Properties

Label 460.2.i.b.137.4
Level $460$
Weight $2$
Character 460.137
Analytic conductor $3.673$
Analytic rank $0$
Dimension $16$
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] = [460,2,Mod(137,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.i (of order \(4\), 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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.4
Root \(2.23266 + 0.123447i\) of defining polynomial
Character \(\chi\) \(=\) 460.137
Dual form 460.2.i.b.413.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.237125 - 0.237125i) q^{3} +(2.23266 - 0.123447i) q^{5} +(-1.69421 + 1.69421i) q^{7} +2.88754i q^{9} +O(q^{10})\) \(q+(0.237125 - 0.237125i) q^{3} +(2.23266 - 0.123447i) q^{5} +(-1.69421 + 1.69421i) q^{7} +2.88754i q^{9} +5.55617i q^{11} +(2.65542 - 2.65542i) q^{13} +(0.500147 - 0.558692i) q^{15} +(-0.435244 + 0.435244i) q^{17} -1.92086 q^{19} +0.803478i q^{21} +(4.76463 - 0.546205i) q^{23} +(4.96952 - 0.551231i) q^{25} +(1.39608 + 1.39608i) q^{27} -7.41329i q^{29} -0.525750 q^{31} +(1.31751 + 1.31751i) q^{33} +(-3.57344 + 3.99173i) q^{35} +(1.94110 - 1.94110i) q^{37} -1.25933i q^{39} -6.78508 q^{41} +(-1.88256 - 1.88256i) q^{43} +(0.356459 + 6.44690i) q^{45} +(7.54296 + 7.54296i) q^{47} +1.25933i q^{49} +0.206415i q^{51} +(0.467258 + 0.467258i) q^{53} +(0.685894 + 12.4050i) q^{55} +(-0.455484 + 0.455484i) q^{57} -7.72412i q^{59} -0.714153i q^{61} +(-4.89209 - 4.89209i) q^{63} +(5.60083 - 6.25644i) q^{65} +(4.97934 - 4.97934i) q^{67} +(1.00029 - 1.25933i) q^{69} -4.93904 q^{71} +(0.547956 - 0.547956i) q^{73} +(1.04769 - 1.30911i) q^{75} +(-9.41329 - 9.41329i) q^{77} -9.07438 q^{79} -8.00054 q^{81} +(-10.7036 - 10.7036i) q^{83} +(-0.918022 + 1.02548i) q^{85} +(-1.75788 - 1.75788i) q^{87} +13.0332 q^{89} +8.99764i q^{91} +(-0.124668 + 0.124668i) q^{93} +(-4.28862 + 0.237125i) q^{95} +(3.99177 - 3.99177i) q^{97} -16.0437 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} + 12 q^{13} + 12 q^{23} + 36 q^{25} + 52 q^{27} - 8 q^{31} + 16 q^{35} - 48 q^{41} + 4 q^{47} + 24 q^{55} + 8 q^{71} - 52 q^{73} - 56 q^{75} - 64 q^{77} - 152 q^{81} + 28 q^{85} + 28 q^{87} + 84 q^{93} - 68 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) 0.237125 0.237125i 0.136904 0.136904i −0.635334 0.772238i \(-0.719137\pi\)
0.772238 + 0.635334i \(0.219137\pi\)
\(4\) 0 0
\(5\) 2.23266 0.123447i 0.998475 0.0552073i
\(6\) 0 0
\(7\) −1.69421 + 1.69421i −0.640350 + 0.640350i −0.950641 0.310292i \(-0.899573\pi\)
0.310292 + 0.950641i \(0.399573\pi\)
\(8\) 0 0
\(9\) 2.88754i 0.962514i
\(10\) 0 0
\(11\) 5.55617i 1.67525i 0.546248 + 0.837624i \(0.316056\pi\)
−0.546248 + 0.837624i \(0.683944\pi\)
\(12\) 0 0
\(13\) 2.65542 2.65542i 0.736480 0.736480i −0.235415 0.971895i \(-0.575645\pi\)
0.971895 + 0.235415i \(0.0756450\pi\)
\(14\) 0 0
\(15\) 0.500147 0.558692i 0.129137 0.144254i
\(16\) 0 0
\(17\) −0.435244 + 0.435244i −0.105562 + 0.105562i −0.757915 0.652353i \(-0.773782\pi\)
0.652353 + 0.757915i \(0.273782\pi\)
\(18\) 0 0
\(19\) −1.92086 −0.440676 −0.220338 0.975424i \(-0.570716\pi\)
−0.220338 + 0.975424i \(0.570716\pi\)
\(20\) 0 0
\(21\) 0.803478i 0.175333i
\(22\) 0 0
\(23\) 4.76463 0.546205i 0.993493 0.113892i
\(24\) 0 0
\(25\) 4.96952 0.551231i 0.993904 0.110246i
\(26\) 0 0
\(27\) 1.39608 + 1.39608i 0.268677 + 0.268677i
\(28\) 0 0
\(29\) 7.41329i 1.37661i −0.725420 0.688307i \(-0.758354\pi\)
0.725420 0.688307i \(-0.241646\pi\)
\(30\) 0 0
\(31\) −0.525750 −0.0944275 −0.0472137 0.998885i \(-0.515034\pi\)
−0.0472137 + 0.998885i \(0.515034\pi\)
\(32\) 0 0
\(33\) 1.31751 + 1.31751i 0.229348 + 0.229348i
\(34\) 0 0
\(35\) −3.57344 + 3.99173i −0.604021 + 0.674725i
\(36\) 0 0
\(37\) 1.94110 1.94110i 0.319115 0.319115i −0.529312 0.848427i \(-0.677550\pi\)
0.848427 + 0.529312i \(0.177550\pi\)
\(38\) 0 0
\(39\) 1.25933i 0.201654i
\(40\) 0 0
\(41\) −6.78508 −1.05965 −0.529826 0.848106i \(-0.677743\pi\)
−0.529826 + 0.848106i \(0.677743\pi\)
\(42\) 0 0
\(43\) −1.88256 1.88256i −0.287087 0.287087i 0.548840 0.835927i \(-0.315070\pi\)
−0.835927 + 0.548840i \(0.815070\pi\)
\(44\) 0 0
\(45\) 0.356459 + 6.44690i 0.0531378 + 0.961047i
\(46\) 0 0
\(47\) 7.54296 + 7.54296i 1.10025 + 1.10025i 0.994380 + 0.105874i \(0.0337640\pi\)
0.105874 + 0.994380i \(0.466236\pi\)
\(48\) 0 0
\(49\) 1.25933i 0.179904i
\(50\) 0 0
\(51\) 0.206415i 0.0289038i
\(52\) 0 0
\(53\) 0.467258 + 0.467258i 0.0641828 + 0.0641828i 0.738470 0.674287i \(-0.235549\pi\)
−0.674287 + 0.738470i \(0.735549\pi\)
\(54\) 0 0
\(55\) 0.685894 + 12.4050i 0.0924859 + 1.67269i
\(56\) 0 0
\(57\) −0.455484 + 0.455484i −0.0603304 + 0.0603304i
\(58\) 0 0
\(59\) 7.72412i 1.00560i −0.864404 0.502798i \(-0.832304\pi\)
0.864404 0.502798i \(-0.167696\pi\)
\(60\) 0 0
\(61\) 0.714153i 0.0914379i −0.998954 0.0457189i \(-0.985442\pi\)
0.998954 0.0457189i \(-0.0145579\pi\)
\(62\) 0 0
\(63\) −4.89209 4.89209i −0.616346 0.616346i
\(64\) 0 0
\(65\) 5.60083 6.25644i 0.694697 0.776016i
\(66\) 0 0
\(67\) 4.97934 4.97934i 0.608324 0.608324i −0.334184 0.942508i \(-0.608461\pi\)
0.942508 + 0.334184i \(0.108461\pi\)
\(68\) 0 0
\(69\) 1.00029 1.25933i 0.120421 0.151606i
\(70\) 0 0
\(71\) −4.93904 −0.586157 −0.293078 0.956088i \(-0.594680\pi\)
−0.293078 + 0.956088i \(0.594680\pi\)
\(72\) 0 0
\(73\) 0.547956 0.547956i 0.0641334 0.0641334i −0.674313 0.738446i \(-0.735560\pi\)
0.738446 + 0.674313i \(0.235560\pi\)
\(74\) 0 0
\(75\) 1.04769 1.30911i 0.120977 0.151163i
\(76\) 0 0
\(77\) −9.41329 9.41329i −1.07274 1.07274i
\(78\) 0 0
\(79\) −9.07438 −1.02095 −0.510474 0.859893i \(-0.670530\pi\)
−0.510474 + 0.859893i \(0.670530\pi\)
\(80\) 0 0
\(81\) −8.00054 −0.888949
\(82\) 0 0
\(83\) −10.7036 10.7036i −1.17488 1.17488i −0.981033 0.193842i \(-0.937905\pi\)
−0.193842 0.981033i \(-0.562095\pi\)
\(84\) 0 0
\(85\) −0.918022 + 1.02548i −0.0995735 + 0.111229i
\(86\) 0 0
\(87\) −1.75788 1.75788i −0.188464 0.188464i
\(88\) 0 0
\(89\) 13.0332 1.38152 0.690758 0.723086i \(-0.257277\pi\)
0.690758 + 0.723086i \(0.257277\pi\)
\(90\) 0 0
\(91\) 8.99764i 0.943209i
\(92\) 0 0
\(93\) −0.124668 + 0.124668i −0.0129275 + 0.0129275i
\(94\) 0 0
\(95\) −4.28862 + 0.237125i −0.440004 + 0.0243285i
\(96\) 0 0
\(97\) 3.99177 3.99177i 0.405302 0.405302i −0.474794 0.880097i \(-0.657478\pi\)
0.880097 + 0.474794i \(0.157478\pi\)
\(98\) 0 0
\(99\) −16.0437 −1.61245
\(100\) 0 0
\(101\) 2.20492 0.219398 0.109699 0.993965i \(-0.465011\pi\)
0.109699 + 0.993965i \(0.465011\pi\)
\(102\) 0 0
\(103\) −2.08268 2.08268i −0.205212 0.205212i 0.597016 0.802229i \(-0.296353\pi\)
−0.802229 + 0.597016i \(0.796353\pi\)
\(104\) 0 0
\(105\) 0.0991871 + 1.79389i 0.00967967 + 0.175066i
\(106\) 0 0
\(107\) 2.68603 2.68603i 0.259669 0.259669i −0.565251 0.824919i \(-0.691221\pi\)
0.824919 + 0.565251i \(0.191221\pi\)
\(108\) 0 0
\(109\) −7.99735 −0.766007 −0.383004 0.923747i \(-0.625110\pi\)
−0.383004 + 0.923747i \(0.625110\pi\)
\(110\) 0 0
\(111\) 0.920567i 0.0873764i
\(112\) 0 0
\(113\) −12.9629 12.9629i −1.21944 1.21944i −0.967827 0.251618i \(-0.919037\pi\)
−0.251618 0.967827i \(-0.580963\pi\)
\(114\) 0 0
\(115\) 10.5704 1.80767i 0.985690 0.168566i
\(116\) 0 0
\(117\) 7.66763 + 7.66763i 0.708872 + 0.708872i
\(118\) 0 0
\(119\) 1.47479i 0.135194i
\(120\) 0 0
\(121\) −19.8710 −1.80645
\(122\) 0 0
\(123\) −1.60891 + 1.60891i −0.145071 + 0.145071i
\(124\) 0 0
\(125\) 11.0272 1.84418i 0.986302 0.164949i
\(126\) 0 0
\(127\) −3.75288 3.75288i −0.333014 0.333014i 0.520716 0.853730i \(-0.325665\pi\)
−0.853730 + 0.520716i \(0.825665\pi\)
\(128\) 0 0
\(129\) −0.892802 −0.0786069
\(130\) 0 0
\(131\) 5.77562 0.504619 0.252309 0.967647i \(-0.418810\pi\)
0.252309 + 0.967647i \(0.418810\pi\)
\(132\) 0 0
\(133\) 3.25433 3.25433i 0.282187 0.282187i
\(134\) 0 0
\(135\) 3.28932 + 2.94464i 0.283100 + 0.253434i
\(136\) 0 0
\(137\) 10.7484 10.7484i 0.918294 0.918294i −0.0786117 0.996905i \(-0.525049\pi\)
0.996905 + 0.0786117i \(0.0250487\pi\)
\(138\) 0 0
\(139\) 10.9390i 0.927838i 0.885878 + 0.463919i \(0.153557\pi\)
−0.885878 + 0.463919i \(0.846443\pi\)
\(140\) 0 0
\(141\) 3.57725 0.301259
\(142\) 0 0
\(143\) 14.7539 + 14.7539i 1.23379 + 1.23379i
\(144\) 0 0
\(145\) −0.915151 16.5513i −0.0759992 1.37451i
\(146\) 0 0
\(147\) 0.298619 + 0.298619i 0.0246297 + 0.0246297i
\(148\) 0 0
\(149\) 13.5132 1.10704 0.553521 0.832835i \(-0.313284\pi\)
0.553521 + 0.832835i \(0.313284\pi\)
\(150\) 0 0
\(151\) −18.2333 −1.48381 −0.741904 0.670507i \(-0.766077\pi\)
−0.741904 + 0.670507i \(0.766077\pi\)
\(152\) 0 0
\(153\) −1.25679 1.25679i −0.101605 0.101605i
\(154\) 0 0
\(155\) −1.17382 + 0.0649024i −0.0942834 + 0.00521309i
\(156\) 0 0
\(157\) −13.6686 + 13.6686i −1.09087 + 1.09087i −0.0954344 + 0.995436i \(0.530424\pi\)
−0.995436 + 0.0954344i \(0.969576\pi\)
\(158\) 0 0
\(159\) 0.221597 0.0175738
\(160\) 0 0
\(161\) −7.14687 + 8.99764i −0.563253 + 0.709114i
\(162\) 0 0
\(163\) −10.0122 + 10.0122i −0.784217 + 0.784217i −0.980539 0.196322i \(-0.937100\pi\)
0.196322 + 0.980539i \(0.437100\pi\)
\(164\) 0 0
\(165\) 3.10418 + 2.77890i 0.241660 + 0.216337i
\(166\) 0 0
\(167\) 7.66763 + 7.66763i 0.593339 + 0.593339i 0.938532 0.345193i \(-0.112187\pi\)
−0.345193 + 0.938532i \(0.612187\pi\)
\(168\) 0 0
\(169\) 1.10246i 0.0848048i
\(170\) 0 0
\(171\) 5.54657i 0.424157i
\(172\) 0 0
\(173\) 16.2449 16.2449i 1.23508 1.23508i 0.273085 0.961990i \(-0.411956\pi\)
0.961990 0.273085i \(-0.0880441\pi\)
\(174\) 0 0
\(175\) −7.48549 + 9.35329i −0.565850 + 0.707042i
\(176\) 0 0
\(177\) −1.83158 1.83158i −0.137670 0.137670i
\(178\) 0 0
\(179\) 14.7851i 1.10509i −0.833483 0.552544i \(-0.813657\pi\)
0.833483 0.552544i \(-0.186343\pi\)
\(180\) 0 0
\(181\) 24.2349i 1.80136i 0.434480 + 0.900681i \(0.356932\pi\)
−0.434480 + 0.900681i \(0.643068\pi\)
\(182\) 0 0
\(183\) −0.169344 0.169344i −0.0125182 0.0125182i
\(184\) 0 0
\(185\) 4.09419 4.57344i 0.301011 0.336246i
\(186\) 0 0
\(187\) −2.41829 2.41829i −0.176843 0.176843i
\(188\) 0 0
\(189\) −4.73051 −0.344094
\(190\) 0 0
\(191\) 18.3062i 1.32459i −0.749243 0.662295i \(-0.769582\pi\)
0.749243 0.662295i \(-0.230418\pi\)
\(192\) 0 0
\(193\) 15.6339 15.6339i 1.12535 1.12535i 0.134427 0.990923i \(-0.457080\pi\)
0.990923 0.134427i \(-0.0429195\pi\)
\(194\) 0 0
\(195\) −0.155461 2.81166i −0.0111328 0.201347i
\(196\) 0 0
\(197\) 14.2621 + 14.2621i 1.01613 + 1.01613i 0.999868 + 0.0162638i \(0.00517716\pi\)
0.0162638 + 0.999868i \(0.494823\pi\)
\(198\) 0 0
\(199\) −20.5102 −1.45393 −0.726966 0.686674i \(-0.759070\pi\)
−0.726966 + 0.686674i \(0.759070\pi\)
\(200\) 0 0
\(201\) 2.36146i 0.166564i
\(202\) 0 0
\(203\) 12.5596 + 12.5596i 0.881514 + 0.881514i
\(204\) 0 0
\(205\) −15.1488 + 0.837600i −1.05804 + 0.0585005i
\(206\) 0 0
\(207\) 1.57719 + 13.7581i 0.109622 + 0.956252i
\(208\) 0 0
\(209\) 10.6726i 0.738241i
\(210\) 0 0
\(211\) 15.1978 1.04626 0.523131 0.852252i \(-0.324764\pi\)
0.523131 + 0.852252i \(0.324764\pi\)
\(212\) 0 0
\(213\) −1.17117 + 1.17117i −0.0802473 + 0.0802473i
\(214\) 0 0
\(215\) −4.43550 3.97071i −0.302499 0.270800i
\(216\) 0 0
\(217\) 0.890729 0.890729i 0.0604666 0.0604666i
\(218\) 0 0
\(219\) 0.259868i 0.0175603i
\(220\) 0 0
\(221\) 2.31151i 0.155489i
\(222\) 0 0
\(223\) 11.3103 11.3103i 0.757393 0.757393i −0.218454 0.975847i \(-0.570101\pi\)
0.975847 + 0.218454i \(0.0701013\pi\)
\(224\) 0 0
\(225\) 1.59170 + 14.3497i 0.106114 + 0.956647i
\(226\) 0 0
\(227\) 1.68026 1.68026i 0.111523 0.111523i −0.649143 0.760666i \(-0.724873\pi\)
0.760666 + 0.649143i \(0.224873\pi\)
\(228\) 0 0
\(229\) 18.8881 1.24816 0.624080 0.781360i \(-0.285474\pi\)
0.624080 + 0.781360i \(0.285474\pi\)
\(230\) 0 0
\(231\) −4.46426 −0.293726
\(232\) 0 0
\(233\) −5.07371 + 5.07371i −0.332390 + 0.332390i −0.853493 0.521104i \(-0.825520\pi\)
0.521104 + 0.853493i \(0.325520\pi\)
\(234\) 0 0
\(235\) 17.7720 + 15.9097i 1.15932 + 1.03783i
\(236\) 0 0
\(237\) −2.15176 + 2.15176i −0.139772 + 0.139772i
\(238\) 0 0
\(239\) 18.9325i 1.22464i −0.790610 0.612321i \(-0.790236\pi\)
0.790610 0.612321i \(-0.209764\pi\)
\(240\) 0 0
\(241\) 9.89044i 0.637100i 0.947906 + 0.318550i \(0.103196\pi\)
−0.947906 + 0.318550i \(0.896804\pi\)
\(242\) 0 0
\(243\) −6.08538 + 6.08538i −0.390377 + 0.390377i
\(244\) 0 0
\(245\) 0.155461 + 2.81166i 0.00993204 + 0.179630i
\(246\) 0 0
\(247\) −5.10068 + 5.10068i −0.324549 + 0.324549i
\(248\) 0 0
\(249\) −5.07619 −0.321691
\(250\) 0 0
\(251\) 3.62272i 0.228664i 0.993443 + 0.114332i \(0.0364728\pi\)
−0.993443 + 0.114332i \(0.963527\pi\)
\(252\) 0 0
\(253\) 3.03481 + 26.4731i 0.190797 + 1.66435i
\(254\) 0 0
\(255\) 0.0254813 + 0.460853i 0.00159570 + 0.0288598i
\(256\) 0 0
\(257\) −5.86034 5.86034i −0.365558 0.365558i 0.500296 0.865854i \(-0.333224\pi\)
−0.865854 + 0.500296i \(0.833224\pi\)
\(258\) 0 0
\(259\) 6.57725i 0.408690i
\(260\) 0 0
\(261\) 21.4062 1.32501
\(262\) 0 0
\(263\) 15.9661 + 15.9661i 0.984513 + 0.984513i 0.999882 0.0153685i \(-0.00489214\pi\)
−0.0153685 + 0.999882i \(0.504892\pi\)
\(264\) 0 0
\(265\) 1.10091 + 0.985546i 0.0676283 + 0.0605416i
\(266\) 0 0
\(267\) 3.09050 3.09050i 0.189135 0.189135i
\(268\) 0 0
\(269\) 27.5442i 1.67940i 0.543053 + 0.839699i \(0.317268\pi\)
−0.543053 + 0.839699i \(0.682732\pi\)
\(270\) 0 0
\(271\) 23.2233 1.41072 0.705359 0.708851i \(-0.250786\pi\)
0.705359 + 0.708851i \(0.250786\pi\)
\(272\) 0 0
\(273\) 2.13357 + 2.13357i 0.129129 + 0.129129i
\(274\) 0 0
\(275\) 3.06273 + 27.6115i 0.184690 + 1.66504i
\(276\) 0 0
\(277\) −9.22558 9.22558i −0.554311 0.554311i 0.373371 0.927682i \(-0.378202\pi\)
−0.927682 + 0.373371i \(0.878202\pi\)
\(278\) 0 0
\(279\) 1.51813i 0.0908878i
\(280\) 0 0
\(281\) 4.47913i 0.267203i 0.991035 + 0.133601i \(0.0426542\pi\)
−0.991035 + 0.133601i \(0.957346\pi\)
\(282\) 0 0
\(283\) −13.0277 13.0277i −0.774417 0.774417i 0.204458 0.978875i \(-0.434457\pi\)
−0.978875 + 0.204458i \(0.934457\pi\)
\(284\) 0 0
\(285\) −0.960712 + 1.07317i −0.0569077 + 0.0635690i
\(286\) 0 0
\(287\) 11.4953 11.4953i 0.678548 0.678548i
\(288\) 0 0
\(289\) 16.6211i 0.977713i
\(290\) 0 0
\(291\) 1.89310i 0.110975i
\(292\) 0 0
\(293\) −17.5250 17.5250i −1.02382 1.02382i −0.999709 0.0241147i \(-0.992323\pi\)
−0.0241147 0.999709i \(-0.507677\pi\)
\(294\) 0 0
\(295\) −0.953522 17.2453i −0.0555162 1.00406i
\(296\) 0 0
\(297\) −7.75688 + 7.75688i −0.450100 + 0.450100i
\(298\) 0 0
\(299\) 11.2017 14.1025i 0.647809 0.815566i
\(300\) 0 0
\(301\) 6.37888 0.367672
\(302\) 0 0
\(303\) 0.522843 0.522843i 0.0300365 0.0300365i
\(304\) 0 0
\(305\) −0.0881602 1.59446i −0.00504804 0.0912984i
\(306\) 0 0
\(307\) 10.4598 + 10.4598i 0.596972 + 0.596972i 0.939506 0.342534i \(-0.111285\pi\)
−0.342534 + 0.939506i \(0.611285\pi\)
\(308\) 0 0
\(309\) −0.987711 −0.0561889
\(310\) 0 0
\(311\) 10.5158 0.596294 0.298147 0.954520i \(-0.403631\pi\)
0.298147 + 0.954520i \(0.403631\pi\)
\(312\) 0 0
\(313\) −21.2497 21.2497i −1.20110 1.20110i −0.973832 0.227271i \(-0.927020\pi\)
−0.227271 0.973832i \(-0.572980\pi\)
\(314\) 0 0
\(315\) −11.5263 10.3185i −0.649433 0.581379i
\(316\) 0 0
\(317\) −12.9785 12.9785i −0.728943 0.728943i 0.241466 0.970409i \(-0.422372\pi\)
−0.970409 + 0.241466i \(0.922372\pi\)
\(318\) 0 0
\(319\) 41.1895 2.30617
\(320\) 0 0
\(321\) 1.27385i 0.0710995i
\(322\) 0 0
\(323\) 0.836044 0.836044i 0.0465187 0.0465187i
\(324\) 0 0
\(325\) 11.7324 14.6599i 0.650796 0.813184i
\(326\) 0 0
\(327\) −1.89637 + 1.89637i −0.104870 + 0.104870i
\(328\) 0 0
\(329\) −25.5587 −1.40909
\(330\) 0 0
\(331\) −4.06804 −0.223600 −0.111800 0.993731i \(-0.535662\pi\)
−0.111800 + 0.993731i \(0.535662\pi\)
\(332\) 0 0
\(333\) 5.60501 + 5.60501i 0.307153 + 0.307153i
\(334\) 0 0
\(335\) 10.5025 11.7319i 0.573812 0.640980i
\(336\) 0 0
\(337\) −9.26903 + 9.26903i −0.504916 + 0.504916i −0.912962 0.408046i \(-0.866210\pi\)
0.408046 + 0.912962i \(0.366210\pi\)
\(338\) 0 0
\(339\) −6.14765 −0.333894
\(340\) 0 0
\(341\) 2.92115i 0.158189i
\(342\) 0 0
\(343\) −13.9930 13.9930i −0.755551 0.755551i
\(344\) 0 0
\(345\) 2.07785 2.93514i 0.111868 0.158023i
\(346\) 0 0
\(347\) −16.6211 16.6211i −0.892269 0.892269i 0.102468 0.994736i \(-0.467326\pi\)
−0.994736 + 0.102468i \(0.967326\pi\)
\(348\) 0 0
\(349\) 19.1918i 1.02731i 0.857995 + 0.513657i \(0.171710\pi\)
−0.857995 + 0.513657i \(0.828290\pi\)
\(350\) 0 0
\(351\) 7.41437 0.395750
\(352\) 0 0
\(353\) −23.1662 + 23.1662i −1.23301 + 1.23301i −0.270209 + 0.962802i \(0.587093\pi\)
−0.962802 + 0.270209i \(0.912907\pi\)
\(354\) 0 0
\(355\) −11.0272 + 0.609712i −0.585263 + 0.0323601i
\(356\) 0 0
\(357\) −0.349709 0.349709i −0.0185086 0.0185086i
\(358\) 0 0
\(359\) −22.3170 −1.17785 −0.588923 0.808189i \(-0.700448\pi\)
−0.588923 + 0.808189i \(0.700448\pi\)
\(360\) 0 0
\(361\) −15.3103 −0.805805
\(362\) 0 0
\(363\) −4.71191 + 4.71191i −0.247311 + 0.247311i
\(364\) 0 0
\(365\) 1.15575 1.29104i 0.0604950 0.0675762i
\(366\) 0 0
\(367\) 11.2742 11.2742i 0.588506 0.588506i −0.348721 0.937227i \(-0.613384\pi\)
0.937227 + 0.348721i \(0.113384\pi\)
\(368\) 0 0
\(369\) 19.5922i 1.01993i
\(370\) 0 0
\(371\) −1.58326 −0.0821989
\(372\) 0 0
\(373\) 24.0730 + 24.0730i 1.24645 + 1.24645i 0.957275 + 0.289180i \(0.0933826\pi\)
0.289180 + 0.957275i \(0.406617\pi\)
\(374\) 0 0
\(375\) 2.17752 3.05213i 0.112447 0.157611i
\(376\) 0 0
\(377\) −19.6854 19.6854i −1.01385 1.01385i
\(378\) 0 0
\(379\) 26.7558 1.37435 0.687176 0.726491i \(-0.258850\pi\)
0.687176 + 0.726491i \(0.258850\pi\)
\(380\) 0 0
\(381\) −1.77980 −0.0911821
\(382\) 0 0
\(383\) 3.73641 + 3.73641i 0.190921 + 0.190921i 0.796094 0.605173i \(-0.206896\pi\)
−0.605173 + 0.796094i \(0.706896\pi\)
\(384\) 0 0
\(385\) −22.1787 19.8546i −1.13033 1.01188i
\(386\) 0 0
\(387\) 5.43596 5.43596i 0.276325 0.276325i
\(388\) 0 0
\(389\) 3.31180 0.167915 0.0839575 0.996469i \(-0.473244\pi\)
0.0839575 + 0.996469i \(0.473244\pi\)
\(390\) 0 0
\(391\) −1.83604 + 2.31151i −0.0928527 + 0.116898i
\(392\) 0 0
\(393\) 1.36955 1.36955i 0.0690844 0.0690844i
\(394\) 0 0
\(395\) −20.2600 + 1.12021i −1.01939 + 0.0563638i
\(396\) 0 0
\(397\) 8.65488 + 8.65488i 0.434376 + 0.434376i 0.890114 0.455738i \(-0.150625\pi\)
−0.455738 + 0.890114i \(0.650625\pi\)
\(398\) 0 0
\(399\) 1.54337i 0.0772651i
\(400\) 0 0
\(401\) 22.7142i 1.13430i −0.823616 0.567148i \(-0.808047\pi\)
0.823616 0.567148i \(-0.191953\pi\)
\(402\) 0 0
\(403\) −1.39608 + 1.39608i −0.0695439 + 0.0695439i
\(404\) 0 0
\(405\) −17.8625 + 0.987645i −0.887593 + 0.0490765i
\(406\) 0 0
\(407\) 10.7851 + 10.7851i 0.534597 + 0.534597i
\(408\) 0 0
\(409\) 30.1114i 1.48891i 0.667671 + 0.744457i \(0.267291\pi\)
−0.667671 + 0.744457i \(0.732709\pi\)
\(410\) 0 0
\(411\) 5.09741i 0.251437i
\(412\) 0 0
\(413\) 13.0863 + 13.0863i 0.643933 + 0.643933i
\(414\) 0 0
\(415\) −25.2189 22.5762i −1.23795 1.10822i
\(416\) 0 0
\(417\) 2.59392 + 2.59392i 0.127025 + 0.127025i
\(418\) 0 0
\(419\) 8.68672 0.424374 0.212187 0.977229i \(-0.431941\pi\)
0.212187 + 0.977229i \(0.431941\pi\)
\(420\) 0 0
\(421\) 17.8878i 0.871798i −0.899996 0.435899i \(-0.856431\pi\)
0.899996 0.435899i \(-0.143569\pi\)
\(422\) 0 0
\(423\) −21.7806 + 21.7806i −1.05901 + 1.05901i
\(424\) 0 0
\(425\) −1.92304 + 2.40288i −0.0932809 + 0.116557i
\(426\) 0 0
\(427\) 1.20992 + 1.20992i 0.0585522 + 0.0585522i
\(428\) 0 0
\(429\) 6.99706 0.337821
\(430\) 0 0
\(431\) 32.4013i 1.56071i −0.625335 0.780357i \(-0.715037\pi\)
0.625335 0.780357i \(-0.284963\pi\)
\(432\) 0 0
\(433\) −11.5838 11.5838i −0.556684 0.556684i 0.371678 0.928362i \(-0.378783\pi\)
−0.928362 + 0.371678i \(0.878783\pi\)
\(434\) 0 0
\(435\) −4.14174 3.70773i −0.198581 0.177772i
\(436\) 0 0
\(437\) −9.15218 + 1.04918i −0.437808 + 0.0501893i
\(438\) 0 0
\(439\) 9.72122i 0.463968i −0.972720 0.231984i \(-0.925478\pi\)
0.972720 0.231984i \(-0.0745218\pi\)
\(440\) 0 0
\(441\) −3.63637 −0.173161
\(442\) 0 0
\(443\) 13.9657 13.9657i 0.663531 0.663531i −0.292680 0.956211i \(-0.594547\pi\)
0.956211 + 0.292680i \(0.0945470\pi\)
\(444\) 0 0
\(445\) 29.0987 1.60891i 1.37941 0.0762698i
\(446\) 0 0
\(447\) 3.20431 3.20431i 0.151559 0.151559i
\(448\) 0 0
\(449\) 35.4406i 1.67255i 0.548313 + 0.836273i \(0.315270\pi\)
−0.548313 + 0.836273i \(0.684730\pi\)
\(450\) 0 0
\(451\) 37.6990i 1.77518i
\(452\) 0 0
\(453\) −4.32358 + 4.32358i −0.203139 + 0.203139i
\(454\) 0 0
\(455\) 1.11073 + 20.0887i 0.0520720 + 0.941771i
\(456\) 0 0
\(457\) −3.87672 + 3.87672i −0.181345 + 0.181345i −0.791942 0.610597i \(-0.790930\pi\)
0.610597 + 0.791942i \(0.290930\pi\)
\(458\) 0 0
\(459\) −1.21528 −0.0567242
\(460\) 0 0
\(461\) −2.32738 −0.108397 −0.0541983 0.998530i \(-0.517260\pi\)
−0.0541983 + 0.998530i \(0.517260\pi\)
\(462\) 0 0
\(463\) −4.77509 + 4.77509i −0.221917 + 0.221917i −0.809305 0.587388i \(-0.800156\pi\)
0.587388 + 0.809305i \(0.300156\pi\)
\(464\) 0 0
\(465\) −0.262952 + 0.293732i −0.0121941 + 0.0136215i
\(466\) 0 0
\(467\) −13.6067 + 13.6067i −0.629644 + 0.629644i −0.947978 0.318335i \(-0.896876\pi\)
0.318335 + 0.947978i \(0.396876\pi\)
\(468\) 0 0
\(469\) 16.8721i 0.779080i
\(470\) 0 0
\(471\) 6.48232i 0.298689i
\(472\) 0 0
\(473\) 10.4598 10.4598i 0.480942 0.480942i
\(474\) 0 0
\(475\) −9.54576 + 1.05884i −0.437989 + 0.0485828i
\(476\) 0 0
\(477\) −1.34923 + 1.34923i −0.0617769 + 0.0617769i
\(478\) 0 0
\(479\) 2.31151 0.105616 0.0528078 0.998605i \(-0.483183\pi\)
0.0528078 + 0.998605i \(0.483183\pi\)
\(480\) 0 0
\(481\) 10.3089i 0.470043i
\(482\) 0 0
\(483\) 0.438863 + 3.82827i 0.0199690 + 0.174192i
\(484\) 0 0
\(485\) 8.41948 9.40502i 0.382309 0.427060i
\(486\) 0 0
\(487\) −22.0687 22.0687i −1.00003 1.00003i −1.00000 2.86468e-5i \(-0.999991\pi\)
−2.86468e−5 1.00000i \(-0.500009\pi\)
\(488\) 0 0
\(489\) 4.74829i 0.214725i
\(490\) 0 0
\(491\) 2.01945 0.0911366 0.0455683 0.998961i \(-0.485490\pi\)
0.0455683 + 0.998961i \(0.485490\pi\)
\(492\) 0 0
\(493\) 3.22659 + 3.22659i 0.145318 + 0.145318i
\(494\) 0 0
\(495\) −35.8200 + 1.98055i −1.60999 + 0.0890190i
\(496\) 0 0
\(497\) 8.36776 8.36776i 0.375345 0.375345i
\(498\) 0 0
\(499\) 11.8266i 0.529431i −0.964327 0.264715i \(-0.914722\pi\)
0.964327 0.264715i \(-0.0852780\pi\)
\(500\) 0 0
\(501\) 3.63637 0.162461
\(502\) 0 0
\(503\) 17.6409 + 17.6409i 0.786569 + 0.786569i 0.980930 0.194361i \(-0.0622634\pi\)
−0.194361 + 0.980930i \(0.562263\pi\)
\(504\) 0 0
\(505\) 4.92284 0.272192i 0.219064 0.0121124i
\(506\) 0 0
\(507\) −0.261421 0.261421i −0.0116101 0.0116101i
\(508\) 0 0
\(509\) 3.36888i 0.149323i 0.997209 + 0.0746615i \(0.0237876\pi\)
−0.997209 + 0.0746615i \(0.976212\pi\)
\(510\) 0 0
\(511\) 1.85670i 0.0821356i
\(512\) 0 0
\(513\) −2.68168 2.68168i −0.118399 0.118399i
\(514\) 0 0
\(515\) −4.90701 4.39281i −0.216229 0.193570i
\(516\) 0 0
\(517\) −41.9099 + 41.9099i −1.84320 + 1.84320i
\(518\) 0 0
\(519\) 7.70414i 0.338174i
\(520\) 0 0
\(521\) 38.1023i 1.66929i −0.550787 0.834646i \(-0.685673\pi\)
0.550787 0.834646i \(-0.314327\pi\)
\(522\) 0 0
\(523\) −10.2246 10.2246i −0.447090 0.447090i 0.447296 0.894386i \(-0.352387\pi\)
−0.894386 + 0.447296i \(0.852387\pi\)
\(524\) 0 0
\(525\) 0.442902 + 3.99290i 0.0193298 + 0.174264i
\(526\) 0 0
\(527\) 0.228830 0.228830i 0.00996797 0.00996797i
\(528\) 0 0
\(529\) 22.4033 5.20492i 0.974057 0.226301i
\(530\) 0 0
\(531\) 22.3037 0.967900
\(532\) 0 0
\(533\) −18.0172 + 18.0172i −0.780412 + 0.780412i
\(534\) 0 0
\(535\) 5.66541 6.32858i 0.244937 0.273608i
\(536\) 0 0
\(537\) −3.50591 3.50591i −0.151291 0.151291i
\(538\) 0 0
\(539\) −6.99706 −0.301385
\(540\) 0 0
\(541\) 14.6046 0.627900 0.313950 0.949440i \(-0.398348\pi\)
0.313950 + 0.949440i \(0.398348\pi\)
\(542\) 0 0
\(543\) 5.74669 + 5.74669i 0.246614 + 0.246614i
\(544\) 0 0
\(545\) −17.8553 + 0.987251i −0.764839 + 0.0422892i
\(546\) 0 0
\(547\) −0.136753 0.136753i −0.00584714 0.00584714i 0.704177 0.710024i \(-0.251316\pi\)
−0.710024 + 0.704177i \(0.751316\pi\)
\(548\) 0 0
\(549\) 2.06215 0.0880103
\(550\) 0 0
\(551\) 14.2399i 0.606640i
\(552\) 0 0
\(553\) 15.3739 15.3739i 0.653764 0.653764i
\(554\) 0 0
\(555\) −0.113642 2.05531i −0.00482382 0.0872431i
\(556\) 0 0
\(557\) 27.2348 27.2348i 1.15398 1.15398i 0.168227 0.985748i \(-0.446196\pi\)
0.985748 0.168227i \(-0.0538042\pi\)
\(558\) 0 0
\(559\) −9.99794 −0.422868
\(560\) 0 0
\(561\) −1.14687 −0.0484211
\(562\) 0 0
\(563\) 11.7093 + 11.7093i 0.493487 + 0.493487i 0.909403 0.415916i \(-0.136539\pi\)
−0.415916 + 0.909403i \(0.636539\pi\)
\(564\) 0 0
\(565\) −30.5419 27.3414i −1.28491 1.15026i
\(566\) 0 0
\(567\) 13.5546 13.5546i 0.569238 0.569238i
\(568\) 0 0
\(569\) −28.0392 −1.17546 −0.587732 0.809055i \(-0.699979\pi\)
−0.587732 + 0.809055i \(0.699979\pi\)
\(570\) 0 0
\(571\) 5.76557i 0.241281i −0.992696 0.120641i \(-0.961505\pi\)
0.992696 0.120641i \(-0.0384949\pi\)
\(572\) 0 0
\(573\) −4.34086 4.34086i −0.181342 0.181342i
\(574\) 0 0
\(575\) 23.3768 5.34079i 0.974881 0.222726i
\(576\) 0 0
\(577\) −12.9662 12.9662i −0.539792 0.539792i 0.383676 0.923468i \(-0.374658\pi\)
−0.923468 + 0.383676i \(0.874658\pi\)
\(578\) 0 0
\(579\) 7.41437i 0.308131i
\(580\) 0 0
\(581\) 36.2683 1.50466
\(582\) 0 0
\(583\) −2.59616 + 2.59616i −0.107522 + 0.107522i
\(584\) 0 0
\(585\) 18.0657 + 16.1726i 0.746926 + 0.668656i
\(586\) 0 0
\(587\) 22.6919 + 22.6919i 0.936596 + 0.936596i 0.998106 0.0615105i \(-0.0195918\pi\)
−0.0615105 + 0.998106i \(0.519592\pi\)
\(588\) 0 0
\(589\) 1.00989 0.0416119
\(590\) 0 0
\(591\) 6.76380 0.278225
\(592\) 0 0
\(593\) −12.5163 + 12.5163i −0.513983 + 0.513983i −0.915744 0.401762i \(-0.868398\pi\)
0.401762 + 0.915744i \(0.368398\pi\)
\(594\) 0 0
\(595\) −0.182058 3.29270i −0.00746367 0.134987i
\(596\) 0 0
\(597\) −4.86349 + 4.86349i −0.199049 + 0.199049i
\(598\) 0 0
\(599\) 20.5607i 0.840088i 0.907504 + 0.420044i \(0.137985\pi\)
−0.907504 + 0.420044i \(0.862015\pi\)
\(600\) 0 0
\(601\) −4.89045 −0.199486 −0.0997428 0.995013i \(-0.531802\pi\)
−0.0997428 + 0.995013i \(0.531802\pi\)
\(602\) 0 0
\(603\) 14.3781 + 14.3781i 0.585520 + 0.585520i
\(604\) 0 0
\(605\) −44.3651 + 2.45302i −1.80370 + 0.0997295i
\(606\) 0 0
\(607\) −1.79663 1.79663i −0.0729229 0.0729229i 0.669705 0.742628i \(-0.266421\pi\)
−0.742628 + 0.669705i \(0.766421\pi\)
\(608\) 0 0
\(609\) 5.95641 0.241366
\(610\) 0 0
\(611\) 40.0594 1.62063
\(612\) 0 0
\(613\) −14.4325 14.4325i −0.582922 0.582922i 0.352783 0.935705i \(-0.385235\pi\)
−0.935705 + 0.352783i \(0.885235\pi\)
\(614\) 0 0
\(615\) −3.39354 + 3.79077i −0.136841 + 0.152859i
\(616\) 0 0
\(617\) −11.3571 + 11.3571i −0.457218 + 0.457218i −0.897741 0.440523i \(-0.854793\pi\)
0.440523 + 0.897741i \(0.354793\pi\)
\(618\) 0 0
\(619\) −10.4230 −0.418934 −0.209467 0.977816i \(-0.567173\pi\)
−0.209467 + 0.977816i \(0.567173\pi\)
\(620\) 0 0
\(621\) 7.41437 + 5.88927i 0.297528 + 0.236328i
\(622\) 0 0
\(623\) −22.0809 + 22.0809i −0.884653 + 0.884653i
\(624\) 0 0
\(625\) 24.3923 5.47871i 0.975692 0.219148i
\(626\) 0 0
\(627\) −2.53075 2.53075i −0.101068 0.101068i
\(628\) 0 0
\(629\) 1.68971i 0.0673730i
\(630\) 0 0
\(631\) 40.8170i 1.62490i 0.583031 + 0.812450i \(0.301867\pi\)
−0.583031 + 0.812450i \(0.698133\pi\)
\(632\) 0 0
\(633\) 3.60379 3.60379i 0.143238 0.143238i
\(634\) 0 0
\(635\) −8.84218 7.91562i −0.350891 0.314122i
\(636\) 0 0
\(637\) 3.34405 + 3.34405i 0.132496 + 0.132496i
\(638\) 0 0
\(639\) 14.2617i 0.564184i
\(640\) 0 0
\(641\) 42.1898i 1.66640i 0.552974 + 0.833198i \(0.313493\pi\)
−0.552974 + 0.833198i \(0.686507\pi\)
\(642\) 0 0
\(643\) 21.0360 + 21.0360i 0.829580 + 0.829580i 0.987459 0.157879i \(-0.0504655\pi\)
−0.157879 + 0.987459i \(0.550465\pi\)
\(644\) 0 0
\(645\) −1.99332 + 0.110214i −0.0784870 + 0.00433967i
\(646\) 0 0
\(647\) 22.2306 + 22.2306i 0.873974 + 0.873974i 0.992903 0.118929i \(-0.0379460\pi\)
−0.118929 + 0.992903i \(0.537946\pi\)
\(648\) 0 0
\(649\) 42.9165 1.68462
\(650\) 0 0
\(651\) 0.422428i 0.0165563i
\(652\) 0 0
\(653\) −11.7442 + 11.7442i −0.459588 + 0.459588i −0.898520 0.438932i \(-0.855357\pi\)
0.438932 + 0.898520i \(0.355357\pi\)
\(654\) 0 0
\(655\) 12.8950 0.712985i 0.503849 0.0278586i
\(656\) 0 0
\(657\) 1.58225 + 1.58225i 0.0617293 + 0.0617293i
\(658\) 0 0
\(659\) −23.4036 −0.911675 −0.455838 0.890063i \(-0.650660\pi\)
−0.455838 + 0.890063i \(0.650660\pi\)
\(660\) 0 0
\(661\) 22.3907i 0.870899i 0.900213 + 0.435449i \(0.143411\pi\)
−0.900213 + 0.435449i \(0.856589\pi\)
\(662\) 0 0
\(663\) 0.548117 + 0.548117i 0.0212871 + 0.0212871i
\(664\) 0 0
\(665\) 6.86408 7.66755i 0.266177 0.297335i
\(666\) 0 0
\(667\) −4.04918 35.3216i −0.156785 1.36766i
\(668\) 0 0
\(669\) 5.36391i 0.207381i
\(670\) 0 0
\(671\) 3.96795 0.153181
\(672\) 0 0
\(673\) −11.5824 + 11.5824i −0.446468 + 0.446468i −0.894178 0.447711i \(-0.852239\pi\)
0.447711 + 0.894178i \(0.352239\pi\)
\(674\) 0 0
\(675\) 7.70744 + 6.16831i 0.296659 + 0.237418i
\(676\) 0 0
\(677\) −6.64702 + 6.64702i −0.255466 + 0.255466i −0.823207 0.567741i \(-0.807817\pi\)
0.567741 + 0.823207i \(0.307817\pi\)
\(678\) 0 0
\(679\) 13.5257i 0.519071i
\(680\) 0 0
\(681\) 0.796862i 0.0305358i
\(682\) 0 0
\(683\) 12.1341 12.1341i 0.464299 0.464299i −0.435762 0.900062i \(-0.643521\pi\)
0.900062 + 0.435762i \(0.143521\pi\)
\(684\) 0 0
\(685\) 22.6705 25.3242i 0.866197 0.967590i
\(686\) 0 0
\(687\) 4.47884 4.47884i 0.170878 0.170878i
\(688\) 0 0
\(689\) 2.48153 0.0945387
\(690\) 0 0
\(691\) −22.2848 −0.847755 −0.423877 0.905720i \(-0.639331\pi\)
−0.423877 + 0.905720i \(0.639331\pi\)
\(692\) 0 0
\(693\) 27.1813 27.1813i 1.03253 1.03253i
\(694\) 0 0
\(695\) 1.35040 + 24.4231i 0.0512234 + 0.926422i
\(696\) 0 0
\(697\) 2.95317 2.95317i 0.111859 0.111859i
\(698\) 0 0
\(699\) 2.40621i 0.0910111i
\(700\) 0 0
\(701\) 1.20372i 0.0454640i −0.999742 0.0227320i \(-0.992764\pi\)
0.999742 0.0227320i \(-0.00723645\pi\)
\(702\) 0 0
\(703\) −3.72858 + 3.72858i −0.140626 + 0.140626i
\(704\) 0 0
\(705\) 7.98677 0.441602i 0.300799 0.0166317i
\(706\) 0 0
\(707\) −3.73560 + 3.73560i −0.140492 + 0.140492i
\(708\) 0 0
\(709\) 36.5049 1.37097 0.685486 0.728086i \(-0.259590\pi\)
0.685486 + 0.728086i \(0.259590\pi\)
\(710\) 0 0
\(711\) 26.2027i 0.982677i
\(712\) 0 0
\(713\) −2.50500 + 0.287167i −0.0938130 + 0.0107545i
\(714\) 0 0
\(715\) 34.7618 + 31.1192i 1.30002 + 1.16379i
\(716\) 0 0
\(717\) −4.48937 4.48937i −0.167659 0.167659i
\(718\) 0 0
\(719\) 11.5502i 0.430749i 0.976532 + 0.215374i \(0.0690972\pi\)
−0.976532 + 0.215374i \(0.930903\pi\)
\(720\) 0 0
\(721\) 7.05698 0.262816
\(722\) 0 0
\(723\) 2.34527 + 2.34527i 0.0872216 + 0.0872216i
\(724\) 0 0
\(725\) −4.08644 36.8405i −0.151767 1.36822i
\(726\) 0 0
\(727\) −27.9819 + 27.9819i −1.03779 + 1.03779i −0.0385340 + 0.999257i \(0.512269\pi\)
−0.999257 + 0.0385340i \(0.987731\pi\)
\(728\) 0 0
\(729\) 21.1156i 0.782060i
\(730\) 0 0
\(731\) 1.63874 0.0606111
\(732\) 0 0
\(733\) 32.8100 + 32.8100i 1.21186 + 1.21186i 0.970413 + 0.241451i \(0.0776232\pi\)
0.241451 + 0.970413i \(0.422377\pi\)
\(734\) 0 0
\(735\) 0.703578 + 0.629850i 0.0259519 + 0.0232324i
\(736\) 0 0
\(737\) 27.6661 + 27.6661i 1.01909 + 1.01909i
\(738\) 0 0
\(739\) 4.30428i 0.158336i 0.996861 + 0.0791678i \(0.0252263\pi\)
−0.996861 + 0.0791678i \(0.974774\pi\)
\(740\) 0 0
\(741\) 2.41900i 0.0888642i
\(742\) 0 0
\(743\) −33.8349 33.8349i −1.24128 1.24128i −0.959469 0.281813i \(-0.909064\pi\)
−0.281813 0.959469i \(-0.590936\pi\)
\(744\) 0 0
\(745\) 30.1703 1.66816i 1.10535 0.0611168i
\(746\) 0 0
\(747\) 30.9072 30.9072i 1.13083 1.13083i
\(748\) 0 0
\(749\) 9.10139i 0.332557i
\(750\) 0 0
\(751\) 23.2961i 0.850088i 0.905173 + 0.425044i \(0.139741\pi\)
−0.905173 + 0.425044i \(0.860259\pi\)
\(752\) 0 0
\(753\) 0.859039 + 0.859039i 0.0313051 + 0.0313051i
\(754\) 0 0
\(755\) −40.7088 + 2.25086i −1.48154 + 0.0819170i
\(756\) 0 0
\(757\) −11.0154 + 11.0154i −0.400360 + 0.400360i −0.878360 0.478000i \(-0.841362\pi\)
0.478000 + 0.878360i \(0.341362\pi\)
\(758\) 0 0
\(759\) 6.99706 + 5.55780i 0.253977 + 0.201735i
\(760\) 0 0
\(761\) 2.31975 0.0840909 0.0420455 0.999116i \(-0.486613\pi\)
0.0420455 + 0.999116i \(0.486613\pi\)
\(762\) 0 0
\(763\) 13.5492 13.5492i 0.490512 0.490512i
\(764\) 0 0
\(765\) −2.96112 2.65083i −0.107060 0.0958409i
\(766\) 0 0
\(767\) −20.5108 20.5108i −0.740601 0.740601i
\(768\) 0 0
\(769\) 15.2220 0.548920 0.274460 0.961598i \(-0.411501\pi\)
0.274460 + 0.961598i \(0.411501\pi\)
\(770\) 0 0
\(771\) −2.77927 −0.100093
\(772\) 0 0
\(773\) −0.882262 0.882262i −0.0317328 0.0317328i 0.691062 0.722795i \(-0.257143\pi\)
−0.722795 + 0.691062i \(0.757143\pi\)
\(774\) 0 0
\(775\) −2.61273 + 0.289810i −0.0938519 + 0.0104103i
\(776\) 0 0
\(777\) 1.55963 + 1.55963i 0.0559514 + 0.0559514i
\(778\) 0 0
\(779\) 13.0332 0.466963
\(780\) 0 0
\(781\) 27.4422i 0.981957i
\(782\) 0 0
\(783\) 10.3496 10.3496i 0.369864 0.369864i
\(784\) 0 0
\(785\) −28.8299 + 32.2046i −1.02898 + 1.14943i
\(786\) 0 0
\(787\) −35.5674 + 35.5674i −1.26784 + 1.26784i −0.320641 + 0.947201i \(0.603898\pi\)
−0.947201 + 0.320641i \(0.896102\pi\)
\(788\) 0 0
\(789\) 7.57194 0.269568
\(790\) 0 0
\(791\) 43.9236 1.56174
\(792\) 0 0
\(793\) −1.89637 1.89637i −0.0673421 0.0673421i
\(794\) 0 0
\(795\) 0.494751 0.0273556i 0.0175470 0.000970202i
\(796\) 0 0
\(797\) 7.37158 7.37158i 0.261115 0.261115i −0.564392 0.825507i \(-0.690889\pi\)
0.825507 + 0.564392i \(0.190889\pi\)
\(798\) 0 0
\(799\) −6.56606 −0.232290
\(800\) 0 0
\(801\) 37.6339i 1.32973i
\(802\) 0 0
\(803\) 3.04454 + 3.04454i 0.107439 + 0.107439i
\(804\) 0 0
\(805\) −14.8458 + 20.9709i −0.523245 + 0.739128i
\(806\) 0 0
\(807\) 6.53141 + 6.53141i 0.229917 + 0.229917i
\(808\) 0 0
\(809\) 27.8450i 0.978978i −0.872009 0.489489i \(-0.837183\pi\)
0.872009 0.489489i \(-0.162817\pi\)
\(810\) 0 0
\(811\) −49.9498 −1.75398 −0.876988 0.480513i \(-0.840451\pi\)
−0.876988 + 0.480513i \(0.840451\pi\)
\(812\) 0 0
\(813\) 5.50684 5.50684i 0.193133 0.193133i
\(814\) 0 0
\(815\) −21.1179 + 23.5898i −0.739726 + 0.826315i
\(816\) 0 0
\(817\) 3.61613 + 3.61613i 0.126512 + 0.126512i
\(818\) 0 0
\(819\) −25.9811 −0.907852
\(820\) 0 0
\(821\) −37.0759 −1.29396 −0.646979 0.762508i \(-0.723968\pi\)
−0.646979 + 0.762508i \(0.723968\pi\)
\(822\) 0 0
\(823\) 0.660413 0.660413i 0.0230205 0.0230205i −0.695503 0.718523i \(-0.744818\pi\)
0.718523 + 0.695503i \(0.244818\pi\)
\(824\) 0 0
\(825\) 7.27363 + 5.82113i 0.253235 + 0.202666i
\(826\) 0 0
\(827\) −27.0527 + 27.0527i −0.940716 + 0.940716i −0.998338 0.0576226i \(-0.981648\pi\)
0.0576226 + 0.998338i \(0.481648\pi\)
\(828\) 0 0
\(829\) 37.3353i 1.29671i 0.761339 + 0.648353i \(0.224542\pi\)
−0.761339 + 0.648353i \(0.775458\pi\)
\(830\) 0 0
\(831\) −4.37523 −0.151775
\(832\) 0 0
\(833\) −0.548117 0.548117i −0.0189911 0.0189911i
\(834\) 0 0
\(835\) 18.0657 + 16.1726i 0.625191 + 0.559677i
\(836\) 0 0
\(837\) −0.733991 0.733991i −0.0253704 0.0253704i
\(838\) 0 0
\(839\) −18.0287 −0.622420 −0.311210 0.950341i \(-0.600734\pi\)
−0.311210 + 0.950341i \(0.600734\pi\)
\(840\) 0 0
\(841\) −25.9569 −0.895066
\(842\) 0 0
\(843\) 1.06211 + 1.06211i 0.0365812 + 0.0365812i
\(844\) 0 0
\(845\) −0.136096 2.46142i −0.00468184 0.0846755i
\(846\) 0 0
\(847\) 33.6656 33.6656i 1.15676 1.15676i
\(848\) 0 0
\(849\) −6.17840 −0.212042
\(850\) 0 0
\(851\) 8.18838 10.3089i 0.280694 0.353383i
\(852\) 0 0
\(853\) −33.9885 + 33.9885i −1.16374 + 1.16374i −0.180093 + 0.983650i \(0.557640\pi\)
−0.983650 + 0.180093i \(0.942360\pi\)
\(854\) 0 0
\(855\) −0.684709 12.3836i −0.0234166 0.423510i
\(856\) 0 0
\(857\) 5.41108 + 5.41108i 0.184839 + 0.184839i 0.793461 0.608622i \(-0.208277\pi\)
−0.608622 + 0.793461i \(0.708277\pi\)
\(858\) 0 0
\(859\) 34.2433i 1.16837i −0.811622 0.584184i \(-0.801415\pi\)
0.811622 0.584184i \(-0.198585\pi\)
\(860\) 0 0
\(861\) 5.45166i 0.185792i
\(862\) 0 0
\(863\) 25.9583 25.9583i 0.883632 0.883632i −0.110269 0.993902i \(-0.535171\pi\)
0.993902 + 0.110269i \(0.0351714\pi\)
\(864\) 0 0
\(865\) 34.2639 38.2746i 1.16501 1.30138i
\(866\) 0 0
\(867\) 3.94129 + 3.94129i 0.133853 + 0.133853i
\(868\) 0 0
\(869\) 50.4188i 1.71034i
\(870\) 0 0
\(871\) 26.4445i 0.896036i
\(872\) 0 0
\(873\) 11.5264 + 11.5264i 0.390109 + 0.390109i
\(874\) 0 0
\(875\) −15.5579 + 21.8068i −0.525953 + 0.737203i
\(876\) 0 0
\(877\) −24.3502 24.3502i −0.822249 0.822249i 0.164181 0.986430i \(-0.447502\pi\)
−0.986430 + 0.164181i \(0.947502\pi\)
\(878\) 0 0
\(879\) −8.31125 −0.280332
\(880\) 0 0
\(881\) 0.218997i 0.00737820i 0.999993 + 0.00368910i \(0.00117428\pi\)
−0.999993 + 0.00368910i \(0.998826\pi\)
\(882\) 0 0
\(883\) −14.5113 + 14.5113i −0.488344 + 0.488344i −0.907783 0.419439i \(-0.862227\pi\)
0.419439 + 0.907783i \(0.362227\pi\)
\(884\) 0 0
\(885\) −4.31540 3.86320i −0.145061 0.129860i
\(886\) 0 0
\(887\) −12.6798 12.6798i −0.425747 0.425747i 0.461430 0.887177i \(-0.347337\pi\)
−0.887177 + 0.461430i \(0.847337\pi\)
\(888\) 0 0
\(889\) 12.7163 0.426491
\(890\) 0 0
\(891\) 44.4523i 1.48921i
\(892\) 0 0
\(893\) −14.4890 14.4890i −0.484855 0.484855i
\(894\) 0 0
\(895\) −1.82518 33.0100i −0.0610090 1.10340i
\(896\) 0 0
\(897\) −0.687853 6.00024i −0.0229667 0.200342i
\(898\) 0 0
\(899\) 3.89754i 0.129990i
\(900\) 0 0
\(901\) −0.406743 −0.0135506
\(902\) 0 0
\(903\) 1.51259 1.51259i 0.0503359 0.0503359i
\(904\) 0 0
\(905\) 2.99173 + 54.1081i 0.0994484 + 1.79862i
\(906\) 0 0
\(907\) 26.1326 26.1326i 0.867719 0.867719i −0.124501 0.992219i \(-0.539733\pi\)
0.992219 + 0.124501i \(0.0397330\pi\)
\(908\) 0 0
\(909\) 6.36682i 0.211174i
\(910\) 0 0
\(911\) 24.8223i 0.822400i 0.911545 + 0.411200i \(0.134890\pi\)
−0.911545 + 0.411200i \(0.865110\pi\)
\(912\) 0 0
\(913\) 59.4711 59.4711i 1.96821 1.96821i
\(914\) 0 0
\(915\) −0.398991 0.357181i −0.0131902 0.0118080i
\(916\) 0 0
\(917\) −9.78510 + 9.78510i −0.323132 + 0.323132i
\(918\) 0 0
\(919\) −10.3089 −0.340058 −0.170029 0.985439i \(-0.554386\pi\)
−0.170029 + 0.985439i \(0.554386\pi\)
\(920\) 0 0
\(921\) 4.96056 0.163456
\(922\) 0 0
\(923\) −13.1152 + 13.1152i −0.431692 + 0.431692i
\(924\) 0 0
\(925\) 8.57635 10.7163i 0.281989 0.352351i
\(926\) 0 0
\(927\) 6.01383 6.01383i 0.197520 0.197520i
\(928\) 0 0
\(929\) 27.6117i 0.905909i −0.891534 0.452955i \(-0.850370\pi\)
0.891534 0.452955i \(-0.149630\pi\)
\(930\) 0 0
\(931\) 2.41900i 0.0792795i
\(932\) 0 0
\(933\) 2.49355 2.49355i 0.0816351 0.0816351i
\(934\) 0 0
\(935\) −5.69775 5.10068i −0.186336 0.166810i
\(936\) 0 0
\(937\) 5.72139 5.72139i 0.186910 0.186910i −0.607449 0.794359i \(-0.707807\pi\)
0.794359 + 0.607449i \(0.207807\pi\)
\(938\) 0 0
\(939\) −10.0777 −0.328872
\(940\) 0 0
\(941\) 4.31871i 0.140786i −0.997519 0.0703930i \(-0.977575\pi\)
0.997519 0.0703930i \(-0.0224253\pi\)
\(942\) 0 0
\(943\) −32.3284 + 3.70604i −1.05276 + 0.120685i
\(944\) 0 0
\(945\) −10.5616 + 0.583969i −0.343569 + 0.0189965i
\(946\) 0 0
\(947\) 41.4515 + 41.4515i 1.34699 + 1.34699i 0.888908 + 0.458086i \(0.151465\pi\)
0.458086 + 0.888908i \(0.348535\pi\)
\(948\) 0 0
\(949\) 2.91010i 0.0944659i
\(950\) 0 0
\(951\) −6.15504 −0.199591
\(952\) 0 0
\(953\) −27.8044 27.8044i −0.900672 0.900672i 0.0948219 0.995494i \(-0.469772\pi\)
−0.995494 + 0.0948219i \(0.969772\pi\)
\(954\) 0 0
\(955\) −2.25985 40.8715i −0.0731271 1.32257i
\(956\) 0 0
\(957\) 9.76706 9.76706i 0.315724 0.315724i
\(958\) 0 0
\(959\) 36.4198i 1.17606i
\(960\) 0 0
\(961\) −30.7236 −0.991083
\(962\) 0 0
\(963\) 7.75604 + 7.75604i 0.249935 + 0.249935i
\(964\) 0 0
\(965\) 32.9751 36.8350i 1.06151 1.18576i
\(966\) 0 0
\(967\) 17.4321 + 17.4321i 0.560577 + 0.560577i 0.929471 0.368894i \(-0.120264\pi\)
−0.368894 + 0.929471i \(0.620264\pi\)
\(968\) 0 0
\(969\) 0.396494i 0.0127372i
\(970\) 0 0
\(971\) 30.0252i 0.963555i 0.876294 + 0.481777i \(0.160009\pi\)
−0.876294 + 0.481777i \(0.839991\pi\)
\(972\) 0 0
\(973\) −18.5330 18.5330i −0.594141 0.594141i
\(974\) 0 0
\(975\) −0.694183 6.25827i −0.0222316 0.200425i
\(976\) 0 0
\(977\) −1.18647 + 1.18647i −0.0379585 + 0.0379585i −0.725831 0.687873i \(-0.758545\pi\)
0.687873 + 0.725831i \(0.258545\pi\)
\(978\) 0 0
\(979\) 72.4146i 2.31438i
\(980\) 0 0
\(981\) 23.0927i 0.737293i
\(982\) 0 0
\(983\) −12.7320 12.7320i −0.406088 0.406088i 0.474284 0.880372i \(-0.342707\pi\)
−0.880372 + 0.474284i \(0.842707\pi\)
\(984\) 0 0
\(985\) 33.6030 + 30.0817i 1.07068 + 0.958484i
\(986\) 0 0
\(987\) −6.06060 + 6.06060i −0.192911 + 0.192911i
\(988\) 0 0
\(989\) −9.99794 7.94141i −0.317916 0.252522i
\(990\) 0 0
\(991\) −3.95248 −0.125555 −0.0627774 0.998028i \(-0.519996\pi\)
−0.0627774 + 0.998028i \(0.519996\pi\)
\(992\) 0 0
\(993\) −0.964635 + 0.964635i −0.0306118 + 0.0306118i
\(994\) 0 0
\(995\) −45.7923 + 2.53193i −1.45171 + 0.0802676i
\(996\) 0 0
\(997\) −27.3544 27.3544i −0.866323 0.866323i 0.125740 0.992063i \(-0.459870\pi\)
−0.992063 + 0.125740i \(0.959870\pi\)
\(998\) 0 0
\(999\) 5.41988 0.171477
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 460.2.i.b.137.4 yes 16
5.2 odd 4 2300.2.i.d.1793.5 16
5.3 odd 4 inner 460.2.i.b.413.3 yes 16
5.4 even 2 2300.2.i.d.1057.6 16
23.22 odd 2 inner 460.2.i.b.137.3 16
115.22 even 4 2300.2.i.d.1793.6 16
115.68 even 4 inner 460.2.i.b.413.4 yes 16
115.114 odd 2 2300.2.i.d.1057.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.i.b.137.3 16 23.22 odd 2 inner
460.2.i.b.137.4 yes 16 1.1 even 1 trivial
460.2.i.b.413.3 yes 16 5.3 odd 4 inner
460.2.i.b.413.4 yes 16 115.68 even 4 inner
2300.2.i.d.1057.5 16 115.114 odd 2
2300.2.i.d.1057.6 16 5.4 even 2
2300.2.i.d.1793.5 16 5.2 odd 4
2300.2.i.d.1793.6 16 115.22 even 4