
\(802\) 0 0
\(803\) −7.21979 7.21979i −0.254781 0.254781i
\(804\) 0 0
\(805\) 8.00000 8.00000i 0.281963 0.281963i
\(806\) 0 0
\(807\) 6.65775 1.30362i 0.234364 0.0458896i
\(808\) 0 0
\(809\) 6.62090 0.232778 0.116389 0.993204i \(-0.462868\pi\)
0.116389 + 0.993204i \(0.462868\pi\)
\(810\) 0 0
\(811\) −4.01299 4.01299i −0.140915 0.140915i 0.633130 0.774045i \(-0.281770\pi\)
−0.774045 + 0.633130i \(0.781770\pi\)
\(812\) 0 0
\(813\) 27.7267 41.2284i 0.972417 1.44594i
\(814\) 0 0
\(815\) 27.5859 0.966291
\(816\) 0 0
\(817\) −28.5535 −0.998960
\(818\) 0 0
\(819\) −12.0360 + 28.5727i −0.420571 + 0.998411i
\(820\) 0 0
\(821\) 7.91085 + 7.91085i 0.276090 + 0.276090i 0.831546 0.555456i \(-0.187456\pi\)
−0.555456 + 0.831546i \(0.687456\pi\)
\(822\) 0 0
\(823\) 44.9751 1.56773 0.783866 0.620930i \(-0.213245\pi\)
0.783866 + 0.620930i \(0.213245\pi\)
\(824\) 0 0
\(825\) −0.0312253 0.159472i −0.00108713 0.00555210i
\(826\) 0 0
\(827\) 9.75631 9.75631i 0.339260 0.339260i −0.516829 0.856089i \(-0.672888\pi\)
0.856089 + 0.516829i \(0.172888\pi\)
\(828\) 0 0
\(829\) 18.6336 + 18.6336i 0.647171 + 0.647171i 0.952308 0.305137i \(-0.0987023\pi\)
−0.305137 + 0.952308i \(0.598702\pi\)
\(830\) 0 0
\(831\) −1.93763 + 0.379397i −0.0672156 + 0.0131611i
\(832\) 0 0
\(833\) 3.23044i 0.111928i
\(834\) 0 0
\(835\) 26.6707 26.6707i 0.922979 0.922979i
\(836\) 0 0
\(837\) 6.65266 + 1.38547i 0.229949 + 0.0478888i
\(838\) 0 0
\(839\) 37.8109i 1.30538i 0.757626 + 0.652689i \(0.226359\pi\)
−0.757626 + 0.652689i \(0.773641\pi\)
\(840\) 0 0
\(841\) 8.05863i 0.277884i
\(842\) 0 0
\(843\) 32.0837 + 21.5767i 1.10502 + 0.743142i
\(844\) 0 0
\(845\) −12.7536 + 12.7536i −0.438739 + 0.438739i
\(846\) 0 0
\(847\) 18.9820i 0.652228i
\(848\) 0 0
\(849\) −3.80444 19.4298i −0.130568 0.666828i
\(850\) 0 0
\(851\) 5.22225 + 5.22225i 0.179017 + 0.179017i
\(852\) 0 0
\(853\) 24.0992 24.0992i 0.825142 0.825142i −0.161699 0.986840i \(-0.551697\pi\)
0.986840 + 0.161699i \(0.0516972\pi\)
\(854\) 0 0
\(855\) 13.4396 + 33.0033i 0.459626 + 1.12869i
\(856\) 0 0
\(857\) 0.794026 0.0271234 0.0135617 0.999908i \(-0.495683\pi\)
0.0135617 + 0.999908i \(0.495683\pi\)
\(858\) 0 0
\(859\) −2.65775 2.65775i −0.0906814 0.0906814i 0.660311 0.750992i \(-0.270424\pi\)
−0.750992 + 0.660311i \(0.770424\pi\)
\(860\) 0 0
\(861\) 33.0187 + 22.2055i 1.12527 + 0.756762i
\(862\) 0 0
\(863\) 21.4069 0.728699 0.364349 0.931262i \(-0.381291\pi\)
0.364349 + 0.931262i \(0.381291\pi\)
\(864\) 0 0
\(865\) −51.2794 −1.74355
\(866\) 0 0
\(867\) 20.4538 + 13.7555i 0.694648 + 0.467161i
\(868\) 0 0
\(869\) −3.04612 3.04612i −0.103332 0.103332i
\(870\) 0 0
\(871\) 52.1656 1.76756
\(872\) 0 0
\(873\) −6.42004 15.7655i −0.217286 0.533580i
\(874\) 0 0
\(875\) −17.8837 + 17.8837i −0.604581 + 0.604581i
\(876\) 0 0
\(877\) −20.0923 20.0923i −0.678470 0.678470i 0.281184 0.959654i \(-0.409273\pi\)
−0.959654 + 0.281184i \(0.909273\pi\)
\(878\) 0 0
\(879\) −3.00274 15.3354i −0.101280 0.517249i
\(880\) 0 0
\(881\) 10.8132i 0.364305i −0.983270 0.182152i \(-0.941694\pi\)
0.983270 0.182152i \(-0.0583064\pi\)
\(882\) 0 0
\(883\) −12.5665 + 12.5665i −0.422895 + 0.422895i −0.886199 0.463304i \(-0.846664\pi\)
0.463304 + 0.886199i \(0.346664\pi\)
\(884\) 0 0
\(885\) 17.5469 + 11.8005i 0.589833 + 0.396671i
\(886\) 0 0
\(887\) 12.0977i 0.406201i −0.979158 0.203101i \(-0.934898\pi\)
0.979158 0.203101i \(-0.0651018\pi\)
\(888\) 0 0
\(889\) 16.7000i 0.560099i
\(890\) 0 0
\(891\) −0.172302 + 14.4000i −0.00577235 + 0.482417i
\(892\) 0 0
\(893\) −14.1412 + 14.1412i −0.473216 + 0.473216i
\(894\) 0 0
\(895\) 4.20855i 0.140676i
\(896\) 0 0
\(897\) 17.6742 3.46069i 0.590124 0.115549i
\(898\) 0 0
\(899\) −4.23175 4.23175i −0.141137 0.141137i
\(900\) 0 0
\(901\) 1.61899 1.61899i 0.0539362 0.0539362i
\(902\) 0 0
\(903\) 4.00000 + 20.4285i 0.133112 + 0.679819i
\(904\) 0 0
\(905\) 32.2092 1.07067
\(906\) 0 0
\(907\) 14.3388 + 14.3388i 0.476112 + 0.476112i 0.903886 0.427774i \(-0.140702\pi\)
−0.427774 + 0.903886i \(0.640702\pi\)
\(908\) 0 0
\(909\) −16.9748 + 40.2972i −0.563018 + 1.33657i
\(910\) 0 0
\(911\) −28.0629 −0.929765 −0.464882 0.885372i \(-0.653903\pi\)
−0.464882 + 0.885372i \(0.653903\pi\)
\(912\) 0 0
\(913\) 5.90871 0.195550
\(914\) 0 0
\(915\) −12.7334 + 18.9340i −0.420952 + 0.625937i
\(916\) 0 0
\(917\) 8.73458 + 8.73458i 0.288441 + 0.288441i
\(918\) 0 0
\(919\) −25.4734 −0.840289 −0.420144 0.907457i \(-0.638021\pi\)
−0.420144 + 0.907457i \(0.638021\pi\)
\(920\) 0 0
\(921\) 15.8364 3.10084i 0.521827 0.102176i
\(922\) 0 0
\(923\) 36.0494 36.0494i 1.18658 1.18658i
\(924\) 0 0
\(925\) −0.135312 0.135312i −0.00444903 0.00444903i
\(926\) 0 0
\(927\) 9.20080 + 22.5941i 0.302194 + 0.742086i
\(928\) 0 0
\(929\) 43.3502i 1.42227i 0.703054 + 0.711137i \(0.251819\pi\)
−0.703054 + 0.711137i \(0.748181\pi\)
\(930\) 0 0
\(931\) −7.33537 + 7.33537i −0.240407 + 0.240407i
\(932\) 0 0
\(933\) −9.21016 + 13.6951i −0.301527 + 0.448358i
\(934\) 0 0
\(935\) 5.91872i 0.193563i
\(936\) 0 0
\(937\) 15.5500i 0.507998i 0.967205 + 0.253999i \(0.0817459\pi\)
−0.967205 + 0.253999i \(0.918254\pi\)
\(938\) 0 0
\(939\) 24.3279 36.1745i 0.793910 1.18051i
\(940\) 0 0
\(941\) −1.85373 + 1.85373i −0.0604299 + 0.0604299i −0.736676 0.676246i \(-0.763606\pi\)
0.676246 + 0.736676i \(0.263606\pi\)
\(942\) 0 0
\(943\) 23.1138i 0.752690i
\(944\) 0 0
\(945\) 21.7294 14.2387i 0.706857 0.463186i
\(946\) 0 0
\(947\) 31.2499 + 31.2499i 1.01549 + 1.01549i 0.999878 + 0.0156087i \(0.00496860\pi\)
0.0156087 + 0.999878i \(0.495031\pi\)
\(948\) 0 0
\(949\) −20.7328 + 20.7328i −0.673016 + 0.673016i
\(950\) 0 0
\(951\) −37.3285 + 7.30909i −1.21046 + 0.237014i
\(952\) 0 0
\(953\) 48.9411 1.58536 0.792679 0.609639i \(-0.208686\pi\)
0.792679 + 0.609639i \(0.208686\pi\)
\(954\) 0 0
\(955\) −38.5726 38.5726i −1.24818 1.24818i
\(956\) 0 0
\(957\) 7.07766 10.5242i 0.228788 0.340199i
\(958\) 0 0
\(959\) 6.13815 0.198211
\(960\) 0 0
\(961\) 29.2897 0.944830
\(962\) 0 0
\(963\) −9.39880 3.95915i −0.302872 0.127582i
\(964\) 0 0
\(965\) 13.1736 + 13.1736i 0.424074 + 0.424074i
\(966\) 0 0
\(967\) −18.5129 −0.595334 −0.297667 0.954670i \(-0.596209\pi\)
−0.297667 + 0.954670i \(0.596209\pi\)
\(968\) 0 0
\(969\) 2.95936 + 15.1138i 0.0950683 + 0.485526i
\(970\) 0 0
\(971\) −0.0663404 + 0.0663404i −0.00212897 + 0.00212897i −0.708170 0.706041i \(-0.750479\pi\)
0.706041 + 0.708170i \(0.250479\pi\)
\(972\) 0 0
\(973\) −0.0620710 0.0620710i −0.00198991 0.00198991i
\(974\) 0 0
\(975\) −0.457950 + 0.0896687i −0.0146661 + 0.00287170i
\(976\) 0 0
\(977\) 3.42557i 0.109594i 0.998498 + 0.0547969i \(0.0174511\pi\)
−0.998498 + 0.0547969i \(0.982549\pi\)
\(978\) 0 0
\(979\) 8.31894 8.31894i 0.265875 0.265875i
\(980\) 0 0
\(981\) −14.6968 + 34.8893i −0.469232 + 1.11393i
\(982\) 0 0
\(983\) 30.5911i 0.975706i −0.872926 0.487853i \(-0.837780\pi\)
0.872926 0.487853i \(-0.162220\pi\)
\(984\) 0 0
\(985\) 4.05520i 0.129209i
\(986\) 0 0
\(987\) 12.0983 + 8.13626i 0.385092 + 0.258980i
\(988\) 0 0
\(989\) 8.55026 8.55026i 0.271882 0.271882i
\(990\) 0 0
\(991\) 24.2975i 0.771834i 0.922533 + 0.385917i \(0.126115\pi\)
−0.922533 + 0.385917i \(0.873885\pi\)
\(992\) 0 0
\(993\) −1.21811 6.22102i −0.0386554 0.197418i
\(994\) 0 0
\(995\) 20.5899 + 20.5899i 0.652743 + 0.652743i
\(996\) 0 0
\(997\) 10.3078 10.3078i 0.326450 0.326450i −0.524785 0.851235i \(-0.675854\pi\)
0.851235 + 0.524785i \(0.175854\pi\)
\(998\) 0 0
\(999\) 9.29478 + 14.1845i 0.294074 + 0.448779i
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 192.2.k.a.47.5 12
3.2 odd 2 inner 192.2.k.a.47.2 12
4.3 odd 2 48.2.k.a.35.4 yes 12
8.3 odd 2 384.2.k.b.95.5 12
8.5 even 2 384.2.k.a.95.2 12
12.11 even 2 48.2.k.a.35.3 yes 12
16.3 odd 4 384.2.k.a.287.5 12
16.5 even 4 48.2.k.a.11.3 12
16.11 odd 4 inner 192.2.k.a.143.2 12
16.13 even 4 384.2.k.b.287.2 12
24.5 odd 2 384.2.k.a.95.5 12
24.11 even 2 384.2.k.b.95.2 12
48.5 odd 4 48.2.k.a.11.4 yes 12
48.11 even 4 inner 192.2.k.a.143.5 12
48.29 odd 4 384.2.k.b.287.5 12
48.35 even 4 384.2.k.a.287.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.k.a.11.3 12 16.5 even 4
48.2.k.a.11.4 yes 12 48.5 odd 4
48.2.k.a.35.3 yes 12 12.11 even 2
48.2.k.a.35.4 yes 12 4.3 odd 2
192.2.k.a.47.2 12 3.2 odd 2 inner
192.2.k.a.47.5 12 1.1 even 1 trivial
192.2.k.a.143.2 12 16.11 odd 4 inner
192.2.k.a.143.5 12 48.11 even 4 inner
384.2.k.a.95.2 12 8.5 even 2
384.2.k.a.95.5 12 24.5 odd 2
384.2.k.a.287.2 12 48.35 even 4
384.2.k.a.287.5 12 16.3 odd 4
384.2.k.b.95.2 12 24.11 even 2
384.2.k.b.95.5 12 8.3 odd 2
384.2.k.b.287.2 12 16.13 even 4
384.2.k.b.287.5 12 48.29 odd 4