Properties

Label 624.2.q.h.529.1
Level $624$
Weight $2$
Character 624.529
Analytic conductor $4.983$
Analytic rank $0$
Dimension $4$
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] = [624,2,Mod(289,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.q (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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(1.28078 + 2.21837i\) of defining polynomial
Character \(\chi\) \(=\) 624.529
Dual form 624.2.q.h.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} -3.56155 q^{5} +(0.280776 + 0.486319i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} -3.56155 q^{5} +(0.280776 + 0.486319i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +(0.500000 + 3.57071i) q^{13} +(-1.78078 + 3.08440i) q^{15} +(0.780776 + 1.35234i) q^{17} +(3.56155 + 6.16879i) q^{19} +0.561553 q^{21} +(1.00000 - 1.73205i) q^{23} +7.68466 q^{25} -1.00000 q^{27} +(-3.34233 + 5.78908i) q^{29} -2.56155 q^{31} +(1.00000 + 1.73205i) q^{33} +(-1.00000 - 1.73205i) q^{35} +(-3.78078 + 6.54850i) q^{37} +(3.34233 + 1.35234i) q^{39} +(0.780776 - 1.35234i) q^{41} +(2.28078 + 3.95042i) q^{43} +(1.78078 + 3.08440i) q^{45} -8.24621 q^{47} +(3.34233 - 5.78908i) q^{49} +1.56155 q^{51} -0.684658 q^{53} +(3.56155 - 6.16879i) q^{55} +7.12311 q^{57} +(-1.43845 - 2.49146i) q^{59} +(-1.93845 - 3.35749i) q^{61} +(0.280776 - 0.486319i) q^{63} +(-1.78078 - 12.7173i) q^{65} +(2.28078 - 3.95042i) q^{67} +(-1.00000 - 1.73205i) q^{69} +(7.00000 + 12.1244i) q^{71} -10.1231 q^{73} +(3.84233 - 6.65511i) q^{75} -1.12311 q^{77} -5.43845 q^{79} +(-0.500000 + 0.866025i) q^{81} +0.876894 q^{83} +(-2.78078 - 4.81645i) q^{85} +(3.34233 + 5.78908i) q^{87} +(-2.43845 + 4.22351i) q^{89} +(-1.59612 + 1.24573i) q^{91} +(-1.28078 + 2.21837i) q^{93} +(-12.6847 - 21.9705i) q^{95} +(4.28078 + 7.41452i) q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{5} - 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 6 q^{5} - 3 q^{7} - 2 q^{9} - 4 q^{11} + 2 q^{13} - 3 q^{15} - q^{17} + 6 q^{19} - 6 q^{21} + 4 q^{23} + 6 q^{25} - 4 q^{27} - q^{29} - 2 q^{31} + 4 q^{33} - 4 q^{35} - 11 q^{37} + q^{39} - q^{41} + 5 q^{43} + 3 q^{45} + q^{49} - 2 q^{51} + 22 q^{53} + 6 q^{55} + 12 q^{57} - 14 q^{59} - 16 q^{61} - 3 q^{63} - 3 q^{65} + 5 q^{67} - 4 q^{69} + 28 q^{71} - 24 q^{73} + 3 q^{75} + 12 q^{77} - 30 q^{79} - 2 q^{81} + 20 q^{83} - 7 q^{85} + q^{87} - 18 q^{89} - 27 q^{91} - q^{93} - 26 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 0.280776 + 0.486319i 0.106124 + 0.183811i 0.914197 0.405271i \(-0.132823\pi\)
−0.808073 + 0.589082i \(0.799489\pi\)
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) −1.00000 + 1.73205i −0.301511 + 0.522233i −0.976478 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0 0
\(13\) 0.500000 + 3.57071i 0.138675 + 0.990338i
\(14\) 0 0
\(15\) −1.78078 + 3.08440i −0.459794 + 0.796387i
\(16\) 0 0
\(17\) 0.780776 + 1.35234i 0.189366 + 0.327992i 0.945039 0.326957i \(-0.106023\pi\)
−0.755673 + 0.654949i \(0.772690\pi\)
\(18\) 0 0
\(19\) 3.56155 + 6.16879i 0.817076 + 1.41522i 0.907827 + 0.419344i \(0.137740\pi\)
−0.0907512 + 0.995874i \(0.528927\pi\)
\(20\) 0 0
\(21\) 0.561553 0.122541
\(22\) 0 0
\(23\) 1.00000 1.73205i 0.208514 0.361158i −0.742732 0.669588i \(-0.766471\pi\)
0.951247 + 0.308431i \(0.0998038\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.34233 + 5.78908i −0.620655 + 1.07501i 0.368709 + 0.929545i \(0.379800\pi\)
−0.989364 + 0.145461i \(0.953533\pi\)
\(30\) 0 0
\(31\) −2.56155 −0.460068 −0.230034 0.973183i \(-0.573884\pi\)
−0.230034 + 0.973183i \(0.573884\pi\)
\(32\) 0 0
\(33\) 1.00000 + 1.73205i 0.174078 + 0.301511i
\(34\) 0 0
\(35\) −1.00000 1.73205i −0.169031 0.292770i
\(36\) 0 0
\(37\) −3.78078 + 6.54850i −0.621556 + 1.07657i 0.367640 + 0.929968i \(0.380166\pi\)
−0.989196 + 0.146598i \(0.953168\pi\)
\(38\) 0 0
\(39\) 3.34233 + 1.35234i 0.535201 + 0.216548i
\(40\) 0 0
\(41\) 0.780776 1.35234i 0.121937 0.211201i −0.798595 0.601869i \(-0.794423\pi\)
0.920531 + 0.390669i \(0.127756\pi\)
\(42\) 0 0
\(43\) 2.28078 + 3.95042i 0.347815 + 0.602433i 0.985861 0.167565i \(-0.0535903\pi\)
−0.638046 + 0.769998i \(0.720257\pi\)
\(44\) 0 0
\(45\) 1.78078 + 3.08440i 0.265462 + 0.459794i
\(46\) 0 0
\(47\) −8.24621 −1.20283 −0.601417 0.798935i \(-0.705397\pi\)
−0.601417 + 0.798935i \(0.705397\pi\)
\(48\) 0 0
\(49\) 3.34233 5.78908i 0.477476 0.827012i
\(50\) 0 0
\(51\) 1.56155 0.218661
\(52\) 0 0
\(53\) −0.684658 −0.0940451 −0.0470225 0.998894i \(-0.514973\pi\)
−0.0470225 + 0.998894i \(0.514973\pi\)
\(54\) 0 0
\(55\) 3.56155 6.16879i 0.480240 0.831800i
\(56\) 0 0
\(57\) 7.12311 0.943478
\(58\) 0 0
\(59\) −1.43845 2.49146i −0.187270 0.324361i 0.757069 0.653335i \(-0.226631\pi\)
−0.944339 + 0.328974i \(0.893297\pi\)
\(60\) 0 0
\(61\) −1.93845 3.35749i −0.248193 0.429882i 0.714832 0.699297i \(-0.246503\pi\)
−0.963024 + 0.269414i \(0.913170\pi\)
\(62\) 0 0
\(63\) 0.280776 0.486319i 0.0353745 0.0612704i
\(64\) 0 0
\(65\) −1.78078 12.7173i −0.220878 1.57739i
\(66\) 0 0
\(67\) 2.28078 3.95042i 0.278641 0.482621i −0.692406 0.721508i \(-0.743449\pi\)
0.971047 + 0.238887i \(0.0767826\pi\)
\(68\) 0 0
\(69\) −1.00000 1.73205i −0.120386 0.208514i
\(70\) 0 0
\(71\) 7.00000 + 12.1244i 0.830747 + 1.43890i 0.897447 + 0.441123i \(0.145420\pi\)
−0.0666994 + 0.997773i \(0.521247\pi\)
\(72\) 0 0
\(73\) −10.1231 −1.18482 −0.592410 0.805637i \(-0.701823\pi\)
−0.592410 + 0.805637i \(0.701823\pi\)
\(74\) 0 0
\(75\) 3.84233 6.65511i 0.443674 0.768466i
\(76\) 0 0
\(77\) −1.12311 −0.127990
\(78\) 0 0
\(79\) −5.43845 −0.611873 −0.305937 0.952052i \(-0.598970\pi\)
−0.305937 + 0.952052i \(0.598970\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 0.876894 0.0962517 0.0481258 0.998841i \(-0.484675\pi\)
0.0481258 + 0.998841i \(0.484675\pi\)
\(84\) 0 0
\(85\) −2.78078 4.81645i −0.301618 0.522417i
\(86\) 0 0
\(87\) 3.34233 + 5.78908i 0.358335 + 0.620655i
\(88\) 0 0
\(89\) −2.43845 + 4.22351i −0.258475 + 0.447692i −0.965834 0.259163i \(-0.916553\pi\)
0.707359 + 0.706855i \(0.249887\pi\)
\(90\) 0 0
\(91\) −1.59612 + 1.24573i −0.167319 + 0.130588i
\(92\) 0 0
\(93\) −1.28078 + 2.21837i −0.132810 + 0.230034i
\(94\) 0 0
\(95\) −12.6847 21.9705i −1.30142 2.25412i
\(96\) 0 0
\(97\) 4.28078 + 7.41452i 0.434647 + 0.752831i 0.997267 0.0738851i \(-0.0235398\pi\)
−0.562620 + 0.826716i \(0.690206\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 3.78078 6.54850i 0.376201 0.651600i −0.614305 0.789069i \(-0.710563\pi\)
0.990506 + 0.137469i \(0.0438968\pi\)
\(102\) 0 0
\(103\) 3.43845 0.338800 0.169400 0.985547i \(-0.445817\pi\)
0.169400 + 0.985547i \(0.445817\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) −4.12311 + 7.14143i −0.398596 + 0.690388i −0.993553 0.113369i \(-0.963836\pi\)
0.594957 + 0.803757i \(0.297169\pi\)
\(108\) 0 0
\(109\) −2.80776 −0.268935 −0.134468 0.990918i \(-0.542932\pi\)
−0.134468 + 0.990918i \(0.542932\pi\)
\(110\) 0 0
\(111\) 3.78078 + 6.54850i 0.358855 + 0.621556i
\(112\) 0 0
\(113\) −2.90388 5.02967i −0.273174 0.473152i 0.696499 0.717558i \(-0.254740\pi\)
−0.969673 + 0.244406i \(0.921407\pi\)
\(114\) 0 0
\(115\) −3.56155 + 6.16879i −0.332117 + 0.575243i
\(116\) 0 0
\(117\) 2.84233 2.21837i 0.262773 0.205088i
\(118\) 0 0
\(119\) −0.438447 + 0.759413i −0.0401924 + 0.0696153i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) −0.780776 1.35234i −0.0704002 0.121937i
\(124\) 0 0
\(125\) −9.56155 −0.855211
\(126\) 0 0
\(127\) 2.71922 4.70983i 0.241292 0.417930i −0.719791 0.694191i \(-0.755762\pi\)
0.961083 + 0.276261i \(0.0890955\pi\)
\(128\) 0 0
\(129\) 4.56155 0.401622
\(130\) 0 0
\(131\) −7.36932 −0.643860 −0.321930 0.946763i \(-0.604332\pi\)
−0.321930 + 0.946763i \(0.604332\pi\)
\(132\) 0 0
\(133\) −2.00000 + 3.46410i −0.173422 + 0.300376i
\(134\) 0 0
\(135\) 3.56155 0.306530
\(136\) 0 0
\(137\) 2.78078 + 4.81645i 0.237578 + 0.411497i 0.960019 0.279936i \(-0.0903132\pi\)
−0.722441 + 0.691433i \(0.756980\pi\)
\(138\) 0 0
\(139\) −8.96543 15.5286i −0.760438 1.31712i −0.942625 0.333854i \(-0.891651\pi\)
0.182187 0.983264i \(-0.441682\pi\)
\(140\) 0 0
\(141\) −4.12311 + 7.14143i −0.347228 + 0.601417i
\(142\) 0 0
\(143\) −6.68466 2.70469i −0.558999 0.226177i
\(144\) 0 0
\(145\) 11.9039 20.6181i 0.988564 1.71224i
\(146\) 0 0
\(147\) −3.34233 5.78908i −0.275671 0.477476i
\(148\) 0 0
\(149\) 1.21922 + 2.11176i 0.0998827 + 0.173002i 0.911636 0.410999i \(-0.134820\pi\)
−0.811753 + 0.584001i \(0.801487\pi\)
\(150\) 0 0
\(151\) 9.36932 0.762464 0.381232 0.924479i \(-0.375500\pi\)
0.381232 + 0.924479i \(0.375500\pi\)
\(152\) 0 0
\(153\) 0.780776 1.35234i 0.0631220 0.109331i
\(154\) 0 0
\(155\) 9.12311 0.732785
\(156\) 0 0
\(157\) 20.3693 1.62565 0.812824 0.582509i \(-0.197929\pi\)
0.812824 + 0.582509i \(0.197929\pi\)
\(158\) 0 0
\(159\) −0.342329 + 0.592932i −0.0271485 + 0.0470225i
\(160\) 0 0
\(161\) 1.12311 0.0885131
\(162\) 0 0
\(163\) −2.40388 4.16365i −0.188287 0.326122i 0.756393 0.654118i \(-0.226960\pi\)
−0.944679 + 0.327996i \(0.893627\pi\)
\(164\) 0 0
\(165\) −3.56155 6.16879i −0.277267 0.480240i
\(166\) 0 0
\(167\) 5.12311 8.87348i 0.396438 0.686650i −0.596846 0.802356i \(-0.703580\pi\)
0.993284 + 0.115706i \(0.0369129\pi\)
\(168\) 0 0
\(169\) −12.5000 + 3.57071i −0.961538 + 0.274670i
\(170\) 0 0
\(171\) 3.56155 6.16879i 0.272359 0.471739i
\(172\) 0 0
\(173\) −10.1231 17.5337i −0.769645 1.33307i −0.937755 0.347297i \(-0.887100\pi\)
0.168110 0.985768i \(-0.446234\pi\)
\(174\) 0 0
\(175\) 2.15767 + 3.73720i 0.163105 + 0.282505i
\(176\) 0 0
\(177\) −2.87689 −0.216241
\(178\) 0 0
\(179\) −2.43845 + 4.22351i −0.182258 + 0.315680i −0.942649 0.333785i \(-0.891674\pi\)
0.760391 + 0.649466i \(0.225007\pi\)
\(180\) 0 0
\(181\) −2.68466 −0.199549 −0.0997745 0.995010i \(-0.531812\pi\)
−0.0997745 + 0.995010i \(0.531812\pi\)
\(182\) 0 0
\(183\) −3.87689 −0.286588
\(184\) 0 0
\(185\) 13.4654 23.3228i 0.989998 1.71473i
\(186\) 0 0
\(187\) −3.12311 −0.228384
\(188\) 0 0
\(189\) −0.280776 0.486319i −0.0204235 0.0353745i
\(190\) 0 0
\(191\) −4.56155 7.90084i −0.330062 0.571685i 0.652461 0.757822i \(-0.273736\pi\)
−0.982524 + 0.186137i \(0.940403\pi\)
\(192\) 0 0
\(193\) 6.74621 11.6848i 0.485603 0.841089i −0.514260 0.857634i \(-0.671933\pi\)
0.999863 + 0.0165453i \(0.00526677\pi\)
\(194\) 0 0
\(195\) −11.9039 4.81645i −0.852455 0.344913i
\(196\) 0 0
\(197\) −6.68466 + 11.5782i −0.476262 + 0.824910i −0.999630 0.0271965i \(-0.991342\pi\)
0.523368 + 0.852107i \(0.324675\pi\)
\(198\) 0 0
\(199\) 11.0885 + 19.2059i 0.786046 + 1.36147i 0.928372 + 0.371651i \(0.121208\pi\)
−0.142327 + 0.989820i \(0.545458\pi\)
\(200\) 0 0
\(201\) −2.28078 3.95042i −0.160874 0.278641i
\(202\) 0 0
\(203\) −3.75379 −0.263464
\(204\) 0 0
\(205\) −2.78078 + 4.81645i −0.194218 + 0.336395i
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −14.2462 −0.985431
\(210\) 0 0
\(211\) 9.84233 17.0474i 0.677574 1.17359i −0.298136 0.954524i \(-0.596365\pi\)
0.975709 0.219069i \(-0.0703019\pi\)
\(212\) 0 0
\(213\) 14.0000 0.959264
\(214\) 0 0
\(215\) −8.12311 14.0696i −0.553991 0.959541i
\(216\) 0 0
\(217\) −0.719224 1.24573i −0.0488241 0.0845658i
\(218\) 0 0
\(219\) −5.06155 + 8.76687i −0.342028 + 0.592410i
\(220\) 0 0
\(221\) −4.43845 + 3.46410i −0.298562 + 0.233021i
\(222\) 0 0
\(223\) 4.00000 6.92820i 0.267860 0.463947i −0.700449 0.713702i \(-0.747017\pi\)
0.968309 + 0.249756i \(0.0803503\pi\)
\(224\) 0 0
\(225\) −3.84233 6.65511i −0.256155 0.443674i
\(226\) 0 0
\(227\) 3.56155 + 6.16879i 0.236389 + 0.409437i 0.959675 0.281111i \(-0.0907028\pi\)
−0.723287 + 0.690548i \(0.757370\pi\)
\(228\) 0 0
\(229\) −16.2462 −1.07358 −0.536790 0.843716i \(-0.680363\pi\)
−0.536790 + 0.843716i \(0.680363\pi\)
\(230\) 0 0
\(231\) −0.561553 + 0.972638i −0.0369475 + 0.0639949i
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 29.3693 1.91584
\(236\) 0 0
\(237\) −2.71922 + 4.70983i −0.176633 + 0.305937i
\(238\) 0 0
\(239\) 25.3693 1.64100 0.820502 0.571643i \(-0.193694\pi\)
0.820502 + 0.571643i \(0.193694\pi\)
\(240\) 0 0
\(241\) 8.90388 + 15.4220i 0.573549 + 0.993417i 0.996198 + 0.0871229i \(0.0277673\pi\)
−0.422648 + 0.906294i \(0.638899\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) −11.9039 + 20.6181i −0.760511 + 1.31724i
\(246\) 0 0
\(247\) −20.2462 + 15.8017i −1.28824 + 1.00544i
\(248\) 0 0
\(249\) 0.438447 0.759413i 0.0277855 0.0481258i
\(250\) 0 0
\(251\) 9.36932 + 16.2281i 0.591386 + 1.02431i 0.994046 + 0.108961i \(0.0347524\pi\)
−0.402660 + 0.915350i \(0.631914\pi\)
\(252\) 0 0
\(253\) 2.00000 + 3.46410i 0.125739 + 0.217786i
\(254\) 0 0
\(255\) −5.56155 −0.348278
\(256\) 0 0
\(257\) −14.5885 + 25.2681i −0.910008 + 1.57618i −0.0959583 + 0.995385i \(0.530592\pi\)
−0.814050 + 0.580795i \(0.802742\pi\)
\(258\) 0 0
\(259\) −4.24621 −0.263847
\(260\) 0 0
\(261\) 6.68466 0.413770
\(262\) 0 0
\(263\) 4.68466 8.11407i 0.288868 0.500335i −0.684672 0.728852i \(-0.740054\pi\)
0.973540 + 0.228517i \(0.0733877\pi\)
\(264\) 0 0
\(265\) 2.43845 0.149793
\(266\) 0 0
\(267\) 2.43845 + 4.22351i 0.149231 + 0.258475i
\(268\) 0 0
\(269\) 10.6847 + 18.5064i 0.651455 + 1.12835i 0.982770 + 0.184833i \(0.0591745\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(270\) 0 0
\(271\) −14.9654 + 25.9209i −0.909085 + 1.57458i −0.0937481 + 0.995596i \(0.529885\pi\)
−0.815337 + 0.578986i \(0.803449\pi\)
\(272\) 0 0
\(273\) 0.280776 + 2.00514i 0.0169934 + 0.121357i
\(274\) 0 0
\(275\) −7.68466 + 13.3102i −0.463402 + 0.802636i
\(276\) 0 0
\(277\) −2.65767 4.60322i −0.159684 0.276581i 0.775071 0.631874i \(-0.217714\pi\)
−0.934755 + 0.355294i \(0.884381\pi\)
\(278\) 0 0
\(279\) 1.28078 + 2.21837i 0.0766781 + 0.132810i
\(280\) 0 0
\(281\) 17.8078 1.06232 0.531161 0.847271i \(-0.321756\pi\)
0.531161 + 0.847271i \(0.321756\pi\)
\(282\) 0 0
\(283\) −6.84233 + 11.8513i −0.406734 + 0.704484i −0.994522 0.104531i \(-0.966666\pi\)
0.587787 + 0.809015i \(0.299999\pi\)
\(284\) 0 0
\(285\) −25.3693 −1.50275
\(286\) 0 0
\(287\) 0.876894 0.0517614
\(288\) 0 0
\(289\) 7.28078 12.6107i 0.428281 0.741804i
\(290\) 0 0
\(291\) 8.56155 0.501887
\(292\) 0 0
\(293\) −10.2192 17.7002i −0.597013 1.03406i −0.993259 0.115913i \(-0.963021\pi\)
0.396246 0.918144i \(-0.370313\pi\)
\(294\) 0 0
\(295\) 5.12311 + 8.87348i 0.298279 + 0.516634i
\(296\) 0 0
\(297\) 1.00000 1.73205i 0.0580259 0.100504i
\(298\) 0 0
\(299\) 6.68466 + 2.70469i 0.386584 + 0.156416i
\(300\) 0 0
\(301\) −1.28078 + 2.21837i −0.0738227 + 0.127865i
\(302\) 0 0
\(303\) −3.78078 6.54850i −0.217200 0.376201i
\(304\) 0 0
\(305\) 6.90388 + 11.9579i 0.395315 + 0.684706i
\(306\) 0 0
\(307\) −30.8078 −1.75829 −0.879146 0.476553i \(-0.841886\pi\)
−0.879146 + 0.476553i \(0.841886\pi\)
\(308\) 0 0
\(309\) 1.71922 2.97778i 0.0978032 0.169400i
\(310\) 0 0
\(311\) 19.1231 1.08437 0.542186 0.840259i \(-0.317597\pi\)
0.542186 + 0.840259i \(0.317597\pi\)
\(312\) 0 0
\(313\) −13.6847 −0.773503 −0.386751 0.922184i \(-0.626403\pi\)
−0.386751 + 0.922184i \(0.626403\pi\)
\(314\) 0 0
\(315\) −1.00000 + 1.73205i −0.0563436 + 0.0975900i
\(316\) 0 0
\(317\) 14.0540 0.789350 0.394675 0.918821i \(-0.370857\pi\)
0.394675 + 0.918821i \(0.370857\pi\)
\(318\) 0 0
\(319\) −6.68466 11.5782i −0.374269 0.648253i
\(320\) 0 0
\(321\) 4.12311 + 7.14143i 0.230129 + 0.398596i
\(322\) 0 0
\(323\) −5.56155 + 9.63289i −0.309453 + 0.535988i
\(324\) 0 0
\(325\) 3.84233 + 27.4397i 0.213134 + 1.52208i
\(326\) 0 0
\(327\) −1.40388 + 2.43160i −0.0776349 + 0.134468i
\(328\) 0 0
\(329\) −2.31534 4.01029i −0.127649 0.221094i
\(330\) 0 0
\(331\) −1.59612 2.76456i −0.0877306 0.151954i 0.818821 0.574049i \(-0.194628\pi\)
−0.906552 + 0.422095i \(0.861295\pi\)
\(332\) 0 0
\(333\) 7.56155 0.414371
\(334\) 0 0
\(335\) −8.12311 + 14.0696i −0.443813 + 0.768706i
\(336\) 0 0
\(337\) −6.12311 −0.333547 −0.166773 0.985995i \(-0.553335\pi\)
−0.166773 + 0.985995i \(0.553335\pi\)
\(338\) 0 0
\(339\) −5.80776 −0.315434
\(340\) 0 0
\(341\) 2.56155 4.43674i 0.138716 0.240263i
\(342\) 0 0
\(343\) 7.68466 0.414933
\(344\) 0 0
\(345\) 3.56155 + 6.16879i 0.191748 + 0.332117i
\(346\) 0 0
\(347\) 13.8078 + 23.9157i 0.741240 + 1.28386i 0.951931 + 0.306312i \(0.0990951\pi\)
−0.210692 + 0.977553i \(0.567572\pi\)
\(348\) 0 0
\(349\) −3.40388 + 5.89570i −0.182206 + 0.315589i −0.942631 0.333836i \(-0.891657\pi\)
0.760426 + 0.649425i \(0.224990\pi\)
\(350\) 0 0
\(351\) −0.500000 3.57071i −0.0266880 0.190591i
\(352\) 0 0
\(353\) −2.65767 + 4.60322i −0.141454 + 0.245005i −0.928044 0.372470i \(-0.878511\pi\)
0.786591 + 0.617475i \(0.211844\pi\)
\(354\) 0 0
\(355\) −24.9309 43.1815i −1.32319 2.29184i
\(356\) 0 0
\(357\) 0.438447 + 0.759413i 0.0232051 + 0.0401924i
\(358\) 0 0
\(359\) 9.36932 0.494494 0.247247 0.968953i \(-0.420474\pi\)
0.247247 + 0.968953i \(0.420474\pi\)
\(360\) 0 0
\(361\) −15.8693 + 27.4865i −0.835227 + 1.44666i
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 36.0540 1.88715
\(366\) 0 0
\(367\) −8.52699 + 14.7692i −0.445105 + 0.770945i −0.998060 0.0622668i \(-0.980167\pi\)
0.552954 + 0.833212i \(0.313500\pi\)
\(368\) 0 0
\(369\) −1.56155 −0.0812912
\(370\) 0 0
\(371\) −0.192236 0.332962i −0.00998039 0.0172865i
\(372\) 0 0
\(373\) −14.1847 24.5685i −0.734454 1.27211i −0.954963 0.296726i \(-0.904105\pi\)
0.220509 0.975385i \(-0.429228\pi\)
\(374\) 0 0
\(375\) −4.78078 + 8.28055i −0.246878 + 0.427606i
\(376\) 0 0
\(377\) −22.3423 9.03996i −1.15069 0.465582i
\(378\) 0 0
\(379\) 11.8423 20.5115i 0.608300 1.05361i −0.383221 0.923657i \(-0.625185\pi\)
0.991521 0.129949i \(-0.0414814\pi\)
\(380\) 0 0
\(381\) −2.71922 4.70983i −0.139310 0.241292i
\(382\) 0 0
\(383\) −11.3693 19.6922i −0.580945 1.00623i −0.995368 0.0961417i \(-0.969350\pi\)
0.414423 0.910085i \(-0.363984\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 2.28078 3.95042i 0.115938 0.200811i
\(388\) 0 0
\(389\) 34.0540 1.72661 0.863303 0.504687i \(-0.168392\pi\)
0.863303 + 0.504687i \(0.168392\pi\)
\(390\) 0 0
\(391\) 3.12311 0.157942
\(392\) 0 0
\(393\) −3.68466 + 6.38202i −0.185866 + 0.321930i
\(394\) 0 0
\(395\) 19.3693 0.974576
\(396\) 0 0
\(397\) −12.5270 21.6974i −0.628711 1.08896i −0.987811 0.155661i \(-0.950249\pi\)
0.359099 0.933299i \(-0.383084\pi\)
\(398\) 0 0
\(399\) 2.00000 + 3.46410i 0.100125 + 0.173422i
\(400\) 0 0
\(401\) −7.21922 + 12.5041i −0.360511 + 0.624423i −0.988045 0.154166i \(-0.950731\pi\)
0.627534 + 0.778589i \(0.284064\pi\)
\(402\) 0 0
\(403\) −1.28078 9.14657i −0.0638000 0.455623i
\(404\) 0 0
\(405\) 1.78078 3.08440i 0.0884875 0.153265i
\(406\) 0 0
\(407\) −7.56155 13.0970i −0.374812 0.649194i
\(408\) 0 0
\(409\) −3.18466 5.51599i −0.157471 0.272748i 0.776485 0.630136i \(-0.217001\pi\)
−0.933956 + 0.357388i \(0.883667\pi\)
\(410\) 0 0
\(411\) 5.56155 0.274331
\(412\) 0 0
\(413\) 0.807764 1.39909i 0.0397475 0.0688446i
\(414\) 0 0
\(415\) −3.12311 −0.153307
\(416\) 0 0
\(417\) −17.9309 −0.878078
\(418\) 0 0
\(419\) −17.1231 + 29.6581i −0.836518 + 1.44889i 0.0562697 + 0.998416i \(0.482079\pi\)
−0.892788 + 0.450477i \(0.851254\pi\)
\(420\) 0 0
\(421\) 31.2462 1.52285 0.761424 0.648255i \(-0.224501\pi\)
0.761424 + 0.648255i \(0.224501\pi\)
\(422\) 0 0
\(423\) 4.12311 + 7.14143i 0.200472 + 0.347228i
\(424\) 0 0
\(425\) 6.00000 + 10.3923i 0.291043 + 0.504101i
\(426\) 0 0
\(427\) 1.08854 1.88541i 0.0526782 0.0912413i
\(428\) 0 0
\(429\) −5.68466 + 4.43674i −0.274458 + 0.214208i
\(430\) 0 0
\(431\) −5.56155 + 9.63289i −0.267891 + 0.464000i −0.968317 0.249725i \(-0.919660\pi\)
0.700426 + 0.713725i \(0.252993\pi\)
\(432\) 0 0
\(433\) −4.37689 7.58100i −0.210340 0.364320i 0.741481 0.670974i \(-0.234124\pi\)
−0.951821 + 0.306654i \(0.900790\pi\)
\(434\) 0 0
\(435\) −11.9039 20.6181i −0.570747 0.988564i
\(436\) 0 0
\(437\) 14.2462 0.681489
\(438\) 0 0
\(439\) −6.84233 + 11.8513i −0.326567 + 0.565630i −0.981828 0.189772i \(-0.939225\pi\)
0.655262 + 0.755402i \(0.272558\pi\)
\(440\) 0 0
\(441\) −6.68466 −0.318317
\(442\) 0 0
\(443\) 34.7386 1.65048 0.825241 0.564781i \(-0.191039\pi\)
0.825241 + 0.564781i \(0.191039\pi\)
\(444\) 0 0
\(445\) 8.68466 15.0423i 0.411692 0.713072i
\(446\) 0 0
\(447\) 2.43845 0.115335
\(448\) 0 0
\(449\) 4.12311 + 7.14143i 0.194581 + 0.337025i 0.946763 0.321931i \(-0.104332\pi\)
−0.752182 + 0.658956i \(0.770998\pi\)
\(450\) 0 0
\(451\) 1.56155 + 2.70469i 0.0735307 + 0.127359i
\(452\) 0 0
\(453\) 4.68466 8.11407i 0.220104 0.381232i
\(454\) 0 0
\(455\) 5.68466 4.43674i 0.266501 0.207998i
\(456\) 0 0
\(457\) −6.30776 + 10.9254i −0.295065 + 0.511067i −0.975000 0.222205i \(-0.928675\pi\)
0.679935 + 0.733272i \(0.262008\pi\)
\(458\) 0 0
\(459\) −0.780776 1.35234i −0.0364435 0.0631220i
\(460\) 0 0
\(461\) 8.09612 + 14.0229i 0.377074 + 0.653111i 0.990635 0.136536i \(-0.0435970\pi\)
−0.613561 + 0.789647i \(0.710264\pi\)
\(462\) 0 0
\(463\) 14.3153 0.665290 0.332645 0.943052i \(-0.392059\pi\)
0.332645 + 0.943052i \(0.392059\pi\)
\(464\) 0 0
\(465\) 4.56155 7.90084i 0.211537 0.366393i
\(466\) 0 0
\(467\) 26.0000 1.20314 0.601568 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) 0 0
\(469\) 2.56155 0.118282
\(470\) 0 0
\(471\) 10.1847 17.6403i 0.469284 0.812824i
\(472\) 0 0
\(473\) −9.12311 −0.419481
\(474\) 0 0
\(475\) 27.3693 + 47.4050i 1.25579 + 2.17509i
\(476\) 0 0
\(477\) 0.342329 + 0.592932i 0.0156742 + 0.0271485i
\(478\) 0 0
\(479\) 5.12311 8.87348i 0.234081 0.405440i −0.724924 0.688828i \(-0.758125\pi\)
0.959005 + 0.283389i \(0.0914587\pi\)
\(480\) 0 0
\(481\) −25.2732 10.2258i −1.15236 0.466257i
\(482\) 0 0
\(483\) 0.561553 0.972638i 0.0255515 0.0442566i
\(484\) 0 0
\(485\) −15.2462 26.4072i −0.692295 1.19909i
\(486\) 0 0
\(487\) 3.56155 + 6.16879i 0.161389 + 0.279535i 0.935367 0.353678i \(-0.115069\pi\)
−0.773978 + 0.633213i \(0.781736\pi\)
\(488\) 0 0
\(489\) −4.80776 −0.217415
\(490\) 0 0
\(491\) −18.1231 + 31.3901i −0.817884 + 1.41662i 0.0893539 + 0.996000i \(0.471520\pi\)
−0.907238 + 0.420617i \(0.861814\pi\)
\(492\) 0 0
\(493\) −10.4384 −0.470124
\(494\) 0 0
\(495\) −7.12311 −0.320160
\(496\) 0 0
\(497\) −3.93087 + 6.80847i −0.176324 + 0.305401i
\(498\) 0 0
\(499\) −4.49242 −0.201108 −0.100554 0.994932i \(-0.532062\pi\)
−0.100554 + 0.994932i \(0.532062\pi\)
\(500\) 0 0
\(501\) −5.12311 8.87348i −0.228883 0.396438i
\(502\) 0 0
\(503\) −14.1231 24.4619i −0.629718 1.09070i −0.987608 0.156940i \(-0.949837\pi\)
0.357890 0.933764i \(-0.383496\pi\)
\(504\) 0 0
\(505\) −13.4654 + 23.3228i −0.599204 + 1.03785i
\(506\) 0 0
\(507\) −3.15767 + 12.6107i −0.140237 + 0.560060i
\(508\) 0 0
\(509\) 6.90388 11.9579i 0.306009 0.530023i −0.671476 0.741026i \(-0.734340\pi\)
0.977486 + 0.211003i \(0.0676728\pi\)
\(510\) 0 0
\(511\) −2.84233 4.92306i −0.125737 0.217783i
\(512\) 0 0
\(513\) −3.56155 6.16879i −0.157246 0.272359i
\(514\) 0 0
\(515\) −12.2462 −0.539633
\(516\) 0 0
\(517\) 8.24621 14.2829i 0.362668 0.628159i
\(518\) 0 0
\(519\) −20.2462 −0.888710
\(520\) 0 0
\(521\) −9.06913 −0.397326 −0.198663 0.980068i \(-0.563660\pi\)
−0.198663 + 0.980068i \(0.563660\pi\)
\(522\) 0 0
\(523\) 16.9309 29.3251i 0.740335 1.28230i −0.212007 0.977268i \(-0.568000\pi\)
0.952343 0.305030i \(-0.0986666\pi\)
\(524\) 0 0
\(525\) 4.31534 0.188337
\(526\) 0 0
\(527\) −2.00000 3.46410i −0.0871214 0.150899i
\(528\) 0 0
\(529\) 9.50000 + 16.4545i 0.413043 + 0.715412i
\(530\) 0 0
\(531\) −1.43845 + 2.49146i −0.0624233 + 0.108120i
\(532\) 0 0
\(533\) 5.21922 + 2.11176i 0.226070 + 0.0914704i
\(534\) 0 0
\(535\) 14.6847 25.4346i 0.634873 1.09963i
\(536\) 0 0
\(537\) 2.43845 + 4.22351i 0.105227 + 0.182258i
\(538\) 0 0
\(539\) 6.68466 + 11.5782i 0.287929 + 0.498707i
\(540\) 0 0
\(541\) 19.7386 0.848630 0.424315 0.905515i \(-0.360515\pi\)
0.424315 + 0.905515i \(0.360515\pi\)
\(542\) 0 0
\(543\) −1.34233 + 2.32498i −0.0576049 + 0.0997745i
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −3.93087 −0.168072 −0.0840359 0.996463i \(-0.526781\pi\)
−0.0840359 + 0.996463i \(0.526781\pi\)
\(548\) 0 0
\(549\) −1.93845 + 3.35749i −0.0827309 + 0.143294i
\(550\) 0 0
\(551\) −47.6155 −2.02849
\(552\) 0 0
\(553\) −1.52699 2.64482i −0.0649341 0.112469i
\(554\) 0 0
\(555\) −13.4654 23.3228i −0.571576 0.989998i
\(556\) 0 0
\(557\) 21.4654 37.1792i 0.909520 1.57533i 0.0947869 0.995498i \(-0.469783\pi\)
0.814733 0.579837i \(-0.196884\pi\)
\(558\) 0 0
\(559\) −12.9654 + 10.1192i −0.548379 + 0.427997i
\(560\) 0 0
\(561\) −1.56155 + 2.70469i −0.0659288 + 0.114192i
\(562\) 0 0
\(563\) −11.6847 20.2384i −0.492450 0.852948i 0.507513 0.861644i \(-0.330565\pi\)
−0.999962 + 0.00869657i \(0.997232\pi\)
\(564\) 0 0
\(565\) 10.3423 + 17.9134i 0.435105 + 0.753624i
\(566\) 0 0
\(567\) −0.561553 −0.0235830
\(568\) 0 0
\(569\) 4.36932 7.56788i 0.183171 0.317262i −0.759787 0.650171i \(-0.774697\pi\)
0.942959 + 0.332910i \(0.108030\pi\)
\(570\) 0 0
\(571\) 5.36932 0.224699 0.112349 0.993669i \(-0.464162\pi\)
0.112349 + 0.993669i \(0.464162\pi\)
\(572\) 0 0
\(573\) −9.12311 −0.381123
\(574\) 0 0
\(575\) 7.68466 13.3102i 0.320472 0.555074i
\(576\) 0 0
\(577\) −17.3153 −0.720847 −0.360424 0.932789i \(-0.617368\pi\)
−0.360424 + 0.932789i \(0.617368\pi\)
\(578\) 0 0
\(579\) −6.74621 11.6848i −0.280363 0.485603i
\(580\) 0 0
\(581\) 0.246211 + 0.426450i 0.0102146 + 0.0176921i
\(582\) 0 0
\(583\) 0.684658 1.18586i 0.0283557 0.0491134i
\(584\) 0 0
\(585\) −10.1231 + 7.90084i −0.418539 + 0.326660i
\(586\) 0 0
\(587\) 19.6847 34.0948i 0.812473 1.40724i −0.0986556 0.995122i \(-0.531454\pi\)
0.911128 0.412123i \(-0.135212\pi\)
\(588\) 0 0
\(589\) −9.12311 15.8017i −0.375911 0.651097i
\(590\) 0 0
\(591\) 6.68466 + 11.5782i 0.274970 + 0.476262i
\(592\) 0 0
\(593\) −17.4233 −0.715489 −0.357744 0.933820i \(-0.616454\pi\)
−0.357744 + 0.933820i \(0.616454\pi\)
\(594\) 0 0
\(595\) 1.56155 2.70469i 0.0640174 0.110881i
\(596\) 0 0
\(597\) 22.1771 0.907648
\(598\) 0 0
\(599\) 41.6155 1.70036 0.850182 0.526489i \(-0.176492\pi\)
0.850182 + 0.526489i \(0.176492\pi\)
\(600\) 0 0
\(601\) 3.53457 6.12205i 0.144178 0.249723i −0.784888 0.619638i \(-0.787280\pi\)
0.929066 + 0.369914i \(0.120613\pi\)
\(602\) 0 0
\(603\) −4.56155 −0.185761
\(604\) 0 0
\(605\) −12.4654 21.5908i −0.506792 0.877789i
\(606\) 0 0
\(607\) −8.00000 13.8564i −0.324710 0.562414i 0.656744 0.754114i \(-0.271933\pi\)
−0.981454 + 0.191700i \(0.938600\pi\)
\(608\) 0 0
\(609\) −1.87689 + 3.25088i −0.0760556 + 0.131732i
\(610\) 0 0
\(611\) −4.12311 29.4449i −0.166803 1.19121i
\(612\) 0 0
\(613\) 17.4309 30.1912i 0.704026 1.21941i −0.263016 0.964792i \(-0.584717\pi\)
0.967042 0.254618i \(-0.0819496\pi\)
\(614\) 0 0
\(615\) 2.78078 + 4.81645i 0.112132 + 0.194218i
\(616\) 0 0
\(617\) −4.90388 8.49377i −0.197423 0.341946i 0.750269 0.661132i \(-0.229924\pi\)
−0.947692 + 0.319186i \(0.896591\pi\)
\(618\) 0 0
\(619\) −29.3002 −1.17767 −0.588837 0.808252i \(-0.700414\pi\)
−0.588837 + 0.808252i \(0.700414\pi\)
\(620\) 0 0
\(621\) −1.00000 + 1.73205i −0.0401286 + 0.0695048i
\(622\) 0 0
\(623\) −2.73863 −0.109721
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) −7.12311 + 12.3376i −0.284469 + 0.492716i
\(628\) 0 0
\(629\) −11.8078 −0.470806
\(630\) 0 0
\(631\) −9.28078 16.0748i −0.369462 0.639927i 0.620020 0.784586i \(-0.287125\pi\)
−0.989481 + 0.144660i \(0.953791\pi\)
\(632\) 0 0
\(633\) −9.84233 17.0474i −0.391197 0.677574i
\(634\) 0 0
\(635\) −9.68466 + 16.7743i −0.384324 + 0.665669i
\(636\) 0 0
\(637\) 22.3423 + 9.03996i 0.885235 + 0.358176i
\(638\) 0 0
\(639\) 7.00000 12.1244i 0.276916 0.479632i
\(640\) 0 0
\(641\) 9.58854 + 16.6078i 0.378725 + 0.655970i 0.990877 0.134770i \(-0.0430294\pi\)
−0.612152 + 0.790740i \(0.709696\pi\)
\(642\) 0 0
\(643\) −15.7732 27.3200i −0.622034 1.07739i −0.989106 0.147202i \(-0.952973\pi\)
0.367072 0.930192i \(-0.380360\pi\)
\(644\) 0 0
\(645\) −16.2462 −0.639694
\(646\) 0 0
\(647\) −3.19224 + 5.52911i −0.125500 + 0.217372i −0.921928 0.387361i \(-0.873387\pi\)
0.796428 + 0.604733i \(0.206720\pi\)
\(648\) 0 0
\(649\) 5.75379 0.225856
\(650\) 0 0
\(651\) −1.43845 −0.0563772
\(652\) 0 0
\(653\) −11.5616 + 20.0252i −0.452439 + 0.783647i −0.998537 0.0540745i \(-0.982779\pi\)
0.546098 + 0.837721i \(0.316112\pi\)
\(654\) 0 0
\(655\) 26.2462 1.02552
\(656\) 0 0
\(657\) 5.06155 + 8.76687i 0.197470 + 0.342028i
\(658\) 0 0
\(659\) −1.12311 1.94528i −0.0437500 0.0757772i 0.843321 0.537410i \(-0.180597\pi\)
−0.887071 + 0.461633i \(0.847264\pi\)
\(660\) 0 0
\(661\) −2.81534 + 4.87631i −0.109504 + 0.189667i −0.915569 0.402160i \(-0.868260\pi\)
0.806065 + 0.591827i \(0.201593\pi\)
\(662\) 0 0
\(663\) 0.780776 + 5.57586i 0.0303228 + 0.216548i
\(664\) 0 0
\(665\) 7.12311 12.3376i 0.276222 0.478431i
\(666\) 0 0
\(667\) 6.68466 + 11.5782i 0.258831 + 0.448308i
\(668\) 0 0
\(669\) −4.00000 6.92820i −0.154649 0.267860i
\(670\) 0 0
\(671\) 7.75379 0.299332
\(672\) 0 0
\(673\) 11.6231 20.1318i 0.448038 0.776024i −0.550220 0.835019i \(-0.685456\pi\)
0.998258 + 0.0589952i \(0.0187897\pi\)
\(674\) 0 0
\(675\) −7.68466 −0.295783
\(676\) 0 0
\(677\) −15.6155 −0.600153 −0.300077 0.953915i \(-0.597012\pi\)
−0.300077 + 0.953915i \(0.597012\pi\)
\(678\) 0 0
\(679\) −2.40388 + 4.16365i −0.0922525 + 0.159786i
\(680\) 0 0
\(681\) 7.12311 0.272958
\(682\) 0 0
\(683\) −19.0540 33.0025i −0.729080 1.26280i −0.957272 0.289188i \(-0.906615\pi\)
0.228192 0.973616i \(-0.426719\pi\)
\(684\) 0 0
\(685\) −9.90388 17.1540i −0.378408 0.655422i
\(686\) 0 0
\(687\) −8.12311 + 14.0696i −0.309916 + 0.536790i
\(688\) 0 0
\(689\) −0.342329 2.44472i −0.0130417 0.0931364i
\(690\) 0 0
\(691\) −25.6501 + 44.4273i −0.975776 + 1.69009i −0.298424 + 0.954433i \(0.596461\pi\)
−0.677352 + 0.735659i \(0.736872\pi\)
\(692\) 0 0
\(693\) 0.561553 + 0.972638i 0.0213316 + 0.0369475i
\(694\) 0 0
\(695\) 31.9309 + 55.3059i 1.21121 + 2.09787i
\(696\) 0 0
\(697\) 2.43845 0.0923628
\(698\) 0 0
\(699\) 13.0000 22.5167i 0.491705 0.851658i
\(700\) 0 0
\(701\) −5.36932 −0.202796 −0.101398 0.994846i \(-0.532332\pi\)
−0.101398 + 0.994846i \(0.532332\pi\)
\(702\) 0 0
\(703\) −53.8617 −2.03143
\(704\) 0 0
\(705\) 14.6847 25.4346i 0.553056 0.957921i
\(706\) 0 0
\(707\) 4.24621 0.159695
\(708\) 0 0
\(709\) 3.74621 + 6.48863i 0.140692 + 0.243686i 0.927757 0.373184i \(-0.121734\pi\)
−0.787065 + 0.616869i \(0.788401\pi\)
\(710\) 0 0
\(711\) 2.71922 + 4.70983i 0.101979 + 0.176633i
\(712\) 0 0
\(713\) −2.56155 + 4.43674i −0.0959309 + 0.166157i
\(714\) 0 0
\(715\) 23.8078 + 9.63289i 0.890360 + 0.360250i
\(716\) 0 0
\(717\) 12.6847 21.9705i 0.473717 0.820502i
\(718\) 0 0
\(719\) −11.6847 20.2384i −0.435764 0.754766i 0.561593 0.827413i \(-0.310189\pi\)
−0.997358 + 0.0726475i \(0.976855\pi\)
\(720\) 0 0
\(721\) 0.965435 + 1.67218i 0.0359547 + 0.0622753i
\(722\) 0 0
\(723\) 17.8078 0.662278
\(724\) 0 0
\(725\) −25.6847 + 44.4871i −0.953904 + 1.65221i
\(726\) 0 0
\(727\) 38.6695 1.43417 0.717086 0.696984i \(-0.245475\pi\)
0.717086 + 0.696984i \(0.245475\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.56155 + 6.16879i −0.131729 + 0.228161i
\(732\) 0 0
\(733\) 20.5076 0.757465 0.378732 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(734\) 0 0
\(735\) 11.9039 + 20.6181i 0.439081 + 0.760511i
\(736\) 0 0
\(737\) 4.56155 + 7.90084i 0.168027 + 0.291031i
\(738\) 0 0
\(739\) 5.12311 8.87348i 0.188456 0.326416i −0.756279 0.654249i \(-0.772985\pi\)
0.944736 + 0.327833i \(0.106318\pi\)
\(740\) 0 0
\(741\) 3.56155 + 25.4346i 0.130837 + 0.934362i
\(742\) 0 0
\(743\) −6.31534 + 10.9385i −0.231687 + 0.401294i −0.958305 0.285748i \(-0.907758\pi\)
0.726617 + 0.687042i \(0.241091\pi\)
\(744\) 0 0
\(745\) −4.34233 7.52113i −0.159091 0.275553i
\(746\) 0 0
\(747\) −0.438447 0.759413i −0.0160419 0.0277855i
\(748\) 0 0
\(749\) −4.63068 −0.169201
\(750\) 0 0
\(751\) 22.0540 38.1986i 0.804761 1.39389i −0.111691 0.993743i \(-0.535627\pi\)
0.916452 0.400144i \(-0.131040\pi\)
\(752\) 0 0
\(753\) 18.7386 0.682874
\(754\) 0 0
\(755\) −33.3693 −1.21443
\(756\) 0 0
\(757\) −15.0000 + 25.9808i −0.545184 + 0.944287i 0.453411 + 0.891302i \(0.350207\pi\)
−0.998595 + 0.0529853i \(0.983126\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −4.68466 8.11407i −0.169819 0.294135i 0.768537 0.639805i \(-0.220985\pi\)
−0.938356 + 0.345670i \(0.887652\pi\)
\(762\) 0 0
\(763\) −0.788354 1.36547i −0.0285403 0.0494333i
\(764\) 0 0
\(765\) −2.78078 + 4.81645i −0.100539 + 0.174139i
\(766\) 0 0
\(767\) 8.17708 6.38202i 0.295257 0.230441i
\(768\) 0 0
\(769\) 9.00000 15.5885i 0.324548 0.562134i −0.656873 0.754002i \(-0.728121\pi\)
0.981421 + 0.191867i \(0.0614544\pi\)
\(770\) 0 0
\(771\) 14.5885 + 25.2681i 0.525393 + 0.910008i
\(772\) 0 0
\(773\) 12.1231 + 20.9978i 0.436038 + 0.755240i 0.997380 0.0723444i \(-0.0230481\pi\)
−0.561342 + 0.827584i \(0.689715\pi\)
\(774\) 0 0
\(775\) −19.6847 −0.707094
\(776\) 0 0
\(777\) −2.12311 + 3.67733i −0.0761660 + 0.131923i
\(778\) 0 0
\(779\) 11.1231 0.398527
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 0 0
\(783\) 3.34233 5.78908i 0.119445 0.206885i
\(784\) 0 0
\(785\) −72.5464 −2.58929
\(786\) 0 0
\(787\) 22.0885 + 38.2585i 0.787371 + 1.36377i 0.927572 + 0.373644i \(0.121892\pi\)
−0.140201 + 0.990123i \(0.544775\pi\)
\(788\) 0 0
\(789\) −4.68466 8.11407i −0.166778 0.288868i
\(790\) 0 0
\(791\) 1.63068 2.82443i 0.0579804 0.100425i
\(792\) 0 0
\(793\) 11.0194 8.60039i 0.391311 0.305409i
\(794\) 0 0
\(795\) 1.21922 2.11176i 0.0432414 0.0748963i
\(796\) 0 0
\(797\) 0.192236 + 0.332962i 0.00680935 + 0.0117941i 0.869410 0.494091i \(-0.164499\pi\)
−0.862601 + 0.505885i \(0.831166\pi\)
\(798\) 0 0
\(799\) −6.43845 11.1517i −0.227776 0.394519i
\(800\) 0 0
\(801\) 4.87689 0.172317
\(802\) 0 0
\(803\) 10.1231 17.5337i 0.357237 0.618752i
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 21.3693 0.752236
\(808\) 0 0
\(809\) 8.15009 14.1164i 0.286542 0.496305i −0.686440 0.727187i \(-0.740828\pi\)
0.972982 + 0.230881i \(0.0741609\pi\)
\(810\) 0 0
\(811\) −2.56155 −0.0899483 −0.0449741 0.998988i \(-0.514321\pi\)
−0.0449741 + 0.998988i \(0.514321\pi\)
\(812\) 0 0
\(813\) 14.9654 + 25.9209i 0.524861 + 0.909085i
\(814\) 0 0
\(815\) 8.56155 + 14.8290i 0.299898 + 0.519439i
\(816\) 0 0
\(817\) −16.2462 + 28.1393i −0.568383 + 0.984468i
\(818\) 0 0
\(819\) 1.87689 + 0.759413i 0.0655840 + 0.0265360i
\(820\) 0 0
\(821\) −3.24621 + 5.62260i −0.113294 + 0.196230i −0.917096 0.398666i \(-0.869473\pi\)
0.803803 + 0.594896i \(0.202807\pi\)
\(822\) 0 0
\(823\) 4.00000 + 6.92820i 0.139431 + 0.241502i 0.927281 0.374365i \(-0.122139\pi\)
−0.787850 + 0.615867i \(0.788806\pi\)
\(824\) 0 0
\(825\) 7.68466 + 13.3102i 0.267545 + 0.463402i
\(826\) 0 0
\(827\) 14.7386 0.512513 0.256256 0.966609i \(-0.417511\pi\)
0.256256 + 0.966609i \(0.417511\pi\)
\(828\) 0 0
\(829\) −6.74621 + 11.6848i −0.234306 + 0.405829i −0.959071 0.283167i \(-0.908615\pi\)
0.724765 + 0.688996i \(0.241948\pi\)
\(830\) 0 0
\(831\) −5.31534 −0.184387
\(832\) 0 0
\(833\) 10.4384 0.361671
\(834\) 0 0
\(835\) −18.2462 + 31.6034i −0.631436 + 1.09368i
\(836\) 0 0
\(837\) 2.56155 0.0885402
\(838\) 0 0
\(839\) 10.8078 + 18.7196i 0.373125 + 0.646272i 0.990045 0.140754i \(-0.0449528\pi\)
−0.616919 + 0.787027i \(0.711619\pi\)
\(840\) 0 0
\(841\) −7.84233 13.5833i −0.270425 0.468390i
\(842\) 0 0
\(843\) 8.90388 15.4220i 0.306666 0.531161i
\(844\) 0 0
\(845\) 44.5194 12.7173i 1.53151 0.437488i
\(846\) 0 0
\(847\) −1.96543 + 3.40423i −0.0675331 + 0.116971i
\(848\) 0 0
\(849\) 6.84233 + 11.8513i 0.234828 + 0.406734i
\(850\) 0 0
\(851\) 7.56155 + 13.0970i 0.259207 + 0.448959i
\(852\) 0 0
\(853\) 2.12311 0.0726938 0.0363469 0.999339i \(-0.488428\pi\)
0.0363469 + 0.999339i \(0.488428\pi\)
\(854\) 0 0
\(855\) −12.6847 + 21.9705i −0.433806 + 0.751374i
\(856\) 0 0
\(857\) 35.5616 1.21476 0.607380 0.794412i \(-0.292221\pi\)
0.607380 + 0.794412i \(0.292221\pi\)
\(858\) 0 0
\(859\) −24.5616 −0.838029 −0.419015 0.907979i \(-0.637624\pi\)
−0.419015 + 0.907979i \(0.637624\pi\)
\(860\) 0 0
\(861\) 0.438447 0.759413i 0.0149422 0.0258807i
\(862\) 0 0
\(863\) −30.4924 −1.03797 −0.518987 0.854782i \(-0.673691\pi\)
−0.518987 + 0.854782i \(0.673691\pi\)
\(864\) 0 0
\(865\) 36.0540 + 62.4473i 1.22587 + 2.12327i
\(866\) 0 0
\(867\) −7.28078 12.6107i −0.247268 0.428281i
\(868\) 0 0
\(869\) 5.43845 9.41967i 0.184487 0.319540i
\(870\) 0 0
\(871\) 15.2462 + 6.16879i 0.516598 + 0.209021i
\(872\) 0 0
\(873\) 4.28078 7.41452i 0.144882 0.250944i
\(874\) 0 0
\(875\) −2.68466 4.64996i −0.0907580 0.157198i
\(876\) 0 0
\(877\) −11.7808 20.4049i −0.397809 0.689025i 0.595647 0.803247i \(-0.296896\pi\)
−0.993455 + 0.114222i \(0.963563\pi\)
\(878\) 0 0
\(879\) −20.4384 −0.689372
\(880\) 0 0
\(881\) 4.53457 7.85410i 0.152773 0.264611i −0.779473 0.626436i \(-0.784513\pi\)
0.932246 + 0.361825i \(0.117846\pi\)
\(882\) 0 0
\(883\) −8.80776 −0.296405 −0.148202 0.988957i \(-0.547349\pi\)
−0.148202 + 0.988957i \(0.547349\pi\)
\(884\) 0 0
\(885\) 10.2462 0.344423
\(886\) 0 0
\(887\) 12.3153 21.3308i 0.413509 0.716218i −0.581762 0.813359i \(-0.697636\pi\)
0.995271 + 0.0971410i \(0.0309698\pi\)
\(888\) 0 0
\(889\) 3.05398 0.102427
\(890\) 0 0
\(891\) −1.00000 1.73205i −0.0335013 0.0580259i
\(892\) 0 0
\(893\) −29.3693 50.8691i −0.982807 1.70227i
\(894\) 0 0
\(895\) 8.68466 15.0423i 0.290296 0.502808i
\(896\) 0 0
\(897\) 5.68466 4.43674i 0.189805 0.148138i
\(898\) 0 0
\(899\) 8.56155 14.8290i 0.285544 0.494576i
\(900\) 0 0
\(901\) −0.534565 0.925894i −0.0178089 0.0308460i
\(902\) 0 0
\(903\) 1.28078 + 2.21837i 0.0426216 + 0.0738227i
\(904\) 0 0
\(905\) 9.56155 0.317837
\(906\) 0 0
\(907\) 14.0000 24.2487i 0.464862 0.805165i −0.534333 0.845274i \(-0.679437\pi\)
0.999195 + 0.0401089i \(0.0127705\pi\)
\(908\) 0 0
\(909\) −7.56155 −0.250801
\(910\) 0 0
\(911\) −38.7386 −1.28347 −0.641734 0.766927i \(-0.721785\pi\)
−0.641734 + 0.766927i \(0.721785\pi\)
\(912\) 0 0
\(913\) −0.876894 + 1.51883i −0.0290210 + 0.0502658i
\(914\) 0 0
\(915\) 13.8078 0.456471
\(916\) 0 0
\(917\) −2.06913 3.58384i −0.0683287 0.118349i
\(918\) 0 0
\(919\) 5.75379 + 9.96585i 0.189800 + 0.328743i 0.945183 0.326540i \(-0.105883\pi\)
−0.755383 + 0.655283i \(0.772549\pi\)
\(920\) 0 0
\(921\) −15.4039 + 26.6803i −0.507575 + 0.879146i
\(922\) 0 0
\(923\) −39.7926 + 31.0572i −1.30979 + 1.02226i
\(924\) 0 0
\(925\) −29.0540 + 50.3230i −0.955289 + 1.65461i
\(926\) 0 0
\(927\) −1.71922 2.97778i −0.0564667 0.0978032i
\(928\) 0 0
\(929\) 3.90388 + 6.76172i 0.128082 + 0.221845i 0.922934 0.384959i \(-0.125785\pi\)
−0.794851 + 0.606804i \(0.792451\pi\)
\(930\) 0 0
\(931\) 47.6155 1.56054
\(932\) 0 0
\(933\) 9.56155 16.5611i 0.313031 0.542186i
\(934\) 0 0
\(935\) 11.1231 0.363764
\(936\) 0 0
\(937\) −7.56155 −0.247025 −0.123513 0.992343i \(-0.539416\pi\)
−0.123513 + 0.992343i \(0.539416\pi\)
\(938\) 0 0
\(939\) −6.84233 + 11.8513i −0.223291 + 0.386751i
\(940\) 0 0
\(941\) 30.4924 0.994025 0.497012 0.867744i \(-0.334430\pi\)
0.497012 + 0.867744i \(0.334430\pi\)
\(942\) 0 0
\(943\) −1.56155 2.70469i −0.0508512 0.0880768i
\(944\) 0 0
\(945\) 1.00000 + 1.73205i 0.0325300 + 0.0563436i
\(946\) 0 0
\(947\) −19.3693 + 33.5486i −0.629418 + 1.09018i 0.358250 + 0.933626i \(0.383373\pi\)
−0.987669 + 0.156559i \(0.949960\pi\)
\(948\) 0 0
\(949\) −5.06155 36.1467i −0.164305 1.17337i
\(950\) 0 0
\(951\) 7.02699 12.1711i 0.227866 0.394675i
\(952\) 0 0
\(953\) −15.4924 26.8337i −0.501849 0.869228i −0.999998 0.00213612i \(-0.999320\pi\)
0.498149 0.867091i \(-0.334013\pi\)
\(954\) 0 0
\(955\) 16.2462 + 28.1393i 0.525715 + 0.910565i
\(956\) 0 0
\(957\) −13.3693 −0.432169
\(958\) 0 0
\(959\) −1.56155 + 2.70469i −0.0504252 + 0.0873390i
\(960\) 0 0
\(961\) −24.4384 −0.788337
\(962\) 0 0
\(963\) 8.24621 0.265730
\(964\) 0 0
\(965\) −24.0270 + 41.6160i −0.773456 + 1.33967i
\(966\) 0 0
\(967\) 0.876894 0.0281990 0.0140995 0.999901i \(-0.495512\pi\)
0.0140995 + 0.999901i \(0.495512\pi\)
\(968\) 0 0
\(969\) 5.56155 + 9.63289i 0.178663 + 0.309453i
\(970\) 0 0
\(971\) −6.49242 11.2452i −0.208352 0.360876i 0.742844 0.669465i \(-0.233477\pi\)
−0.951195 + 0.308589i \(0.900143\pi\)
\(972\) 0 0
\(973\) 5.03457 8.72012i 0.161401 0.279554i
\(974\) 0 0
\(975\) 25.6847 + 10.3923i 0.822567 + 0.332820i
\(976\) 0 0
\(977\) −30.5885 + 52.9809i −0.978614 + 1.69501i −0.311162 + 0.950357i \(0.600718\pi\)
−0.667453 + 0.744652i \(0.732615\pi\)
\(978\) 0 0
\(979\) −4.87689 8.44703i −0.155866 0.269968i
\(980\) 0 0
\(981\) 1.40388 + 2.43160i 0.0448225 + 0.0776349i
\(982\) 0 0
\(983\) −13.6155 −0.434268 −0.217134 0.976142i \(-0.569671\pi\)
−0.217134 + 0.976142i \(0.569671\pi\)
\(984\) 0 0
\(985\) 23.8078 41.2363i 0.758578 1.31390i
\(986\) 0 0
\(987\) −4.63068 −0.147396
\(988\) 0 0
\(989\) 9.12311 0.290098
\(990\) 0 0
\(991\) −25.1771 + 43.6080i −0.799776 + 1.38525i 0.119985 + 0.992776i \(0.461715\pi\)
−0.919762 + 0.392478i \(0.871618\pi\)
\(992\) 0 0
\(993\) −3.19224 −0.101303
\(994\) 0 0
\(995\) −39.4924 68.4029i −1.25199 2.16852i
\(996\) 0 0
\(997\) 10.3078 + 17.8536i 0.326450 + 0.565428i 0.981805 0.189893i \(-0.0608141\pi\)
−0.655355 + 0.755321i \(0.727481\pi\)
\(998\) 0 0
\(999\) 3.78078 6.54850i 0.119618 0.207185i
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 624.2.q.h.529.1 4
3.2 odd 2 1872.2.t.r.1153.2 4
4.3 odd 2 39.2.e.b.22.2 yes 4
12.11 even 2 117.2.g.c.100.1 4
13.3 even 3 inner 624.2.q.h.289.1 4
13.4 even 6 8112.2.a.bo.1.2 2
13.9 even 3 8112.2.a.bk.1.1 2
20.3 even 4 975.2.bb.i.724.2 8
20.7 even 4 975.2.bb.i.724.3 8
20.19 odd 2 975.2.i.k.451.1 4
39.29 odd 6 1872.2.t.r.289.2 4
52.3 odd 6 39.2.e.b.16.2 4
52.7 even 12 507.2.b.d.337.2 4
52.11 even 12 507.2.j.g.361.2 8
52.15 even 12 507.2.j.g.361.3 8
52.19 even 12 507.2.b.d.337.3 4
52.23 odd 6 507.2.e.g.484.1 4
52.31 even 4 507.2.j.g.316.2 8
52.35 odd 6 507.2.a.g.1.1 2
52.43 odd 6 507.2.a.d.1.2 2
52.47 even 4 507.2.j.g.316.3 8
52.51 odd 2 507.2.e.g.22.1 4
156.35 even 6 1521.2.a.g.1.2 2
156.59 odd 12 1521.2.b.h.1351.3 4
156.71 odd 12 1521.2.b.h.1351.2 4
156.95 even 6 1521.2.a.m.1.1 2
156.107 even 6 117.2.g.c.55.1 4
260.3 even 12 975.2.bb.i.874.3 8
260.107 even 12 975.2.bb.i.874.2 8
260.159 odd 6 975.2.i.k.601.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.2 4 52.3 odd 6
39.2.e.b.22.2 yes 4 4.3 odd 2
117.2.g.c.55.1 4 156.107 even 6
117.2.g.c.100.1 4 12.11 even 2
507.2.a.d.1.2 2 52.43 odd 6
507.2.a.g.1.1 2 52.35 odd 6
507.2.b.d.337.2 4 52.7 even 12
507.2.b.d.337.3 4 52.19 even 12
507.2.e.g.22.1 4 52.51 odd 2
507.2.e.g.484.1 4 52.23 odd 6
507.2.j.g.316.2 8 52.31 even 4
507.2.j.g.316.3 8 52.47 even 4
507.2.j.g.361.2 8 52.11 even 12
507.2.j.g.361.3 8 52.15 even 12
624.2.q.h.289.1 4 13.3 even 3 inner
624.2.q.h.529.1 4 1.1 even 1 trivial
975.2.i.k.451.1 4 20.19 odd 2
975.2.i.k.601.1 4 260.159 odd 6
975.2.bb.i.724.2 8 20.3 even 4
975.2.bb.i.724.3 8 20.7 even 4
975.2.bb.i.874.2 8 260.107 even 12
975.2.bb.i.874.3 8 260.3 even 12
1521.2.a.g.1.2 2 156.35 even 6
1521.2.a.m.1.1 2 156.95 even 6
1521.2.b.h.1351.2 4 156.71 odd 12
1521.2.b.h.1351.3 4 156.59 odd 12
1872.2.t.r.289.2 4 39.29 odd 6
1872.2.t.r.1153.2 4 3.2 odd 2
8112.2.a.bk.1.1 2 13.9 even 3
8112.2.a.bo.1.2 2 13.4 even 6