Properties

Label 1216.2.s.g.31.5
Level $1216$
Weight $2$
Character 1216.31
Analytic conductor $9.710$
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] = [1216,2,Mod(31,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.s (of order \(6\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} - 18 x^{9} + 11 x^{8} - 36 x^{7} + 180 x^{6} - 120 x^{5} + 31 x^{4} + 18 x^{3} + 162 x^{2} - 54 x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.5
Root \(0.163156 - 0.163156i\) of defining polynomial
Character \(\chi\) \(=\) 1216.31
Dual form 1216.2.s.g.863.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.96854 - 1.13654i) q^{3} +(-1.50000 + 0.866025i) q^{5} +2.00000i q^{7} +(1.08343 - 1.87656i) q^{9} +O(q^{10})\) \(q+(1.96854 - 1.13654i) q^{3} +(-1.50000 + 0.866025i) q^{5} +2.00000i q^{7} +(1.08343 - 1.87656i) q^{9} +(-2.85650 + 4.94761i) q^{13} +(-1.96854 + 3.40961i) q^{15} +(-0.583430 - 1.01053i) q^{17} +(-0.144505 + 4.35650i) q^{19} +(2.27307 + 3.93708i) q^{21} +(4.31807 + 2.49304i) q^{23} +(-1.00000 + 1.73205i) q^{25} +1.89379i q^{27} +(0.583430 - 1.01053i) q^{29} +0.289010 q^{31} +(-1.73205 - 3.00000i) q^{35} +2.71301 q^{37} +12.9861i q^{39} +(8.56951 - 4.94761i) q^{41} +(0.380993 + 0.659900i) q^{43} +3.75311i q^{45} +(-5.90562 - 3.40961i) q^{47} +3.00000 q^{49} +(-2.29701 - 1.32618i) q^{51} +(-0.333140 + 0.577015i) q^{53} +(4.66686 + 8.74018i) q^{57} +(-9.51422 + 5.49304i) q^{59} +(-5.31922 - 3.07105i) q^{61} +(3.75311 + 2.16686i) q^{63} -9.89522i q^{65} +(-9.18575 - 5.30340i) q^{67} +11.3337 q^{69} +(5.57715 + 9.65990i) q^{71} +(2.50000 + 4.33013i) q^{73} +4.54615i q^{75} +(-3.84509 - 6.65990i) q^{79} +(5.40265 + 9.35766i) q^{81} +0.289010 q^{83} +(1.75029 + 1.01053i) q^{85} -2.65236i q^{87} +(14.0689 + 8.12270i) q^{89} +(-9.89522 - 5.71301i) q^{91} +(0.568928 - 0.328471i) q^{93} +(-3.55608 - 6.65990i) q^{95} +(8.56951 - 4.94761i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{5} + 12 q^{9} - 2 q^{13} - 6 q^{17} - 4 q^{21} - 12 q^{25} + 6 q^{29} - 32 q^{37} + 6 q^{41} + 36 q^{49} - 6 q^{53} + 54 q^{57} + 30 q^{61} + 132 q^{69} + 30 q^{73} - 30 q^{81} + 18 q^{85} + 78 q^{89} - 84 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) 1.96854 1.13654i 1.13654 0.656180i 0.190966 0.981597i \(-0.438838\pi\)
0.945571 + 0.325417i \(0.105505\pi\)
\(4\) 0 0
\(5\) −1.50000 + 0.866025i −0.670820 + 0.387298i −0.796387 0.604787i \(-0.793258\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 1.08343 1.87656i 0.361143 0.625519i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.85650 + 4.94761i −0.792251 + 1.37222i 0.132319 + 0.991207i \(0.457758\pi\)
−0.924570 + 0.381012i \(0.875576\pi\)
\(14\) 0 0
\(15\) −1.96854 + 3.40961i −0.508275 + 0.880357i
\(16\) 0 0
\(17\) −0.583430 1.01053i −0.141503 0.245090i 0.786560 0.617514i \(-0.211860\pi\)
−0.928063 + 0.372424i \(0.878527\pi\)
\(18\) 0 0
\(19\) −0.144505 + 4.35650i −0.0331518 + 0.999450i
\(20\) 0 0
\(21\) 2.27307 + 3.93708i 0.496025 + 0.859141i
\(22\) 0 0
\(23\) 4.31807 + 2.49304i 0.900380 + 0.519835i 0.877323 0.479900i \(-0.159327\pi\)
0.0230566 + 0.999734i \(0.492660\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 1.89379i 0.364460i
\(28\) 0 0
\(29\) 0.583430 1.01053i 0.108340 0.187651i −0.806758 0.590882i \(-0.798780\pi\)
0.915098 + 0.403232i \(0.132113\pi\)
\(30\) 0 0
\(31\) 0.289010 0.0519078 0.0259539 0.999663i \(-0.491738\pi\)
0.0259539 + 0.999663i \(0.491738\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.73205 3.00000i −0.292770 0.507093i
\(36\) 0 0
\(37\) 2.71301 0.446015 0.223008 0.974817i \(-0.428412\pi\)
0.223008 + 0.974817i \(0.428412\pi\)
\(38\) 0 0
\(39\) 12.9861i 2.07944i
\(40\) 0 0
\(41\) 8.56951 4.94761i 1.33833 0.772687i 0.351772 0.936086i \(-0.385579\pi\)
0.986560 + 0.163399i \(0.0522458\pi\)
\(42\) 0 0
\(43\) 0.380993 + 0.659900i 0.0581009 + 0.100634i 0.893613 0.448838i \(-0.148162\pi\)
−0.835512 + 0.549472i \(0.814829\pi\)
\(44\) 0 0
\(45\) 3.75311i 0.559481i
\(46\) 0 0
\(47\) −5.90562 3.40961i −0.861423 0.497343i 0.00306575 0.999995i \(-0.499024\pi\)
−0.864488 + 0.502653i \(0.832357\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.29701 1.32618i −0.321646 0.185702i
\(52\) 0 0
\(53\) −0.333140 + 0.577015i −0.0457603 + 0.0792591i −0.887998 0.459847i \(-0.847904\pi\)
0.842238 + 0.539106i \(0.181238\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.66686 + 8.74018i 0.618141 + 1.15767i
\(58\) 0 0
\(59\) −9.51422 + 5.49304i −1.23865 + 0.715133i −0.968817 0.247775i \(-0.920300\pi\)
−0.269829 + 0.962908i \(0.586967\pi\)
\(60\) 0 0
\(61\) −5.31922 3.07105i −0.681056 0.393208i 0.119197 0.992871i \(-0.461968\pi\)
−0.800253 + 0.599663i \(0.795301\pi\)
\(62\) 0 0
\(63\) 3.75311 + 2.16686i 0.472848 + 0.272999i
\(64\) 0 0
\(65\) 9.89522i 1.22735i
\(66\) 0 0
\(67\) −9.18575 5.30340i −1.12222 0.647913i −0.180252 0.983620i \(-0.557691\pi\)
−0.941966 + 0.335707i \(0.891025\pi\)
\(68\) 0 0
\(69\) 11.3337 1.36442
\(70\) 0 0
\(71\) 5.57715 + 9.65990i 0.661885 + 1.14642i 0.980120 + 0.198407i \(0.0635768\pi\)
−0.318234 + 0.948012i \(0.603090\pi\)
\(72\) 0 0
\(73\) 2.50000 + 4.33013i 0.292603 + 0.506803i 0.974424 0.224716i \(-0.0721453\pi\)
−0.681822 + 0.731519i \(0.738812\pi\)
\(74\) 0 0
\(75\) 4.54615i 0.524944i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.84509 6.65990i −0.432607 0.749297i 0.564490 0.825440i \(-0.309073\pi\)
−0.997097 + 0.0761427i \(0.975740\pi\)
\(80\) 0 0
\(81\) 5.40265 + 9.35766i 0.600294 + 1.03974i
\(82\) 0 0
\(83\) 0.289010 0.0317230 0.0158615 0.999874i \(-0.494951\pi\)
0.0158615 + 0.999874i \(0.494951\pi\)
\(84\) 0 0
\(85\) 1.75029 + 1.01053i 0.189846 + 0.109607i
\(86\) 0 0
\(87\) 2.65236i 0.284363i
\(88\) 0 0
\(89\) 14.0689 + 8.12270i 1.49130 + 0.861004i 0.999950 0.00995611i \(-0.00316918\pi\)
0.491353 + 0.870961i \(0.336503\pi\)
\(90\) 0 0
\(91\) −9.89522 5.71301i −1.03730 0.598886i
\(92\) 0 0
\(93\) 0.568928 0.328471i 0.0589951 0.0340608i
\(94\) 0 0
\(95\) −3.55608 6.65990i −0.364847 0.683291i
\(96\) 0 0
\(97\) 8.56951 4.94761i 0.870102 0.502354i 0.00271974 0.999996i \(-0.499134\pi\)
0.867382 + 0.497643i \(0.165801\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.75029 + 4.47463i 0.771183 + 0.445243i 0.833296 0.552827i \(-0.186451\pi\)
−0.0621136 + 0.998069i \(0.519784\pi\)
\(102\) 0 0
\(103\) 4.41005 0.434536 0.217268 0.976112i \(-0.430286\pi\)
0.217268 + 0.976112i \(0.430286\pi\)
\(104\) 0 0
\(105\) −6.81922 3.93708i −0.665488 0.384219i
\(106\) 0 0
\(107\) 2.33372i 0.225609i −0.993617 0.112805i \(-0.964017\pi\)
0.993617 0.112805i \(-0.0359834\pi\)
\(108\) 0 0
\(109\) −9.96272 17.2559i −0.954255 1.65282i −0.736064 0.676912i \(-0.763318\pi\)
−0.218191 0.975906i \(-0.570016\pi\)
\(110\) 0 0
\(111\) 5.34066 3.08343i 0.506913 0.292666i
\(112\) 0 0
\(113\) 12.5732i 1.18279i −0.806382 0.591395i \(-0.798578\pi\)
0.806382 0.591395i \(-0.201422\pi\)
\(114\) 0 0
\(115\) −8.63614 −0.805324
\(116\) 0 0
\(117\) 6.18964 + 10.7208i 0.572233 + 0.991136i
\(118\) 0 0
\(119\) 2.02106 1.16686i 0.185270 0.106966i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 11.2463 19.4791i 1.01404 1.75637i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) −4.31807 + 7.47912i −0.383167 + 0.663664i −0.991513 0.130007i \(-0.958500\pi\)
0.608346 + 0.793672i \(0.291833\pi\)
\(128\) 0 0
\(129\) 1.50000 + 0.866025i 0.132068 + 0.0762493i
\(130\) 0 0
\(131\) 6.19463 + 10.7294i 0.541227 + 0.937433i 0.998834 + 0.0482781i \(0.0153734\pi\)
−0.457607 + 0.889155i \(0.651293\pi\)
\(132\) 0 0
\(133\) −8.71301 0.289010i −0.755513 0.0250604i
\(134\) 0 0
\(135\) −1.64007 2.84068i −0.141155 0.244487i
\(136\) 0 0
\(137\) 5.41657 9.38177i 0.462769 0.801539i −0.536329 0.844009i \(-0.680189\pi\)
0.999098 + 0.0424700i \(0.0135227\pi\)
\(138\) 0 0
\(139\) −0.0919828 + 0.159319i −0.00780188 + 0.0135133i −0.869900 0.493228i \(-0.835817\pi\)
0.862098 + 0.506741i \(0.169150\pi\)
\(140\) 0 0
\(141\) −15.5006 −1.30538
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.02106i 0.167840i
\(146\) 0 0
\(147\) 5.90562 3.40961i 0.487087 0.281220i
\(148\) 0 0
\(149\) 7.75029 4.47463i 0.634929 0.366576i −0.147730 0.989028i \(-0.547197\pi\)
0.782658 + 0.622452i \(0.213863\pi\)
\(150\) 0 0
\(151\) 18.2665 1.48650 0.743252 0.669012i \(-0.233282\pi\)
0.743252 + 0.669012i \(0.233282\pi\)
\(152\) 0 0
\(153\) −2.52842 −0.204411
\(154\) 0 0
\(155\) −0.433516 + 0.250290i −0.0348208 + 0.0201038i
\(156\) 0 0
\(157\) −5.81980 + 3.36006i −0.464471 + 0.268162i −0.713922 0.700225i \(-0.753083\pi\)
0.249452 + 0.968387i \(0.419750\pi\)
\(158\) 0 0
\(159\) 1.51450i 0.120108i
\(160\) 0 0
\(161\) −4.98608 + 8.63614i −0.392958 + 0.680623i
\(162\) 0 0
\(163\) 11.3383 0.888081 0.444040 0.896007i \(-0.353545\pi\)
0.444040 + 0.896007i \(0.353545\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.96854 + 3.40961i −0.152330 + 0.263843i −0.932084 0.362243i \(-0.882011\pi\)
0.779754 + 0.626087i \(0.215344\pi\)
\(168\) 0 0
\(169\) −9.81922 17.0074i −0.755325 1.30826i
\(170\) 0 0
\(171\) 8.01866 + 4.99114i 0.613202 + 0.381682i
\(172\) 0 0
\(173\) 1.50000 + 2.59808i 0.114043 + 0.197528i 0.917397 0.397974i \(-0.130287\pi\)
−0.803354 + 0.595502i \(0.796953\pi\)
\(174\) 0 0
\(175\) −3.46410 2.00000i −0.261861 0.151186i
\(176\) 0 0
\(177\) −12.4861 + 21.6265i −0.938511 + 1.62555i
\(178\) 0 0
\(179\) 25.6384i 1.91631i 0.286255 + 0.958153i \(0.407589\pi\)
−0.286255 + 0.958153i \(0.592411\pi\)
\(180\) 0 0
\(181\) 4.21301 7.29714i 0.313150 0.542392i −0.665892 0.746048i \(-0.731949\pi\)
0.979043 + 0.203656i \(0.0652823\pi\)
\(182\) 0 0
\(183\) −13.9615 −1.03206
\(184\) 0 0
\(185\) −4.06951 + 2.34953i −0.299196 + 0.172741i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.78757 −0.275506
\(190\) 0 0
\(191\) 17.3047i 1.25213i 0.779773 + 0.626063i \(0.215335\pi\)
−0.779773 + 0.626063i \(0.784665\pi\)
\(192\) 0 0
\(193\) −12.1384 + 7.00813i −0.873744 + 0.504456i −0.868591 0.495530i \(-0.834974\pi\)
−0.00515328 + 0.999987i \(0.501640\pi\)
\(194\) 0 0
\(195\) −11.2463 19.4791i −0.805362 1.39493i
\(196\) 0 0
\(197\) 3.17509i 0.226216i 0.993583 + 0.113108i \(0.0360806\pi\)
−0.993583 + 0.113108i \(0.963919\pi\)
\(198\) 0 0
\(199\) −19.4489 11.2288i −1.37870 0.795991i −0.386693 0.922208i \(-0.626383\pi\)
−0.992003 + 0.126218i \(0.959716\pi\)
\(200\) 0 0
\(201\) −24.1100 −1.70059
\(202\) 0 0
\(203\) 2.02106 + 1.16686i 0.141851 + 0.0818975i
\(204\) 0 0
\(205\) −8.56951 + 14.8428i −0.598520 + 1.03667i
\(206\) 0 0
\(207\) 9.35666 5.40207i 0.650333 0.375470i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5.40853 3.12262i 0.372338 0.214970i −0.302141 0.953263i \(-0.597701\pi\)
0.674480 + 0.738294i \(0.264368\pi\)
\(212\) 0 0
\(213\) 21.9577 + 12.6773i 1.50451 + 0.868631i
\(214\) 0 0
\(215\) −1.14298 0.659900i −0.0779506 0.0450048i
\(216\) 0 0
\(217\) 0.578021i 0.0392386i
\(218\) 0 0
\(219\) 9.84269 + 5.68268i 0.665108 + 0.384000i
\(220\) 0 0
\(221\) 6.66628 0.448422
\(222\) 0 0
\(223\) −9.65873 16.7294i −0.646796 1.12028i −0.983884 0.178811i \(-0.942775\pi\)
0.337087 0.941473i \(-0.390558\pi\)
\(224\) 0 0
\(225\) 2.16686 + 3.75311i 0.144457 + 0.250207i
\(226\) 0 0
\(227\) 9.66628i 0.641574i 0.947151 + 0.320787i \(0.103947\pi\)
−0.947151 + 0.320787i \(0.896053\pi\)
\(228\) 0 0
\(229\) 18.9234i 1.25049i 0.780427 + 0.625247i \(0.215002\pi\)
−0.780427 + 0.625247i \(0.784998\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.56951 14.8428i −0.561407 0.972386i −0.997374 0.0724230i \(-0.976927\pi\)
0.435967 0.899963i \(-0.356407\pi\)
\(234\) 0 0
\(235\) 11.8112 0.770480
\(236\) 0 0
\(237\) −15.1384 8.74018i −0.983347 0.567736i
\(238\) 0 0
\(239\) 22.4727i 1.45364i −0.686828 0.726820i \(-0.740997\pi\)
0.686828 0.726820i \(-0.259003\pi\)
\(240\) 0 0
\(241\) 18.3887 + 10.6167i 1.18452 + 0.683884i 0.957056 0.289902i \(-0.0936227\pi\)
0.227466 + 0.973786i \(0.426956\pi\)
\(242\) 0 0
\(243\) 16.3504 + 9.43993i 1.04888 + 0.605572i
\(244\) 0 0
\(245\) −4.50000 + 2.59808i −0.287494 + 0.165985i
\(246\) 0 0
\(247\) −21.1415 13.1593i −1.34520 0.837307i
\(248\) 0 0
\(249\) 0.568928 0.328471i 0.0360544 0.0208160i
\(250\) 0 0
\(251\) −2.58602 + 4.47912i −0.163228 + 0.282719i −0.936025 0.351934i \(-0.885524\pi\)
0.772797 + 0.634654i \(0.218857\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.59402 0.287689
\(256\) 0 0
\(257\) −20.3192 11.7313i −1.26748 0.731779i −0.292968 0.956122i \(-0.594643\pi\)
−0.974510 + 0.224343i \(0.927976\pi\)
\(258\) 0 0
\(259\) 5.42601i 0.337156i
\(260\) 0 0
\(261\) −1.26421 2.18968i −0.0782527 0.135538i
\(262\) 0 0
\(263\) 16.2979 9.40961i 1.00497 0.580221i 0.0952570 0.995453i \(-0.469633\pi\)
0.909716 + 0.415231i \(0.136299\pi\)
\(264\) 0 0
\(265\) 1.15403i 0.0708915i
\(266\) 0 0
\(267\) 36.9270 2.25989
\(268\) 0 0
\(269\) −5.56951 9.64667i −0.339579 0.588168i 0.644775 0.764373i \(-0.276951\pi\)
−0.984354 + 0.176205i \(0.943618\pi\)
\(270\) 0 0
\(271\) −8.50470 + 4.91019i −0.516624 + 0.298273i −0.735552 0.677468i \(-0.763077\pi\)
0.218929 + 0.975741i \(0.429744\pi\)
\(272\) 0 0
\(273\) −25.9722 −1.57191
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 30.8397i 1.85298i −0.376323 0.926488i \(-0.622812\pi\)
0.376323 0.926488i \(-0.377188\pi\)
\(278\) 0 0
\(279\) 0.313123 0.542344i 0.0187462 0.0324693i
\(280\) 0 0
\(281\) 9.13844 + 5.27608i 0.545153 + 0.314744i 0.747165 0.664639i \(-0.231415\pi\)
−0.202011 + 0.979383i \(0.564748\pi\)
\(282\) 0 0
\(283\) −7.92668 13.7294i −0.471192 0.816129i 0.528265 0.849080i \(-0.322843\pi\)
−0.999457 + 0.0329510i \(0.989509\pi\)
\(284\) 0 0
\(285\) −14.5695 9.06865i −0.863023 0.537181i
\(286\) 0 0
\(287\) 9.89522 + 17.1390i 0.584096 + 1.01168i
\(288\) 0 0
\(289\) 7.81922 13.5433i 0.459954 0.796664i
\(290\) 0 0
\(291\) 11.2463 19.4791i 0.659268 1.14189i
\(292\) 0 0
\(293\) 24.1112 1.40859 0.704295 0.709907i \(-0.251263\pi\)
0.704295 + 0.709907i \(0.251263\pi\)
\(294\) 0 0
\(295\) 9.51422 16.4791i 0.553939 0.959451i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.6692 + 14.2427i −1.42665 + 0.823679i
\(300\) 0 0
\(301\) −1.31980 + 0.761986i −0.0760720 + 0.0439202i
\(302\) 0 0
\(303\) 20.3423 1.16864
\(304\) 0 0
\(305\) 10.6384 0.609155
\(306\) 0 0
\(307\) 17.2285 9.94689i 0.983284 0.567699i 0.0800239 0.996793i \(-0.474500\pi\)
0.903260 + 0.429094i \(0.141167\pi\)
\(308\) 0 0
\(309\) 8.68136 5.01219i 0.493865 0.285133i
\(310\) 0 0
\(311\) 10.4727i 0.593855i −0.954900 0.296927i \(-0.904038\pi\)
0.954900 0.296927i \(-0.0959619\pi\)
\(312\) 0 0
\(313\) −16.3887 + 28.3861i −0.926346 + 1.60448i −0.136963 + 0.990576i \(0.543734\pi\)
−0.789383 + 0.613902i \(0.789599\pi\)
\(314\) 0 0
\(315\) −7.50622 −0.422928
\(316\) 0 0
\(317\) −1.40265 + 2.42946i −0.0787806 + 0.136452i −0.902724 0.430220i \(-0.858436\pi\)
0.823943 + 0.566672i \(0.191769\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.65236 4.59402i −0.148040 0.256413i
\(322\) 0 0
\(323\) 4.48669 2.39569i 0.249646 0.133300i
\(324\) 0 0
\(325\) −5.71301 9.89522i −0.316901 0.548888i
\(326\) 0 0
\(327\) −39.2240 22.6460i −2.16909 1.25233i
\(328\) 0 0
\(329\) 6.81922 11.8112i 0.375956 0.651174i
\(330\) 0 0
\(331\) 22.4249i 1.23258i 0.787519 + 0.616291i \(0.211365\pi\)
−0.787519 + 0.616291i \(0.788635\pi\)
\(332\) 0 0
\(333\) 2.93935 5.09111i 0.161076 0.278991i
\(334\) 0 0
\(335\) 18.3715 1.00374
\(336\) 0 0
\(337\) −11.0689 + 6.39065i −0.602963 + 0.348121i −0.770206 0.637795i \(-0.779847\pi\)
0.167243 + 0.985916i \(0.446513\pi\)
\(338\) 0 0
\(339\) −14.2899 24.7509i −0.776122 1.34428i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) −17.0006 + 9.81529i −0.915280 + 0.528437i
\(346\) 0 0
\(347\) −14.7104 25.4791i −0.789694 1.36779i −0.926154 0.377145i \(-0.876906\pi\)
0.136460 0.990646i \(-0.456428\pi\)
\(348\) 0 0
\(349\) 19.5014i 1.04389i −0.852980 0.521944i \(-0.825207\pi\)
0.852980 0.521944i \(-0.174793\pi\)
\(350\) 0 0
\(351\) −9.36972 5.40961i −0.500119 0.288744i
\(352\) 0 0
\(353\) 7.83314 0.416916 0.208458 0.978031i \(-0.433156\pi\)
0.208458 + 0.978031i \(0.433156\pi\)
\(354\) 0 0
\(355\) −16.7314 9.65990i −0.888012 0.512694i
\(356\) 0 0
\(357\) 2.65236 4.59402i 0.140378 0.243141i
\(358\) 0 0
\(359\) 30.9972 17.8963i 1.63597 0.944529i 0.653772 0.756692i \(-0.273186\pi\)
0.982200 0.187837i \(-0.0601478\pi\)
\(360\) 0 0
\(361\) −18.9582 1.25907i −0.997802 0.0662671i
\(362\) 0 0
\(363\) −21.6539 + 12.5019i −1.13654 + 0.656180i
\(364\) 0 0
\(365\) −7.50000 4.33013i −0.392568 0.226649i
\(366\) 0 0
\(367\) −2.44151 1.40961i −0.127446 0.0735810i 0.434922 0.900468i \(-0.356776\pi\)
−0.562368 + 0.826887i \(0.690109\pi\)
\(368\) 0 0
\(369\) 21.4416i 1.11620i
\(370\) 0 0
\(371\) −1.15403 0.666280i −0.0599143 0.0345915i
\(372\) 0 0
\(373\) 8.21243 0.425223 0.212612 0.977137i \(-0.431803\pi\)
0.212612 + 0.977137i \(0.431803\pi\)
\(374\) 0 0
\(375\) −13.7798 23.8673i −0.711584 1.23250i
\(376\) 0 0
\(377\) 3.33314 + 5.77317i 0.171665 + 0.297333i
\(378\) 0 0
\(379\) 16.8520i 0.865630i 0.901483 + 0.432815i \(0.142480\pi\)
−0.901483 + 0.432815i \(0.857520\pi\)
\(380\) 0 0
\(381\) 19.6306i 1.00571i
\(382\) 0 0
\(383\) 14.7104 + 25.4791i 0.751665 + 1.30192i 0.947015 + 0.321189i \(0.104082\pi\)
−0.195350 + 0.980734i \(0.562584\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.65112 0.0839311
\(388\) 0 0
\(389\) −19.4582 11.2342i −0.986572 0.569597i −0.0823240 0.996606i \(-0.526234\pi\)
−0.904248 + 0.427008i \(0.859568\pi\)
\(390\) 0 0
\(391\) 5.81806i 0.294232i
\(392\) 0 0
\(393\) 24.3887 + 14.0808i 1.23025 + 0.710284i
\(394\) 0 0
\(395\) 11.5353 + 6.65990i 0.580403 + 0.335096i
\(396\) 0 0
\(397\) 20.3192 11.7313i 1.01979 0.588777i 0.105749 0.994393i \(-0.466276\pi\)
0.914044 + 0.405616i \(0.132943\pi\)
\(398\) 0 0
\(399\) −17.4804 + 9.33372i −0.875113 + 0.467270i
\(400\) 0 0
\(401\) 3.74913 2.16456i 0.187223 0.108093i −0.403459 0.914998i \(-0.632192\pi\)
0.590682 + 0.806905i \(0.298859\pi\)
\(402\) 0 0
\(403\) −0.825559 + 1.42991i −0.0411240 + 0.0712289i
\(404\) 0 0
\(405\) −16.2079 9.35766i −0.805379 0.464986i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 23.8198 + 13.7524i 1.17781 + 0.680011i 0.955508 0.294966i \(-0.0953084\pi\)
0.222306 + 0.974977i \(0.428642\pi\)
\(410\) 0 0
\(411\) 24.6245i 1.21464i
\(412\) 0 0
\(413\) −10.9861 19.0284i −0.540590 0.936329i
\(414\) 0 0
\(415\) −0.433516 + 0.250290i −0.0212804 + 0.0122863i
\(416\) 0 0
\(417\) 0.418167i 0.0204777i
\(418\) 0 0
\(419\) −27.6646 −1.35150 −0.675752 0.737129i \(-0.736181\pi\)
−0.675752 + 0.737129i \(0.736181\pi\)
\(420\) 0 0
\(421\) 16.2130 + 28.0818i 0.790174 + 1.36862i 0.925859 + 0.377869i \(0.123343\pi\)
−0.135686 + 0.990752i \(0.543324\pi\)
\(422\) 0 0
\(423\) −12.7966 + 7.38815i −0.622194 + 0.359224i
\(424\) 0 0
\(425\) 2.33372 0.113202
\(426\) 0 0
\(427\) 6.14210 10.6384i 0.297237 0.514830i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.53502 + 2.65874i −0.0739395 + 0.128067i −0.900625 0.434598i \(-0.856890\pi\)
0.826685 + 0.562665i \(0.190224\pi\)
\(432\) 0 0
\(433\) 18.9577 + 10.9452i 0.911047 + 0.525993i 0.880768 0.473548i \(-0.157027\pi\)
0.0302790 + 0.999541i \(0.490360\pi\)
\(434\) 0 0
\(435\) 2.29701 + 3.97854i 0.110133 + 0.190756i
\(436\) 0 0
\(437\) −11.4849 + 18.4514i −0.549398 + 0.882652i
\(438\) 0 0
\(439\) 0.564959 + 0.978538i 0.0269640 + 0.0467030i 0.879193 0.476467i \(-0.158083\pi\)
−0.852229 + 0.523170i \(0.824749\pi\)
\(440\) 0 0
\(441\) 3.25029 5.62967i 0.154776 0.268079i
\(442\) 0 0
\(443\) 7.20415 12.4780i 0.342280 0.592846i −0.642576 0.766222i \(-0.722134\pi\)
0.984856 + 0.173376i \(0.0554677\pi\)
\(444\) 0 0
\(445\) −28.1379 −1.33386
\(446\) 0 0
\(447\) 10.1712 17.6170i 0.481080 0.833254i
\(448\) 0 0
\(449\) 29.5258i 1.39341i −0.717358 0.696704i \(-0.754649\pi\)
0.717358 0.696704i \(-0.245351\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 35.9582 20.7605i 1.68947 0.975413i
\(454\) 0 0
\(455\) 19.7904 0.927790
\(456\) 0 0
\(457\) 6.63960 0.310587 0.155294 0.987868i \(-0.450368\pi\)
0.155294 + 0.987868i \(0.450368\pi\)
\(458\) 0 0
\(459\) 1.91373 1.10489i 0.0893253 0.0515720i
\(460\) 0 0
\(461\) 31.7085 18.3069i 1.47681 0.852639i 0.477156 0.878819i \(-0.341668\pi\)
0.999657 + 0.0261800i \(0.00833430\pi\)
\(462\) 0 0
\(463\) 18.1390i 0.842992i 0.906830 + 0.421496i \(0.138495\pi\)
−0.906830 + 0.421496i \(0.861505\pi\)
\(464\) 0 0
\(465\) −0.568928 + 0.985413i −0.0263834 + 0.0456974i
\(466\) 0 0
\(467\) 6.87998 0.318367 0.159184 0.987249i \(-0.449114\pi\)
0.159184 + 0.987249i \(0.449114\pi\)
\(468\) 0 0
\(469\) 10.6068 18.3715i 0.489776 0.848317i
\(470\) 0 0
\(471\) −7.63767 + 13.2288i −0.351925 + 0.609552i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −7.40118 4.60679i −0.339589 0.211374i
\(476\) 0 0
\(477\) 0.721867 + 1.25031i 0.0330520 + 0.0572478i
\(478\) 0 0
\(479\) 28.1092 + 16.2288i 1.28434 + 0.741514i 0.977639 0.210292i \(-0.0674414\pi\)
0.306701 + 0.951806i \(0.400775\pi\)
\(480\) 0 0
\(481\) −7.74971 + 13.4229i −0.353356 + 0.612031i
\(482\) 0 0
\(483\) 22.6674i 1.03140i
\(484\) 0 0
\(485\) −8.56951 + 14.8428i −0.389121 + 0.673978i
\(486\) 0 0
\(487\) −23.8326 −1.07996 −0.539978 0.841679i \(-0.681567\pi\)
−0.539978 + 0.841679i \(0.681567\pi\)
\(488\) 0 0
\(489\) 22.3198 12.8863i 1.00934 0.582741i
\(490\) 0 0
\(491\) 6.48364 + 11.2300i 0.292602 + 0.506802i 0.974424 0.224716i \(-0.0721454\pi\)
−0.681822 + 0.731518i \(0.738812\pi\)
\(492\) 0 0
\(493\) −1.36156 −0.0613217
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.3198 + 11.1543i −0.866611 + 0.500338i
\(498\) 0 0
\(499\) 0.380993 + 0.659900i 0.0170556 + 0.0295412i 0.874427 0.485157i \(-0.161237\pi\)
−0.857372 + 0.514698i \(0.827904\pi\)
\(500\) 0 0
\(501\) 8.94926i 0.399824i
\(502\) 0 0
\(503\) 8.36019 + 4.82676i 0.372763 + 0.215215i 0.674665 0.738124i \(-0.264288\pi\)
−0.301902 + 0.953339i \(0.597622\pi\)
\(504\) 0 0
\(505\) −15.5006 −0.689767
\(506\) 0 0
\(507\) −38.6590 22.3198i −1.71691 0.991257i
\(508\) 0 0
\(509\) −17.0689 + 29.5643i −0.756567 + 1.31041i 0.188025 + 0.982164i \(0.439792\pi\)
−0.944592 + 0.328248i \(0.893542\pi\)
\(510\) 0 0
\(511\) −8.66025 + 5.00000i −0.383107 + 0.221187i
\(512\) 0 0
\(513\) −8.25029 0.273662i −0.364259 0.0120825i
\(514\) 0 0
\(515\) −6.61508 + 3.81922i −0.291495 + 0.168295i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 5.90562 + 3.40961i 0.259228 + 0.149665i
\(520\) 0 0
\(521\) 39.5809i 1.73407i −0.498248 0.867035i \(-0.666023\pi\)
0.498248 0.867035i \(-0.333977\pi\)
\(522\) 0 0
\(523\) 29.4733 + 17.0164i 1.28878 + 0.744075i 0.978436 0.206550i \(-0.0662237\pi\)
0.310340 + 0.950626i \(0.399557\pi\)
\(524\) 0 0
\(525\) −9.09229 −0.396820
\(526\) 0 0
\(527\) −0.168617 0.292054i −0.00734509 0.0127221i
\(528\) 0 0
\(529\) 0.930491 + 1.61166i 0.0404561 + 0.0700721i
\(530\) 0 0
\(531\) 23.8053i 1.03306i
\(532\) 0 0
\(533\) 56.5314i 2.44865i
\(534\) 0 0
\(535\) 2.02106 + 3.50058i 0.0873781 + 0.151343i
\(536\) 0 0
\(537\) 29.1390 + 50.4703i 1.25744 + 2.17795i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.931072 0.537555i −0.0400299 0.0231113i 0.479851 0.877350i \(-0.340691\pi\)
−0.519881 + 0.854238i \(0.674024\pi\)
\(542\) 0 0
\(543\) 19.1529i 0.821931i
\(544\) 0 0
\(545\) 29.8881 + 17.2559i 1.28027 + 0.739163i
\(546\) 0 0
\(547\) 12.0324 + 6.94689i 0.514467 + 0.297028i 0.734668 0.678427i \(-0.237338\pi\)
−0.220201 + 0.975455i \(0.570671\pi\)
\(548\) 0 0
\(549\) −11.5260 + 6.65454i −0.491918 + 0.284009i
\(550\) 0 0
\(551\) 4.31807 + 2.68774i 0.183956 + 0.114502i
\(552\) 0 0
\(553\) 13.3198 7.69019i 0.566415 0.327020i
\(554\) 0 0
\(555\) −5.34066 + 9.25029i −0.226698 + 0.392653i
\(556\) 0 0
\(557\) 33.4571 + 19.3165i 1.41762 + 0.818464i 0.996090 0.0883491i \(-0.0281591\pi\)
0.421532 + 0.906813i \(0.361492\pi\)
\(558\) 0 0
\(559\) −4.35323 −0.184122
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.63960i 0.111246i 0.998452 + 0.0556229i \(0.0177145\pi\)
−0.998452 + 0.0556229i \(0.982286\pi\)
\(564\) 0 0
\(565\) 10.8887 + 18.8598i 0.458092 + 0.793439i
\(566\) 0 0
\(567\) −18.7153 + 10.8053i −0.785970 + 0.453780i
\(568\) 0 0
\(569\) 41.4421i 1.73734i −0.495389 0.868671i \(-0.664975\pi\)
0.495389 0.868671i \(-0.335025\pi\)
\(570\) 0 0
\(571\) 4.45828 0.186573 0.0932866 0.995639i \(-0.470263\pi\)
0.0932866 + 0.995639i \(0.470263\pi\)
\(572\) 0 0
\(573\) 19.6674 + 34.0650i 0.821619 + 1.42309i
\(574\) 0 0
\(575\) −8.63614 + 4.98608i −0.360152 + 0.207934i
\(576\) 0 0
\(577\) −11.1379 −0.463675 −0.231838 0.972755i \(-0.574474\pi\)
−0.231838 + 0.972755i \(0.574474\pi\)
\(578\) 0 0
\(579\) −15.9300 + 27.5916i −0.662028 + 1.14667i
\(580\) 0 0
\(581\) 0.578021i 0.0239803i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −18.5689 10.7208i −0.767731 0.443250i
\(586\) 0 0
\(587\) 12.8338 + 22.2288i 0.529708 + 0.917482i 0.999399 + 0.0346511i \(0.0110320\pi\)
−0.469691 + 0.882831i \(0.655635\pi\)
\(588\) 0 0
\(589\) −0.0417635 + 1.25907i −0.00172084 + 0.0518793i
\(590\) 0 0
\(591\) 3.60861 + 6.25029i 0.148438 + 0.257103i
\(592\) 0 0
\(593\) 11.5695 20.0390i 0.475103 0.822902i −0.524491 0.851416i \(-0.675744\pi\)
0.999593 + 0.0285141i \(0.00907754\pi\)
\(594\) 0 0
\(595\) −2.02106 + 3.50058i −0.0828554 + 0.143510i
\(596\) 0 0
\(597\) −51.0479 −2.08925
\(598\) 0 0
\(599\) −8.75224 + 15.1593i −0.357607 + 0.619393i −0.987560 0.157240i \(-0.949740\pi\)
0.629954 + 0.776633i \(0.283074\pi\)
\(600\) 0 0
\(601\) 17.0315i 0.694729i 0.937730 + 0.347365i \(0.112923\pi\)
−0.937730 + 0.347365i \(0.887077\pi\)
\(602\) 0 0
\(603\) −19.9042 + 11.4917i −0.810563 + 0.467979i
\(604\) 0 0
\(605\) 16.5000 9.52628i 0.670820 0.387298i
\(606\) 0 0
\(607\) −29.6047 −1.20162 −0.600809 0.799392i \(-0.705155\pi\)
−0.600809 + 0.799392i \(0.705155\pi\)
\(608\) 0 0
\(609\) 5.30472 0.214958
\(610\) 0 0
\(611\) 33.7388 19.4791i 1.36493 0.788041i
\(612\) 0 0
\(613\) 19.2497 11.1138i 0.777488 0.448883i −0.0580510 0.998314i \(-0.518489\pi\)
0.835539 + 0.549430i \(0.185155\pi\)
\(614\) 0 0
\(615\) 38.9582i 1.57095i
\(616\) 0 0
\(617\) −8.06893 + 13.9758i −0.324843 + 0.562644i −0.981481 0.191562i \(-0.938645\pi\)
0.656638 + 0.754206i \(0.271978\pi\)
\(618\) 0 0
\(619\) 9.97615 0.400975 0.200488 0.979696i \(-0.435747\pi\)
0.200488 + 0.979696i \(0.435747\pi\)
\(620\) 0 0
\(621\) −4.72129 + 8.17751i −0.189459 + 0.328152i
\(622\) 0 0
\(623\) −16.2454 + 28.1379i −0.650858 + 1.12732i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.58285 2.74158i −0.0631123 0.109314i
\(630\) 0 0
\(631\) 28.6631 + 16.5486i 1.14106 + 0.658790i 0.946693 0.322139i \(-0.104402\pi\)
0.194366 + 0.980929i \(0.437735\pi\)
\(632\) 0 0
\(633\) 7.09793 12.2940i 0.282117 0.488642i
\(634\) 0 0
\(635\) 14.9582i 0.593600i
\(636\) 0 0
\(637\) −8.56951 + 14.8428i −0.339536 + 0.588094i
\(638\) 0 0
\(639\) 24.1698 0.956142
\(640\) 0 0
\(641\) 3.18020 1.83609i 0.125610 0.0725212i −0.435878 0.900006i \(-0.643562\pi\)
0.561489 + 0.827484i \(0.310229\pi\)
\(642\) 0 0
\(643\) −19.8015 34.2972i −0.780894 1.35255i −0.931421 0.363943i \(-0.881430\pi\)
0.150527 0.988606i \(-0.451903\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) 0 0
\(647\) 41.6106i 1.63588i 0.575303 + 0.817941i \(0.304884\pi\)
−0.575303 + 0.817941i \(0.695116\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.656942 + 1.13786i 0.0257476 + 0.0445961i
\(652\) 0 0
\(653\) 44.1181i 1.72647i 0.504799 + 0.863237i \(0.331566\pi\)
−0.504799 + 0.863237i \(0.668434\pi\)
\(654\) 0 0
\(655\) −18.5839 10.7294i −0.726132 0.419233i
\(656\) 0 0
\(657\) 10.8343 0.422686
\(658\) 0 0
\(659\) −14.2769 8.24275i −0.556148 0.321092i 0.195450 0.980714i \(-0.437383\pi\)
−0.751598 + 0.659622i \(0.770717\pi\)
\(660\) 0 0
\(661\) 0.962716 1.66747i 0.0374453 0.0648572i −0.846695 0.532078i \(-0.821411\pi\)
0.884141 + 0.467221i \(0.154745\pi\)
\(662\) 0 0
\(663\) 13.1228 7.57647i 0.509648 0.294246i
\(664\) 0 0
\(665\) 13.3198 7.11217i 0.516520 0.275798i
\(666\) 0 0
\(667\) 5.03859 2.90903i 0.195095 0.112638i
\(668\) 0 0
\(669\) −38.0272 21.9550i −1.47022 0.848829i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19.5496i 0.753584i −0.926298 0.376792i \(-0.877027\pi\)
0.926298 0.376792i \(-0.122973\pi\)
\(674\) 0 0
\(675\) −3.28014 1.89379i −0.126252 0.0728919i
\(676\) 0 0
\(677\) −12.1947 −0.468680 −0.234340 0.972155i \(-0.575293\pi\)
−0.234340 + 0.972155i \(0.575293\pi\)
\(678\) 0 0
\(679\) 9.89522 + 17.1390i 0.379744 + 0.657735i
\(680\) 0 0
\(681\) 10.9861 + 19.0284i 0.420987 + 0.729172i
\(682\) 0 0
\(683\) 2.13902i 0.0818472i −0.999162 0.0409236i \(-0.986970\pi\)
0.999162 0.0409236i \(-0.0130300\pi\)
\(684\) 0 0
\(685\) 18.7635i 0.716918i
\(686\) 0 0
\(687\) 21.5071 + 37.2515i 0.820548 + 1.42123i
\(688\) 0 0
\(689\) −1.90323 3.29649i −0.0725073 0.125586i
\(690\) 0 0
\(691\) 20.7846 0.790684 0.395342 0.918534i \(-0.370626\pi\)
0.395342 + 0.918534i \(0.370626\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.318638i 0.0120866i
\(696\) 0 0
\(697\) −9.99942 5.77317i −0.378755 0.218674i
\(698\) 0 0
\(699\) −33.7388 19.4791i −1.27612 0.736768i
\(700\) 0 0
\(701\) 43.1662 24.9220i 1.63036 0.941291i 0.646384 0.763012i \(-0.276280\pi\)
0.983980 0.178279i \(-0.0570530\pi\)
\(702\) 0 0
\(703\) −0.392043 + 11.8192i −0.0147862 + 0.445770i
\(704\) 0 0
\(705\) 23.2509 13.4239i 0.875678 0.505573i
\(706\) 0 0
\(707\) −8.94926 + 15.5006i −0.336572 + 0.582959i
\(708\) 0 0
\(709\) −11.8198 6.82416i −0.443902 0.256287i 0.261349 0.965244i \(-0.415833\pi\)
−0.705251 + 0.708957i \(0.749166\pi\)
\(710\) 0 0
\(711\) −16.6636 −0.624933
\(712\) 0 0
\(713\) 1.24797 + 0.720514i 0.0467367 + 0.0269835i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −25.5411 44.2385i −0.953849 1.65212i
\(718\) 0 0
\(719\) 9.94774 5.74333i 0.370988 0.214190i −0.302902 0.953022i \(-0.597955\pi\)
0.673890 + 0.738832i \(0.264622\pi\)
\(720\) 0 0
\(721\) 8.82011i 0.328478i
\(722\) 0 0
\(723\) 48.2652 1.79500
\(724\) 0 0
\(725\) 1.16686 + 2.02106i 0.0433361 + 0.0750603i
\(726\) 0 0
\(727\) 23.4669 13.5486i 0.870340 0.502491i 0.00287848 0.999996i \(-0.499084\pi\)
0.867461 + 0.497505i \(0.165750\pi\)
\(728\) 0 0
\(729\) 10.4994 0.388867
\(730\) 0 0
\(731\) 0.444566 0.770011i 0.0164429 0.0284799i
\(732\) 0 0
\(733\) 45.2741i 1.67224i 0.548549 + 0.836119i \(0.315181\pi\)
−0.548549 + 0.836119i \(0.684819\pi\)
\(734\) 0 0
\(735\) −5.90562 + 10.2288i −0.217832 + 0.377296i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −12.7944 22.1605i −0.470648 0.815186i 0.528788 0.848754i \(-0.322647\pi\)
−0.999436 + 0.0335673i \(0.989313\pi\)
\(740\) 0 0
\(741\) −56.5739 1.87656i −2.07829 0.0689370i
\(742\) 0 0
\(743\) 12.3608 + 21.4096i 0.453475 + 0.785442i 0.998599 0.0529132i \(-0.0168507\pi\)
−0.545124 + 0.838356i \(0.683517\pi\)
\(744\) 0 0
\(745\) −7.75029 + 13.4239i −0.283949 + 0.491814i
\(746\) 0 0
\(747\) 0.313123 0.542344i 0.0114566 0.0198433i
\(748\) 0 0
\(749\) 4.66744 0.170545
\(750\) 0 0
\(751\) −12.4812 + 21.6181i −0.455447 + 0.788857i −0.998714 0.0507030i \(-0.983854\pi\)
0.543267 + 0.839560i \(0.317187\pi\)
\(752\) 0 0
\(753\) 11.7564i 0.428428i
\(754\) 0 0
\(755\) −27.3997 + 15.8192i −0.997177 + 0.575720i
\(756\) 0 0
\(757\) −16.6402 + 9.60721i −0.604798 + 0.349180i −0.770927 0.636924i \(-0.780207\pi\)
0.166129 + 0.986104i \(0.446873\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.8065 −1.33423 −0.667117 0.744953i \(-0.732472\pi\)
−0.667117 + 0.744953i \(0.732472\pi\)
\(762\) 0 0
\(763\) 34.5119 19.9254i 1.24941 0.721349i
\(764\) 0 0
\(765\) 3.79263 2.18968i 0.137123 0.0791680i
\(766\) 0 0
\(767\) 62.7635i 2.26626i
\(768\) 0 0
\(769\) 0.819799 1.41993i 0.0295627 0.0512041i −0.850866 0.525384i \(-0.823922\pi\)
0.880428 + 0.474179i \(0.157255\pi\)
\(770\) 0 0
\(771\) −53.3322 −1.92071
\(772\) 0 0
\(773\) 11.1663 19.3406i 0.401623 0.695632i −0.592299 0.805718i \(-0.701780\pi\)
0.993922 + 0.110087i \(0.0351128\pi\)
\(774\) 0 0
\(775\) −0.289010 + 0.500581i −0.0103816 + 0.0179814i
\(776\) 0 0
\(777\) 6.16686 + 10.6813i 0.221235 + 0.383190i
\(778\) 0 0
\(779\) 20.3159 + 38.0480i 0.727894 + 1.36321i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.91373 + 1.10489i 0.0683911 + 0.0394856i
\(784\) 0 0
\(785\) 5.81980 10.0802i 0.207718 0.359777i
\(786\) 0 0
\(787\) 20.2124i 0.720495i 0.932857 + 0.360248i \(0.117308\pi\)
−0.932857 + 0.360248i \(0.882692\pi\)
\(788\) 0 0
\(789\) 21.3887 37.0464i 0.761459 1.31889i
\(790\) 0 0
\(791\) 25.1464 0.894104
\(792\) 0 0
\(793\) 30.3887 17.5449i 1.07914 0.623039i
\(794\) 0 0
\(795\) −1.31160 2.27175i −0.0465176 0.0805708i
\(796\) 0 0
\(797\) 48.8065 1.72881 0.864407 0.502793i \(-0.167694\pi\)
0.864407 + 0.502793i \(0.167694\pi\)
\(798\) 0 0
\(799\) 7.95708i 0.281501i
\(800\) 0 0
\(801\) 30.4854 17.6008i 1.07715 0.621892i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 17.2723i 0.608768i
\(806\) 0 0
\(807\) −21.9276 12.6599i −0.771888 0.445650i
\(808\) 0 0
\(809\) 23.8053 0.836950 0.418475 0.908228i \(-0.362565\pi\)
0.418475 + 0.908228i \(0.362565\pi\)
\(810\) 0 0
\(811\) −44.1947 25.5158i −1.55189 0.895982i −0.997988 0.0633968i \(-0.979807\pi\)
−0.553897 0.832585i \(-0.686860\pi\)
\(812\) 0 0
\(813\) −11.1612 + 19.3318i −0.391441 + 0.677996i
\(814\) 0 0
\(815\) −17.0074 + 9.81922i −0.595743 + 0.343952i
\(816\) 0 0
\(817\) −2.92991 + 1.56444i −0.102505 + 0.0547328i
\(818\) 0 0
\(819\) −21.4416 + 12.3793i −0.749228 + 0.432567i
\(820\) 0 0
\(821\) 1.70853 + 0.986418i 0.0596280 + 0.0344262i 0.529518 0.848299i \(-0.322373\pi\)
−0.469890 + 0.882725i \(0.655706\pi\)
\(822\) 0 0
\(823\) 18.3431 + 10.5904i 0.639400 + 0.369158i 0.784383 0.620276i \(-0.212980\pi\)
−0.144983 + 0.989434i \(0.546313\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −21.5904 12.4652i −0.750770 0.433457i 0.0752019 0.997168i \(-0.476040\pi\)
−0.825972 + 0.563711i \(0.809373\pi\)
\(828\) 0 0
\(829\) 16.8520 0.585295 0.292647 0.956220i \(-0.405464\pi\)
0.292647 + 0.956220i \(0.405464\pi\)
\(830\) 0 0
\(831\) −35.0504 60.7091i −1.21589 2.10598i
\(832\) 0 0
\(833\) −1.75029 3.03159i −0.0606440 0.105038i
\(834\) 0 0
\(835\) 6.81922i 0.235989i
\(836\) 0 0
\(837\) 0.547324i 0.0189183i
\(838\) 0 0
\(839\) 0.444566 + 0.770011i 0.0153481 + 0.0265837i 0.873597 0.486649i \(-0.161781\pi\)
−0.858249 + 0.513233i \(0.828448\pi\)
\(840\) 0 0
\(841\) 13.8192 + 23.9356i 0.476525 + 0.825365i
\(842\) 0 0
\(843\) 23.9858 0.826116
\(844\) 0 0
\(845\) 29.4577 + 17.0074i 1.01337 + 0.585072i
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) −31.2079 18.0179i −1.07105 0.618373i
\(850\) 0 0
\(851\) 11.7150 + 6.76363i 0.401583 + 0.231854i
\(852\) 0 0
\(853\) 0.680781 0.393049i 0.0233095 0.0134578i −0.488300 0.872676i \(-0.662383\pi\)
0.511609 + 0.859218i \(0.329049\pi\)
\(854\) 0 0
\(855\) −16.3504 0.542344i −0.559173 0.0185478i
\(856\) 0 0
\(857\) −4.50000 + 2.59808i −0.153717 + 0.0887486i −0.574886 0.818234i \(-0.694953\pi\)
0.421168 + 0.906982i \(0.361620\pi\)
\(858\) 0 0
\(859\) −0.236488 + 0.409609i −0.00806887 + 0.0139757i −0.870032 0.492996i \(-0.835902\pi\)
0.861963 + 0.506972i \(0.169235\pi\)
\(860\) 0 0
\(861\) 38.9582 + 22.4925i 1.32769 + 0.766544i
\(862\) 0 0
\(863\) −22.9655 −0.781756 −0.390878 0.920443i \(-0.627828\pi\)
−0.390878 + 0.920443i \(0.627828\pi\)
\(864\) 0 0
\(865\) −4.50000 2.59808i −0.153005 0.0883372i
\(866\) 0 0
\(867\) 35.5473i 1.20725i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 52.4783 30.2983i 1.77816 1.02662i
\(872\) 0 0
\(873\) 21.4416i 0.725687i
\(874\) 0 0
\(875\) 24.2487 0.819756
\(876\) 0 0
\(877\) −1.78193 3.08640i −0.0601716 0.104220i 0.834370 0.551204i \(-0.185831\pi\)
−0.894542 + 0.446984i \(0.852498\pi\)
\(878\) 0 0
\(879\) 47.4638 27.4032i 1.60091 0.924288i
\(880\) 0 0
\(881\) 47.4159 1.59748 0.798741 0.601675i \(-0.205500\pi\)
0.798741 + 0.601675i \(0.205500\pi\)
\(882\) 0 0
\(883\) −4.52615 + 7.83952i −0.152317 + 0.263821i −0.932079 0.362255i \(-0.882007\pi\)
0.779762 + 0.626076i \(0.215340\pi\)
\(884\) 0 0
\(885\) 43.2530i 1.45394i
\(886\) 0 0
\(887\) 13.5149 23.4084i 0.453785 0.785979i −0.544832 0.838545i \(-0.683407\pi\)
0.998617 + 0.0525662i \(0.0167401\pi\)
\(888\) 0 0
\(889\) −14.9582 8.63614i −0.501683 0.289647i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.7074 25.2351i 0.525627 0.844461i
\(894\) 0 0
\(895\) −22.2035 38.4577i −0.742182 1.28550i
\(896\) 0 0
\(897\) −32.3748 + 56.0748i −1.08096 + 1.87228i
\(898\) 0 0
\(899\) 0.168617 0.292054i 0.00562370 0.00974054i
\(900\) 0 0
\(901\) 0.777455 0.0259008
\(902\) 0 0
\(903\) −1.73205 + 3.00000i −0.0576390 + 0.0998337i
\(904\) 0 0
\(905\) 14.5943i 0.485130i
\(906\) 0 0
\(907\) −21.5576 + 12.4463i −0.715810 + 0.413273i −0.813209 0.581972i \(-0.802281\pi\)
0.0973986 + 0.995245i \(0.468948\pi\)
\(908\) 0 0
\(909\) 16.7938 9.69590i 0.557015 0.321593i
\(910\) 0 0
\(911\) −39.0511 −1.29382 −0.646910 0.762567i \(-0.723939\pi\)
−0.646910 + 0.762567i \(0.723939\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 20.9422 12.0910i 0.692327 0.399715i
\(916\) 0 0
\(917\) −21.4588 + 12.3893i −0.708633 + 0.409129i
\(918\) 0 0
\(919\) 18.3616i 0.605692i 0.953039 + 0.302846i \(0.0979368\pi\)
−0.953039 + 0.302846i \(0.902063\pi\)
\(920\) 0 0
\(921\) 22.6100 39.1617i 0.745025 1.29042i
\(922\) 0 0
\(923\) −63.7245 −2.09752
\(924\) 0 0
\(925\) −2.71301 + 4.69906i −0.0892031 + 0.154504i
\(926\) 0 0
\(927\) 4.77799 8.27571i 0.156930 0.271810i
\(928\) 0 0
\(929\) 10.6529 + 18.4514i 0.349512 + 0.605372i 0.986163 0.165780i \(-0.0530142\pi\)
−0.636651 + 0.771152i \(0.719681\pi\)
\(930\) 0 0
\(931\) −0.433516 + 13.0695i −0.0142079 + 0.428336i
\(932\) 0 0
\(933\) −11.9026 20.6160i −0.389675 0.674937i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24.1396 + 41.8110i −0.788606 + 1.36591i 0.138215 + 0.990402i \(0.455864\pi\)
−0.926821 + 0.375504i \(0.877470\pi\)
\(938\) 0 0
\(939\) 74.5055i 2.43140i
\(940\) 0 0
\(941\) 11.9727 20.7374i 0.390300 0.676020i −0.602189 0.798354i \(-0.705705\pi\)
0.992489 + 0.122334i \(0.0390379\pi\)
\(942\) 0 0
\(943\) 49.3383 1.60668
\(944\) 0 0
\(945\) 5.68136 3.28014i 0.184815 0.106703i
\(946\) 0 0
\(947\) −21.3496 36.9785i −0.693768 1.20164i −0.970594 0.240721i \(-0.922616\pi\)
0.276827 0.960920i \(-0.410717\pi\)
\(948\) 0 0
\(949\) −28.5650 −0.927260
\(950\) 0 0
\(951\) 6.37665i 0.206777i
\(952\) 0 0
\(953\) −21.4571 + 12.3882i −0.695063 + 0.401295i −0.805506 0.592588i \(-0.798106\pi\)
0.110443 + 0.993882i \(0.464773\pi\)
\(954\) 0 0
\(955\) −14.9863 25.9571i −0.484946 0.839951i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.7635 + 10.8331i 0.605907 + 0.349820i
\(960\) 0 0
\(961\) −30.9165 −0.997306
\(962\) 0 0
\(963\) −4.37936 2.52842i −0.141123 0.0814773i
\(964\) 0 0
\(965\) 12.1384 21.0244i 0.390750 0.676799i
\(966\) 0 0
\(967\) −53.3453 + 30.7989i −1.71547 + 0.990426i −0.788714 + 0.614761i \(0.789253\pi\)
−0.926755 + 0.375666i \(0.877414\pi\)
\(968\) 0 0
\(969\) 6.10943 9.81529i 0.196263 0.315313i
\(970\) 0 0
\(971\) 42.6399 24.6181i 1.36838 0.790034i 0.377657 0.925945i \(-0.376730\pi\)
0.990721 + 0.135912i \(0.0433964\pi\)
\(972\) 0 0
\(973\) −0.318638 0.183966i −0.0102151 0.00589767i
\(974\) 0 0
\(975\) −22.4925 12.9861i −0.720338 0.415887i
\(976\) 0 0
\(977\) 3.70489i 0.118530i 0.998242 + 0.0592649i \(0.0188757\pi\)
−0.998242 + 0.0592649i \(0.981124\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −43.1756 −1.37849
\(982\) 0 0
\(983\) 9.01714 + 15.6181i 0.287602 + 0.498141i 0.973237 0.229804i \(-0.0738087\pi\)
−0.685635 + 0.727946i \(0.740475\pi\)
\(984\) 0 0
\(985\) −2.74971 4.76264i −0.0876130 0.151750i
\(986\) 0 0
\(987\) 31.0012i 0.986778i
\(988\) 0 0
\(989\) 3.79932i 0.120811i
\(990\) 0 0
\(991\) −14.5023 25.1187i −0.460681 0.797922i 0.538314 0.842744i \(-0.319061\pi\)
−0.998995 + 0.0448218i \(0.985728\pi\)
\(992\) 0 0
\(993\) 25.4867 + 44.1442i 0.808795 + 1.40087i
\(994\) 0 0
\(995\) 38.8978 1.23314
\(996\) 0 0
\(997\) −27.1384 15.6684i −0.859483 0.496223i 0.00435617 0.999991i \(-0.498613\pi\)
−0.863839 + 0.503768i \(0.831947\pi\)
\(998\) 0 0
\(999\) 5.13786i 0.162555i
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 1216.2.s.g.31.5 yes 12
4.3 odd 2 inner 1216.2.s.g.31.2 12
8.3 odd 2 1216.2.s.h.31.5 yes 12
8.5 even 2 1216.2.s.h.31.2 yes 12
19.8 odd 6 1216.2.s.h.863.5 yes 12
76.27 even 6 1216.2.s.h.863.2 yes 12
152.27 even 6 inner 1216.2.s.g.863.5 yes 12
152.141 odd 6 inner 1216.2.s.g.863.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.s.g.31.2 12 4.3 odd 2 inner
1216.2.s.g.31.5 yes 12 1.1 even 1 trivial
1216.2.s.g.863.2 yes 12 152.141 odd 6 inner
1216.2.s.g.863.5 yes 12 152.27 even 6 inner
1216.2.s.h.31.2 yes 12 8.5 even 2
1216.2.s.h.31.5 yes 12 8.3 odd 2
1216.2.s.h.863.2 yes 12 76.27 even 6
1216.2.s.h.863.5 yes 12 19.8 odd 6