Properties

Label 624.2.d.i.287.3
Level $624$
Weight $2$
Character 624.287
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(287,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.d (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 287.3
Root \(1.68614 - 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 624.287
Dual form 624.2.d.i.287.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.68614 - 0.396143i) q^{3} +4.25639i q^{5} +0.792287i q^{7} +(2.68614 - 1.33591i) q^{9} +O(q^{10})\) \(q+(1.68614 - 0.396143i) q^{3} +4.25639i q^{5} +0.792287i q^{7} +(2.68614 - 1.33591i) q^{9} -4.00000 q^{11} +1.00000 q^{13} +(1.68614 + 7.17687i) q^{15} +5.84096i q^{17} -3.46410i q^{19} +(0.313859 + 1.33591i) q^{21} +6.74456 q^{23} -13.1168 q^{25} +(4.00000 - 3.31662i) q^{27} +6.63325i q^{31} +(-6.74456 + 1.58457i) q^{33} -3.37228 q^{35} -5.37228 q^{37} +(1.68614 - 0.396143i) q^{39} +1.58457i q^{41} -9.30506i q^{43} +(5.68614 + 11.4333i) q^{45} +0.627719 q^{47} +6.37228 q^{49} +(2.31386 + 9.84868i) q^{51} -5.34363i q^{53} -17.0256i q^{55} +(-1.37228 - 5.84096i) q^{57} +12.0000 q^{59} +11.4891 q^{61} +(1.05842 + 2.12819i) q^{63} +4.25639i q^{65} -3.46410i q^{67} +(11.3723 - 2.67181i) q^{69} +6.11684 q^{71} +0.744563 q^{73} +(-22.1168 + 5.19615i) q^{75} -3.16915i q^{77} -5.04868i q^{79} +(5.43070 - 7.17687i) q^{81} -12.0000 q^{83} -24.8614 q^{85} -1.58457i q^{89} +0.792287i q^{91} +(2.62772 + 11.1846i) q^{93} +14.7446 q^{95} +8.74456 q^{97} +(-10.7446 + 5.34363i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 5 q^{9} - 16 q^{11} + 4 q^{13} + q^{15} + 7 q^{21} + 4 q^{23} - 18 q^{25} + 16 q^{27} - 4 q^{33} - 2 q^{35} - 10 q^{37} + q^{39} + 17 q^{45} + 14 q^{47} + 14 q^{49} + 15 q^{51} + 6 q^{57} + 48 q^{59} - 13 q^{63} + 34 q^{69} - 10 q^{71} - 20 q^{73} - 54 q^{75} - 7 q^{81} - 48 q^{83} - 42 q^{85} + 22 q^{93} + 36 q^{95} + 12 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.68614 0.396143i 0.973494 0.228714i
\(4\) 0 0
\(5\) 4.25639i 1.90351i 0.306851 + 0.951757i \(0.400725\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 0.792287i 0.299456i 0.988727 + 0.149728i \(0.0478399\pi\)
−0.988727 + 0.149728i \(0.952160\pi\)
\(8\) 0 0
\(9\) 2.68614 1.33591i 0.895380 0.445302i
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.68614 + 7.17687i 0.435360 + 1.85306i
\(16\) 0 0
\(17\) 5.84096i 1.41664i 0.705891 + 0.708321i \(0.250547\pi\)
−0.705891 + 0.708321i \(0.749453\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0.313859 + 1.33591i 0.0684897 + 0.291519i
\(22\) 0 0
\(23\) 6.74456 1.40634 0.703169 0.711022i \(-0.251768\pi\)
0.703169 + 0.711022i \(0.251768\pi\)
\(24\) 0 0
\(25\) −13.1168 −2.62337
\(26\) 0 0
\(27\) 4.00000 3.31662i 0.769800 0.638285i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 6.63325i 1.19137i 0.803219 + 0.595683i \(0.203119\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) −6.74456 + 1.58457i −1.17408 + 0.275839i
\(34\) 0 0
\(35\) −3.37228 −0.570020
\(36\) 0 0
\(37\) −5.37228 −0.883198 −0.441599 0.897213i \(-0.645589\pi\)
−0.441599 + 0.897213i \(0.645589\pi\)
\(38\) 0 0
\(39\) 1.68614 0.396143i 0.269999 0.0634337i
\(40\) 0 0
\(41\) 1.58457i 0.247469i 0.992315 + 0.123734i \(0.0394871\pi\)
−0.992315 + 0.123734i \(0.960513\pi\)
\(42\) 0 0
\(43\) 9.30506i 1.41901i −0.704701 0.709504i \(-0.748919\pi\)
0.704701 0.709504i \(-0.251081\pi\)
\(44\) 0 0
\(45\) 5.68614 + 11.4333i 0.847640 + 1.70437i
\(46\) 0 0
\(47\) 0.627719 0.0915622 0.0457811 0.998951i \(-0.485422\pi\)
0.0457811 + 0.998951i \(0.485422\pi\)
\(48\) 0 0
\(49\) 6.37228 0.910326
\(50\) 0 0
\(51\) 2.31386 + 9.84868i 0.324005 + 1.37909i
\(52\) 0 0
\(53\) 5.34363i 0.734004i −0.930220 0.367002i \(-0.880384\pi\)
0.930220 0.367002i \(-0.119616\pi\)
\(54\) 0 0
\(55\) 17.0256i 2.29573i
\(56\) 0 0
\(57\) −1.37228 5.84096i −0.181763 0.773654i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 11.4891 1.47103 0.735516 0.677507i \(-0.236940\pi\)
0.735516 + 0.677507i \(0.236940\pi\)
\(62\) 0 0
\(63\) 1.05842 + 2.12819i 0.133349 + 0.268127i
\(64\) 0 0
\(65\) 4.25639i 0.527940i
\(66\) 0 0
\(67\) 3.46410i 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 11.3723 2.67181i 1.36906 0.321649i
\(70\) 0 0
\(71\) 6.11684 0.725936 0.362968 0.931802i \(-0.381763\pi\)
0.362968 + 0.931802i \(0.381763\pi\)
\(72\) 0 0
\(73\) 0.744563 0.0871445 0.0435722 0.999050i \(-0.486126\pi\)
0.0435722 + 0.999050i \(0.486126\pi\)
\(74\) 0 0
\(75\) −22.1168 + 5.19615i −2.55383 + 0.600000i
\(76\) 0 0
\(77\) 3.16915i 0.361158i
\(78\) 0 0
\(79\) 5.04868i 0.568020i −0.958821 0.284010i \(-0.908335\pi\)
0.958821 0.284010i \(-0.0916650\pi\)
\(80\) 0 0
\(81\) 5.43070 7.17687i 0.603411 0.797430i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −24.8614 −2.69660
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.58457i 0.167965i −0.996467 0.0839823i \(-0.973236\pi\)
0.996467 0.0839823i \(-0.0267639\pi\)
\(90\) 0 0
\(91\) 0.792287i 0.0830542i
\(92\) 0 0
\(93\) 2.62772 + 11.1846i 0.272482 + 1.15979i
\(94\) 0 0
\(95\) 14.7446 1.51276
\(96\) 0 0
\(97\) 8.74456 0.887876 0.443938 0.896058i \(-0.353581\pi\)
0.443938 + 0.896058i \(0.353581\pi\)
\(98\) 0 0
\(99\) −10.7446 + 5.34363i −1.07987 + 0.537055i
\(100\) 0 0
\(101\) 3.16915i 0.315342i −0.987492 0.157671i \(-0.949601\pi\)
0.987492 0.157671i \(-0.0503985\pi\)
\(102\) 0 0
\(103\) 1.87953i 0.185195i −0.995704 0.0925977i \(-0.970483\pi\)
0.995704 0.0925977i \(-0.0295170\pi\)
\(104\) 0 0
\(105\) −5.68614 + 1.33591i −0.554911 + 0.130371i
\(106\) 0 0
\(107\) −5.48913 −0.530654 −0.265327 0.964159i \(-0.585480\pi\)
−0.265327 + 0.964159i \(0.585480\pi\)
\(108\) 0 0
\(109\) −6.62772 −0.634820 −0.317410 0.948288i \(-0.602813\pi\)
−0.317410 + 0.948288i \(0.602813\pi\)
\(110\) 0 0
\(111\) −9.05842 + 2.12819i −0.859787 + 0.201999i
\(112\) 0 0
\(113\) 6.33830i 0.596257i −0.954526 0.298128i \(-0.903638\pi\)
0.954526 0.298128i \(-0.0963623\pi\)
\(114\) 0 0
\(115\) 28.7075i 2.67699i
\(116\) 0 0
\(117\) 2.68614 1.33591i 0.248334 0.123505i
\(118\) 0 0
\(119\) −4.62772 −0.424222
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0.627719 + 2.67181i 0.0565995 + 0.240909i
\(124\) 0 0
\(125\) 34.5484i 3.09011i
\(126\) 0 0
\(127\) 6.63325i 0.588606i 0.955712 + 0.294303i \(0.0950874\pi\)
−0.955712 + 0.294303i \(0.904913\pi\)
\(128\) 0 0
\(129\) −3.68614 15.6896i −0.324547 1.38140i
\(130\) 0 0
\(131\) −10.1168 −0.883913 −0.441956 0.897037i \(-0.645715\pi\)
−0.441956 + 0.897037i \(0.645715\pi\)
\(132\) 0 0
\(133\) 2.74456 0.237984
\(134\) 0 0
\(135\) 14.1168 + 17.0256i 1.21498 + 1.46533i
\(136\) 0 0
\(137\) 4.75372i 0.406138i 0.979164 + 0.203069i \(0.0650916\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(138\) 0 0
\(139\) 15.6434i 1.32685i −0.748242 0.663426i \(-0.769102\pi\)
0.748242 0.663426i \(-0.230898\pi\)
\(140\) 0 0
\(141\) 1.05842 0.248667i 0.0891352 0.0209415i
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 10.7446 2.52434i 0.886197 0.208204i
\(148\) 0 0
\(149\) 3.75906i 0.307954i −0.988074 0.153977i \(-0.950792\pi\)
0.988074 0.153977i \(-0.0492081\pi\)
\(150\) 0 0
\(151\) 10.8896i 0.886186i −0.896476 0.443093i \(-0.853881\pi\)
0.896476 0.443093i \(-0.146119\pi\)
\(152\) 0 0
\(153\) 7.80298 + 15.6896i 0.630834 + 1.26843i
\(154\) 0 0
\(155\) −28.2337 −2.26778
\(156\) 0 0
\(157\) −8.74456 −0.697892 −0.348946 0.937143i \(-0.613460\pi\)
−0.348946 + 0.937143i \(0.613460\pi\)
\(158\) 0 0
\(159\) −2.11684 9.01011i −0.167877 0.714548i
\(160\) 0 0
\(161\) 5.34363i 0.421137i
\(162\) 0 0
\(163\) 20.4897i 1.60487i −0.596737 0.802437i \(-0.703536\pi\)
0.596737 0.802437i \(-0.296464\pi\)
\(164\) 0 0
\(165\) −6.74456 28.7075i −0.525063 2.23487i
\(166\) 0 0
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.62772 9.30506i −0.353890 0.711576i
\(172\) 0 0
\(173\) 20.1947i 1.53537i 0.640825 + 0.767687i \(0.278593\pi\)
−0.640825 + 0.767687i \(0.721407\pi\)
\(174\) 0 0
\(175\) 10.3923i 0.785584i
\(176\) 0 0
\(177\) 20.2337 4.75372i 1.52086 0.357312i
\(178\) 0 0
\(179\) 11.3723 0.850004 0.425002 0.905192i \(-0.360273\pi\)
0.425002 + 0.905192i \(0.360273\pi\)
\(180\) 0 0
\(181\) 10.2337 0.760664 0.380332 0.924850i \(-0.375810\pi\)
0.380332 + 0.924850i \(0.375810\pi\)
\(182\) 0 0
\(183\) 19.3723 4.55134i 1.43204 0.336445i
\(184\) 0 0
\(185\) 22.8665i 1.68118i
\(186\) 0 0
\(187\) 23.3639i 1.70853i
\(188\) 0 0
\(189\) 2.62772 + 3.16915i 0.191138 + 0.230522i
\(190\) 0 0
\(191\) −21.4891 −1.55490 −0.777449 0.628946i \(-0.783487\pi\)
−0.777449 + 0.628946i \(0.783487\pi\)
\(192\) 0 0
\(193\) 11.2554 0.810184 0.405092 0.914276i \(-0.367239\pi\)
0.405092 + 0.914276i \(0.367239\pi\)
\(194\) 0 0
\(195\) 1.68614 + 7.17687i 0.120747 + 0.513946i
\(196\) 0 0
\(197\) 4.25639i 0.303255i 0.988438 + 0.151628i \(0.0484514\pi\)
−0.988438 + 0.151628i \(0.951549\pi\)
\(198\) 0 0
\(199\) 9.80240i 0.694874i 0.937703 + 0.347437i \(0.112948\pi\)
−0.937703 + 0.347437i \(0.887052\pi\)
\(200\) 0 0
\(201\) −1.37228 5.84096i −0.0967933 0.411990i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.74456 −0.471061
\(206\) 0 0
\(207\) 18.1168 9.01011i 1.25921 0.626246i
\(208\) 0 0
\(209\) 13.8564i 0.958468i
\(210\) 0 0
\(211\) 16.2333i 1.11754i 0.829321 + 0.558772i \(0.188727\pi\)
−0.829321 + 0.558772i \(0.811273\pi\)
\(212\) 0 0
\(213\) 10.3139 2.42315i 0.706694 0.166031i
\(214\) 0 0
\(215\) 39.6060 2.70110
\(216\) 0 0
\(217\) −5.25544 −0.356762
\(218\) 0 0
\(219\) 1.25544 0.294954i 0.0848346 0.0199311i
\(220\) 0 0
\(221\) 5.84096i 0.392906i
\(222\) 0 0
\(223\) 26.3306i 1.76323i 0.471971 + 0.881614i \(0.343543\pi\)
−0.471971 + 0.881614i \(0.656457\pi\)
\(224\) 0 0
\(225\) −35.2337 + 17.5229i −2.34891 + 1.16819i
\(226\) 0 0
\(227\) −2.74456 −0.182163 −0.0910815 0.995843i \(-0.529032\pi\)
−0.0910815 + 0.995843i \(0.529032\pi\)
\(228\) 0 0
\(229\) −1.13859 −0.0752404 −0.0376202 0.999292i \(-0.511978\pi\)
−0.0376202 + 0.999292i \(0.511978\pi\)
\(230\) 0 0
\(231\) −1.25544 5.34363i −0.0826017 0.351585i
\(232\) 0 0
\(233\) 0.497333i 0.0325814i 0.999867 + 0.0162907i \(0.00518572\pi\)
−0.999867 + 0.0162907i \(0.994814\pi\)
\(234\) 0 0
\(235\) 2.67181i 0.174290i
\(236\) 0 0
\(237\) −2.00000 8.51278i −0.129914 0.552964i
\(238\) 0 0
\(239\) −20.8614 −1.34941 −0.674706 0.738086i \(-0.735730\pi\)
−0.674706 + 0.738086i \(0.735730\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 6.31386 14.2525i 0.405034 0.914302i
\(244\) 0 0
\(245\) 27.1229i 1.73282i
\(246\) 0 0
\(247\) 3.46410i 0.220416i
\(248\) 0 0
\(249\) −20.2337 + 4.75372i −1.28226 + 0.301255i
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) −26.9783 −1.69611
\(254\) 0 0
\(255\) −41.9198 + 9.84868i −2.62512 + 0.616749i
\(256\) 0 0
\(257\) 22.8665i 1.42637i −0.700974 0.713187i \(-0.747251\pi\)
0.700974 0.713187i \(-0.252749\pi\)
\(258\) 0 0
\(259\) 4.25639i 0.264479i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.25544 0.0774136 0.0387068 0.999251i \(-0.487676\pi\)
0.0387068 + 0.999251i \(0.487676\pi\)
\(264\) 0 0
\(265\) 22.7446 1.39719
\(266\) 0 0
\(267\) −0.627719 2.67181i −0.0384158 0.163512i
\(268\) 0 0
\(269\) 5.34363i 0.325807i 0.986642 + 0.162903i \(0.0520859\pi\)
−0.986642 + 0.162903i \(0.947914\pi\)
\(270\) 0 0
\(271\) 14.0588i 0.854010i −0.904249 0.427005i \(-0.859569\pi\)
0.904249 0.427005i \(-0.140431\pi\)
\(272\) 0 0
\(273\) 0.313859 + 1.33591i 0.0189956 + 0.0808528i
\(274\) 0 0
\(275\) 52.4674 3.16390
\(276\) 0 0
\(277\) −20.9783 −1.26046 −0.630230 0.776408i \(-0.717040\pi\)
−0.630230 + 0.776408i \(0.717040\pi\)
\(278\) 0 0
\(279\) 8.86141 + 17.8178i 0.530519 + 1.06673i
\(280\) 0 0
\(281\) 24.9484i 1.48830i −0.668014 0.744149i \(-0.732855\pi\)
0.668014 0.744149i \(-0.267145\pi\)
\(282\) 0 0
\(283\) 27.4179i 1.62982i 0.579586 + 0.814911i \(0.303214\pi\)
−0.579586 + 0.814911i \(0.696786\pi\)
\(284\) 0 0
\(285\) 24.8614 5.84096i 1.47266 0.345989i
\(286\) 0 0
\(287\) −1.25544 −0.0741061
\(288\) 0 0
\(289\) −17.1168 −1.00687
\(290\) 0 0
\(291\) 14.7446 3.46410i 0.864342 0.203069i
\(292\) 0 0
\(293\) 30.7894i 1.79874i 0.437193 + 0.899368i \(0.355973\pi\)
−0.437193 + 0.899368i \(0.644027\pi\)
\(294\) 0 0
\(295\) 51.0767i 2.97380i
\(296\) 0 0
\(297\) −16.0000 + 13.2665i −0.928414 + 0.769800i
\(298\) 0 0
\(299\) 6.74456 0.390048
\(300\) 0 0
\(301\) 7.37228 0.424931
\(302\) 0 0
\(303\) −1.25544 5.34363i −0.0721230 0.306983i
\(304\) 0 0
\(305\) 48.9022i 2.80013i
\(306\) 0 0
\(307\) 25.8333i 1.47438i −0.675684 0.737192i \(-0.736151\pi\)
0.675684 0.737192i \(-0.263849\pi\)
\(308\) 0 0
\(309\) −0.744563 3.16915i −0.0423567 0.180287i
\(310\) 0 0
\(311\) −9.25544 −0.524828 −0.262414 0.964955i \(-0.584519\pi\)
−0.262414 + 0.964955i \(0.584519\pi\)
\(312\) 0 0
\(313\) 14.6277 0.826808 0.413404 0.910548i \(-0.364340\pi\)
0.413404 + 0.910548i \(0.364340\pi\)
\(314\) 0 0
\(315\) −9.05842 + 4.50506i −0.510384 + 0.253831i
\(316\) 0 0
\(317\) 3.75906i 0.211130i −0.994412 0.105565i \(-0.966335\pi\)
0.994412 0.105565i \(-0.0336650\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −9.25544 + 2.17448i −0.516588 + 0.121368i
\(322\) 0 0
\(323\) 20.2337 1.12583
\(324\) 0 0
\(325\) −13.1168 −0.727592
\(326\) 0 0
\(327\) −11.1753 + 2.62553i −0.617994 + 0.145192i
\(328\) 0 0
\(329\) 0.497333i 0.0274189i
\(330\) 0 0
\(331\) 22.0742i 1.21331i 0.794965 + 0.606655i \(0.207489\pi\)
−0.794965 + 0.606655i \(0.792511\pi\)
\(332\) 0 0
\(333\) −14.4307 + 7.17687i −0.790798 + 0.393290i
\(334\) 0 0
\(335\) 14.7446 0.805582
\(336\) 0 0
\(337\) 20.1168 1.09583 0.547917 0.836533i \(-0.315421\pi\)
0.547917 + 0.836533i \(0.315421\pi\)
\(338\) 0 0
\(339\) −2.51087 10.6873i −0.136372 0.580452i
\(340\) 0 0
\(341\) 26.5330i 1.43684i
\(342\) 0 0
\(343\) 10.5947i 0.572059i
\(344\) 0 0
\(345\) 11.3723 + 48.4048i 0.612263 + 2.60603i
\(346\) 0 0
\(347\) −23.6060 −1.26723 −0.633617 0.773647i \(-0.718431\pi\)
−0.633617 + 0.773647i \(0.718431\pi\)
\(348\) 0 0
\(349\) 32.1168 1.71918 0.859588 0.510988i \(-0.170720\pi\)
0.859588 + 0.510988i \(0.170720\pi\)
\(350\) 0 0
\(351\) 4.00000 3.31662i 0.213504 0.177028i
\(352\) 0 0
\(353\) 19.6048i 1.04346i 0.853111 + 0.521729i \(0.174713\pi\)
−0.853111 + 0.521729i \(0.825287\pi\)
\(354\) 0 0
\(355\) 26.0357i 1.38183i
\(356\) 0 0
\(357\) −7.80298 + 1.83324i −0.412978 + 0.0970254i
\(358\) 0 0
\(359\) −1.48913 −0.0785930 −0.0392965 0.999228i \(-0.512512\pi\)
−0.0392965 + 0.999228i \(0.512512\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 8.43070 1.98072i 0.442497 0.103961i
\(364\) 0 0
\(365\) 3.16915i 0.165881i
\(366\) 0 0
\(367\) 6.04334i 0.315460i −0.987482 0.157730i \(-0.949582\pi\)
0.987482 0.157730i \(-0.0504175\pi\)
\(368\) 0 0
\(369\) 2.11684 + 4.25639i 0.110198 + 0.221579i
\(370\) 0 0
\(371\) 4.23369 0.219802
\(372\) 0 0
\(373\) 12.7446 0.659888 0.329944 0.944000i \(-0.392970\pi\)
0.329944 + 0.944000i \(0.392970\pi\)
\(374\) 0 0
\(375\) −13.6861 58.2535i −0.706749 3.00820i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 17.3205i 0.889695i −0.895606 0.444847i \(-0.853258\pi\)
0.895606 0.444847i \(-0.146742\pi\)
\(380\) 0 0
\(381\) 2.62772 + 11.1846i 0.134622 + 0.573004i
\(382\) 0 0
\(383\) 22.1168 1.13012 0.565059 0.825051i \(-0.308853\pi\)
0.565059 + 0.825051i \(0.308853\pi\)
\(384\) 0 0
\(385\) 13.4891 0.687469
\(386\) 0 0
\(387\) −12.4307 24.9947i −0.631888 1.27055i
\(388\) 0 0
\(389\) 24.5437i 1.24441i 0.782853 + 0.622207i \(0.213764\pi\)
−0.782853 + 0.622207i \(0.786236\pi\)
\(390\) 0 0
\(391\) 39.3947i 1.99228i
\(392\) 0 0
\(393\) −17.0584 + 4.00772i −0.860484 + 0.202163i
\(394\) 0 0
\(395\) 21.4891 1.08124
\(396\) 0 0
\(397\) 27.4891 1.37964 0.689820 0.723981i \(-0.257690\pi\)
0.689820 + 0.723981i \(0.257690\pi\)
\(398\) 0 0
\(399\) 4.62772 1.08724i 0.231676 0.0544301i
\(400\) 0 0
\(401\) 12.2718i 0.612826i −0.951899 0.306413i \(-0.900871\pi\)
0.951899 0.306413i \(-0.0991289\pi\)
\(402\) 0 0
\(403\) 6.63325i 0.330426i
\(404\) 0 0
\(405\) 30.5475 + 23.1152i 1.51792 + 1.14860i
\(406\) 0 0
\(407\) 21.4891 1.06518
\(408\) 0 0
\(409\) −32.9783 −1.63067 −0.815335 0.578990i \(-0.803447\pi\)
−0.815335 + 0.578990i \(0.803447\pi\)
\(410\) 0 0
\(411\) 1.88316 + 8.01544i 0.0928892 + 0.395373i
\(412\) 0 0
\(413\) 9.50744i 0.467831i
\(414\) 0 0
\(415\) 51.0767i 2.50725i
\(416\) 0 0
\(417\) −6.19702 26.3769i −0.303469 1.29168i
\(418\) 0 0
\(419\) 0.861407 0.0420825 0.0210412 0.999779i \(-0.493302\pi\)
0.0210412 + 0.999779i \(0.493302\pi\)
\(420\) 0 0
\(421\) −28.1168 −1.37033 −0.685166 0.728387i \(-0.740270\pi\)
−0.685166 + 0.728387i \(0.740270\pi\)
\(422\) 0 0
\(423\) 1.68614 0.838574i 0.0819830 0.0407729i
\(424\) 0 0
\(425\) 76.6150i 3.71637i
\(426\) 0 0
\(427\) 9.10268i 0.440510i
\(428\) 0 0
\(429\) −6.74456 + 1.58457i −0.325631 + 0.0765040i
\(430\) 0 0
\(431\) −18.3505 −0.883914 −0.441957 0.897036i \(-0.645716\pi\)
−0.441957 + 0.897036i \(0.645716\pi\)
\(432\) 0 0
\(433\) −20.3505 −0.977984 −0.488992 0.872288i \(-0.662635\pi\)
−0.488992 + 0.872288i \(0.662635\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 23.3639i 1.11764i
\(438\) 0 0
\(439\) 31.5817i 1.50731i −0.657270 0.753656i \(-0.728289\pi\)
0.657270 0.753656i \(-0.271711\pi\)
\(440\) 0 0
\(441\) 17.1168 8.51278i 0.815088 0.405370i
\(442\) 0 0
\(443\) 3.37228 0.160222 0.0801110 0.996786i \(-0.474473\pi\)
0.0801110 + 0.996786i \(0.474473\pi\)
\(444\) 0 0
\(445\) 6.74456 0.319723
\(446\) 0 0
\(447\) −1.48913 6.33830i −0.0704332 0.299791i
\(448\) 0 0
\(449\) 6.92820i 0.326962i −0.986546 0.163481i \(-0.947728\pi\)
0.986546 0.163481i \(-0.0522723\pi\)
\(450\) 0 0
\(451\) 6.33830i 0.298459i
\(452\) 0 0
\(453\) −4.31386 18.3615i −0.202683 0.862697i
\(454\) 0 0
\(455\) −3.37228 −0.158095
\(456\) 0 0
\(457\) −16.9783 −0.794209 −0.397105 0.917773i \(-0.629985\pi\)
−0.397105 + 0.917773i \(0.629985\pi\)
\(458\) 0 0
\(459\) 19.3723 + 23.3639i 0.904221 + 1.09053i
\(460\) 0 0
\(461\) 19.1075i 0.889923i −0.895550 0.444962i \(-0.853217\pi\)
0.895550 0.444962i \(-0.146783\pi\)
\(462\) 0 0
\(463\) 10.9822i 0.510387i 0.966890 + 0.255193i \(0.0821391\pi\)
−0.966890 + 0.255193i \(0.917861\pi\)
\(464\) 0 0
\(465\) −47.6060 + 11.1846i −2.20767 + 0.518673i
\(466\) 0 0
\(467\) 2.51087 0.116189 0.0580947 0.998311i \(-0.481497\pi\)
0.0580947 + 0.998311i \(0.481497\pi\)
\(468\) 0 0
\(469\) 2.74456 0.126732
\(470\) 0 0
\(471\) −14.7446 + 3.46410i −0.679394 + 0.159617i
\(472\) 0 0
\(473\) 37.2203i 1.71139i
\(474\) 0 0
\(475\) 45.4381i 2.08484i
\(476\) 0 0
\(477\) −7.13859 14.3537i −0.326854 0.657213i
\(478\) 0 0
\(479\) 9.88316 0.451573 0.225786 0.974177i \(-0.427505\pi\)
0.225786 + 0.974177i \(0.427505\pi\)
\(480\) 0 0
\(481\) −5.37228 −0.244955
\(482\) 0 0
\(483\) 2.11684 + 9.01011i 0.0963197 + 0.409974i
\(484\) 0 0
\(485\) 37.2203i 1.69008i
\(486\) 0 0
\(487\) 0.884861i 0.0400969i −0.999799 0.0200484i \(-0.993618\pi\)
0.999799 0.0200484i \(-0.00638204\pi\)
\(488\) 0 0
\(489\) −8.11684 34.5484i −0.367056 1.56233i
\(490\) 0 0
\(491\) −22.3505 −1.00867 −0.504333 0.863509i \(-0.668261\pi\)
−0.504333 + 0.863509i \(0.668261\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −22.7446 45.7330i −1.02229 2.05555i
\(496\) 0 0
\(497\) 4.84630i 0.217386i
\(498\) 0 0
\(499\) 2.87419i 0.128667i 0.997928 + 0.0643333i \(0.0204921\pi\)
−0.997928 + 0.0643333i \(0.979508\pi\)
\(500\) 0 0
\(501\) −6.74456 + 1.58457i −0.301325 + 0.0707935i
\(502\) 0 0
\(503\) −28.2337 −1.25888 −0.629439 0.777050i \(-0.716715\pi\)
−0.629439 + 0.777050i \(0.716715\pi\)
\(504\) 0 0
\(505\) 13.4891 0.600258
\(506\) 0 0
\(507\) 1.68614 0.396143i 0.0748841 0.0175934i
\(508\) 0 0
\(509\) 17.6155i 0.780792i 0.920647 + 0.390396i \(0.127662\pi\)
−0.920647 + 0.390396i \(0.872338\pi\)
\(510\) 0 0
\(511\) 0.589907i 0.0260960i
\(512\) 0 0
\(513\) −11.4891 13.8564i −0.507257 0.611775i
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −2.51087 −0.110428
\(518\) 0 0
\(519\) 8.00000 + 34.0511i 0.351161 + 1.49468i
\(520\) 0 0
\(521\) 30.3846i 1.33117i 0.746320 + 0.665587i \(0.231819\pi\)
−0.746320 + 0.665587i \(0.768181\pi\)
\(522\) 0 0
\(523\) 23.0689i 1.00873i 0.863490 + 0.504366i \(0.168274\pi\)
−0.863490 + 0.504366i \(0.831726\pi\)
\(524\) 0 0
\(525\) −4.11684 17.5229i −0.179674 0.764762i
\(526\) 0 0
\(527\) −38.7446 −1.68774
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) 0 0
\(531\) 32.2337 16.0309i 1.39882 0.695681i
\(532\) 0 0
\(533\) 1.58457i 0.0686355i
\(534\) 0 0
\(535\) 23.3639i 1.01011i
\(536\) 0 0
\(537\) 19.1753 4.50506i 0.827474 0.194407i
\(538\) 0 0
\(539\) −25.4891 −1.09789
\(540\) 0 0
\(541\) −22.6277 −0.972842 −0.486421 0.873725i \(-0.661698\pi\)
−0.486421 + 0.873725i \(0.661698\pi\)
\(542\) 0 0
\(543\) 17.2554 4.05401i 0.740502 0.173974i
\(544\) 0 0
\(545\) 28.2101i 1.20839i
\(546\) 0 0
\(547\) 43.9461i 1.87900i 0.342551 + 0.939499i \(0.388709\pi\)
−0.342551 + 0.939499i \(0.611291\pi\)
\(548\) 0 0
\(549\) 30.8614 15.3484i 1.31713 0.655054i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) −9.05842 38.5562i −0.384509 1.63662i
\(556\) 0 0
\(557\) 43.6511i 1.84956i −0.380505 0.924779i \(-0.624250\pi\)
0.380505 0.924779i \(-0.375750\pi\)
\(558\) 0 0
\(559\) 9.30506i 0.393562i
\(560\) 0 0
\(561\) −9.25544 39.3947i −0.390765 1.66325i
\(562\) 0 0
\(563\) 29.0951 1.22621 0.613106 0.790001i \(-0.289920\pi\)
0.613106 + 0.790001i \(0.289920\pi\)
\(564\) 0 0
\(565\) 26.9783 1.13498
\(566\) 0 0
\(567\) 5.68614 + 4.30268i 0.238795 + 0.180695i
\(568\) 0 0
\(569\) 22.8665i 0.958614i −0.877647 0.479307i \(-0.840888\pi\)
0.877647 0.479307i \(-0.159112\pi\)
\(570\) 0 0
\(571\) 15.6434i 0.654654i −0.944911 0.327327i \(-0.893852\pi\)
0.944911 0.327327i \(-0.106148\pi\)
\(572\) 0 0
\(573\) −36.2337 + 8.51278i −1.51368 + 0.355626i
\(574\) 0 0
\(575\) −88.4674 −3.68934
\(576\) 0 0
\(577\) 8.74456 0.364041 0.182020 0.983295i \(-0.441736\pi\)
0.182020 + 0.983295i \(0.441736\pi\)
\(578\) 0 0
\(579\) 18.9783 4.45877i 0.788709 0.185300i
\(580\) 0 0
\(581\) 9.50744i 0.394435i
\(582\) 0 0
\(583\) 21.3745i 0.885242i
\(584\) 0 0
\(585\) 5.68614 + 11.4333i 0.235093 + 0.472707i
\(586\) 0 0
\(587\) 10.7446 0.443476 0.221738 0.975106i \(-0.428827\pi\)
0.221738 + 0.975106i \(0.428827\pi\)
\(588\) 0 0
\(589\) 22.9783 0.946802
\(590\) 0 0
\(591\) 1.68614 + 7.17687i 0.0693586 + 0.295217i
\(592\) 0 0
\(593\) 2.57924i 0.105917i −0.998597 0.0529584i \(-0.983135\pi\)
0.998597 0.0529584i \(-0.0168651\pi\)
\(594\) 0 0
\(595\) 19.6974i 0.807513i
\(596\) 0 0
\(597\) 3.88316 + 16.5282i 0.158927 + 0.676455i
\(598\) 0 0
\(599\) −21.4891 −0.878022 −0.439011 0.898482i \(-0.644671\pi\)
−0.439011 + 0.898482i \(0.644671\pi\)
\(600\) 0 0
\(601\) 29.8397 1.21719 0.608593 0.793483i \(-0.291734\pi\)
0.608593 + 0.793483i \(0.291734\pi\)
\(602\) 0 0
\(603\) −4.62772 9.30506i −0.188455 0.378932i
\(604\) 0 0
\(605\) 21.2819i 0.865234i
\(606\) 0 0
\(607\) 5.63858i 0.228863i 0.993431 + 0.114432i \(0.0365046\pi\)
−0.993431 + 0.114432i \(0.963495\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.627719 0.0253948
\(612\) 0 0
\(613\) −7.48913 −0.302483 −0.151241 0.988497i \(-0.548327\pi\)
−0.151241 + 0.988497i \(0.548327\pi\)
\(614\) 0 0
\(615\) −11.3723 + 2.67181i −0.458575 + 0.107738i
\(616\) 0 0
\(617\) 17.6155i 0.709172i 0.935023 + 0.354586i \(0.115378\pi\)
−0.935023 + 0.354586i \(0.884622\pi\)
\(618\) 0 0
\(619\) 6.04334i 0.242902i 0.992597 + 0.121451i \(0.0387548\pi\)
−0.992597 + 0.121451i \(0.961245\pi\)
\(620\) 0 0
\(621\) 26.9783 22.3692i 1.08260 0.897644i
\(622\) 0 0
\(623\) 1.25544 0.0502980
\(624\) 0 0
\(625\) 81.4674 3.25870
\(626\) 0 0
\(627\) 5.48913 + 23.3639i 0.219215 + 0.933062i
\(628\) 0 0
\(629\) 31.3793i 1.25117i
\(630\) 0 0
\(631\) 12.4742i 0.496591i 0.968684 + 0.248295i \(0.0798703\pi\)
−0.968684 + 0.248295i \(0.920130\pi\)
\(632\) 0 0
\(633\) 6.43070 + 27.3716i 0.255598 + 1.08792i
\(634\) 0 0
\(635\) −28.2337 −1.12042
\(636\) 0 0
\(637\) 6.37228 0.252479
\(638\) 0 0
\(639\) 16.4307 8.17154i 0.649989 0.323261i
\(640\) 0 0
\(641\) 24.5437i 0.969416i −0.874676 0.484708i \(-0.838926\pi\)
0.874676 0.484708i \(-0.161074\pi\)
\(642\) 0 0
\(643\) 43.8535i 1.72941i −0.502277 0.864707i \(-0.667504\pi\)
0.502277 0.864707i \(-0.332496\pi\)
\(644\) 0 0
\(645\) 66.7812 15.6896i 2.62951 0.617779i
\(646\) 0 0
\(647\) 10.9783 0.431600 0.215800 0.976438i \(-0.430764\pi\)
0.215800 + 0.976438i \(0.430764\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) −8.86141 + 2.08191i −0.347306 + 0.0815964i
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 43.0612i 1.68254i
\(656\) 0 0
\(657\) 2.00000 0.994667i 0.0780274 0.0388056i
\(658\) 0 0
\(659\) 5.48913 0.213826 0.106913 0.994268i \(-0.465903\pi\)
0.106913 + 0.994268i \(0.465903\pi\)
\(660\) 0 0
\(661\) −23.4891 −0.913621 −0.456811 0.889564i \(-0.651008\pi\)
−0.456811 + 0.889564i \(0.651008\pi\)
\(662\) 0 0
\(663\) 2.31386 + 9.84868i 0.0898629 + 0.382491i
\(664\) 0 0
\(665\) 11.6819i 0.453006i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 10.4307 + 44.3971i 0.403274 + 1.71649i
\(670\) 0 0
\(671\) −45.9565 −1.77413
\(672\) 0 0
\(673\) 15.8832 0.612251 0.306125 0.951991i \(-0.400967\pi\)
0.306125 + 0.951991i \(0.400967\pi\)
\(674\) 0 0
\(675\) −52.4674 + 43.5036i −2.01947 + 1.67446i
\(676\) 0 0
\(677\) 34.0511i 1.30869i −0.756196 0.654345i \(-0.772944\pi\)
0.756196 0.654345i \(-0.227056\pi\)
\(678\) 0 0
\(679\) 6.92820i 0.265880i
\(680\) 0 0
\(681\) −4.62772 + 1.08724i −0.177335 + 0.0416632i
\(682\) 0 0
\(683\) 22.9783 0.879238 0.439619 0.898184i \(-0.355113\pi\)
0.439619 + 0.898184i \(0.355113\pi\)
\(684\) 0 0
\(685\) −20.2337 −0.773089
\(686\) 0 0
\(687\) −1.91983 + 0.451046i −0.0732460 + 0.0172085i
\(688\) 0 0
\(689\) 5.34363i 0.203576i
\(690\) 0 0
\(691\) 8.80773i 0.335062i −0.985867 0.167531i \(-0.946421\pi\)
0.985867 0.167531i \(-0.0535794\pi\)
\(692\) 0 0
\(693\) −4.23369 8.51278i −0.160825 0.323374i
\(694\) 0 0
\(695\) 66.5842 2.52568
\(696\) 0 0
\(697\) −9.25544 −0.350575
\(698\) 0 0
\(699\) 0.197015 + 0.838574i 0.00745181 + 0.0317178i
\(700\) 0 0
\(701\) 42.5639i 1.60762i −0.594889 0.803808i \(-0.702804\pi\)
0.594889 0.803808i \(-0.297196\pi\)
\(702\) 0 0
\(703\) 18.6101i 0.701894i
\(704\) 0 0
\(705\) 1.05842 + 4.50506i 0.0398625 + 0.169670i
\(706\) 0 0
\(707\) 2.51087 0.0944312
\(708\) 0 0
\(709\) −20.9783 −0.787855 −0.393927 0.919142i \(-0.628884\pi\)
−0.393927 + 0.919142i \(0.628884\pi\)
\(710\) 0 0
\(711\) −6.74456 13.5615i −0.252941 0.508594i
\(712\) 0 0
\(713\) 44.7384i 1.67547i
\(714\) 0 0
\(715\) 17.0256i 0.636720i
\(716\) 0 0
\(717\) −35.1753 + 8.26411i −1.31364 + 0.308629i
\(718\) 0 0
\(719\) −49.7228 −1.85435 −0.927174 0.374631i \(-0.877769\pi\)
−0.927174 + 0.374631i \(0.877769\pi\)
\(720\) 0 0
\(721\) 1.48913 0.0554579
\(722\) 0 0
\(723\) 30.3505 7.13058i 1.12875 0.265189i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 44.8482i 1.66333i 0.555280 + 0.831663i \(0.312611\pi\)
−0.555280 + 0.831663i \(0.687389\pi\)
\(728\) 0 0
\(729\) 5.00000 26.5330i 0.185185 0.982704i
\(730\) 0 0
\(731\) 54.3505 2.01023
\(732\) 0 0
\(733\) −40.3505 −1.49038 −0.745190 0.666852i \(-0.767641\pi\)
−0.745190 + 0.666852i \(0.767641\pi\)
\(734\) 0 0
\(735\) 10.7446 + 45.7330i 0.396319 + 1.68689i
\(736\) 0 0
\(737\) 13.8564i 0.510407i
\(738\) 0 0
\(739\) 19.8997i 0.732024i 0.930610 + 0.366012i \(0.119277\pi\)
−0.930610 + 0.366012i \(0.880723\pi\)
\(740\) 0 0
\(741\) −1.37228 5.84096i −0.0504120 0.214573i
\(742\) 0 0
\(743\) −7.37228 −0.270463 −0.135231 0.990814i \(-0.543178\pi\)
−0.135231 + 0.990814i \(0.543178\pi\)
\(744\) 0 0
\(745\) 16.0000 0.586195
\(746\) 0 0
\(747\) −32.2337 + 16.0309i −1.17937 + 0.586540i
\(748\) 0 0
\(749\) 4.34896i 0.158908i
\(750\) 0 0
\(751\) 40.6844i 1.48459i 0.670071 + 0.742297i \(0.266263\pi\)
−0.670071 + 0.742297i \(0.733737\pi\)
\(752\) 0 0
\(753\) −26.9783 + 6.33830i −0.983142 + 0.230980i
\(754\) 0 0
\(755\) 46.3505 1.68687
\(756\) 0 0
\(757\) 16.9783 0.617085 0.308543 0.951211i \(-0.400159\pi\)
0.308543 + 0.951211i \(0.400159\pi\)
\(758\) 0 0
\(759\) −45.4891 + 10.6873i −1.65115 + 0.387923i
\(760\) 0 0
\(761\) 14.4463i 0.523678i 0.965112 + 0.261839i \(0.0843290\pi\)
−0.965112 + 0.261839i \(0.915671\pi\)
\(762\) 0 0
\(763\) 5.25106i 0.190101i
\(764\) 0 0
\(765\) −66.7812 + 33.2125i −2.41448 + 1.20080i
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −47.7228 −1.72093 −0.860464 0.509511i \(-0.829826\pi\)
−0.860464 + 0.509511i \(0.829826\pi\)
\(770\) 0 0
\(771\) −9.05842 38.5562i −0.326231 1.38857i
\(772\) 0 0
\(773\) 5.25106i 0.188867i 0.995531 + 0.0944337i \(0.0301040\pi\)
−0.995531 + 0.0944337i \(0.969896\pi\)
\(774\) 0 0
\(775\) 87.0073i 3.12539i
\(776\) 0 0
\(777\) −1.68614 7.17687i −0.0604900 0.257469i
\(778\) 0 0
\(779\) 5.48913 0.196668
\(780\) 0 0
\(781\) −24.4674 −0.875512
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 37.2203i 1.32845i
\(786\) 0 0
\(787\) 14.1514i 0.504442i −0.967670 0.252221i \(-0.918839\pi\)
0.967670 0.252221i \(-0.0811609\pi\)
\(788\) 0 0
\(789\) 2.11684 0.497333i 0.0753616 0.0177055i
\(790\) 0 0
\(791\) 5.02175 0.178553
\(792\) 0 0
\(793\) 11.4891 0.407991
\(794\) 0 0
\(795\) 38.3505 9.01011i 1.36015 0.319556i
\(796\) 0 0
\(797\) 19.2000i 0.680100i 0.940407 + 0.340050i \(0.110444\pi\)
−0.940407 + 0.340050i \(0.889556\pi\)
\(798\) 0 0
\(799\) 3.66648i 0.129711i
\(800\) 0 0
\(801\) −2.11684 4.25639i −0.0747950 0.150392i
\(802\) 0 0
\(803\) −2.97825 −0.105100
\(804\) 0 0
\(805\) −22.7446 −0.801640
\(806\) 0 0
\(807\) 2.11684 + 9.01011i 0.0745164 + 0.317171i
\(808\) 0 0
\(809\) 3.66648i 0.128907i 0.997921 + 0.0644533i \(0.0205304\pi\)
−0.997921 + 0.0644533i \(0.979470\pi\)
\(810\) 0 0
\(811\) 39.6897i 1.39369i −0.717220 0.696847i \(-0.754586\pi\)
0.717220 0.696847i \(-0.245414\pi\)
\(812\) 0 0
\(813\) −5.56930 23.7051i −0.195324 0.831374i
\(814\) 0 0
\(815\) 87.2119 3.05490
\(816\) 0 0
\(817\) −32.2337 −1.12771
\(818\) 0 0
\(819\) 1.05842 + 2.12819i 0.0369843 + 0.0743651i
\(820\) 0 0
\(821\) 34.1437i 1.19162i −0.803124 0.595811i \(-0.796831\pi\)
0.803124 0.595811i \(-0.203169\pi\)
\(822\) 0 0
\(823\) 18.9051i 0.658990i −0.944157 0.329495i \(-0.893122\pi\)
0.944157 0.329495i \(-0.106878\pi\)
\(824\) 0 0
\(825\) 88.4674 20.7846i 3.08004 0.723627i
\(826\) 0 0
\(827\) −29.2554 −1.01731 −0.508656 0.860970i \(-0.669857\pi\)
−0.508656 + 0.860970i \(0.669857\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −35.3723 + 8.31040i −1.22705 + 0.288284i
\(832\) 0 0
\(833\) 37.2203i 1.28961i
\(834\) 0 0
\(835\) 17.0256i 0.589194i
\(836\) 0 0
\(837\) 22.0000 + 26.5330i 0.760431 + 0.917115i
\(838\) 0 0
\(839\) −46.9783 −1.62187 −0.810935 0.585137i \(-0.801041\pi\)
−0.810935 + 0.585137i \(0.801041\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) −9.88316 42.0666i −0.340394 1.44885i
\(844\) 0 0
\(845\) 4.25639i 0.146424i
\(846\) 0 0
\(847\) 3.96143i 0.136117i
\(848\) 0 0
\(849\) 10.8614 + 46.2304i 0.372763 + 1.58662i
\(850\) 0 0
\(851\) −36.2337 −1.24207
\(852\) 0 0
\(853\) −25.6060 −0.876732 −0.438366 0.898797i \(-0.644443\pi\)
−0.438366 + 0.898797i \(0.644443\pi\)
\(854\) 0 0
\(855\) 39.6060 19.6974i 1.35450 0.673636i
\(856\) 0 0
\(857\) 3.16915i 0.108256i −0.998534 0.0541280i \(-0.982762\pi\)
0.998534 0.0541280i \(-0.0172379\pi\)
\(858\) 0 0
\(859\) 12.9715i 0.442583i −0.975208 0.221292i \(-0.928973\pi\)
0.975208 0.221292i \(-0.0710273\pi\)
\(860\) 0 0
\(861\) −2.11684 + 0.497333i −0.0721418 + 0.0169491i
\(862\) 0 0
\(863\) 42.3505 1.44163 0.720814 0.693128i \(-0.243768\pi\)
0.720814 + 0.693128i \(0.243768\pi\)
\(864\) 0 0
\(865\) −85.9565 −2.92261
\(866\) 0 0
\(867\) −28.8614 + 6.78073i −0.980185 + 0.230286i
\(868\) 0 0
\(869\) 20.1947i 0.685058i
\(870\) 0 0
\(871\) 3.46410i 0.117377i
\(872\) 0 0
\(873\) 23.4891 11.6819i 0.794986 0.395373i
\(874\) 0 0
\(875\) 27.3723 0.925352
\(876\) 0 0
\(877\) 16.1168 0.544227 0.272114 0.962265i \(-0.412277\pi\)
0.272114 + 0.962265i \(0.412277\pi\)
\(878\) 0 0
\(879\) 12.1970 + 51.9152i 0.411395 + 1.75106i
\(880\) 0 0
\(881\) 28.2101i 0.950424i 0.879871 + 0.475212i \(0.157629\pi\)
−0.879871 + 0.475212i \(0.842371\pi\)
\(882\) 0 0
\(883\) 11.4795i 0.386317i −0.981168 0.193159i \(-0.938127\pi\)
0.981168 0.193159i \(-0.0618732\pi\)
\(884\) 0 0
\(885\) 20.2337 + 86.1224i 0.680148 + 2.89497i
\(886\) 0 0
\(887\) 3.76631 0.126460 0.0632302 0.997999i \(-0.479860\pi\)
0.0632302 + 0.997999i \(0.479860\pi\)
\(888\) 0 0
\(889\) −5.25544 −0.176262
\(890\) 0 0
\(891\) −21.7228 + 28.7075i −0.727742 + 0.961737i
\(892\) 0 0
\(893\) 2.17448i 0.0727662i
\(894\) 0 0
\(895\) 48.4048i 1.61800i
\(896\) 0 0
\(897\) 11.3723 2.67181i 0.379709 0.0892093i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 31.2119 1.03982
\(902\) 0 0
\(903\) 12.4307 2.92048i 0.413668 0.0971875i
\(904\) 0 0
\(905\) 43.5586i 1.44794i
\(906\) 0 0
\(907\) 13.0641i 0.433787i 0.976195 + 0.216893i \(0.0695924\pi\)
−0.976195 + 0.216893i \(0.930408\pi\)
\(908\) 0 0
\(909\) −4.23369 8.51278i −0.140423 0.282351i
\(910\) 0 0
\(911\) 34.9783 1.15888 0.579441 0.815014i \(-0.303271\pi\)
0.579441 + 0.815014i \(0.303271\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 0 0
\(915\) 19.3723 + 82.4560i 0.640428 + 2.72591i
\(916\) 0 0
\(917\) 8.01544i 0.264693i
\(918\) 0 0
\(919\) 23.2540i 0.767080i −0.923524 0.383540i \(-0.874705\pi\)
0.923524 0.383540i \(-0.125295\pi\)
\(920\) 0 0
\(921\) −10.2337 43.5586i −0.337211 1.43530i
\(922\) 0 0
\(923\) 6.11684 0.201338
\(924\) 0 0
\(925\) 70.4674 2.31695
\(926\) 0 0
\(927\) −2.51087 5.04868i −0.0824679 0.165820i
\(928\) 0 0
\(929\) 14.4463i 0.473968i −0.971514 0.236984i \(-0.923841\pi\)
0.971514 0.236984i \(-0.0761589\pi\)
\(930\) 0 0
\(931\) 22.0742i 0.723454i
\(932\) 0 0
\(933\) −15.6060 + 3.66648i −0.510916 + 0.120035i
\(934\) 0 0
\(935\) 99.4456 3.25222
\(936\) 0 0
\(937\) −35.4891 −1.15938 −0.579690 0.814837i \(-0.696826\pi\)
−0.579690 + 0.814837i \(0.696826\pi\)
\(938\) 0 0
\(939\) 24.6644 5.79468i 0.804892 0.189102i
\(940\) 0 0
\(941\) 15.9383i 0.519574i 0.965666 + 0.259787i \(0.0836524\pi\)
−0.965666 + 0.259787i \(0.916348\pi\)
\(942\) 0 0
\(943\) 10.6873i 0.348025i
\(944\) 0 0
\(945\) −13.4891 + 11.1846i −0.438801 + 0.363835i
\(946\) 0 0
\(947\) 22.9783 0.746693 0.373346 0.927692i \(-0.378210\pi\)
0.373346 + 0.927692i \(0.378210\pi\)
\(948\) 0 0
\(949\) 0.744563 0.0241695
\(950\) 0 0
\(951\) −1.48913 6.33830i −0.0482882 0.205533i
\(952\) 0 0
\(953\) 39.8921i 1.29223i 0.763240 + 0.646115i \(0.223608\pi\)
−0.763240 + 0.646115i \(0.776392\pi\)
\(954\) 0 0
\(955\) 91.4661i 2.95977i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.76631 −0.121621
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 0 0
\(963\) −14.7446 + 7.33296i −0.475137 + 0.236301i
\(964\) 0 0
\(965\) 47.9075i 1.54220i
\(966\) 0 0
\(967\) 4.55134i 0.146361i −0.997319 0.0731806i \(-0.976685\pi\)
0.997319 0.0731806i \(-0.0233150\pi\)
\(968\) 0 0
\(969\) 34.1168 8.01544i 1.09599 0.257493i
\(970\) 0 0
\(971\) −16.8614 −0.541108 −0.270554 0.962705i \(-0.587207\pi\)
−0.270554 + 0.962705i \(0.587207\pi\)
\(972\) 0 0
\(973\) 12.3940 0.397334
\(974\) 0 0
\(975\) −22.1168 + 5.19615i −0.708306 + 0.166410i
\(976\) 0 0
\(977\) 29.2974i 0.937306i 0.883382 + 0.468653i \(0.155261\pi\)
−0.883382 + 0.468653i \(0.844739\pi\)
\(978\) 0 0
\(979\) 6.33830i 0.202573i
\(980\) 0 0
\(981\) −17.8030 + 8.85402i −0.568406 + 0.282687i
\(982\) 0 0
\(983\) −33.8832 −1.08070 −0.540352 0.841439i \(-0.681709\pi\)
−0.540352 + 0.841439i \(0.681709\pi\)
\(984\) 0 0
\(985\) −18.1168 −0.577251
\(986\) 0 0
\(987\) 0.197015 + 0.838574i 0.00627107 + 0.0266921i
\(988\) 0 0
\(989\) 62.7586i 1.99561i
\(990\) 0 0
\(991\) 33.7562i 1.07230i −0.844123 0.536150i \(-0.819878\pi\)
0.844123 0.536150i \(-0.180122\pi\)
\(992\) 0 0
\(993\) 8.74456 + 37.2203i 0.277500 + 1.18115i
\(994\) 0 0
\(995\) −41.7228 −1.32270
\(996\) 0 0
\(997\) 31.2554 0.989870 0.494935 0.868930i \(-0.335192\pi\)
0.494935 + 0.868930i \(0.335192\pi\)
\(998\) 0 0
\(999\) −21.4891 + 17.8178i −0.679886 + 0.563732i
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.d.i.287.3 yes 4
3.2 odd 2 624.2.d.g.287.1 4
4.3 odd 2 624.2.d.g.287.2 yes 4
8.3 odd 2 2496.2.d.l.1535.3 4
8.5 even 2 2496.2.d.i.1535.2 4
12.11 even 2 inner 624.2.d.i.287.4 yes 4
24.5 odd 2 2496.2.d.l.1535.4 4
24.11 even 2 2496.2.d.i.1535.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
624.2.d.g.287.1 4 3.2 odd 2
624.2.d.g.287.2 yes 4 4.3 odd 2
624.2.d.i.287.3 yes 4 1.1 even 1 trivial
624.2.d.i.287.4 yes 4 12.11 even 2 inner
2496.2.d.i.1535.1 4 24.11 even 2
2496.2.d.i.1535.2 4 8.5 even 2
2496.2.d.l.1535.3 4 8.3 odd 2
2496.2.d.l.1535.4 4 24.5 odd 2