Properties

Label 1444.2.e.c.653.1
Level $1444$
Weight $2$
Character 1444.653
Analytic conductor $11.530$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1444,2,Mod(429,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.429");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1444.e (of order \(3\), 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: \(11.5303980519\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 653.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1444.653
Dual form 1444.2.e.c.429.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{3} +(0.500000 + 0.866025i) q^{5} -3.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(1.00000 + 1.73205i) q^{3} +(0.500000 + 0.866025i) q^{5} -3.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +5.00000 q^{11} +(-2.00000 + 3.46410i) q^{13} +(-1.00000 + 1.73205i) q^{15} +(1.50000 + 2.59808i) q^{17} +(-3.00000 - 5.19615i) q^{21} +(-4.00000 + 6.92820i) q^{23} +(2.00000 - 3.46410i) q^{25} +4.00000 q^{27} +(-1.00000 + 1.73205i) q^{29} -4.00000 q^{31} +(5.00000 + 8.66025i) q^{33} +(-1.50000 - 2.59808i) q^{35} -10.0000 q^{37} -8.00000 q^{39} +(5.00000 + 8.66025i) q^{41} +(-0.500000 - 0.866025i) q^{43} -1.00000 q^{45} +(0.500000 - 0.866025i) q^{47} +2.00000 q^{49} +(-3.00000 + 5.19615i) q^{51} +(-2.00000 + 3.46410i) q^{53} +(2.50000 + 4.33013i) q^{55} +(3.00000 + 5.19615i) q^{59} +(6.50000 - 11.2583i) q^{61} +(1.50000 - 2.59808i) q^{63} -4.00000 q^{65} +(-6.00000 + 10.3923i) q^{67} -16.0000 q^{69} +(1.00000 + 1.73205i) q^{71} +(-4.50000 - 7.79423i) q^{73} +8.00000 q^{75} -15.0000 q^{77} +(4.00000 + 6.92820i) q^{79} +(5.50000 + 9.52628i) q^{81} -12.0000 q^{83} +(-1.50000 + 2.59808i) q^{85} -4.00000 q^{87} +(6.00000 - 10.3923i) q^{89} +(6.00000 - 10.3923i) q^{91} +(-4.00000 - 6.92820i) q^{93} +(-4.00000 - 6.92820i) q^{97} +(-2.50000 + 4.33013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} - 6 q^{7} - q^{9} + 10 q^{11} - 4 q^{13} - 2 q^{15} + 3 q^{17} - 6 q^{21} - 8 q^{23} + 4 q^{25} + 8 q^{27} - 2 q^{29} - 8 q^{31} + 10 q^{33} - 3 q^{35} - 20 q^{37} - 16 q^{39} + 10 q^{41} - q^{43} - 2 q^{45} + q^{47} + 4 q^{49} - 6 q^{51} - 4 q^{53} + 5 q^{55} + 6 q^{59} + 13 q^{61} + 3 q^{63} - 8 q^{65} - 12 q^{67} - 32 q^{69} + 2 q^{71} - 9 q^{73} + 16 q^{75} - 30 q^{77} + 8 q^{79} + 11 q^{81} - 24 q^{83} - 3 q^{85} - 8 q^{87} + 12 q^{89} + 12 q^{91} - 8 q^{93} - 8 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.00000 + 1.73205i 0.577350 + 1.00000i 0.995782 + 0.0917517i \(0.0292466\pi\)
−0.418432 + 0.908248i \(0.637420\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) −1.00000 + 1.73205i −0.258199 + 0.447214i
\(16\) 0 0
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −3.00000 5.19615i −0.654654 1.13389i
\(22\) 0 0
\(23\) −4.00000 + 6.92820i −0.834058 + 1.44463i 0.0607377 + 0.998154i \(0.480655\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 0 0
\(25\) 2.00000 3.46410i 0.400000 0.692820i
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −1.00000 + 1.73205i −0.185695 + 0.321634i −0.943811 0.330487i \(-0.892787\pi\)
0.758115 + 0.652121i \(0.226120\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 5.00000 + 8.66025i 0.870388 + 1.50756i
\(34\) 0 0
\(35\) −1.50000 2.59808i −0.253546 0.439155i
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 5.00000 + 8.66025i 0.780869 + 1.35250i 0.931436 + 0.363905i \(0.118557\pi\)
−0.150567 + 0.988600i \(0.548110\pi\)
\(42\) 0 0
\(43\) −0.500000 0.866025i −0.0762493 0.132068i 0.825380 0.564578i \(-0.190961\pi\)
−0.901629 + 0.432511i \(0.857628\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0.500000 0.866025i 0.0729325 0.126323i −0.827253 0.561830i \(-0.810098\pi\)
0.900185 + 0.435507i \(0.143431\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 0 0
\(53\) −2.00000 + 3.46410i −0.274721 + 0.475831i −0.970065 0.242846i \(-0.921919\pi\)
0.695344 + 0.718677i \(0.255252\pi\)
\(54\) 0 0
\(55\) 2.50000 + 4.33013i 0.337100 + 0.583874i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 + 5.19615i 0.390567 + 0.676481i 0.992524 0.122047i \(-0.0389457\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(60\) 0 0
\(61\) 6.50000 11.2583i 0.832240 1.44148i −0.0640184 0.997949i \(-0.520392\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 1.50000 2.59808i 0.188982 0.327327i
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −6.00000 + 10.3923i −0.733017 + 1.26962i 0.222571 + 0.974916i \(0.428555\pi\)
−0.955588 + 0.294706i \(0.904778\pi\)
\(68\) 0 0
\(69\) −16.0000 −1.92617
\(70\) 0 0
\(71\) 1.00000 + 1.73205i 0.118678 + 0.205557i 0.919244 0.393688i \(-0.128801\pi\)
−0.800566 + 0.599245i \(0.795468\pi\)
\(72\) 0 0
\(73\) −4.50000 7.79423i −0.526685 0.912245i −0.999517 0.0310925i \(-0.990101\pi\)
0.472831 0.881153i \(-0.343232\pi\)
\(74\) 0 0
\(75\) 8.00000 0.923760
\(76\) 0 0
\(77\) −15.0000 −1.70941
\(78\) 0 0
\(79\) 4.00000 + 6.92820i 0.450035 + 0.779484i 0.998388 0.0567635i \(-0.0180781\pi\)
−0.548352 + 0.836247i \(0.684745\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −1.50000 + 2.59808i −0.162698 + 0.281801i
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 6.00000 10.3923i 0.635999 1.10158i −0.350304 0.936636i \(-0.613922\pi\)
0.986303 0.164946i \(-0.0527450\pi\)
\(90\) 0 0
\(91\) 6.00000 10.3923i 0.628971 1.08941i
\(92\) 0 0
\(93\) −4.00000 6.92820i −0.414781 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.00000 6.92820i −0.406138 0.703452i 0.588315 0.808632i \(-0.299792\pi\)
−0.994453 + 0.105180i \(0.966458\pi\)
\(98\) 0 0
\(99\) −2.50000 + 4.33013i −0.251259 + 0.435194i
\(100\) 0 0
\(101\) 5.00000 8.66025i 0.497519 0.861727i −0.502477 0.864590i \(-0.667578\pi\)
0.999996 + 0.00286291i \(0.000911295\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 3.00000 5.19615i 0.292770 0.507093i
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(110\) 0 0
\(111\) −10.0000 17.3205i −0.949158 1.64399i
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −2.00000 3.46410i −0.184900 0.320256i
\(118\) 0 0
\(119\) −4.50000 7.79423i −0.412514 0.714496i
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −10.0000 + 17.3205i −0.901670 + 1.56174i
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 3.00000 5.19615i 0.266207 0.461084i −0.701672 0.712500i \(-0.747563\pi\)
0.967879 + 0.251416i \(0.0808962\pi\)
\(128\) 0 0
\(129\) 1.00000 1.73205i 0.0880451 0.152499i
\(130\) 0 0
\(131\) 4.50000 + 7.79423i 0.393167 + 0.680985i 0.992865 0.119241i \(-0.0380462\pi\)
−0.599699 + 0.800226i \(0.704713\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.00000 + 3.46410i 0.172133 + 0.298142i
\(136\) 0 0
\(137\) 5.50000 9.52628i 0.469897 0.813885i −0.529511 0.848303i \(-0.677624\pi\)
0.999408 + 0.0344182i \(0.0109578\pi\)
\(138\) 0 0
\(139\) 1.50000 2.59808i 0.127228 0.220366i −0.795373 0.606120i \(-0.792725\pi\)
0.922602 + 0.385754i \(0.126059\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) −10.0000 + 17.3205i −0.836242 + 1.44841i
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 2.00000 + 3.46410i 0.164957 + 0.285714i
\(148\) 0 0
\(149\) 7.50000 + 12.9904i 0.614424 + 1.06421i 0.990485 + 0.137619i \(0.0439449\pi\)
−0.376061 + 0.926595i \(0.622722\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −2.00000 3.46410i −0.160644 0.278243i
\(156\) 0 0
\(157\) 1.00000 + 1.73205i 0.0798087 + 0.138233i 0.903167 0.429289i \(-0.141236\pi\)
−0.823359 + 0.567521i \(0.807902\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 12.0000 20.7846i 0.945732 1.63806i
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −5.00000 + 8.66025i −0.389249 + 0.674200i
\(166\) 0 0
\(167\) −3.00000 + 5.19615i −0.232147 + 0.402090i −0.958440 0.285295i \(-0.907908\pi\)
0.726293 + 0.687386i \(0.241242\pi\)
\(168\) 0 0
\(169\) −1.50000 2.59808i −0.115385 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.00000 + 5.19615i 0.228086 + 0.395056i 0.957241 0.289292i \(-0.0934200\pi\)
−0.729155 + 0.684349i \(0.760087\pi\)
\(174\) 0 0
\(175\) −6.00000 + 10.3923i −0.453557 + 0.785584i
\(176\) 0 0
\(177\) −6.00000 + 10.3923i −0.450988 + 0.781133i
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 5.00000 8.66025i 0.371647 0.643712i −0.618172 0.786043i \(-0.712126\pi\)
0.989819 + 0.142331i \(0.0454598\pi\)
\(182\) 0 0
\(183\) 26.0000 1.92198
\(184\) 0 0
\(185\) −5.00000 8.66025i −0.367607 0.636715i
\(186\) 0 0
\(187\) 7.50000 + 12.9904i 0.548454 + 0.949951i
\(188\) 0 0
\(189\) −12.0000 −0.872872
\(190\) 0 0
\(191\) 25.0000 1.80894 0.904468 0.426541i \(-0.140268\pi\)
0.904468 + 0.426541i \(0.140268\pi\)
\(192\) 0 0
\(193\) 6.00000 + 10.3923i 0.431889 + 0.748054i 0.997036 0.0769360i \(-0.0245137\pi\)
−0.565147 + 0.824991i \(0.691180\pi\)
\(194\) 0 0
\(195\) −4.00000 6.92820i −0.286446 0.496139i
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 3.50000 6.06218i 0.248108 0.429736i −0.714893 0.699234i \(-0.753524\pi\)
0.963001 + 0.269498i \(0.0868577\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) 3.00000 5.19615i 0.210559 0.364698i
\(204\) 0 0
\(205\) −5.00000 + 8.66025i −0.349215 + 0.604858i
\(206\) 0 0
\(207\) −4.00000 6.92820i −0.278019 0.481543i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.00000 + 15.5885i 0.619586 + 1.07315i 0.989561 + 0.144112i \(0.0460326\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(212\) 0 0
\(213\) −2.00000 + 3.46410i −0.137038 + 0.237356i
\(214\) 0 0
\(215\) 0.500000 0.866025i 0.0340997 0.0590624i
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) 9.00000 15.5885i 0.608164 1.05337i
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 1.00000 + 1.73205i 0.0669650 + 0.115987i 0.897564 0.440884i \(-0.145335\pi\)
−0.830599 + 0.556871i \(0.812002\pi\)
\(224\) 0 0
\(225\) 2.00000 + 3.46410i 0.133333 + 0.230940i
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) −15.0000 25.9808i −0.986928 1.70941i
\(232\) 0 0
\(233\) −1.50000 2.59808i −0.0982683 0.170206i 0.812700 0.582683i \(-0.197997\pi\)
−0.910968 + 0.412477i \(0.864664\pi\)
\(234\) 0 0
\(235\) 1.00000 0.0652328
\(236\) 0 0
\(237\) −8.00000 + 13.8564i −0.519656 + 0.900070i
\(238\) 0 0
\(239\) 21.0000 1.35838 0.679189 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) −13.0000 + 22.5167i −0.837404 + 1.45043i 0.0546547 + 0.998505i \(0.482594\pi\)
−0.892058 + 0.451920i \(0.850739\pi\)
\(242\) 0 0
\(243\) −5.00000 + 8.66025i −0.320750 + 0.555556i
\(244\) 0 0
\(245\) 1.00000 + 1.73205i 0.0638877 + 0.110657i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −12.0000 20.7846i −0.760469 1.31717i
\(250\) 0 0
\(251\) −5.50000 + 9.52628i −0.347157 + 0.601293i −0.985743 0.168257i \(-0.946186\pi\)
0.638586 + 0.769550i \(0.279520\pi\)
\(252\) 0 0
\(253\) −20.0000 + 34.6410i −1.25739 + 2.17786i
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) 16.0000 27.7128i 0.998053 1.72868i 0.445005 0.895528i \(-0.353202\pi\)
0.553047 0.833150i \(-0.313465\pi\)
\(258\) 0 0
\(259\) 30.0000 1.86411
\(260\) 0 0
\(261\) −1.00000 1.73205i −0.0618984 0.107211i
\(262\) 0 0
\(263\) 10.5000 + 18.1865i 0.647458 + 1.12143i 0.983728 + 0.179664i \(0.0575011\pi\)
−0.336270 + 0.941766i \(0.609166\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 24.0000 1.46878
\(268\) 0 0
\(269\) −12.0000 20.7846i −0.731653 1.26726i −0.956176 0.292791i \(-0.905416\pi\)
0.224523 0.974469i \(-0.427917\pi\)
\(270\) 0 0
\(271\) 4.00000 + 6.92820i 0.242983 + 0.420858i 0.961563 0.274586i \(-0.0885408\pi\)
−0.718580 + 0.695444i \(0.755208\pi\)
\(272\) 0 0
\(273\) 24.0000 1.45255
\(274\) 0 0
\(275\) 10.0000 17.3205i 0.603023 1.04447i
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 0 0
\(279\) 2.00000 3.46410i 0.119737 0.207390i
\(280\) 0 0
\(281\) 11.0000 19.0526i 0.656205 1.13658i −0.325385 0.945582i \(-0.605494\pi\)
0.981590 0.190999i \(-0.0611727\pi\)
\(282\) 0 0
\(283\) 1.50000 + 2.59808i 0.0891657 + 0.154440i 0.907159 0.420789i \(-0.138247\pi\)
−0.817993 + 0.575228i \(0.804913\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −15.0000 25.9808i −0.885422 1.53360i
\(288\) 0 0
\(289\) 4.00000 6.92820i 0.235294 0.407541i
\(290\) 0 0
\(291\) 8.00000 13.8564i 0.468968 0.812277i
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −3.00000 + 5.19615i −0.174667 + 0.302532i
\(296\) 0 0
\(297\) 20.0000 1.16052
\(298\) 0 0
\(299\) −16.0000 27.7128i −0.925304 1.60267i
\(300\) 0 0
\(301\) 1.50000 + 2.59808i 0.0864586 + 0.149751i
\(302\) 0 0
\(303\) 20.0000 1.14897
\(304\) 0 0
\(305\) 13.0000 0.744378
\(306\) 0 0
\(307\) 6.00000 + 10.3923i 0.342438 + 0.593120i 0.984885 0.173210i \(-0.0554140\pi\)
−0.642447 + 0.766330i \(0.722081\pi\)
\(308\) 0 0
\(309\) 6.00000 + 10.3923i 0.341328 + 0.591198i
\(310\) 0 0
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) 0 0
\(313\) 5.00000 8.66025i 0.282617 0.489506i −0.689412 0.724370i \(-0.742131\pi\)
0.972028 + 0.234863i \(0.0754642\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) 15.0000 25.9808i 0.842484 1.45922i −0.0453045 0.998973i \(-0.514426\pi\)
0.887788 0.460252i \(-0.152241\pi\)
\(318\) 0 0
\(319\) −5.00000 + 8.66025i −0.279946 + 0.484881i
\(320\) 0 0
\(321\) −2.00000 3.46410i −0.111629 0.193347i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 8.00000 + 13.8564i 0.443760 + 0.768615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.50000 + 2.59808i −0.0826977 + 0.143237i
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 5.00000 8.66025i 0.273998 0.474579i
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −16.0000 27.7128i −0.871576 1.50961i −0.860366 0.509676i \(-0.829765\pi\)
−0.0112091 0.999937i \(-0.503568\pi\)
\(338\) 0 0
\(339\) 10.0000 + 17.3205i 0.543125 + 0.940721i
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −8.00000 13.8564i −0.430706 0.746004i
\(346\) 0 0
\(347\) −9.50000 16.4545i −0.509987 0.883323i −0.999933 0.0115703i \(-0.996317\pi\)
0.489946 0.871753i \(-0.337016\pi\)
\(348\) 0 0
\(349\) −11.0000 −0.588817 −0.294408 0.955680i \(-0.595123\pi\)
−0.294408 + 0.955680i \(0.595123\pi\)
\(350\) 0 0
\(351\) −8.00000 + 13.8564i −0.427008 + 0.739600i
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −1.00000 + 1.73205i −0.0530745 + 0.0919277i
\(356\) 0 0
\(357\) 9.00000 15.5885i 0.476331 0.825029i
\(358\) 0 0
\(359\) −10.5000 18.1865i −0.554169 0.959849i −0.997968 0.0637221i \(-0.979703\pi\)
0.443799 0.896126i \(-0.353630\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 14.0000 + 24.2487i 0.734809 + 1.27273i
\(364\) 0 0
\(365\) 4.50000 7.79423i 0.235541 0.407969i
\(366\) 0 0
\(367\) −8.00000 + 13.8564i −0.417597 + 0.723299i −0.995697 0.0926670i \(-0.970461\pi\)
0.578101 + 0.815966i \(0.303794\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 6.00000 10.3923i 0.311504 0.539542i
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 9.00000 + 15.5885i 0.464758 + 0.804984i
\(376\) 0 0
\(377\) −4.00000 6.92820i −0.206010 0.356821i
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) 2.00000 + 3.46410i 0.102195 + 0.177007i 0.912589 0.408879i \(-0.134080\pi\)
−0.810394 + 0.585886i \(0.800747\pi\)
\(384\) 0 0
\(385\) −7.50000 12.9904i −0.382235 0.662051i
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) 10.5000 18.1865i 0.532371 0.922094i −0.466915 0.884302i \(-0.654634\pi\)
0.999286 0.0377914i \(-0.0120322\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) −9.00000 + 15.5885i −0.453990 + 0.786334i
\(394\) 0 0
\(395\) −4.00000 + 6.92820i −0.201262 + 0.348596i
\(396\) 0 0
\(397\) −2.50000 4.33013i −0.125471 0.217323i 0.796446 0.604710i \(-0.206711\pi\)
−0.921917 + 0.387387i \(0.873378\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.0000 + 24.2487i 0.699127 + 1.21092i 0.968770 + 0.247962i \(0.0797610\pi\)
−0.269643 + 0.962960i \(0.586906\pi\)
\(402\) 0 0
\(403\) 8.00000 13.8564i 0.398508 0.690237i
\(404\) 0 0
\(405\) −5.50000 + 9.52628i −0.273297 + 0.473365i
\(406\) 0 0
\(407\) −50.0000 −2.47841
\(408\) 0 0
\(409\) −10.0000 + 17.3205i −0.494468 + 0.856444i −0.999980 0.00637586i \(-0.997970\pi\)
0.505511 + 0.862820i \(0.331304\pi\)
\(410\) 0 0
\(411\) 22.0000 1.08518
\(412\) 0 0
\(413\) −9.00000 15.5885i −0.442861 0.767058i
\(414\) 0 0
\(415\) −6.00000 10.3923i −0.294528 0.510138i
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −20.0000 34.6410i −0.974740 1.68830i −0.680789 0.732479i \(-0.738363\pi\)
−0.293951 0.955820i \(-0.594970\pi\)
\(422\) 0 0
\(423\) 0.500000 + 0.866025i 0.0243108 + 0.0421076i
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −19.5000 + 33.7750i −0.943671 + 1.63449i
\(428\) 0 0
\(429\) −40.0000 −1.93122
\(430\) 0 0
\(431\) 12.0000 20.7846i 0.578020 1.00116i −0.417687 0.908591i \(-0.637159\pi\)
0.995706 0.0925683i \(-0.0295076\pi\)
\(432\) 0 0
\(433\) 1.00000 1.73205i 0.0480569 0.0832370i −0.840996 0.541041i \(-0.818030\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) −2.00000 3.46410i −0.0958927 0.166091i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.00000 + 1.73205i 0.0477274 + 0.0826663i 0.888902 0.458097i \(-0.151469\pi\)
−0.841175 + 0.540763i \(0.818135\pi\)
\(440\) 0 0
\(441\) −1.00000 + 1.73205i −0.0476190 + 0.0824786i
\(442\) 0 0
\(443\) 2.50000 4.33013i 0.118779 0.205731i −0.800505 0.599326i \(-0.795435\pi\)
0.919284 + 0.393595i \(0.128769\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) −15.0000 + 25.9808i −0.709476 + 1.22885i
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 25.0000 + 43.3013i 1.17720 + 2.03898i
\(452\) 0 0
\(453\) −2.00000 3.46410i −0.0939682 0.162758i
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) 0 0
\(459\) 6.00000 + 10.3923i 0.280056 + 0.485071i
\(460\) 0 0
\(461\) 9.50000 + 16.4545i 0.442459 + 0.766362i 0.997871 0.0652135i \(-0.0207728\pi\)
−0.555412 + 0.831575i \(0.687440\pi\)
\(462\) 0 0
\(463\) 19.0000 0.883005 0.441502 0.897260i \(-0.354446\pi\)
0.441502 + 0.897260i \(0.354446\pi\)
\(464\) 0 0
\(465\) 4.00000 6.92820i 0.185496 0.321288i
\(466\) 0 0
\(467\) −5.00000 −0.231372 −0.115686 0.993286i \(-0.536907\pi\)
−0.115686 + 0.993286i \(0.536907\pi\)
\(468\) 0 0
\(469\) 18.0000 31.1769i 0.831163 1.43962i
\(470\) 0 0
\(471\) −2.00000 + 3.46410i −0.0921551 + 0.159617i
\(472\) 0 0
\(473\) −2.50000 4.33013i −0.114950 0.199099i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 3.46410i −0.0915737 0.158610i
\(478\) 0 0
\(479\) −2.00000 + 3.46410i −0.0913823 + 0.158279i −0.908093 0.418769i \(-0.862462\pi\)
0.816711 + 0.577047i \(0.195795\pi\)
\(480\) 0 0
\(481\) 20.0000 34.6410i 0.911922 1.57949i
\(482\) 0 0
\(483\) 48.0000 2.18408
\(484\) 0 0
\(485\) 4.00000 6.92820i 0.181631 0.314594i
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) −4.00000 6.92820i −0.180886 0.313304i
\(490\) 0 0
\(491\) 14.0000 + 24.2487i 0.631811 + 1.09433i 0.987181 + 0.159603i \(0.0510215\pi\)
−0.355370 + 0.934726i \(0.615645\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) −3.00000 5.19615i −0.134568 0.233079i
\(498\) 0 0
\(499\) −9.50000 16.4545i −0.425278 0.736604i 0.571168 0.820833i \(-0.306490\pi\)
−0.996446 + 0.0842294i \(0.973157\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 18.0000 31.1769i 0.802580 1.39011i −0.115332 0.993327i \(-0.536793\pi\)
0.917912 0.396783i \(-0.129873\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 3.00000 5.19615i 0.133235 0.230769i
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 13.5000 + 23.3827i 0.597205 + 1.03439i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.00000 + 5.19615i 0.132196 + 0.228970i
\(516\) 0 0
\(517\) 2.50000 4.33013i 0.109950 0.190439i
\(518\) 0 0
\(519\) −6.00000 + 10.3923i −0.263371 + 0.456172i
\(520\) 0 0
\(521\) −32.0000 −1.40195 −0.700973 0.713188i \(-0.747251\pi\)
−0.700973 + 0.713188i \(0.747251\pi\)
\(522\) 0 0
\(523\) 13.0000 22.5167i 0.568450 0.984585i −0.428269 0.903651i \(-0.640876\pi\)
0.996719 0.0809336i \(-0.0257902\pi\)
\(524\) 0 0
\(525\) −24.0000 −1.04745
\(526\) 0 0
\(527\) −6.00000 10.3923i −0.261364 0.452696i
\(528\) 0 0
\(529\) −20.5000 35.5070i −0.891304 1.54378i
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) −40.0000 −1.73259
\(534\) 0 0
\(535\) −1.00000 1.73205i −0.0432338 0.0748831i
\(536\) 0 0
\(537\) −18.0000 31.1769i −0.776757 1.34538i
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) −5.50000 + 9.52628i −0.236463 + 0.409567i −0.959697 0.281037i \(-0.909322\pi\)
0.723234 + 0.690604i \(0.242655\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.0000 24.2487i 0.598597 1.03680i −0.394432 0.918925i \(-0.629059\pi\)
0.993028 0.117875i \(-0.0376081\pi\)
\(548\) 0 0
\(549\) 6.50000 + 11.2583i 0.277413 + 0.480494i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −12.0000 20.7846i −0.510292 0.883852i
\(554\) 0 0
\(555\) 10.0000 17.3205i 0.424476 0.735215i
\(556\) 0 0
\(557\) −0.500000 + 0.866025i −0.0211857 + 0.0366947i −0.876424 0.481540i \(-0.840077\pi\)
0.855238 + 0.518235i \(0.173411\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −15.0000 + 25.9808i −0.633300 + 1.09691i
\(562\) 0 0
\(563\) −42.0000 −1.77009 −0.885044 0.465506i \(-0.845872\pi\)
−0.885044 + 0.465506i \(0.845872\pi\)
\(564\) 0 0
\(565\) 5.00000 + 8.66025i 0.210352 + 0.364340i
\(566\) 0 0
\(567\) −16.5000 28.5788i −0.692935 1.20020i
\(568\) 0 0
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 25.0000 + 43.3013i 1.04439 + 1.80894i
\(574\) 0 0
\(575\) 16.0000 + 27.7128i 0.667246 + 1.15570i
\(576\) 0 0
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 0 0
\(579\) −12.0000 + 20.7846i −0.498703 + 0.863779i
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) −10.0000 + 17.3205i −0.414158 + 0.717342i
\(584\) 0 0
\(585\) 2.00000 3.46410i 0.0826898 0.143223i
\(586\) 0 0
\(587\) −1.50000 2.59808i −0.0619116 0.107234i 0.833408 0.552658i \(-0.186386\pi\)
−0.895320 + 0.445424i \(0.853053\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 + 3.46410i 0.0822690 + 0.142494i
\(592\) 0 0
\(593\) −11.0000 + 19.0526i −0.451716 + 0.782395i −0.998493 0.0548835i \(-0.982521\pi\)
0.546777 + 0.837278i \(0.315855\pi\)
\(594\) 0 0
\(595\) 4.50000 7.79423i 0.184482 0.319532i
\(596\) 0 0
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) −6.00000 + 10.3923i −0.245153 + 0.424618i −0.962175 0.272433i \(-0.912172\pi\)
0.717021 + 0.697051i \(0.245505\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −6.00000 10.3923i −0.244339 0.423207i
\(604\) 0 0
\(605\) 7.00000 + 12.1244i 0.284590 + 0.492925i
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 2.00000 + 3.46410i 0.0809113 + 0.140143i
\(612\) 0 0
\(613\) 23.5000 + 40.7032i 0.949156 + 1.64399i 0.747208 + 0.664590i \(0.231394\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) −4.50000 + 7.79423i −0.181163 + 0.313784i −0.942277 0.334835i \(-0.891320\pi\)
0.761114 + 0.648618i \(0.224653\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −16.0000 + 27.7128i −0.642058 + 1.11208i
\(622\) 0 0
\(623\) −18.0000 + 31.1769i −0.721155 + 1.24908i
\(624\) 0 0
\(625\) −5.50000 9.52628i −0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −15.0000 25.9808i −0.598089 1.03592i
\(630\) 0 0
\(631\) 3.50000 6.06218i 0.139333 0.241331i −0.787911 0.615789i \(-0.788838\pi\)
0.927244 + 0.374457i \(0.122171\pi\)
\(632\) 0 0
\(633\) −18.0000 + 31.1769i −0.715436 + 1.23917i
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) 0 0
\(637\) −4.00000 + 6.92820i −0.158486 + 0.274505i
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 5.50000 + 9.52628i 0.216899 + 0.375680i 0.953858 0.300257i \(-0.0970725\pi\)
−0.736959 + 0.675937i \(0.763739\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −7.00000 −0.275198 −0.137599 0.990488i \(-0.543939\pi\)
−0.137599 + 0.990488i \(0.543939\pi\)
\(648\) 0 0
\(649\) 15.0000 + 25.9808i 0.588802 + 1.01983i
\(650\) 0 0
\(651\) 12.0000 + 20.7846i 0.470317 + 0.814613i
\(652\) 0 0
\(653\) −27.0000 −1.05659 −0.528296 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) −4.50000 + 7.79423i −0.175830 + 0.304546i
\(656\) 0 0
\(657\) 9.00000 0.351123
\(658\) 0 0
\(659\) −17.0000 + 29.4449i −0.662226 + 1.14701i 0.317803 + 0.948157i \(0.397055\pi\)
−0.980029 + 0.198852i \(0.936279\pi\)
\(660\) 0 0
\(661\) 8.00000 13.8564i 0.311164 0.538952i −0.667451 0.744654i \(-0.732615\pi\)
0.978615 + 0.205702i \(0.0659478\pi\)
\(662\) 0 0
\(663\) −12.0000 20.7846i −0.466041 0.807207i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 13.8564i −0.309761 0.536522i
\(668\) 0 0
\(669\) −2.00000 + 3.46410i −0.0773245 + 0.133930i
\(670\) 0 0
\(671\) 32.5000 56.2917i 1.25465 2.17312i
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 8.00000 13.8564i 0.307920 0.533333i
\(676\) 0 0
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 12.0000 + 20.7846i 0.460518 + 0.797640i
\(680\) 0 0
\(681\) −4.00000 6.92820i −0.153280 0.265489i
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) 11.0000 0.420288
\(686\) 0 0
\(687\) 17.0000 + 29.4449i 0.648590 + 1.12339i
\(688\) 0 0
\(689\) −8.00000 13.8564i −0.304776 0.527887i
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 0 0
\(693\) 7.50000 12.9904i 0.284901 0.493464i
\(694\) 0 0
\(695\) 3.00000 0.113796
\(696\) 0 0
\(697\) −15.0000 + 25.9808i −0.568166 + 0.984092i
\(698\) 0 0
\(699\) 3.00000 5.19615i 0.113470 0.196537i
\(700\) 0 0
\(701\) 5.00000 + 8.66025i 0.188847 + 0.327093i 0.944866 0.327457i \(-0.106192\pi\)
−0.756019 + 0.654550i \(0.772858\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 1.00000 + 1.73205i 0.0376622 + 0.0652328i
\(706\) 0 0
\(707\) −15.0000 + 25.9808i −0.564133 + 0.977107i
\(708\) 0 0
\(709\) 3.00000 5.19615i 0.112667 0.195146i −0.804178 0.594389i \(-0.797394\pi\)
0.916845 + 0.399244i \(0.130727\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 16.0000 27.7128i 0.599205 1.03785i
\(714\) 0 0
\(715\) −20.0000 −0.747958
\(716\) 0 0
\(717\) 21.0000 + 36.3731i 0.784259 + 1.35838i
\(718\) 0 0
\(719\) −6.50000 11.2583i −0.242409 0.419865i 0.718991 0.695019i \(-0.244604\pi\)
−0.961400 + 0.275155i \(0.911271\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) −52.0000 −1.93390
\(724\) 0 0
\(725\) 4.00000 + 6.92820i 0.148556 + 0.257307i
\(726\) 0 0
\(727\) −3.50000 6.06218i −0.129808 0.224834i 0.793794 0.608186i \(-0.208103\pi\)
−0.923602 + 0.383353i \(0.874769\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 1.50000 2.59808i 0.0554795 0.0960933i
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) −2.00000 + 3.46410i −0.0737711 + 0.127775i
\(736\) 0 0
\(737\) −30.0000 + 51.9615i −1.10506 + 1.91403i
\(738\) 0 0
\(739\) 21.5000 + 37.2391i 0.790890 + 1.36986i 0.925416 + 0.378952i \(0.123715\pi\)
−0.134526 + 0.990910i \(0.542951\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20.0000 34.6410i −0.733729 1.27086i −0.955279 0.295707i \(-0.904445\pi\)
0.221550 0.975149i \(-0.428888\pi\)
\(744\) 0 0
\(745\) −7.50000 + 12.9904i −0.274779 + 0.475931i
\(746\) 0 0
\(747\) 6.00000 10.3923i 0.219529 0.380235i
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −8.00000 + 13.8564i −0.291924 + 0.505627i −0.974265 0.225407i \(-0.927629\pi\)
0.682341 + 0.731034i \(0.260962\pi\)
\(752\) 0 0
\(753\) −22.0000 −0.801725
\(754\) 0 0
\(755\) −1.00000 1.73205i −0.0363937 0.0630358i
\(756\) 0 0
\(757\) 2.50000 + 4.33013i 0.0908640 + 0.157381i 0.907875 0.419241i \(-0.137704\pi\)
−0.817011 + 0.576622i \(0.804370\pi\)
\(758\) 0 0
\(759\) −80.0000 −2.90382
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.50000 2.59808i −0.0542326 0.0939336i
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −15.5000 + 26.8468i −0.558944 + 0.968120i 0.438641 + 0.898663i \(0.355460\pi\)
−0.997585 + 0.0694574i \(0.977873\pi\)
\(770\) 0 0
\(771\) 64.0000 2.30490
\(772\) 0 0
\(773\) 9.00000 15.5885i 0.323708 0.560678i −0.657542 0.753418i \(-0.728404\pi\)
0.981250 + 0.192740i \(0.0617373\pi\)
\(774\) 0 0
\(775\) −8.00000 + 13.8564i −0.287368 + 0.497737i
\(776\) 0 0
\(777\) 30.0000 + 51.9615i 1.07624 + 1.86411i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 5.00000 + 8.66025i 0.178914 + 0.309888i
\(782\) 0 0
\(783\) −4.00000 + 6.92820i −0.142948 + 0.247594i
\(784\) 0 0
\(785\) −1.00000 + 1.73205i −0.0356915 + 0.0618195i
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) −21.0000 + 36.3731i −0.747620 + 1.29492i
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 26.0000 + 45.0333i 0.923287 + 1.59918i
\(794\) 0 0
\(795\) −4.00000 6.92820i −0.141865 0.245718i
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 3.00000 0.106132
\(800\) 0 0
\(801\) 6.00000 + 10.3923i 0.212000 + 0.367194i
\(802\) 0 0
\(803\) −22.5000 38.9711i −0.794008 1.37526i
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 24.0000 41.5692i 0.844840 1.46331i
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(812\) 0 0
\(813\) −8.00000 + 13.8564i −0.280572 + 0.485965i
\(814\) 0 0
\(815\) −2.00000 3.46410i −0.0700569 0.121342i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 6.00000 + 10.3923i 0.209657 + 0.363137i
\(820\) 0 0
\(821\) −22.5000 + 38.9711i −0.785255 + 1.36010i 0.143591 + 0.989637i \(0.454135\pi\)
−0.928846 + 0.370465i \(0.879198\pi\)
\(822\) 0 0
\(823\) −26.5000 + 45.8993i −0.923732 + 1.59995i −0.130144 + 0.991495i \(0.541544\pi\)
−0.793588 + 0.608456i \(0.791789\pi\)
\(824\) 0 0
\(825\) 40.0000 1.39262
\(826\) 0 0
\(827\) −14.0000 + 24.2487i −0.486828 + 0.843210i −0.999885 0.0151439i \(-0.995179\pi\)
0.513058 + 0.858354i \(0.328513\pi\)
\(828\) 0 0
\(829\) 48.0000 1.66711 0.833554 0.552437i \(-0.186302\pi\)
0.833554 + 0.552437i \(0.186302\pi\)
\(830\) 0 0
\(831\) 1.00000 + 1.73205i 0.0346896 + 0.0600842i
\(832\) 0 0
\(833\) 3.00000 + 5.19615i 0.103944 + 0.180036i
\(834\) 0 0
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −5.00000 8.66025i −0.172619 0.298985i 0.766716 0.641987i \(-0.221890\pi\)
−0.939335 + 0.343002i \(0.888556\pi\)
\(840\) 0 0
\(841\) 12.5000 + 21.6506i 0.431034 + 0.746574i
\(842\) 0 0
\(843\) 44.0000 1.51544
\(844\) 0 0
\(845\) 1.50000 2.59808i 0.0516016 0.0893765i
\(846\) 0 0
\(847\) −42.0000 −1.44314
\(848\) 0 0
\(849\) −3.00000 + 5.19615i −0.102960 + 0.178331i
\(850\) 0 0
\(851\) 40.0000 69.2820i 1.37118 2.37496i
\(852\) 0 0
\(853\) −21.0000 36.3731i −0.719026 1.24539i −0.961386 0.275204i \(-0.911255\pi\)
0.242360 0.970186i \(-0.422079\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.00000 + 15.5885i 0.307434 + 0.532492i 0.977800 0.209539i \(-0.0671963\pi\)
−0.670366 + 0.742030i \(0.733863\pi\)
\(858\) 0 0
\(859\) −0.500000 + 0.866025i −0.0170598 + 0.0295484i −0.874429 0.485153i \(-0.838764\pi\)
0.857369 + 0.514701i \(0.172097\pi\)
\(860\) 0 0
\(861\) 30.0000 51.9615i 1.02240 1.77084i
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −3.00000 + 5.19615i −0.102003 + 0.176674i
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 20.0000 + 34.6410i 0.678454 + 1.17512i
\(870\) 0 0
\(871\) −24.0000 41.5692i −0.813209 1.40852i
\(872\) 0 0
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −27.0000 −0.912767
\(876\) 0 0
\(877\) 17.0000 + 29.4449i 0.574049 + 0.994282i 0.996144 + 0.0877308i \(0.0279615\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) 0 0
\(879\) 12.0000 + 20.7846i 0.404750 + 0.701047i
\(880\) 0 0
\(881\) −51.0000 −1.71823 −0.859117 0.511780i \(-0.828986\pi\)
−0.859117 + 0.511780i \(0.828986\pi\)
\(882\) 0 0
\(883\) −0.500000 + 0.866025i −0.0168263 + 0.0291441i −0.874316 0.485357i \(-0.838690\pi\)
0.857490 + 0.514501i \(0.172023\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 11.0000 19.0526i 0.369344 0.639722i −0.620119 0.784508i \(-0.712916\pi\)
0.989463 + 0.144785i \(0.0462491\pi\)
\(888\) 0 0
\(889\) −9.00000 + 15.5885i −0.301850 + 0.522820i
\(890\) 0 0
\(891\) 27.5000 + 47.6314i 0.921285 + 1.59571i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −9.00000 15.5885i −0.300837 0.521065i
\(896\) 0 0
\(897\) 32.0000 55.4256i 1.06845 1.85061i
\(898\) 0 0
\(899\) 4.00000 6.92820i 0.133407 0.231069i
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −3.00000 + 5.19615i −0.0998337 + 0.172917i
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −20.0000 34.6410i −0.664089 1.15024i −0.979531 0.201291i \(-0.935486\pi\)
0.315442 0.948945i \(-0.397847\pi\)
\(908\) 0 0
\(909\) 5.00000 + 8.66025i 0.165840 + 0.287242i
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 13.0000 + 22.5167i 0.429767 + 0.744378i
\(916\) 0 0
\(917\) −13.5000 23.3827i −0.445809 0.772164i
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) 0 0
\(921\) −12.0000 + 20.7846i −0.395413 + 0.684876i
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −20.0000 + 34.6410i −0.657596 + 1.13899i
\(926\) 0 0
\(927\) −3.00000 + 5.19615i −0.0985329 + 0.170664i
\(928\) 0 0
\(929\) −7.00000 12.1244i −0.229663 0.397787i 0.728046 0.685529i \(-0.240429\pi\)
−0.957708 + 0.287742i \(0.907096\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 7.00000 + 12.1244i 0.229170 + 0.396934i
\(934\) 0 0
\(935\) −7.50000 + 12.9904i −0.245276 + 0.424831i
\(936\) 0 0
\(937\) 15.5000 26.8468i 0.506363 0.877046i −0.493610 0.869683i \(-0.664323\pi\)
0.999973 0.00736293i \(-0.00234371\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −13.0000 + 22.5167i −0.423788 + 0.734022i −0.996306 0.0858697i \(-0.972633\pi\)
0.572518 + 0.819892i \(0.305966\pi\)
\(942\) 0 0
\(943\) −80.0000 −2.60516
\(944\) 0 0
\(945\) −6.00000 10.3923i −0.195180 0.338062i
\(946\) 0 0
\(947\) 6.00000 + 10.3923i 0.194974 + 0.337705i 0.946892 0.321552i \(-0.104204\pi\)
−0.751918 + 0.659256i \(0.770871\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 60.0000 1.94563
\(952\) 0 0
\(953\) −8.00000 13.8564i −0.259145 0.448853i 0.706868 0.707346i \(-0.250108\pi\)
−0.966013 + 0.258493i \(0.916774\pi\)
\(954\) 0 0
\(955\) 12.5000 + 21.6506i 0.404491 + 0.700598i
\(956\) 0 0
\(957\) −20.0000 −0.646508
\(958\) 0 0
\(959\) −16.5000 + 28.5788i −0.532813 + 0.922859i
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 1.00000 1.73205i 0.0322245 0.0558146i
\(964\) 0 0
\(965\) −6.00000 + 10.3923i −0.193147 + 0.334540i
\(966\) 0 0
\(967\) −24.0000 41.5692i −0.771788 1.33678i −0.936582 0.350448i \(-0.886029\pi\)
0.164794 0.986328i \(-0.447304\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.0000 + 17.3205i 0.320915 + 0.555842i 0.980677 0.195633i \(-0.0626762\pi\)
−0.659762 + 0.751475i \(0.729343\pi\)
\(972\) 0 0
\(973\) −4.50000 + 7.79423i −0.144263 + 0.249871i
\(974\) 0 0
\(975\) −16.0000 + 27.7128i −0.512410 + 0.887520i
\(976\) 0 0
\(977\) −56.0000 −1.79160 −0.895799 0.444459i \(-0.853396\pi\)
−0.895799 + 0.444459i \(0.853396\pi\)
\(978\) 0 0
\(979\) 30.0000 51.9615i 0.958804 1.66070i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −6.00000 10.3923i −0.191370 0.331463i 0.754334 0.656490i \(-0.227960\pi\)
−0.945705 + 0.325027i \(0.894626\pi\)
\(984\) 0 0
\(985\) 1.00000 + 1.73205i 0.0318626 + 0.0551877i
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −19.0000 32.9090i −0.603555 1.04539i −0.992278 0.124033i \(-0.960417\pi\)
0.388723 0.921355i \(-0.372916\pi\)
\(992\) 0 0
\(993\) 4.00000 + 6.92820i 0.126936 + 0.219860i
\(994\) 0 0
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) −18.5000 + 32.0429i −0.585901 + 1.01481i 0.408862 + 0.912596i \(0.365926\pi\)
−0.994762 + 0.102214i \(0.967407\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 1444.2.e.c.653.1 2
19.7 even 3 1444.2.a.a.1.1 1
19.8 odd 6 1444.2.e.a.429.1 2
19.11 even 3 inner 1444.2.e.c.429.1 2
19.12 odd 6 76.2.a.a.1.1 1
19.18 odd 2 1444.2.e.a.653.1 2
57.50 even 6 684.2.a.b.1.1 1
76.7 odd 6 5776.2.a.p.1.1 1
76.31 even 6 304.2.a.a.1.1 1
95.12 even 12 1900.2.c.b.1749.1 2
95.69 odd 6 1900.2.a.b.1.1 1
95.88 even 12 1900.2.c.b.1749.2 2
133.69 even 6 3724.2.a.a.1.1 1
152.69 odd 6 1216.2.a.c.1.1 1
152.107 even 6 1216.2.a.q.1.1 1
209.164 even 6 9196.2.a.f.1.1 1
228.107 odd 6 2736.2.a.q.1.1 1
380.259 even 6 7600.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.a.a.1.1 1 19.12 odd 6
304.2.a.a.1.1 1 76.31 even 6
684.2.a.b.1.1 1 57.50 even 6
1216.2.a.c.1.1 1 152.69 odd 6
1216.2.a.q.1.1 1 152.107 even 6
1444.2.a.a.1.1 1 19.7 even 3
1444.2.e.a.429.1 2 19.8 odd 6
1444.2.e.a.653.1 2 19.18 odd 2
1444.2.e.c.429.1 2 19.11 even 3 inner
1444.2.e.c.653.1 2 1.1 even 1 trivial
1900.2.a.b.1.1 1 95.69 odd 6
1900.2.c.b.1749.1 2 95.12 even 12
1900.2.c.b.1749.2 2 95.88 even 12
2736.2.a.q.1.1 1 228.107 odd 6
3724.2.a.a.1.1 1 133.69 even 6
5776.2.a.p.1.1 1 76.7 odd 6
7600.2.a.p.1.1 1 380.259 even 6
9196.2.a.f.1.1 1 209.164 even 6