Properties

Label 1008.2.t.l.193.10
Level $1008$
Weight $2$
Character 1008.193
Analytic conductor $8.049$
Analytic rank $0$
Dimension $22$
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] = [1008,2,Mod(193,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.10
Character \(\chi\) \(=\) 1008.193
Dual form 1008.2.t.l.961.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.72608 + 0.143720i) q^{3} +2.77180 q^{5} +(-0.855737 - 2.50354i) q^{7} +(2.95869 + 0.496145i) q^{9} +O(q^{10})\) \(q+(1.72608 + 0.143720i) q^{3} +2.77180 q^{5} +(-0.855737 - 2.50354i) q^{7} +(2.95869 + 0.496145i) q^{9} +3.43944 q^{11} +(-0.429164 + 0.743335i) q^{13} +(4.78434 + 0.398364i) q^{15} +(-0.405132 + 0.701710i) q^{17} +(-0.750215 - 1.29941i) q^{19} +(-1.11726 - 4.44429i) q^{21} -7.64930 q^{23} +2.68286 q^{25} +(5.03562 + 1.28161i) q^{27} +(3.99696 + 6.92294i) q^{29} +(-3.60578 - 6.24540i) q^{31} +(5.93674 + 0.494317i) q^{33} +(-2.37193 - 6.93931i) q^{35} +(0.458211 + 0.793644i) q^{37} +(-0.847604 + 1.22137i) q^{39} +(1.67577 - 2.90251i) q^{41} +(-1.20465 - 2.08652i) q^{43} +(8.20089 + 1.37521i) q^{45} +(-0.307520 + 0.532640i) q^{47} +(-5.53543 + 4.28474i) q^{49} +(-0.800140 + 1.15298i) q^{51} +(6.31646 - 10.9404i) q^{53} +9.53342 q^{55} +(-1.10818 - 2.35070i) q^{57} +(0.734690 + 1.27252i) q^{59} +(-5.71635 + 9.90101i) q^{61} +(-1.28974 - 7.83177i) q^{63} +(-1.18956 + 2.06037i) q^{65} +(8.10061 + 14.0307i) q^{67} +(-13.2033 - 1.09936i) q^{69} -14.4177 q^{71} +(-4.16893 + 7.22079i) q^{73} +(4.63083 + 0.385582i) q^{75} +(-2.94325 - 8.61077i) q^{77} +(-1.37586 + 2.38305i) q^{79} +(8.50768 + 2.93588i) q^{81} +(5.75814 + 9.97340i) q^{83} +(-1.12294 + 1.94500i) q^{85} +(5.90410 + 12.5240i) q^{87} +(-5.11395 - 8.85763i) q^{89} +(2.22822 + 0.438331i) q^{91} +(-5.32627 - 11.2983i) q^{93} +(-2.07944 - 3.60170i) q^{95} +(3.82852 + 6.63119i) q^{97} +(10.1762 + 1.70646i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 2 q^{3} - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 2 q^{3} - 2 q^{5} + q^{7} + 6 q^{11} + 7 q^{13} + q^{15} - q^{17} - 13 q^{19} + 33 q^{21} + 44 q^{25} + 2 q^{27} - 7 q^{29} - 6 q^{31} + 9 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{43} - 17 q^{47} + 29 q^{49} + 25 q^{51} + q^{53} - 2 q^{55} - 21 q^{57} + 21 q^{59} + 31 q^{61} + 7 q^{63} - 3 q^{65} + 26 q^{67} - 40 q^{69} + 32 q^{71} + 17 q^{73} + 16 q^{75} - 4 q^{77} + 16 q^{79} + 36 q^{83} + 28 q^{85} - 7 q^{87} - 2 q^{89} - 15 q^{91} - 56 q^{93} + 24 q^{95} + 19 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.72608 + 0.143720i 0.996551 + 0.0829771i
\(4\) 0 0
\(5\) 2.77180 1.23959 0.619793 0.784766i \(-0.287217\pi\)
0.619793 + 0.784766i \(0.287217\pi\)
\(6\) 0 0
\(7\) −0.855737 2.50354i −0.323438 0.946249i
\(8\) 0 0
\(9\) 2.95869 + 0.496145i 0.986230 + 0.165382i
\(10\) 0 0
\(11\) 3.43944 1.03703 0.518515 0.855069i \(-0.326485\pi\)
0.518515 + 0.855069i \(0.326485\pi\)
\(12\) 0 0
\(13\) −0.429164 + 0.743335i −0.119029 + 0.206164i −0.919383 0.393363i \(-0.871311\pi\)
0.800354 + 0.599527i \(0.204645\pi\)
\(14\) 0 0
\(15\) 4.78434 + 0.398364i 1.23531 + 0.102857i
\(16\) 0 0
\(17\) −0.405132 + 0.701710i −0.0982590 + 0.170190i −0.910964 0.412486i \(-0.864661\pi\)
0.812705 + 0.582675i \(0.197994\pi\)
\(18\) 0 0
\(19\) −0.750215 1.29941i −0.172111 0.298105i 0.767047 0.641591i \(-0.221725\pi\)
−0.939158 + 0.343486i \(0.888392\pi\)
\(20\) 0 0
\(21\) −1.11726 4.44429i −0.243806 0.969824i
\(22\) 0 0
\(23\) −7.64930 −1.59499 −0.797495 0.603326i \(-0.793842\pi\)
−0.797495 + 0.603326i \(0.793842\pi\)
\(24\) 0 0
\(25\) 2.68286 0.536572
\(26\) 0 0
\(27\) 5.03562 + 1.28161i 0.969106 + 0.246646i
\(28\) 0 0
\(29\) 3.99696 + 6.92294i 0.742217 + 1.28556i 0.951484 + 0.307700i \(0.0995592\pi\)
−0.209266 + 0.977859i \(0.567107\pi\)
\(30\) 0 0
\(31\) −3.60578 6.24540i −0.647618 1.12171i −0.983690 0.179871i \(-0.942432\pi\)
0.336073 0.941836i \(-0.390901\pi\)
\(32\) 0 0
\(33\) 5.93674 + 0.494317i 1.03345 + 0.0860496i
\(34\) 0 0
\(35\) −2.37193 6.93931i −0.400929 1.17296i
\(36\) 0 0
\(37\) 0.458211 + 0.793644i 0.0753294 + 0.130474i 0.901229 0.433342i \(-0.142666\pi\)
−0.825900 + 0.563817i \(0.809333\pi\)
\(38\) 0 0
\(39\) −0.847604 + 1.22137i −0.135725 + 0.195576i
\(40\) 0 0
\(41\) 1.67577 2.90251i 0.261711 0.453297i −0.704986 0.709221i \(-0.749047\pi\)
0.966697 + 0.255925i \(0.0823800\pi\)
\(42\) 0 0
\(43\) −1.20465 2.08652i −0.183708 0.318191i 0.759433 0.650586i \(-0.225477\pi\)
−0.943140 + 0.332395i \(0.892143\pi\)
\(44\) 0 0
\(45\) 8.20089 + 1.37521i 1.22252 + 0.205005i
\(46\) 0 0
\(47\) −0.307520 + 0.532640i −0.0448564 + 0.0776935i −0.887582 0.460650i \(-0.847616\pi\)
0.842726 + 0.538343i \(0.180950\pi\)
\(48\) 0 0
\(49\) −5.53543 + 4.28474i −0.790775 + 0.612106i
\(50\) 0 0
\(51\) −0.800140 + 1.15298i −0.112042 + 0.161449i
\(52\) 0 0
\(53\) 6.31646 10.9404i 0.867633 1.50278i 0.00322332 0.999995i \(-0.498974\pi\)
0.864409 0.502789i \(-0.167693\pi\)
\(54\) 0 0
\(55\) 9.53342 1.28549
\(56\) 0 0
\(57\) −1.10818 2.35070i −0.146782 0.311358i
\(58\) 0 0
\(59\) 0.734690 + 1.27252i 0.0956485 + 0.165668i 0.909879 0.414874i \(-0.136174\pi\)
−0.814231 + 0.580542i \(0.802841\pi\)
\(60\) 0 0
\(61\) −5.71635 + 9.90101i −0.731904 + 1.26769i 0.224165 + 0.974551i \(0.428035\pi\)
−0.956069 + 0.293143i \(0.905299\pi\)
\(62\) 0 0
\(63\) −1.28974 7.83177i −0.162492 0.986710i
\(64\) 0 0
\(65\) −1.18956 + 2.06037i −0.147546 + 0.255558i
\(66\) 0 0
\(67\) 8.10061 + 14.0307i 0.989647 + 1.71412i 0.619117 + 0.785299i \(0.287491\pi\)
0.370530 + 0.928820i \(0.379176\pi\)
\(68\) 0 0
\(69\) −13.2033 1.09936i −1.58949 0.132348i
\(70\) 0 0
\(71\) −14.4177 −1.71106 −0.855532 0.517749i \(-0.826770\pi\)
−0.855532 + 0.517749i \(0.826770\pi\)
\(72\) 0 0
\(73\) −4.16893 + 7.22079i −0.487936 + 0.845130i −0.999904 0.0138749i \(-0.995583\pi\)
0.511968 + 0.859005i \(0.328917\pi\)
\(74\) 0 0
\(75\) 4.63083 + 0.385582i 0.534722 + 0.0445232i
\(76\) 0 0
\(77\) −2.94325 8.61077i −0.335415 0.981288i
\(78\) 0 0
\(79\) −1.37586 + 2.38305i −0.154796 + 0.268115i −0.932985 0.359916i \(-0.882805\pi\)
0.778189 + 0.628031i \(0.216139\pi\)
\(80\) 0 0
\(81\) 8.50768 + 2.93588i 0.945298 + 0.326209i
\(82\) 0 0
\(83\) 5.75814 + 9.97340i 0.632038 + 1.09472i 0.987134 + 0.159893i \(0.0511148\pi\)
−0.355096 + 0.934830i \(0.615552\pi\)
\(84\) 0 0
\(85\) −1.12294 + 1.94500i −0.121800 + 0.210965i
\(86\) 0 0
\(87\) 5.90410 + 12.5240i 0.632986 + 1.34271i
\(88\) 0 0
\(89\) −5.11395 8.85763i −0.542078 0.938907i −0.998785 0.0492892i \(-0.984304\pi\)
0.456707 0.889617i \(-0.349029\pi\)
\(90\) 0 0
\(91\) 2.22822 + 0.438331i 0.233581 + 0.0459496i
\(92\) 0 0
\(93\) −5.32627 11.2983i −0.552308 1.17158i
\(94\) 0 0
\(95\) −2.07944 3.60170i −0.213346 0.369527i
\(96\) 0 0
\(97\) 3.82852 + 6.63119i 0.388727 + 0.673296i 0.992279 0.124029i \(-0.0395814\pi\)
−0.603551 + 0.797324i \(0.706248\pi\)
\(98\) 0 0
\(99\) 10.1762 + 1.70646i 1.02275 + 0.171506i
\(100\) 0 0
\(101\) −3.68603 −0.366774 −0.183387 0.983041i \(-0.558706\pi\)
−0.183387 + 0.983041i \(0.558706\pi\)
\(102\) 0 0
\(103\) 16.1205 1.58840 0.794201 0.607656i \(-0.207890\pi\)
0.794201 + 0.607656i \(0.207890\pi\)
\(104\) 0 0
\(105\) −3.09681 12.3187i −0.302218 1.20218i
\(106\) 0 0
\(107\) −3.16767 5.48656i −0.306230 0.530405i 0.671305 0.741182i \(-0.265734\pi\)
−0.977534 + 0.210776i \(0.932401\pi\)
\(108\) 0 0
\(109\) 4.89477 8.47799i 0.468834 0.812044i −0.530532 0.847665i \(-0.678008\pi\)
0.999365 + 0.0356213i \(0.0113410\pi\)
\(110\) 0 0
\(111\) 0.676844 + 1.43575i 0.0642432 + 0.136275i
\(112\) 0 0
\(113\) 4.06963 7.04881i 0.382839 0.663096i −0.608628 0.793456i \(-0.708280\pi\)
0.991467 + 0.130360i \(0.0416132\pi\)
\(114\) 0 0
\(115\) −21.2023 −1.97713
\(116\) 0 0
\(117\) −1.63857 + 1.98637i −0.151485 + 0.183640i
\(118\) 0 0
\(119\) 2.10345 + 0.413786i 0.192823 + 0.0379317i
\(120\) 0 0
\(121\) 0.829725 0.0754295
\(122\) 0 0
\(123\) 3.30966 4.76912i 0.298422 0.430017i
\(124\) 0 0
\(125\) −6.42264 −0.574459
\(126\) 0 0
\(127\) 12.5658 1.11504 0.557518 0.830165i \(-0.311754\pi\)
0.557518 + 0.830165i \(0.311754\pi\)
\(128\) 0 0
\(129\) −1.77945 3.77463i −0.156672 0.332337i
\(130\) 0 0
\(131\) −1.58550 −0.138526 −0.0692631 0.997598i \(-0.522065\pi\)
−0.0692631 + 0.997598i \(0.522065\pi\)
\(132\) 0 0
\(133\) −2.61114 + 2.99015i −0.226415 + 0.259279i
\(134\) 0 0
\(135\) 13.9577 + 3.55236i 1.20129 + 0.305739i
\(136\) 0 0
\(137\) 4.92788 0.421017 0.210508 0.977592i \(-0.432488\pi\)
0.210508 + 0.977592i \(0.432488\pi\)
\(138\) 0 0
\(139\) −4.12999 + 7.15336i −0.350301 + 0.606740i −0.986302 0.164949i \(-0.947254\pi\)
0.636001 + 0.771688i \(0.280588\pi\)
\(140\) 0 0
\(141\) −0.607354 + 0.875181i −0.0511485 + 0.0737035i
\(142\) 0 0
\(143\) −1.47608 + 2.55665i −0.123436 + 0.213798i
\(144\) 0 0
\(145\) 11.0788 + 19.1890i 0.920042 + 1.59356i
\(146\) 0 0
\(147\) −10.1704 + 6.60025i −0.838839 + 0.544379i
\(148\) 0 0
\(149\) −13.9140 −1.13988 −0.569938 0.821688i \(-0.693033\pi\)
−0.569938 + 0.821688i \(0.693033\pi\)
\(150\) 0 0
\(151\) −22.9927 −1.87112 −0.935561 0.353166i \(-0.885105\pi\)
−0.935561 + 0.353166i \(0.885105\pi\)
\(152\) 0 0
\(153\) −1.54681 + 1.87514i −0.125052 + 0.151596i
\(154\) 0 0
\(155\) −9.99450 17.3110i −0.802777 1.39045i
\(156\) 0 0
\(157\) −9.29022 16.0911i −0.741441 1.28421i −0.951839 0.306597i \(-0.900810\pi\)
0.210399 0.977616i \(-0.432524\pi\)
\(158\) 0 0
\(159\) 12.4751 17.9762i 0.989337 1.42561i
\(160\) 0 0
\(161\) 6.54579 + 19.1503i 0.515880 + 1.50926i
\(162\) 0 0
\(163\) 2.45194 + 4.24688i 0.192050 + 0.332641i 0.945930 0.324372i \(-0.105153\pi\)
−0.753879 + 0.657013i \(0.771820\pi\)
\(164\) 0 0
\(165\) 16.4554 + 1.37015i 1.28105 + 0.106666i
\(166\) 0 0
\(167\) 5.47493 9.48286i 0.423663 0.733805i −0.572632 0.819813i \(-0.694077\pi\)
0.996295 + 0.0860073i \(0.0274108\pi\)
\(168\) 0 0
\(169\) 6.13164 + 10.6203i 0.471664 + 0.816947i
\(170\) 0 0
\(171\) −1.57496 4.21677i −0.120440 0.322464i
\(172\) 0 0
\(173\) 6.70271 11.6094i 0.509598 0.882649i −0.490340 0.871531i \(-0.663127\pi\)
0.999938 0.0111184i \(-0.00353917\pi\)
\(174\) 0 0
\(175\) −2.29582 6.71665i −0.173548 0.507731i
\(176\) 0 0
\(177\) 1.08524 + 2.30206i 0.0815720 + 0.173033i
\(178\) 0 0
\(179\) 6.64888 11.5162i 0.496961 0.860761i −0.503033 0.864267i \(-0.667783\pi\)
0.999994 + 0.00350600i \(0.00111600\pi\)
\(180\) 0 0
\(181\) −10.9190 −0.811601 −0.405801 0.913962i \(-0.633007\pi\)
−0.405801 + 0.913962i \(0.633007\pi\)
\(182\) 0 0
\(183\) −11.2898 + 16.2684i −0.834569 + 1.20259i
\(184\) 0 0
\(185\) 1.27007 + 2.19982i 0.0933772 + 0.161734i
\(186\) 0 0
\(187\) −1.39343 + 2.41349i −0.101897 + 0.176492i
\(188\) 0 0
\(189\) −1.10061 13.7036i −0.0800573 0.996790i
\(190\) 0 0
\(191\) −9.30553 + 16.1177i −0.673325 + 1.16623i 0.303631 + 0.952790i \(0.401801\pi\)
−0.976956 + 0.213443i \(0.931532\pi\)
\(192\) 0 0
\(193\) −1.46470 2.53693i −0.105431 0.182613i 0.808483 0.588520i \(-0.200289\pi\)
−0.913914 + 0.405907i \(0.866956\pi\)
\(194\) 0 0
\(195\) −2.34939 + 3.38540i −0.168243 + 0.242434i
\(196\) 0 0
\(197\) −18.5050 −1.31843 −0.659214 0.751956i \(-0.729111\pi\)
−0.659214 + 0.751956i \(0.729111\pi\)
\(198\) 0 0
\(199\) −0.793836 + 1.37496i −0.0562736 + 0.0974687i −0.892790 0.450473i \(-0.851255\pi\)
0.836516 + 0.547942i \(0.184589\pi\)
\(200\) 0 0
\(201\) 11.9658 + 25.3823i 0.844002 + 1.79033i
\(202\) 0 0
\(203\) 13.9115 15.9308i 0.976397 1.11812i
\(204\) 0 0
\(205\) 4.64489 8.04518i 0.324413 0.561900i
\(206\) 0 0
\(207\) −22.6319 3.79517i −1.57303 0.263782i
\(208\) 0 0
\(209\) −2.58032 4.46924i −0.178484 0.309144i
\(210\) 0 0
\(211\) −12.3436 + 21.3798i −0.849770 + 1.47184i 0.0316443 + 0.999499i \(0.489926\pi\)
−0.881414 + 0.472345i \(0.843408\pi\)
\(212\) 0 0
\(213\) −24.8860 2.07212i −1.70516 0.141979i
\(214\) 0 0
\(215\) −3.33905 5.78340i −0.227721 0.394425i
\(216\) 0 0
\(217\) −12.5500 + 14.3716i −0.851950 + 0.975611i
\(218\) 0 0
\(219\) −8.23367 + 11.8645i −0.556380 + 0.801728i
\(220\) 0 0
\(221\) −0.347737 0.602298i −0.0233913 0.0405149i
\(222\) 0 0
\(223\) 9.78468 + 16.9476i 0.655231 + 1.13489i 0.981836 + 0.189732i \(0.0607619\pi\)
−0.326605 + 0.945161i \(0.605905\pi\)
\(224\) 0 0
\(225\) 7.93775 + 1.33109i 0.529183 + 0.0887393i
\(226\) 0 0
\(227\) −8.64808 −0.573993 −0.286996 0.957932i \(-0.592657\pi\)
−0.286996 + 0.957932i \(0.592657\pi\)
\(228\) 0 0
\(229\) 11.5427 0.762765 0.381382 0.924417i \(-0.375448\pi\)
0.381382 + 0.924417i \(0.375448\pi\)
\(230\) 0 0
\(231\) −3.84274 15.2859i −0.252834 1.00574i
\(232\) 0 0
\(233\) −8.12745 14.0772i −0.532447 0.922225i −0.999282 0.0378811i \(-0.987939\pi\)
0.466835 0.884344i \(-0.345394\pi\)
\(234\) 0 0
\(235\) −0.852382 + 1.47637i −0.0556033 + 0.0963077i
\(236\) 0 0
\(237\) −2.71733 + 3.91560i −0.176510 + 0.254345i
\(238\) 0 0
\(239\) −12.4336 + 21.5355i −0.804260 + 1.39302i 0.112530 + 0.993648i \(0.464105\pi\)
−0.916790 + 0.399371i \(0.869229\pi\)
\(240\) 0 0
\(241\) −19.0363 −1.22623 −0.613117 0.789992i \(-0.710085\pi\)
−0.613117 + 0.789992i \(0.710085\pi\)
\(242\) 0 0
\(243\) 14.2630 + 6.29028i 0.914970 + 0.403522i
\(244\) 0 0
\(245\) −15.3431 + 11.8764i −0.980234 + 0.758758i
\(246\) 0 0
\(247\) 1.28786 0.0819447
\(248\) 0 0
\(249\) 8.50562 + 18.0424i 0.539022 + 1.14339i
\(250\) 0 0
\(251\) 0.980433 0.0618844 0.0309422 0.999521i \(-0.490149\pi\)
0.0309422 + 0.999521i \(0.490149\pi\)
\(252\) 0 0
\(253\) −26.3093 −1.65405
\(254\) 0 0
\(255\) −2.21783 + 3.19583i −0.138886 + 0.200130i
\(256\) 0 0
\(257\) 4.36725 0.272421 0.136211 0.990680i \(-0.456508\pi\)
0.136211 + 0.990680i \(0.456508\pi\)
\(258\) 0 0
\(259\) 1.59481 1.82630i 0.0990968 0.113481i
\(260\) 0 0
\(261\) 8.39098 + 22.4659i 0.519389 + 1.39060i
\(262\) 0 0
\(263\) 12.4842 0.769811 0.384905 0.922956i \(-0.374234\pi\)
0.384905 + 0.922956i \(0.374234\pi\)
\(264\) 0 0
\(265\) 17.5079 30.3247i 1.07550 1.86283i
\(266\) 0 0
\(267\) −7.55406 16.0239i −0.462301 0.980649i
\(268\) 0 0
\(269\) 8.29270 14.3634i 0.505615 0.875750i −0.494364 0.869255i \(-0.664599\pi\)
0.999979 0.00649532i \(-0.00206754\pi\)
\(270\) 0 0
\(271\) −12.9814 22.4845i −0.788566 1.36584i −0.926845 0.375444i \(-0.877490\pi\)
0.138279 0.990393i \(-0.455843\pi\)
\(272\) 0 0
\(273\) 3.78308 + 1.07684i 0.228963 + 0.0651730i
\(274\) 0 0
\(275\) 9.22753 0.556441
\(276\) 0 0
\(277\) −1.96075 −0.117810 −0.0589049 0.998264i \(-0.518761\pi\)
−0.0589049 + 0.998264i \(0.518761\pi\)
\(278\) 0 0
\(279\) −7.56976 20.2672i −0.453190 1.21336i
\(280\) 0 0
\(281\) 9.42057 + 16.3169i 0.561984 + 0.973385i 0.997323 + 0.0731185i \(0.0232951\pi\)
−0.435339 + 0.900267i \(0.643372\pi\)
\(282\) 0 0
\(283\) −11.7422 20.3381i −0.698002 1.20898i −0.969158 0.246440i \(-0.920739\pi\)
0.271156 0.962536i \(-0.412594\pi\)
\(284\) 0 0
\(285\) −3.07164 6.51568i −0.181948 0.385955i
\(286\) 0 0
\(287\) −8.70058 1.71156i −0.513579 0.101030i
\(288\) 0 0
\(289\) 8.17174 + 14.1539i 0.480690 + 0.832580i
\(290\) 0 0
\(291\) 5.65529 + 11.9962i 0.331519 + 0.703229i
\(292\) 0 0
\(293\) −1.00384 + 1.73871i −0.0586452 + 0.101576i −0.893857 0.448351i \(-0.852011\pi\)
0.835212 + 0.549928i \(0.185345\pi\)
\(294\) 0 0
\(295\) 2.03641 + 3.52717i 0.118564 + 0.205360i
\(296\) 0 0
\(297\) 17.3197 + 4.40802i 1.00499 + 0.255779i
\(298\) 0 0
\(299\) 3.28281 5.68599i 0.189850 0.328829i
\(300\) 0 0
\(301\) −4.19282 + 4.80140i −0.241670 + 0.276748i
\(302\) 0 0
\(303\) −6.36237 0.529758i −0.365509 0.0304338i
\(304\) 0 0
\(305\) −15.8446 + 27.4436i −0.907257 + 1.57142i
\(306\) 0 0
\(307\) −32.7633 −1.86990 −0.934951 0.354777i \(-0.884557\pi\)
−0.934951 + 0.354777i \(0.884557\pi\)
\(308\) 0 0
\(309\) 27.8253 + 2.31685i 1.58292 + 0.131801i
\(310\) 0 0
\(311\) −7.04979 12.2106i −0.399757 0.692400i 0.593939 0.804510i \(-0.297572\pi\)
−0.993696 + 0.112111i \(0.964239\pi\)
\(312\) 0 0
\(313\) 17.0769 29.5781i 0.965245 1.67185i 0.256291 0.966600i \(-0.417499\pi\)
0.708954 0.705255i \(-0.249167\pi\)
\(314\) 0 0
\(315\) −3.57490 21.7081i −0.201423 1.22311i
\(316\) 0 0
\(317\) 0.790586 1.36933i 0.0444037 0.0769095i −0.842969 0.537962i \(-0.819195\pi\)
0.887373 + 0.461052i \(0.152528\pi\)
\(318\) 0 0
\(319\) 13.7473 + 23.8110i 0.769701 + 1.33316i
\(320\) 0 0
\(321\) −4.67911 9.92548i −0.261162 0.553986i
\(322\) 0 0
\(323\) 1.21575 0.0676459
\(324\) 0 0
\(325\) −1.15139 + 1.99426i −0.0638675 + 0.110622i
\(326\) 0 0
\(327\) 9.66721 13.9302i 0.534598 0.770341i
\(328\) 0 0
\(329\) 1.59664 + 0.314088i 0.0880257 + 0.0173163i
\(330\) 0 0
\(331\) 6.86862 11.8968i 0.377533 0.653907i −0.613169 0.789951i \(-0.710106\pi\)
0.990703 + 0.136044i \(0.0434390\pi\)
\(332\) 0 0
\(333\) 0.961940 + 2.57549i 0.0527140 + 0.141136i
\(334\) 0 0
\(335\) 22.4533 + 38.8902i 1.22675 + 2.12480i
\(336\) 0 0
\(337\) −8.72318 + 15.1090i −0.475182 + 0.823039i −0.999596 0.0284243i \(-0.990951\pi\)
0.524414 + 0.851463i \(0.324284\pi\)
\(338\) 0 0
\(339\) 8.03756 11.5819i 0.436540 0.629042i
\(340\) 0 0
\(341\) −12.4019 21.4807i −0.671598 1.16324i
\(342\) 0 0
\(343\) 15.4639 + 10.1916i 0.834972 + 0.550292i
\(344\) 0 0
\(345\) −36.5968 3.04721i −1.97031 0.164056i
\(346\) 0 0
\(347\) 1.91552 + 3.31778i 0.102830 + 0.178108i 0.912850 0.408296i \(-0.133877\pi\)
−0.810019 + 0.586403i \(0.800543\pi\)
\(348\) 0 0
\(349\) 1.69984 + 2.94421i 0.0909903 + 0.157600i 0.907928 0.419126i \(-0.137663\pi\)
−0.816938 + 0.576726i \(0.804330\pi\)
\(350\) 0 0
\(351\) −3.11377 + 3.19313i −0.166201 + 0.170437i
\(352\) 0 0
\(353\) 12.5568 0.668332 0.334166 0.942514i \(-0.391545\pi\)
0.334166 + 0.942514i \(0.391545\pi\)
\(354\) 0 0
\(355\) −39.9629 −2.12101
\(356\) 0 0
\(357\) 3.57124 + 1.01654i 0.189010 + 0.0538008i
\(358\) 0 0
\(359\) 6.02209 + 10.4306i 0.317834 + 0.550504i 0.980036 0.198821i \(-0.0637113\pi\)
−0.662202 + 0.749325i \(0.730378\pi\)
\(360\) 0 0
\(361\) 8.37435 14.5048i 0.440756 0.763411i
\(362\) 0 0
\(363\) 1.43217 + 0.119248i 0.0751694 + 0.00625892i
\(364\) 0 0
\(365\) −11.5554 + 20.0146i −0.604838 + 1.04761i
\(366\) 0 0
\(367\) 2.02514 0.105711 0.0528557 0.998602i \(-0.483168\pi\)
0.0528557 + 0.998602i \(0.483168\pi\)
\(368\) 0 0
\(369\) 6.39814 7.75621i 0.333074 0.403772i
\(370\) 0 0
\(371\) −32.7950 6.45138i −1.70263 0.334939i
\(372\) 0 0
\(373\) 21.6259 1.11975 0.559874 0.828578i \(-0.310849\pi\)
0.559874 + 0.828578i \(0.310849\pi\)
\(374\) 0 0
\(375\) −11.0860 0.923065i −0.572477 0.0476669i
\(376\) 0 0
\(377\) −6.86142 −0.353381
\(378\) 0 0
\(379\) 6.76701 0.347598 0.173799 0.984781i \(-0.444396\pi\)
0.173799 + 0.984781i \(0.444396\pi\)
\(380\) 0 0
\(381\) 21.6896 + 1.80597i 1.11119 + 0.0925224i
\(382\) 0 0
\(383\) 7.65645 0.391226 0.195613 0.980681i \(-0.437330\pi\)
0.195613 + 0.980681i \(0.437330\pi\)
\(384\) 0 0
\(385\) −8.15810 23.8673i −0.415775 1.21639i
\(386\) 0 0
\(387\) −2.52897 6.77104i −0.128555 0.344191i
\(388\) 0 0
\(389\) −21.1561 −1.07266 −0.536329 0.844009i \(-0.680189\pi\)
−0.536329 + 0.844009i \(0.680189\pi\)
\(390\) 0 0
\(391\) 3.09898 5.36759i 0.156722 0.271451i
\(392\) 0 0
\(393\) −2.73670 0.227869i −0.138048 0.0114945i
\(394\) 0 0
\(395\) −3.81360 + 6.60534i −0.191883 + 0.332351i
\(396\) 0 0
\(397\) 4.02642 + 6.97396i 0.202080 + 0.350013i 0.949199 0.314678i \(-0.101896\pi\)
−0.747118 + 0.664691i \(0.768563\pi\)
\(398\) 0 0
\(399\) −4.93678 + 4.78595i −0.247148 + 0.239597i
\(400\) 0 0
\(401\) −7.77773 −0.388401 −0.194201 0.980962i \(-0.562211\pi\)
−0.194201 + 0.980962i \(0.562211\pi\)
\(402\) 0 0
\(403\) 6.18989 0.308341
\(404\) 0 0
\(405\) 23.5816 + 8.13766i 1.17178 + 0.404364i
\(406\) 0 0
\(407\) 1.57599 + 2.72969i 0.0781188 + 0.135306i
\(408\) 0 0
\(409\) −9.76327 16.9105i −0.482763 0.836170i 0.517041 0.855960i \(-0.327033\pi\)
−0.999804 + 0.0197907i \(0.993700\pi\)
\(410\) 0 0
\(411\) 8.50590 + 0.708237i 0.419565 + 0.0349347i
\(412\) 0 0
\(413\) 2.55710 2.92827i 0.125827 0.144091i
\(414\) 0 0
\(415\) 15.9604 + 27.6442i 0.783466 + 1.35700i
\(416\) 0 0
\(417\) −8.15677 + 11.7537i −0.399439 + 0.575580i
\(418\) 0 0
\(419\) 12.5259 21.6955i 0.611932 1.05990i −0.378983 0.925404i \(-0.623726\pi\)
0.990915 0.134493i \(-0.0429406\pi\)
\(420\) 0 0
\(421\) 18.0746 + 31.3061i 0.880902 + 1.52577i 0.850340 + 0.526234i \(0.176396\pi\)
0.0305620 + 0.999533i \(0.490270\pi\)
\(422\) 0 0
\(423\) −1.17412 + 1.42334i −0.0570878 + 0.0692052i
\(424\) 0 0
\(425\) −1.08691 + 1.88259i −0.0527231 + 0.0913190i
\(426\) 0 0
\(427\) 29.6793 + 5.83845i 1.43628 + 0.282543i
\(428\) 0 0
\(429\) −2.91528 + 4.20084i −0.140751 + 0.202818i
\(430\) 0 0
\(431\) 1.95636 3.38852i 0.0942346 0.163219i −0.815054 0.579385i \(-0.803293\pi\)
0.909289 + 0.416165i \(0.136626\pi\)
\(432\) 0 0
\(433\) −14.2929 −0.686872 −0.343436 0.939176i \(-0.611591\pi\)
−0.343436 + 0.939176i \(0.611591\pi\)
\(434\) 0 0
\(435\) 16.3650 + 34.7139i 0.784640 + 1.66441i
\(436\) 0 0
\(437\) 5.73862 + 9.93958i 0.274515 + 0.475475i
\(438\) 0 0
\(439\) −2.39235 + 4.14367i −0.114180 + 0.197766i −0.917452 0.397847i \(-0.869758\pi\)
0.803271 + 0.595613i \(0.203091\pi\)
\(440\) 0 0
\(441\) −18.5035 + 9.93085i −0.881117 + 0.472898i
\(442\) 0 0
\(443\) −4.13213 + 7.15707i −0.196324 + 0.340042i −0.947334 0.320248i \(-0.896234\pi\)
0.751010 + 0.660291i \(0.229567\pi\)
\(444\) 0 0
\(445\) −14.1748 24.5515i −0.671952 1.16385i
\(446\) 0 0
\(447\) −24.0166 1.99972i −1.13594 0.0945835i
\(448\) 0 0
\(449\) −20.6036 −0.972346 −0.486173 0.873863i \(-0.661608\pi\)
−0.486173 + 0.873863i \(0.661608\pi\)
\(450\) 0 0
\(451\) 5.76370 9.98301i 0.271402 0.470082i
\(452\) 0 0
\(453\) −39.6872 3.30453i −1.86467 0.155260i
\(454\) 0 0
\(455\) 6.17617 + 1.21497i 0.289543 + 0.0569585i
\(456\) 0 0
\(457\) 8.98220 15.5576i 0.420170 0.727755i −0.575786 0.817600i \(-0.695304\pi\)
0.995956 + 0.0898451i \(0.0286372\pi\)
\(458\) 0 0
\(459\) −2.93941 + 3.01432i −0.137200 + 0.140697i
\(460\) 0 0
\(461\) 4.03501 + 6.98885i 0.187929 + 0.325503i 0.944560 0.328340i \(-0.106489\pi\)
−0.756630 + 0.653843i \(0.773156\pi\)
\(462\) 0 0
\(463\) 2.50704 4.34232i 0.116512 0.201805i −0.801871 0.597497i \(-0.796162\pi\)
0.918383 + 0.395692i \(0.129495\pi\)
\(464\) 0 0
\(465\) −14.7633 31.3165i −0.684633 1.45227i
\(466\) 0 0
\(467\) 13.1673 + 22.8063i 0.609308 + 1.05535i 0.991355 + 0.131209i \(0.0418859\pi\)
−0.382047 + 0.924143i \(0.624781\pi\)
\(468\) 0 0
\(469\) 28.1944 32.2868i 1.30189 1.49086i
\(470\) 0 0
\(471\) −13.7230 29.1098i −0.632323 1.34131i
\(472\) 0 0
\(473\) −4.14332 7.17645i −0.190510 0.329973i
\(474\) 0 0
\(475\) −2.01272 3.48614i −0.0923500 0.159955i
\(476\) 0 0
\(477\) 24.1165 29.2354i 1.10422 1.33860i
\(478\) 0 0
\(479\) 25.2445 1.15345 0.576724 0.816939i \(-0.304331\pi\)
0.576724 + 0.816939i \(0.304331\pi\)
\(480\) 0 0
\(481\) −0.786591 −0.0358655
\(482\) 0 0
\(483\) 8.54625 + 33.9957i 0.388868 + 1.54686i
\(484\) 0 0
\(485\) 10.6119 + 18.3803i 0.481861 + 0.834608i
\(486\) 0 0
\(487\) −1.36124 + 2.35774i −0.0616837 + 0.106839i −0.895218 0.445628i \(-0.852980\pi\)
0.833534 + 0.552468i \(0.186314\pi\)
\(488\) 0 0
\(489\) 3.62187 + 7.68283i 0.163787 + 0.347430i
\(490\) 0 0
\(491\) 15.8020 27.3698i 0.713134 1.23518i −0.250541 0.968106i \(-0.580609\pi\)
0.963675 0.267078i \(-0.0860581\pi\)
\(492\) 0 0
\(493\) −6.47720 −0.291718
\(494\) 0 0
\(495\) 28.2064 + 4.72996i 1.26778 + 0.212596i
\(496\) 0 0
\(497\) 12.3377 + 36.0953i 0.553424 + 1.61909i
\(498\) 0 0
\(499\) −5.94890 −0.266309 −0.133155 0.991095i \(-0.542511\pi\)
−0.133155 + 0.991095i \(0.542511\pi\)
\(500\) 0 0
\(501\) 10.8130 15.5813i 0.483091 0.696121i
\(502\) 0 0
\(503\) 27.6905 1.23466 0.617329 0.786705i \(-0.288215\pi\)
0.617329 + 0.786705i \(0.288215\pi\)
\(504\) 0 0
\(505\) −10.2169 −0.454647
\(506\) 0 0
\(507\) 9.05732 + 19.2127i 0.402250 + 0.853267i
\(508\) 0 0
\(509\) 37.2885 1.65278 0.826392 0.563095i \(-0.190389\pi\)
0.826392 + 0.563095i \(0.190389\pi\)
\(510\) 0 0
\(511\) 21.6450 + 4.25798i 0.957521 + 0.188362i
\(512\) 0 0
\(513\) −2.11246 7.50482i −0.0932674 0.331346i
\(514\) 0 0
\(515\) 44.6828 1.96896
\(516\) 0 0
\(517\) −1.05769 + 1.83198i −0.0465174 + 0.0805704i
\(518\) 0 0
\(519\) 13.2379 19.0755i 0.581080 0.837321i
\(520\) 0 0
\(521\) 18.1271 31.3971i 0.794163 1.37553i −0.129207 0.991618i \(-0.541243\pi\)
0.923370 0.383912i \(-0.125423\pi\)
\(522\) 0 0
\(523\) −10.2931 17.8282i −0.450086 0.779572i 0.548305 0.836278i \(-0.315273\pi\)
−0.998391 + 0.0567068i \(0.981940\pi\)
\(524\) 0 0
\(525\) −2.99745 11.9234i −0.130819 0.520380i
\(526\) 0 0
\(527\) 5.84328 0.254537
\(528\) 0 0
\(529\) 35.5118 1.54399
\(530\) 0 0
\(531\) 1.54236 + 4.12951i 0.0669329 + 0.179205i
\(532\) 0 0
\(533\) 1.43836 + 2.49131i 0.0623023 + 0.107911i
\(534\) 0 0
\(535\) −8.78013 15.2076i −0.379598 0.657483i
\(536\) 0 0
\(537\) 13.1316 18.9223i 0.566670 0.816556i
\(538\) 0 0
\(539\) −19.0388 + 14.7371i −0.820057 + 0.634772i
\(540\) 0 0
\(541\) −0.649192 1.12443i −0.0279109 0.0483432i 0.851733 0.523977i \(-0.175552\pi\)
−0.879643 + 0.475634i \(0.842219\pi\)
\(542\) 0 0
\(543\) −18.8470 1.56928i −0.808802 0.0673443i
\(544\) 0 0
\(545\) 13.5673 23.4993i 0.581159 1.00660i
\(546\) 0 0
\(547\) 13.8412 + 23.9736i 0.591805 + 1.02504i 0.993989 + 0.109478i \(0.0349180\pi\)
−0.402184 + 0.915559i \(0.631749\pi\)
\(548\) 0 0
\(549\) −21.8252 + 26.4579i −0.931479 + 1.12919i
\(550\) 0 0
\(551\) 5.99716 10.3874i 0.255488 0.442518i
\(552\) 0 0
\(553\) 7.14344 + 1.40525i 0.303770 + 0.0597571i
\(554\) 0 0
\(555\) 1.87608 + 3.97960i 0.0796350 + 0.168924i
\(556\) 0 0
\(557\) −7.72089 + 13.3730i −0.327145 + 0.566631i −0.981944 0.189172i \(-0.939420\pi\)
0.654799 + 0.755803i \(0.272753\pi\)
\(558\) 0 0
\(559\) 2.06797 0.0874660
\(560\) 0 0
\(561\) −2.75203 + 3.96560i −0.116191 + 0.167428i
\(562\) 0 0
\(563\) 0.956715 + 1.65708i 0.0403207 + 0.0698375i 0.885482 0.464675i \(-0.153829\pi\)
−0.845161 + 0.534512i \(0.820495\pi\)
\(564\) 0 0
\(565\) 11.2802 19.5379i 0.474561 0.821964i
\(566\) 0 0
\(567\) 0.0697573 23.8117i 0.00292953 0.999996i
\(568\) 0 0
\(569\) 7.38138 12.7849i 0.309444 0.535972i −0.668797 0.743445i \(-0.733191\pi\)
0.978241 + 0.207473i \(0.0665239\pi\)
\(570\) 0 0
\(571\) 1.28208 + 2.22063i 0.0536535 + 0.0929306i 0.891605 0.452814i \(-0.149580\pi\)
−0.837951 + 0.545745i \(0.816247\pi\)
\(572\) 0 0
\(573\) −18.3785 + 26.4829i −0.767773 + 1.10634i
\(574\) 0 0
\(575\) −20.5220 −0.855827
\(576\) 0 0
\(577\) 7.01283 12.1466i 0.291948 0.505669i −0.682322 0.731052i \(-0.739030\pi\)
0.974270 + 0.225383i \(0.0723632\pi\)
\(578\) 0 0
\(579\) −2.16358 4.58945i −0.0899151 0.190731i
\(580\) 0 0
\(581\) 20.0413 22.9504i 0.831455 0.952141i
\(582\) 0 0
\(583\) 21.7251 37.6289i 0.899760 1.55843i
\(584\) 0 0
\(585\) −4.54177 + 5.50581i −0.187779 + 0.227637i
\(586\) 0 0
\(587\) −15.7666 27.3085i −0.650756 1.12714i −0.982940 0.183928i \(-0.941119\pi\)
0.332183 0.943215i \(-0.392215\pi\)
\(588\) 0 0
\(589\) −5.41022 + 9.37078i −0.222924 + 0.386116i
\(590\) 0 0
\(591\) −31.9411 2.65955i −1.31388 0.109399i
\(592\) 0 0
\(593\) −5.72311 9.91272i −0.235020 0.407067i 0.724258 0.689529i \(-0.242182\pi\)
−0.959279 + 0.282462i \(0.908849\pi\)
\(594\) 0 0
\(595\) 5.83033 + 1.14693i 0.239020 + 0.0470196i
\(596\) 0 0
\(597\) −1.56783 + 2.25921i −0.0641672 + 0.0924631i
\(598\) 0 0
\(599\) 2.80684 + 4.86159i 0.114684 + 0.198639i 0.917654 0.397381i \(-0.130081\pi\)
−0.802969 + 0.596021i \(0.796748\pi\)
\(600\) 0 0
\(601\) 19.2223 + 33.2940i 0.784094 + 1.35809i 0.929539 + 0.368725i \(0.120205\pi\)
−0.145444 + 0.989366i \(0.546461\pi\)
\(602\) 0 0
\(603\) 17.0059 + 45.5315i 0.692535 + 1.85418i
\(604\) 0 0
\(605\) 2.29983 0.0935013
\(606\) 0 0
\(607\) 16.3437 0.663372 0.331686 0.943390i \(-0.392383\pi\)
0.331686 + 0.943390i \(0.392383\pi\)
\(608\) 0 0
\(609\) 26.3019 25.4984i 1.06581 1.03325i
\(610\) 0 0
\(611\) −0.263953 0.457180i −0.0106784 0.0184955i
\(612\) 0 0
\(613\) −6.19332 + 10.7272i −0.250146 + 0.433266i −0.963566 0.267471i \(-0.913812\pi\)
0.713420 + 0.700737i \(0.247145\pi\)
\(614\) 0 0
\(615\) 9.17369 13.2190i 0.369919 0.533043i
\(616\) 0 0
\(617\) 20.9853 36.3476i 0.844836 1.46330i −0.0409280 0.999162i \(-0.513031\pi\)
0.885764 0.464136i \(-0.153635\pi\)
\(618\) 0 0
\(619\) 35.8945 1.44272 0.721361 0.692559i \(-0.243517\pi\)
0.721361 + 0.692559i \(0.243517\pi\)
\(620\) 0 0
\(621\) −38.5190 9.80342i −1.54571 0.393398i
\(622\) 0 0
\(623\) −17.7992 + 20.3828i −0.713111 + 0.816619i
\(624\) 0 0
\(625\) −31.2166 −1.24866
\(626\) 0 0
\(627\) −3.81151 8.08510i −0.152217 0.322888i
\(628\) 0 0
\(629\) −0.742544 −0.0296072
\(630\) 0 0
\(631\) 19.5519 0.778349 0.389175 0.921164i \(-0.372760\pi\)
0.389175 + 0.921164i \(0.372760\pi\)
\(632\) 0 0
\(633\) −24.3788 + 35.1291i −0.968968 + 1.39626i
\(634\) 0 0
\(635\) 34.8299 1.38218
\(636\) 0 0
\(637\) −0.809390 5.95353i −0.0320692 0.235888i
\(638\) 0 0
\(639\) −42.6574 7.15327i −1.68750 0.282979i
\(640\) 0 0
\(641\) −14.6026 −0.576769 −0.288385 0.957515i \(-0.593118\pi\)
−0.288385 + 0.957515i \(0.593118\pi\)
\(642\) 0 0
\(643\) 5.96942 10.3393i 0.235411 0.407744i −0.723981 0.689820i \(-0.757690\pi\)
0.959392 + 0.282076i \(0.0910231\pi\)
\(644\) 0 0
\(645\) −4.93227 10.4625i −0.194208 0.411960i
\(646\) 0 0
\(647\) 1.92060 3.32658i 0.0755067 0.130781i −0.825800 0.563963i \(-0.809276\pi\)
0.901306 + 0.433182i \(0.142609\pi\)
\(648\) 0 0
\(649\) 2.52692 + 4.37675i 0.0991903 + 0.171803i
\(650\) 0 0
\(651\) −23.7278 + 23.0029i −0.929965 + 0.901554i
\(652\) 0 0
\(653\) 39.9918 1.56500 0.782501 0.622650i \(-0.213944\pi\)
0.782501 + 0.622650i \(0.213944\pi\)
\(654\) 0 0
\(655\) −4.39469 −0.171715
\(656\) 0 0
\(657\) −15.9171 + 19.2957i −0.620986 + 0.752796i
\(658\) 0 0
\(659\) −1.60101 2.77303i −0.0623665 0.108022i 0.833156 0.553038i \(-0.186531\pi\)
−0.895523 + 0.445016i \(0.853198\pi\)
\(660\) 0 0
\(661\) 21.6515 + 37.5015i 0.842146 + 1.45864i 0.888077 + 0.459695i \(0.152041\pi\)
−0.0459311 + 0.998945i \(0.514625\pi\)
\(662\) 0 0
\(663\) −0.513658 1.08959i −0.0199488 0.0423162i
\(664\) 0 0
\(665\) −7.23755 + 8.28808i −0.280660 + 0.321398i
\(666\) 0 0
\(667\) −30.5740 52.9557i −1.18383 2.05045i
\(668\) 0 0
\(669\) 14.4534 + 30.6591i 0.558801 + 1.18535i
\(670\) 0 0
\(671\) −19.6610 + 34.0539i −0.759006 + 1.31464i
\(672\) 0 0
\(673\) 12.2936 + 21.2931i 0.473883 + 0.820790i 0.999553 0.0298991i \(-0.00951859\pi\)
−0.525670 + 0.850689i \(0.676185\pi\)
\(674\) 0 0
\(675\) 13.5099 + 3.43838i 0.519995 + 0.132343i
\(676\) 0 0
\(677\) 19.2094 33.2716i 0.738276 1.27873i −0.214994 0.976615i \(-0.568973\pi\)
0.953271 0.302117i \(-0.0976933\pi\)
\(678\) 0 0
\(679\) 13.3253 15.2594i 0.511376 0.585603i
\(680\) 0 0
\(681\) −14.9273 1.24291i −0.572013 0.0476282i
\(682\) 0 0
\(683\) 0.122464 0.212113i 0.00468594 0.00811629i −0.863673 0.504053i \(-0.831842\pi\)
0.868359 + 0.495936i \(0.165175\pi\)
\(684\) 0 0
\(685\) 13.6591 0.521886
\(686\) 0 0
\(687\) 19.9236 + 1.65893i 0.760135 + 0.0632920i
\(688\) 0 0
\(689\) 5.42160 + 9.39049i 0.206547 + 0.357749i
\(690\) 0 0
\(691\) −3.65146 + 6.32452i −0.138908 + 0.240596i −0.927084 0.374855i \(-0.877693\pi\)
0.788175 + 0.615451i \(0.211026\pi\)
\(692\) 0 0
\(693\) −4.43598 26.9369i −0.168509 1.02325i
\(694\) 0 0
\(695\) −11.4475 + 19.8277i −0.434229 + 0.752106i
\(696\) 0 0
\(697\) 1.35782 + 2.35181i 0.0514309 + 0.0890810i
\(698\) 0 0
\(699\) −12.0054 25.4664i −0.454087 0.963226i
\(700\) 0 0
\(701\) 32.3889 1.22331 0.611656 0.791124i \(-0.290504\pi\)
0.611656 + 0.791124i \(0.290504\pi\)
\(702\) 0 0
\(703\) 0.687513 1.19081i 0.0259301 0.0449122i
\(704\) 0 0
\(705\) −1.68346 + 2.42582i −0.0634029 + 0.0913618i
\(706\) 0 0
\(707\) 3.15427 + 9.22813i 0.118629 + 0.347059i
\(708\) 0 0
\(709\) −1.46137 + 2.53116i −0.0548828 + 0.0950599i −0.892162 0.451716i \(-0.850812\pi\)
0.837279 + 0.546776i \(0.184145\pi\)
\(710\) 0 0
\(711\) −5.25307 + 6.36809i −0.197006 + 0.238822i
\(712\) 0 0
\(713\) 27.5817 + 47.7729i 1.03294 + 1.78911i
\(714\) 0 0
\(715\) −4.09141 + 7.08652i −0.153010 + 0.265021i
\(716\) 0 0
\(717\) −24.5564 + 35.3851i −0.917075 + 1.32148i
\(718\) 0 0
\(719\) −8.78527 15.2165i −0.327635 0.567481i 0.654407 0.756143i \(-0.272918\pi\)
−0.982042 + 0.188662i \(0.939585\pi\)
\(720\) 0 0
\(721\) −13.7949 40.3584i −0.513750 1.50302i
\(722\) 0 0
\(723\) −32.8581 2.73590i −1.22201 0.101749i
\(724\) 0 0
\(725\) 10.7233 + 18.5733i 0.398253 + 0.689795i
\(726\) 0 0
\(727\) −20.0486 34.7252i −0.743561 1.28789i −0.950864 0.309609i \(-0.899802\pi\)
0.207303 0.978277i \(-0.433531\pi\)
\(728\) 0 0
\(729\) 23.7150 + 12.9074i 0.878332 + 0.478052i
\(730\) 0 0
\(731\) 1.95217 0.0722037
\(732\) 0 0
\(733\) −7.12469 −0.263156 −0.131578 0.991306i \(-0.542004\pi\)
−0.131578 + 0.991306i \(0.542004\pi\)
\(734\) 0 0
\(735\) −28.1902 + 18.2945i −1.03981 + 0.674805i
\(736\) 0 0
\(737\) 27.8615 + 48.2576i 1.02629 + 1.77759i
\(738\) 0 0
\(739\) 2.13570 3.69914i 0.0785631 0.136075i −0.824067 0.566492i \(-0.808300\pi\)
0.902630 + 0.430417i \(0.141633\pi\)
\(740\) 0 0
\(741\) 2.22295 + 0.185092i 0.0816621 + 0.00679953i
\(742\) 0 0
\(743\) −0.108257 + 0.187507i −0.00397157 + 0.00687896i −0.868004 0.496557i \(-0.834598\pi\)
0.864033 + 0.503436i \(0.167931\pi\)
\(744\) 0 0
\(745\) −38.5667 −1.41297
\(746\) 0 0
\(747\) 12.0883 + 32.3651i 0.442288 + 1.18418i
\(748\) 0 0
\(749\) −11.0251 + 12.6254i −0.402849 + 0.461323i
\(750\) 0 0
\(751\) −23.5840 −0.860594 −0.430297 0.902687i \(-0.641591\pi\)
−0.430297 + 0.902687i \(0.641591\pi\)
\(752\) 0 0
\(753\) 1.69230 + 0.140908i 0.0616710 + 0.00513499i
\(754\) 0 0
\(755\) −63.7312 −2.31942
\(756\) 0 0
\(757\) −22.0176 −0.800242 −0.400121 0.916462i \(-0.631032\pi\)
−0.400121 + 0.916462i \(0.631032\pi\)
\(758\) 0 0
\(759\) −45.4119 3.78118i −1.64835 0.137248i
\(760\) 0 0
\(761\) −18.1140 −0.656631 −0.328316 0.944568i \(-0.606481\pi\)
−0.328316 + 0.944568i \(0.606481\pi\)
\(762\) 0 0
\(763\) −25.4136 4.99932i −0.920035 0.180988i
\(764\) 0 0
\(765\) −4.28745 + 5.19750i −0.155013 + 0.187916i
\(766\) 0 0
\(767\) −1.26121 −0.0455397
\(768\) 0 0
\(769\) 3.16710 5.48558i 0.114209 0.197815i −0.803255 0.595636i \(-0.796900\pi\)
0.917463 + 0.397821i \(0.130233\pi\)
\(770\) 0 0
\(771\) 7.53821 + 0.627663i 0.271482 + 0.0226047i
\(772\) 0 0
\(773\) 3.10740 5.38218i 0.111765 0.193584i −0.804717 0.593659i \(-0.797683\pi\)
0.916482 + 0.400076i \(0.131016\pi\)
\(774\) 0 0
\(775\) −9.67381 16.7555i −0.347494 0.601876i
\(776\) 0 0
\(777\) 3.01525 2.92313i 0.108171 0.104867i
\(778\) 0 0
\(779\) −5.02874 −0.180173
\(780\) 0 0
\(781\) −49.5887 −1.77442
\(782\) 0 0
\(783\) 11.2547 + 39.9839i 0.402209 + 1.42891i
\(784\) 0 0
\(785\) −25.7506 44.6014i −0.919079 1.59189i
\(786\) 0 0
\(787\) −9.65916 16.7302i −0.344312 0.596366i 0.640917 0.767611i \(-0.278554\pi\)
−0.985229 + 0.171245i \(0.945221\pi\)
\(788\) 0 0
\(789\) 21.5488 + 1.79424i 0.767156 + 0.0638766i
\(790\) 0 0
\(791\) −21.1295 4.15656i −0.751279 0.147790i
\(792\) 0 0
\(793\) −4.90651 8.49832i −0.174235 0.301784i
\(794\) 0 0
\(795\) 34.5784 49.8265i 1.22637 1.76716i
\(796\) 0 0
\(797\) −5.09519 + 8.82513i −0.180481 + 0.312602i −0.942044 0.335488i \(-0.891099\pi\)
0.761563 + 0.648090i \(0.224432\pi\)
\(798\) 0 0
\(799\) −0.249172 0.431579i −0.00881509 0.0152682i
\(800\) 0 0
\(801\) −10.7359 28.7442i −0.379335 1.01563i
\(802\) 0 0
\(803\) −14.3388 + 24.8355i −0.506004 + 0.876424i
\(804\) 0 0
\(805\) 18.1436 + 53.0808i 0.639478 + 1.87085i
\(806\) 0 0
\(807\) 16.3781 23.6005i 0.576538 0.830776i
\(808\) 0 0
\(809\) −23.7068 + 41.0613i −0.833485 + 1.44364i 0.0617729 + 0.998090i \(0.480325\pi\)
−0.895258 + 0.445548i \(0.853009\pi\)
\(810\) 0 0
\(811\) −47.4177 −1.66506 −0.832531 0.553979i \(-0.813109\pi\)
−0.832531 + 0.553979i \(0.813109\pi\)
\(812\) 0 0
\(813\) −19.1755 40.6757i −0.672514 1.42656i
\(814\) 0 0
\(815\) 6.79627 + 11.7715i 0.238063 + 0.412337i
\(816\) 0 0
\(817\) −1.80750 + 3.13067i −0.0632363 + 0.109528i
\(818\) 0 0
\(819\) 6.37513 + 2.40241i 0.222765 + 0.0839469i
\(820\) 0 0
\(821\) −26.8177 + 46.4495i −0.935943 + 1.62110i −0.162998 + 0.986626i \(0.552117\pi\)
−0.772944 + 0.634474i \(0.781217\pi\)
\(822\) 0 0
\(823\) −0.475192 0.823057i −0.0165641 0.0286899i 0.857625 0.514276i \(-0.171939\pi\)
−0.874189 + 0.485586i \(0.838606\pi\)
\(824\) 0 0
\(825\) 15.9274 + 1.32618i 0.554522 + 0.0461718i
\(826\) 0 0
\(827\) −20.7813 −0.722636 −0.361318 0.932443i \(-0.617673\pi\)
−0.361318 + 0.932443i \(0.617673\pi\)
\(828\) 0 0
\(829\) 12.1615 21.0644i 0.422387 0.731596i −0.573785 0.819006i \(-0.694526\pi\)
0.996172 + 0.0874096i \(0.0278589\pi\)
\(830\) 0 0
\(831\) −3.38440 0.281799i −0.117403 0.00977551i
\(832\) 0 0
\(833\) −0.764066 5.62015i −0.0264733 0.194727i
\(834\) 0 0
\(835\) 15.1754 26.2846i 0.525166 0.909615i
\(836\) 0 0
\(837\) −10.1532 36.0707i −0.350946 1.24678i
\(838\) 0 0
\(839\) 15.7367 + 27.2567i 0.543290 + 0.941006i 0.998712 + 0.0507305i \(0.0161550\pi\)
−0.455422 + 0.890276i \(0.650512\pi\)
\(840\) 0 0
\(841\) −17.4514 + 30.2268i −0.601774 + 1.04230i
\(842\) 0 0
\(843\) 13.9156 + 29.5182i 0.479277 + 1.01666i
\(844\) 0 0
\(845\) 16.9957 + 29.4373i 0.584668 + 1.01268i
\(846\) 0 0
\(847\) −0.710026 2.07725i −0.0243968 0.0713751i
\(848\) 0 0
\(849\) −17.3450 36.7928i −0.595278 1.26272i
\(850\) 0 0
\(851\) −3.50499 6.07082i −0.120150 0.208105i
\(852\) 0 0
\(853\) −26.6959 46.2386i −0.914049 1.58318i −0.808287 0.588788i \(-0.799605\pi\)
−0.105762 0.994391i \(-0.533728\pi\)
\(854\) 0 0
\(855\) −4.36546 11.6880i −0.149296 0.399722i
\(856\) 0 0
\(857\) 21.6297 0.738857 0.369428 0.929259i \(-0.379554\pi\)
0.369428 + 0.929259i \(0.379554\pi\)
\(858\) 0 0
\(859\) 34.4288 1.17470 0.587348 0.809334i \(-0.300172\pi\)
0.587348 + 0.809334i \(0.300172\pi\)
\(860\) 0 0
\(861\) −14.7719 4.20474i −0.503425 0.143297i
\(862\) 0 0
\(863\) 13.4262 + 23.2548i 0.457033 + 0.791604i 0.998803 0.0489229i \(-0.0155789\pi\)
−0.541770 + 0.840527i \(0.682246\pi\)
\(864\) 0 0
\(865\) 18.5786 32.1790i 0.631690 1.09412i
\(866\) 0 0
\(867\) 12.0709 + 25.6051i 0.409948 + 0.869595i
\(868\) 0 0
\(869\) −4.73217 + 8.19636i −0.160528 + 0.278043i
\(870\) 0 0
\(871\) −13.9060 −0.471186
\(872\) 0 0
\(873\) 8.03737 + 21.5191i 0.272024 + 0.728313i
\(874\) 0 0
\(875\) 5.49609 + 16.0793i 0.185802 + 0.543581i
\(876\) 0 0
\(877\) 32.4068 1.09430 0.547150 0.837034i \(-0.315713\pi\)
0.547150 + 0.837034i \(0.315713\pi\)
\(878\) 0 0
\(879\) −1.98260 + 2.85687i −0.0668715 + 0.0963599i
\(880\) 0 0
\(881\) −39.0404 −1.31530 −0.657652 0.753322i \(-0.728450\pi\)
−0.657652 + 0.753322i \(0.728450\pi\)
\(882\) 0 0
\(883\) 13.8079 0.464672 0.232336 0.972636i \(-0.425363\pi\)
0.232336 + 0.972636i \(0.425363\pi\)
\(884\) 0 0
\(885\) 3.00808 + 6.38084i 0.101115 + 0.214490i
\(886\) 0 0
\(887\) −23.9555 −0.804348 −0.402174 0.915563i \(-0.631745\pi\)
−0.402174 + 0.915563i \(0.631745\pi\)
\(888\) 0 0
\(889\) −10.7530 31.4590i −0.360645 1.05510i
\(890\) 0 0
\(891\) 29.2616 + 10.0978i 0.980301 + 0.338288i
\(892\) 0 0
\(893\) 0.922824 0.0308811
\(894\) 0 0
\(895\) 18.4293 31.9206i 0.616025 1.06699i
\(896\) 0 0
\(897\) 6.48357 9.34265i 0.216480 0.311942i
\(898\) 0 0
\(899\) 28.8244 49.9253i 0.961346 1.66510i
\(900\) 0 0
\(901\) 5.11801 + 8.86465i 0.170505 + 0.295324i
\(902\) 0 0
\(903\) −7.92719 + 7.68500i −0.263800 + 0.255741i
\(904\) 0 0
\(905\) −30.2652 −1.00605
\(906\) 0 0
\(907\) −4.70128 −0.156103 −0.0780517 0.996949i \(-0.524870\pi\)
−0.0780517 + 0.996949i \(0.524870\pi\)
\(908\) 0 0
\(909\) −10.9058 1.82881i −0.361723 0.0606577i
\(910\) 0 0
\(911\) −21.0884 36.5262i −0.698689 1.21017i −0.968921 0.247370i \(-0.920434\pi\)
0.270232 0.962795i \(-0.412900\pi\)
\(912\) 0 0
\(913\) 19.8048 + 34.3029i 0.655442 + 1.13526i
\(914\) 0 0
\(915\) −31.2932 + 45.0926i −1.03452 + 1.49071i
\(916\) 0 0
\(917\) 1.35677 + 3.96937i 0.0448046 + 0.131080i
\(918\) 0 0
\(919\) 20.1071 + 34.8265i 0.663271 + 1.14882i 0.979751 + 0.200220i \(0.0641656\pi\)
−0.316480 + 0.948599i \(0.602501\pi\)
\(920\) 0 0
\(921\) −56.5521 4.70876i −1.86345 0.155159i
\(922\) 0 0
\(923\) 6.18756 10.7172i 0.203666 0.352760i
\(924\) 0 0
\(925\) 1.22932 + 2.12924i 0.0404196 + 0.0700089i
\(926\) 0 0
\(927\) 47.6956 + 7.99812i 1.56653 + 0.262693i
\(928\) 0 0
\(929\) 15.0712 26.1040i 0.494469 0.856446i −0.505510 0.862820i \(-0.668696\pi\)
0.999980 + 0.00637464i \(0.00202912\pi\)
\(930\) 0 0
\(931\) 9.72040 + 3.97831i 0.318573 + 0.130384i
\(932\) 0 0
\(933\) −10.4136 22.0896i −0.340925 0.723183i
\(934\) 0 0
\(935\) −3.86230 + 6.68970i −0.126311 + 0.218776i
\(936\) 0 0
\(937\) 35.1550 1.14846 0.574231 0.818693i \(-0.305301\pi\)
0.574231 + 0.818693i \(0.305301\pi\)
\(938\) 0 0
\(939\) 33.7271 48.5998i 1.10064 1.58600i
\(940\) 0 0
\(941\) 1.52163 + 2.63554i 0.0496038 + 0.0859162i 0.889761 0.456426i \(-0.150871\pi\)
−0.840157 + 0.542343i \(0.817537\pi\)
\(942\) 0 0
\(943\) −12.8184 + 22.2022i −0.417426 + 0.723003i
\(944\) 0 0
\(945\) −3.05066 37.9836i −0.0992378 1.23561i
\(946\) 0 0
\(947\) 6.33905 10.9796i 0.205991 0.356788i −0.744457 0.667671i \(-0.767291\pi\)
0.950448 + 0.310883i \(0.100625\pi\)
\(948\) 0 0
\(949\) −3.57831 6.19781i −0.116157 0.201190i
\(950\) 0 0
\(951\) 1.56141 2.24996i 0.0506323 0.0729598i
\(952\) 0 0
\(953\) 23.0052 0.745212 0.372606 0.927990i \(-0.378464\pi\)
0.372606 + 0.927990i \(0.378464\pi\)
\(954\) 0 0
\(955\) −25.7930 + 44.6749i −0.834643 + 1.44564i
\(956\) 0 0
\(957\) 20.3068 + 43.0755i 0.656425 + 1.39243i
\(958\) 0 0
\(959\) −4.21697 12.3371i −0.136173 0.398387i
\(960\) 0 0
\(961\) −10.5033 + 18.1923i −0.338817 + 0.586849i
\(962\) 0 0
\(963\) −6.65001 17.8046i −0.214293 0.573746i
\(964\) 0 0
\(965\) −4.05985 7.03187i −0.130691 0.226364i
\(966\) 0 0
\(967\) 0.617767 1.07000i 0.0198660 0.0344090i −0.855921 0.517106i \(-0.827009\pi\)
0.875788 + 0.482697i \(0.160343\pi\)
\(968\) 0 0
\(969\) 2.09847 + 0.174728i 0.0674126 + 0.00561306i
\(970\) 0 0
\(971\) 7.01657 + 12.1530i 0.225172 + 0.390010i 0.956371 0.292155i \(-0.0943722\pi\)
−0.731199 + 0.682164i \(0.761039\pi\)
\(972\) 0 0
\(973\) 21.4429 + 4.21821i 0.687428 + 0.135230i
\(974\) 0 0
\(975\) −2.27400 + 3.27677i −0.0728263 + 0.104941i
\(976\) 0 0
\(977\) −11.1301 19.2780i −0.356084 0.616756i 0.631218 0.775605i \(-0.282555\pi\)
−0.987303 + 0.158849i \(0.949222\pi\)
\(978\) 0 0
\(979\) −17.5891 30.4652i −0.562151 0.973673i
\(980\) 0 0
\(981\) 18.6884 22.6552i 0.596675 0.723325i
\(982\) 0 0
\(983\) −33.7341 −1.07595 −0.537976 0.842960i \(-0.680811\pi\)
−0.537976 + 0.842960i \(0.680811\pi\)
\(984\) 0 0
\(985\) −51.2921 −1.63430
\(986\) 0 0
\(987\) 2.71079 + 0.771611i 0.0862853 + 0.0245607i
\(988\) 0 0
\(989\) 9.21474 + 15.9604i 0.293012 + 0.507511i
\(990\) 0 0
\(991\) −6.87364 + 11.9055i −0.218348 + 0.378191i −0.954303 0.298840i \(-0.903400\pi\)
0.735955 + 0.677031i \(0.236734\pi\)
\(992\) 0 0
\(993\) 13.5656 19.5476i 0.430491 0.620325i
\(994\) 0 0
\(995\) −2.20035 + 3.81112i −0.0697559 + 0.120821i
\(996\) 0 0
\(997\) 37.2565 1.17993 0.589963 0.807430i \(-0.299142\pi\)
0.589963 + 0.807430i \(0.299142\pi\)
\(998\) 0 0
\(999\) 1.29023 + 4.58374i 0.0408212 + 0.145023i
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 1008.2.t.l.193.10 22
3.2 odd 2 3024.2.t.k.1873.3 22
4.3 odd 2 504.2.t.c.193.2 yes 22
7.2 even 3 1008.2.q.l.625.4 22
9.2 odd 6 3024.2.q.l.2881.9 22
9.7 even 3 1008.2.q.l.529.4 22
12.11 even 2 1512.2.t.c.361.3 22
21.2 odd 6 3024.2.q.l.2305.9 22
28.23 odd 6 504.2.q.c.121.8 yes 22
36.7 odd 6 504.2.q.c.25.8 22
36.11 even 6 1512.2.q.d.1369.9 22
63.2 odd 6 3024.2.t.k.289.3 22
63.16 even 3 inner 1008.2.t.l.961.10 22
84.23 even 6 1512.2.q.d.793.9 22
252.79 odd 6 504.2.t.c.457.2 yes 22
252.191 even 6 1512.2.t.c.289.3 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.c.25.8 22 36.7 odd 6
504.2.q.c.121.8 yes 22 28.23 odd 6
504.2.t.c.193.2 yes 22 4.3 odd 2
504.2.t.c.457.2 yes 22 252.79 odd 6
1008.2.q.l.529.4 22 9.7 even 3
1008.2.q.l.625.4 22 7.2 even 3
1008.2.t.l.193.10 22 1.1 even 1 trivial
1008.2.t.l.961.10 22 63.16 even 3 inner
1512.2.q.d.793.9 22 84.23 even 6
1512.2.q.d.1369.9 22 36.11 even 6
1512.2.t.c.289.3 22 252.191 even 6
1512.2.t.c.361.3 22 12.11 even 2
3024.2.q.l.2305.9 22 21.2 odd 6
3024.2.q.l.2881.9 22 9.2 odd 6
3024.2.t.k.289.3 22 63.2 odd 6
3024.2.t.k.1873.3 22 3.2 odd 2