Properties

Label 1216.2.t.d.1185.7
Level $1216$
Weight $2$
Character 1216.1185
Analytic conductor $9.710$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(353,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.353");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.t (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1185.7
Character \(\chi\) \(=\) 1216.1185
Dual form 1216.2.t.d.353.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.495361 - 0.285997i) q^{3} +(3.31453 - 1.91364i) q^{5} -2.37943 q^{7} +(-1.33641 + 2.31473i) q^{9} +O(q^{10})\) \(q+(0.495361 - 0.285997i) q^{3} +(3.31453 - 1.91364i) q^{5} -2.37943 q^{7} +(-1.33641 + 2.31473i) q^{9} -2.62170i q^{11} +(1.36594 + 0.788626i) q^{13} +(1.09459 - 1.89589i) q^{15} +(1.53509 + 2.65885i) q^{17} +(0.0442721 - 4.35867i) q^{19} +(-1.17868 + 0.680510i) q^{21} +(4.47737 - 7.75503i) q^{23} +(4.82405 - 8.35551i) q^{25} +3.24482i q^{27} +(8.05237 + 4.64904i) q^{29} +4.38612 q^{31} +(-0.749798 - 1.29869i) q^{33} +(-7.88669 + 4.55338i) q^{35} -7.00890i q^{37} +0.902177 q^{39} +(-6.22065 - 10.7745i) q^{41} +(2.35778 - 1.36127i) q^{43} +10.2297i q^{45} +(-1.15544 + 2.00129i) q^{47} -1.33830 q^{49} +(1.52085 + 0.878061i) q^{51} +(1.38780 + 0.801249i) q^{53} +(-5.01700 - 8.68970i) q^{55} +(-1.22464 - 2.17178i) q^{57} +(-10.4393 + 6.02711i) q^{59} +(0.477247 + 0.275539i) q^{61} +(3.17990 - 5.50775i) q^{63} +6.03659 q^{65} +(-7.64072 - 4.41137i) q^{67} -5.12205i q^{69} +(3.63270 + 6.29203i) q^{71} +(-1.80857 - 3.13253i) q^{73} -5.51865i q^{75} +6.23817i q^{77} +(1.99460 + 3.45475i) q^{79} +(-3.08123 - 5.33684i) q^{81} +8.25278i q^{83} +(10.1762 + 5.87522i) q^{85} +5.31844 q^{87} +(-4.83103 + 8.36758i) q^{89} +(-3.25016 - 1.87648i) q^{91} +(2.17271 - 1.25442i) q^{93} +(-8.19420 - 14.5317i) q^{95} +(6.67320 + 11.5583i) q^{97} +(6.06854 + 3.50367i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 6 q^{5} + 8 q^{9} + 30 q^{13} + 6 q^{17} + 24 q^{21} + 6 q^{25} + 42 q^{29} - 14 q^{33} - 24 q^{41} + 24 q^{49} - 18 q^{53} - 42 q^{57} + 18 q^{61} - 20 q^{65} - 16 q^{73} + 52 q^{81} - 78 q^{85} + 14 q^{89} + 60 q^{93} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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
\(3\) 0.495361 0.285997i 0.285997 0.165120i −0.350138 0.936698i \(-0.613865\pi\)
0.636135 + 0.771578i \(0.280532\pi\)
\(4\) 0 0
\(5\) 3.31453 1.91364i 1.48230 0.855807i 0.482503 0.875894i \(-0.339728\pi\)
0.999798 + 0.0200876i \(0.00639450\pi\)
\(6\) 0 0
\(7\) −2.37943 −0.899341 −0.449671 0.893194i \(-0.648459\pi\)
−0.449671 + 0.893194i \(0.648459\pi\)
\(8\) 0 0
\(9\) −1.33641 + 2.31473i −0.445471 + 0.771578i
\(10\) 0 0
\(11\) 2.62170i 0.790473i −0.918579 0.395237i \(-0.870663\pi\)
0.918579 0.395237i \(-0.129337\pi\)
\(12\) 0 0
\(13\) 1.36594 + 0.788626i 0.378844 + 0.218725i 0.677315 0.735693i \(-0.263143\pi\)
−0.298471 + 0.954419i \(0.596477\pi\)
\(14\) 0 0
\(15\) 1.09459 1.89589i 0.282622 0.489516i
\(16\) 0 0
\(17\) 1.53509 + 2.65885i 0.372314 + 0.644866i 0.989921 0.141620i \(-0.0452312\pi\)
−0.617607 + 0.786487i \(0.711898\pi\)
\(18\) 0 0
\(19\) 0.0442721 4.35867i 0.0101567 0.999948i
\(20\) 0 0
\(21\) −1.17868 + 0.680510i −0.257209 + 0.148499i
\(22\) 0 0
\(23\) 4.47737 7.75503i 0.933596 1.61704i 0.156477 0.987682i \(-0.449986\pi\)
0.777119 0.629354i \(-0.216680\pi\)
\(24\) 0 0
\(25\) 4.82405 8.35551i 0.964811 1.67110i
\(26\) 0 0
\(27\) 3.24482i 0.624465i
\(28\) 0 0
\(29\) 8.05237 + 4.64904i 1.49529 + 0.863305i 0.999985 0.00541604i \(-0.00172399\pi\)
0.495302 + 0.868721i \(0.335057\pi\)
\(30\) 0 0
\(31\) 4.38612 0.787771 0.393885 0.919160i \(-0.371131\pi\)
0.393885 + 0.919160i \(0.371131\pi\)
\(32\) 0 0
\(33\) −0.749798 1.29869i −0.130523 0.226073i
\(34\) 0 0
\(35\) −7.88669 + 4.55338i −1.33309 + 0.769662i
\(36\) 0 0
\(37\) 7.00890i 1.15226i −0.817359 0.576128i \(-0.804563\pi\)
0.817359 0.576128i \(-0.195437\pi\)
\(38\) 0 0
\(39\) 0.902177 0.144464
\(40\) 0 0
\(41\) −6.22065 10.7745i −0.971502 1.68269i −0.691027 0.722829i \(-0.742841\pi\)
−0.280475 0.959861i \(-0.590492\pi\)
\(42\) 0 0
\(43\) 2.35778 1.36127i 0.359559 0.207591i −0.309329 0.950955i \(-0.600104\pi\)
0.668887 + 0.743364i \(0.266771\pi\)
\(44\) 0 0
\(45\) 10.2297i 1.52495i
\(46\) 0 0
\(47\) −1.15544 + 2.00129i −0.168539 + 0.291918i −0.937906 0.346889i \(-0.887238\pi\)
0.769368 + 0.638806i \(0.220571\pi\)
\(48\) 0 0
\(49\) −1.33830 −0.191185
\(50\) 0 0
\(51\) 1.52085 + 0.878061i 0.212961 + 0.122953i
\(52\) 0 0
\(53\) 1.38780 + 0.801249i 0.190630 + 0.110060i 0.592277 0.805734i \(-0.298229\pi\)
−0.401648 + 0.915794i \(0.631562\pi\)
\(54\) 0 0
\(55\) −5.01700 8.68970i −0.676492 1.17172i
\(56\) 0 0
\(57\) −1.22464 2.17178i −0.162207 0.287659i
\(58\) 0 0
\(59\) −10.4393 + 6.02711i −1.35908 + 0.784663i −0.989499 0.144537i \(-0.953831\pi\)
−0.369577 + 0.929200i \(0.620497\pi\)
\(60\) 0 0
\(61\) 0.477247 + 0.275539i 0.0611052 + 0.0352791i 0.530241 0.847847i \(-0.322101\pi\)
−0.469136 + 0.883126i \(0.655435\pi\)
\(62\) 0 0
\(63\) 3.17990 5.50775i 0.400630 0.693912i
\(64\) 0 0
\(65\) 6.03659 0.748747
\(66\) 0 0
\(67\) −7.64072 4.41137i −0.933463 0.538935i −0.0455578 0.998962i \(-0.514507\pi\)
−0.887905 + 0.460027i \(0.847840\pi\)
\(68\) 0 0
\(69\) 5.12205i 0.616622i
\(70\) 0 0
\(71\) 3.63270 + 6.29203i 0.431123 + 0.746726i 0.996970 0.0777836i \(-0.0247843\pi\)
−0.565848 + 0.824510i \(0.691451\pi\)
\(72\) 0 0
\(73\) −1.80857 3.13253i −0.211677 0.366635i 0.740563 0.671987i \(-0.234559\pi\)
−0.952239 + 0.305352i \(0.901226\pi\)
\(74\) 0 0
\(75\) 5.51865i 0.637239i
\(76\) 0 0
\(77\) 6.23817i 0.710905i
\(78\) 0 0
\(79\) 1.99460 + 3.45475i 0.224410 + 0.388690i 0.956142 0.292903i \(-0.0946212\pi\)
−0.731732 + 0.681592i \(0.761288\pi\)
\(80\) 0 0
\(81\) −3.08123 5.33684i −0.342359 0.592983i
\(82\) 0 0
\(83\) 8.25278i 0.905861i 0.891546 + 0.452930i \(0.149621\pi\)
−0.891546 + 0.452930i \(0.850379\pi\)
\(84\) 0 0
\(85\) 10.1762 + 5.87522i 1.10376 + 0.637257i
\(86\) 0 0
\(87\) 5.31844 0.570196
\(88\) 0 0
\(89\) −4.83103 + 8.36758i −0.512088 + 0.886962i 0.487814 + 0.872948i \(0.337794\pi\)
−0.999902 + 0.0140145i \(0.995539\pi\)
\(90\) 0 0
\(91\) −3.25016 1.87648i −0.340710 0.196709i
\(92\) 0 0
\(93\) 2.17271 1.25442i 0.225300 0.130077i
\(94\) 0 0
\(95\) −8.19420 14.5317i −0.840707 1.49092i
\(96\) 0 0
\(97\) 6.67320 + 11.5583i 0.677561 + 1.17357i 0.975713 + 0.219051i \(0.0702963\pi\)
−0.298153 + 0.954518i \(0.596370\pi\)
\(98\) 0 0
\(99\) 6.06854 + 3.50367i 0.609911 + 0.352133i
\(100\) 0 0
\(101\) −1.57668 0.910298i −0.156886 0.0905780i 0.419502 0.907754i \(-0.362205\pi\)
−0.576388 + 0.817176i \(0.695538\pi\)
\(102\) 0 0
\(103\) −9.94561 −0.979970 −0.489985 0.871731i \(-0.662998\pi\)
−0.489985 + 0.871731i \(0.662998\pi\)
\(104\) 0 0
\(105\) −2.60450 + 4.51113i −0.254174 + 0.440242i
\(106\) 0 0
\(107\) 13.2581i 1.28171i 0.767662 + 0.640855i \(0.221420\pi\)
−0.767662 + 0.640855i \(0.778580\pi\)
\(108\) 0 0
\(109\) 6.54713 3.77999i 0.627102 0.362057i −0.152527 0.988299i \(-0.548741\pi\)
0.779629 + 0.626242i \(0.215408\pi\)
\(110\) 0 0
\(111\) −2.00452 3.47193i −0.190261 0.329541i
\(112\) 0 0
\(113\) 7.07753 0.665798 0.332899 0.942963i \(-0.391973\pi\)
0.332899 + 0.942963i \(0.391973\pi\)
\(114\) 0 0
\(115\) 34.2723i 3.19591i
\(116\) 0 0
\(117\) −3.65092 + 2.10786i −0.337527 + 0.194871i
\(118\) 0 0
\(119\) −3.65264 6.32656i −0.334837 0.579955i
\(120\) 0 0
\(121\) 4.12668 0.375152
\(122\) 0 0
\(123\) −6.16293 3.55817i −0.555692 0.320829i
\(124\) 0 0
\(125\) 17.7896i 1.59115i
\(126\) 0 0
\(127\) 2.91907 5.05597i 0.259025 0.448645i −0.706956 0.707258i \(-0.749932\pi\)
0.965981 + 0.258613i \(0.0832654\pi\)
\(128\) 0 0
\(129\) 0.778635 1.34864i 0.0685550 0.118741i
\(130\) 0 0
\(131\) 12.2631 7.08012i 1.07143 0.618593i 0.142861 0.989743i \(-0.454370\pi\)
0.928573 + 0.371150i \(0.121036\pi\)
\(132\) 0 0
\(133\) −0.105343 + 10.3712i −0.00913436 + 0.899295i
\(134\) 0 0
\(135\) 6.20942 + 10.7550i 0.534422 + 0.925646i
\(136\) 0 0
\(137\) 0.616142 1.06719i 0.0526405 0.0911761i −0.838504 0.544895i \(-0.816570\pi\)
0.891145 + 0.453719i \(0.149903\pi\)
\(138\) 0 0
\(139\) 5.20766 + 3.00664i 0.441708 + 0.255020i 0.704322 0.709881i \(-0.251251\pi\)
−0.262614 + 0.964901i \(0.584585\pi\)
\(140\) 0 0
\(141\) 1.32181i 0.111317i
\(142\) 0 0
\(143\) 2.06754 3.58109i 0.172897 0.299466i
\(144\) 0 0
\(145\) 35.5864 2.95529
\(146\) 0 0
\(147\) −0.662940 + 0.382749i −0.0546784 + 0.0315686i
\(148\) 0 0
\(149\) 0.499112 0.288162i 0.0408888 0.0236072i −0.479416 0.877588i \(-0.659152\pi\)
0.520305 + 0.853980i \(0.325818\pi\)
\(150\) 0 0
\(151\) 13.2393 1.07740 0.538700 0.842498i \(-0.318916\pi\)
0.538700 + 0.842498i \(0.318916\pi\)
\(152\) 0 0
\(153\) −8.20604 −0.663419
\(154\) 0 0
\(155\) 14.5379 8.39347i 1.16771 0.674180i
\(156\) 0 0
\(157\) −15.3009 + 8.83396i −1.22114 + 0.705027i −0.965162 0.261655i \(-0.915732\pi\)
−0.255981 + 0.966682i \(0.582399\pi\)
\(158\) 0 0
\(159\) 0.916618 0.0726925
\(160\) 0 0
\(161\) −10.6536 + 18.4526i −0.839621 + 1.45427i
\(162\) 0 0
\(163\) 18.7739i 1.47049i 0.677803 + 0.735244i \(0.262932\pi\)
−0.677803 + 0.735244i \(0.737068\pi\)
\(164\) 0 0
\(165\) −4.97045 2.86969i −0.386949 0.223405i
\(166\) 0 0
\(167\) −2.37890 + 4.12037i −0.184085 + 0.318844i −0.943268 0.332033i \(-0.892265\pi\)
0.759183 + 0.650877i \(0.225599\pi\)
\(168\) 0 0
\(169\) −5.25614 9.10390i −0.404318 0.700300i
\(170\) 0 0
\(171\) 10.0300 + 5.92746i 0.767013 + 0.453284i
\(172\) 0 0
\(173\) 14.4488 8.34203i 1.09852 0.634233i 0.162690 0.986677i \(-0.447983\pi\)
0.935833 + 0.352445i \(0.114649\pi\)
\(174\) 0 0
\(175\) −11.4785 + 19.8814i −0.867694 + 1.50289i
\(176\) 0 0
\(177\) −3.44747 + 5.97119i −0.259128 + 0.448822i
\(178\) 0 0
\(179\) 17.2627i 1.29027i 0.764067 + 0.645136i \(0.223199\pi\)
−0.764067 + 0.645136i \(0.776801\pi\)
\(180\) 0 0
\(181\) −0.358255 0.206838i −0.0266289 0.0153742i 0.486626 0.873610i \(-0.338227\pi\)
−0.513255 + 0.858236i \(0.671561\pi\)
\(182\) 0 0
\(183\) 0.315213 0.0233012
\(184\) 0 0
\(185\) −13.4125 23.2312i −0.986109 1.70799i
\(186\) 0 0
\(187\) 6.97072 4.02455i 0.509749 0.294304i
\(188\) 0 0
\(189\) 7.72082i 0.561607i
\(190\) 0 0
\(191\) −24.8774 −1.80006 −0.900031 0.435825i \(-0.856457\pi\)
−0.900031 + 0.435825i \(0.856457\pi\)
\(192\) 0 0
\(193\) −4.75804 8.24117i −0.342491 0.593212i 0.642403 0.766367i \(-0.277937\pi\)
−0.984895 + 0.173154i \(0.944604\pi\)
\(194\) 0 0
\(195\) 2.99029 1.72644i 0.214139 0.123633i
\(196\) 0 0
\(197\) 5.85154i 0.416905i 0.978033 + 0.208452i \(0.0668426\pi\)
−0.978033 + 0.208452i \(0.933157\pi\)
\(198\) 0 0
\(199\) −8.39537 + 14.5412i −0.595132 + 1.03080i 0.398397 + 0.917213i \(0.369567\pi\)
−0.993528 + 0.113585i \(0.963767\pi\)
\(200\) 0 0
\(201\) −5.04655 −0.355956
\(202\) 0 0
\(203\) −19.1601 11.0621i −1.34477 0.776405i
\(204\) 0 0
\(205\) −41.2370 23.8082i −2.88012 1.66284i
\(206\) 0 0
\(207\) 11.9672 + 20.7278i 0.831779 + 1.44068i
\(208\) 0 0
\(209\) −11.4271 0.116068i −0.790432 0.00802862i
\(210\) 0 0
\(211\) −3.15685 + 1.82261i −0.217327 + 0.125474i −0.604712 0.796444i \(-0.706712\pi\)
0.387385 + 0.921918i \(0.373378\pi\)
\(212\) 0 0
\(213\) 3.59900 + 2.07788i 0.246599 + 0.142374i
\(214\) 0 0
\(215\) 5.20995 9.02391i 0.355316 0.615425i
\(216\) 0 0
\(217\) −10.4365 −0.708475
\(218\) 0 0
\(219\) −1.79179 1.03449i −0.121078 0.0699042i
\(220\) 0 0
\(221\) 4.84244i 0.325738i
\(222\) 0 0
\(223\) 8.66306 + 15.0049i 0.580121 + 1.00480i 0.995464 + 0.0951349i \(0.0303283\pi\)
−0.415343 + 0.909665i \(0.636338\pi\)
\(224\) 0 0
\(225\) 12.8938 + 22.3328i 0.859590 + 1.48885i
\(226\) 0 0
\(227\) 15.0900i 1.00156i 0.865575 + 0.500779i \(0.166953\pi\)
−0.865575 + 0.500779i \(0.833047\pi\)
\(228\) 0 0
\(229\) 9.04237i 0.597536i 0.954326 + 0.298768i \(0.0965757\pi\)
−0.954326 + 0.298768i \(0.903424\pi\)
\(230\) 0 0
\(231\) 1.78409 + 3.09014i 0.117385 + 0.203316i
\(232\) 0 0
\(233\) −5.64804 9.78269i −0.370015 0.640885i 0.619552 0.784955i \(-0.287314\pi\)
−0.989567 + 0.144070i \(0.953981\pi\)
\(234\) 0 0
\(235\) 8.84442i 0.576946i
\(236\) 0 0
\(237\) 1.97609 + 1.14090i 0.128361 + 0.0741093i
\(238\) 0 0
\(239\) 6.47049 0.418541 0.209271 0.977858i \(-0.432891\pi\)
0.209271 + 0.977858i \(0.432891\pi\)
\(240\) 0 0
\(241\) −10.1454 + 17.5724i −0.653524 + 1.13194i 0.328737 + 0.944421i \(0.393377\pi\)
−0.982262 + 0.187516i \(0.939956\pi\)
\(242\) 0 0
\(243\) −11.4829 6.62967i −0.736630 0.425293i
\(244\) 0 0
\(245\) −4.43582 + 2.56102i −0.283394 + 0.163618i
\(246\) 0 0
\(247\) 3.49784 5.91877i 0.222562 0.376602i
\(248\) 0 0
\(249\) 2.36027 + 4.08810i 0.149576 + 0.259073i
\(250\) 0 0
\(251\) 9.81627 + 5.66742i 0.619597 + 0.357725i 0.776712 0.629856i \(-0.216886\pi\)
−0.157115 + 0.987580i \(0.550219\pi\)
\(252\) 0 0
\(253\) −20.3314 11.7383i −1.27822 0.737982i
\(254\) 0 0
\(255\) 6.72118 0.420896
\(256\) 0 0
\(257\) −2.33679 + 4.04744i −0.145765 + 0.252472i −0.929658 0.368424i \(-0.879898\pi\)
0.783893 + 0.620896i \(0.213231\pi\)
\(258\) 0 0
\(259\) 16.6772i 1.03627i
\(260\) 0 0
\(261\) −21.5226 + 12.4261i −1.33221 + 0.769154i
\(262\) 0 0
\(263\) 5.51632 + 9.55454i 0.340151 + 0.589158i 0.984460 0.175606i \(-0.0561886\pi\)
−0.644310 + 0.764765i \(0.722855\pi\)
\(264\) 0 0
\(265\) 6.13322 0.376761
\(266\) 0 0
\(267\) 5.52663i 0.338224i
\(268\) 0 0
\(269\) 12.9884 7.49889i 0.791920 0.457215i −0.0487182 0.998813i \(-0.515514\pi\)
0.840638 + 0.541597i \(0.182180\pi\)
\(270\) 0 0
\(271\) −8.33304 14.4332i −0.506196 0.876757i −0.999974 0.00716938i \(-0.997718\pi\)
0.493778 0.869588i \(-0.335615\pi\)
\(272\) 0 0
\(273\) −2.14667 −0.129922
\(274\) 0 0
\(275\) −21.9057 12.6472i −1.32096 0.762657i
\(276\) 0 0
\(277\) 15.6486i 0.940235i 0.882604 + 0.470118i \(0.155789\pi\)
−0.882604 + 0.470118i \(0.844211\pi\)
\(278\) 0 0
\(279\) −5.86167 + 10.1527i −0.350929 + 0.607826i
\(280\) 0 0
\(281\) 3.50045 6.06295i 0.208819 0.361685i −0.742524 0.669820i \(-0.766371\pi\)
0.951343 + 0.308134i \(0.0997047\pi\)
\(282\) 0 0
\(283\) 1.72756 0.997408i 0.102693 0.0592897i −0.447774 0.894147i \(-0.647783\pi\)
0.550467 + 0.834857i \(0.314450\pi\)
\(284\) 0 0
\(285\) −8.21509 4.85490i −0.486620 0.287579i
\(286\) 0 0
\(287\) 14.8016 + 25.6371i 0.873712 + 1.51331i
\(288\) 0 0
\(289\) 3.78700 6.55928i 0.222765 0.385840i
\(290\) 0 0
\(291\) 6.61128 + 3.81703i 0.387560 + 0.223758i
\(292\) 0 0
\(293\) 17.9363i 1.04785i −0.851765 0.523924i \(-0.824468\pi\)
0.851765 0.523924i \(-0.175532\pi\)
\(294\) 0 0
\(295\) −23.0675 + 39.9540i −1.34304 + 2.32621i
\(296\) 0 0
\(297\) 8.50695 0.493623
\(298\) 0 0
\(299\) 12.2316 7.06194i 0.707373 0.408402i
\(300\) 0 0
\(301\) −5.61019 + 3.23904i −0.323366 + 0.186695i
\(302\) 0 0
\(303\) −1.04137 −0.0598250
\(304\) 0 0
\(305\) 2.10913 0.120768
\(306\) 0 0
\(307\) 25.5431 14.7473i 1.45782 0.841673i 0.458916 0.888480i \(-0.348238\pi\)
0.998904 + 0.0468072i \(0.0149046\pi\)
\(308\) 0 0
\(309\) −4.92666 + 2.84441i −0.280268 + 0.161813i
\(310\) 0 0
\(311\) −18.2270 −1.03356 −0.516779 0.856119i \(-0.672869\pi\)
−0.516779 + 0.856119i \(0.672869\pi\)
\(312\) 0 0
\(313\) −14.9213 + 25.8444i −0.843401 + 1.46081i 0.0436015 + 0.999049i \(0.486117\pi\)
−0.887003 + 0.461764i \(0.847217\pi\)
\(314\) 0 0
\(315\) 24.3408i 1.37145i
\(316\) 0 0
\(317\) 5.53123 + 3.19346i 0.310665 + 0.179363i 0.647224 0.762300i \(-0.275930\pi\)
−0.336559 + 0.941662i \(0.609263\pi\)
\(318\) 0 0
\(319\) 12.1884 21.1109i 0.682419 1.18198i
\(320\) 0 0
\(321\) 3.79177 + 6.56754i 0.211636 + 0.366565i
\(322\) 0 0
\(323\) 11.6570 6.57324i 0.648615 0.365745i
\(324\) 0 0
\(325\) 13.1787 7.60875i 0.731025 0.422057i
\(326\) 0 0
\(327\) 2.16213 3.74492i 0.119566 0.207094i
\(328\) 0 0
\(329\) 2.74930 4.76193i 0.151574 0.262533i
\(330\) 0 0
\(331\) 6.22210i 0.341997i 0.985271 + 0.170999i \(0.0546994\pi\)
−0.985271 + 0.170999i \(0.945301\pi\)
\(332\) 0 0
\(333\) 16.2237 + 9.36678i 0.889055 + 0.513296i
\(334\) 0 0
\(335\) −33.7672 −1.84490
\(336\) 0 0
\(337\) 14.7925 + 25.6213i 0.805797 + 1.39568i 0.915752 + 0.401745i \(0.131596\pi\)
−0.109954 + 0.993937i \(0.535070\pi\)
\(338\) 0 0
\(339\) 3.50593 2.02415i 0.190416 0.109937i
\(340\) 0 0
\(341\) 11.4991i 0.622712i
\(342\) 0 0
\(343\) 19.8404 1.07128
\(344\) 0 0
\(345\) −9.80177 16.9772i −0.527710 0.914020i
\(346\) 0 0
\(347\) −5.78717 + 3.34122i −0.310671 + 0.179366i −0.647227 0.762297i \(-0.724071\pi\)
0.336555 + 0.941664i \(0.390738\pi\)
\(348\) 0 0
\(349\) 17.1074i 0.915736i 0.889020 + 0.457868i \(0.151387\pi\)
−0.889020 + 0.457868i \(0.848613\pi\)
\(350\) 0 0
\(351\) −2.55895 + 4.43222i −0.136586 + 0.236575i
\(352\) 0 0
\(353\) 0.857173 0.0456227 0.0228114 0.999740i \(-0.492738\pi\)
0.0228114 + 0.999740i \(0.492738\pi\)
\(354\) 0 0
\(355\) 24.0814 + 13.9034i 1.27811 + 0.737915i
\(356\) 0 0
\(357\) −3.61875 2.08929i −0.191525 0.110577i
\(358\) 0 0
\(359\) −17.9279 31.0520i −0.946198 1.63886i −0.753335 0.657637i \(-0.771556\pi\)
−0.192863 0.981226i \(-0.561777\pi\)
\(360\) 0 0
\(361\) −18.9961 0.385936i −0.999794 0.0203124i
\(362\) 0 0
\(363\) 2.04419 1.18022i 0.107292 0.0619452i
\(364\) 0 0
\(365\) −11.9891 6.92190i −0.627537 0.362309i
\(366\) 0 0
\(367\) 3.60358 6.24158i 0.188105 0.325808i −0.756513 0.653978i \(-0.773099\pi\)
0.944618 + 0.328171i \(0.106432\pi\)
\(368\) 0 0
\(369\) 33.2534 1.73110
\(370\) 0 0
\(371\) −3.30219 1.90652i −0.171441 0.0989815i
\(372\) 0 0
\(373\) 1.23956i 0.0641818i 0.999485 + 0.0320909i \(0.0102166\pi\)
−0.999485 + 0.0320909i \(0.989783\pi\)
\(374\) 0 0
\(375\) −5.08778 8.81229i −0.262732 0.455064i
\(376\) 0 0
\(377\) 7.33270 + 12.7006i 0.377653 + 0.654115i
\(378\) 0 0
\(379\) 7.69864i 0.395453i 0.980257 + 0.197726i \(0.0633558\pi\)
−0.980257 + 0.197726i \(0.936644\pi\)
\(380\) 0 0
\(381\) 3.33937i 0.171081i
\(382\) 0 0
\(383\) −6.13360 10.6237i −0.313413 0.542846i 0.665686 0.746232i \(-0.268139\pi\)
−0.979099 + 0.203385i \(0.934806\pi\)
\(384\) 0 0
\(385\) 11.9376 + 20.6766i 0.608397 + 1.05378i
\(386\) 0 0
\(387\) 7.27685i 0.369903i
\(388\) 0 0
\(389\) −23.7105 13.6893i −1.20217 0.694074i −0.241135 0.970492i \(-0.577520\pi\)
−0.961038 + 0.276417i \(0.910853\pi\)
\(390\) 0 0
\(391\) 27.4926 1.39036
\(392\) 0 0
\(393\) 4.04978 7.01443i 0.204284 0.353831i
\(394\) 0 0
\(395\) 13.2223 + 7.63390i 0.665286 + 0.384103i
\(396\) 0 0
\(397\) −2.94101 + 1.69799i −0.147605 + 0.0852197i −0.571984 0.820265i \(-0.693826\pi\)
0.424379 + 0.905485i \(0.360493\pi\)
\(398\) 0 0
\(399\) 2.91394 + 5.16760i 0.145879 + 0.258704i
\(400\) 0 0
\(401\) −12.4586 21.5789i −0.622151 1.07760i −0.989084 0.147350i \(-0.952926\pi\)
0.366934 0.930247i \(-0.380408\pi\)
\(402\) 0 0
\(403\) 5.99118 + 3.45901i 0.298442 + 0.172306i
\(404\) 0 0
\(405\) −20.4256 11.7927i −1.01496 0.585986i
\(406\) 0 0
\(407\) −18.3753 −0.910827
\(408\) 0 0
\(409\) 5.57567 9.65734i 0.275699 0.477525i −0.694612 0.719384i \(-0.744424\pi\)
0.970311 + 0.241860i \(0.0777574\pi\)
\(410\) 0 0
\(411\) 0.704858i 0.0347681i
\(412\) 0 0
\(413\) 24.8395 14.3411i 1.22227 0.705680i
\(414\) 0 0
\(415\) 15.7929 + 27.3541i 0.775242 + 1.34276i
\(416\) 0 0
\(417\) 3.43956 0.168436
\(418\) 0 0
\(419\) 5.54457i 0.270870i −0.990786 0.135435i \(-0.956757\pi\)
0.990786 0.135435i \(-0.0432432\pi\)
\(420\) 0 0
\(421\) 19.9281 11.5055i 0.971235 0.560743i 0.0716223 0.997432i \(-0.477182\pi\)
0.899612 + 0.436689i \(0.143849\pi\)
\(422\) 0 0
\(423\) −3.08830 5.34908i −0.150158 0.260081i
\(424\) 0 0
\(425\) 29.6214 1.43685
\(426\) 0 0
\(427\) −1.13558 0.655626i −0.0549544 0.0317280i
\(428\) 0 0
\(429\) 2.36524i 0.114195i
\(430\) 0 0
\(431\) 3.84478 6.65935i 0.185196 0.320770i −0.758446 0.651736i \(-0.774041\pi\)
0.943643 + 0.330966i \(0.107375\pi\)
\(432\) 0 0
\(433\) −13.8560 + 23.9994i −0.665879 + 1.15334i 0.313167 + 0.949698i \(0.398610\pi\)
−0.979046 + 0.203639i \(0.934723\pi\)
\(434\) 0 0
\(435\) 17.6281 10.1776i 0.845203 0.487978i
\(436\) 0 0
\(437\) −33.6034 19.8587i −1.60747 0.949971i
\(438\) 0 0
\(439\) −17.4148 30.1634i −0.831165 1.43962i −0.897115 0.441797i \(-0.854341\pi\)
0.0659505 0.997823i \(-0.478992\pi\)
\(440\) 0 0
\(441\) 1.78852 3.09780i 0.0851675 0.147514i
\(442\) 0 0
\(443\) −6.78938 3.91985i −0.322573 0.186238i 0.329966 0.943993i \(-0.392963\pi\)
−0.652539 + 0.757755i \(0.726296\pi\)
\(444\) 0 0
\(445\) 36.9794i 1.75299i
\(446\) 0 0
\(447\) 0.164827 0.285488i 0.00779604 0.0135031i
\(448\) 0 0
\(449\) −22.7057 −1.07155 −0.535775 0.844361i \(-0.679980\pi\)
−0.535775 + 0.844361i \(0.679980\pi\)
\(450\) 0 0
\(451\) −28.2475 + 16.3087i −1.33012 + 0.767946i
\(452\) 0 0
\(453\) 6.55823 3.78640i 0.308133 0.177900i
\(454\) 0 0
\(455\) −14.3637 −0.673379
\(456\) 0 0
\(457\) −34.5207 −1.61481 −0.807405 0.589998i \(-0.799129\pi\)
−0.807405 + 0.589998i \(0.799129\pi\)
\(458\) 0 0
\(459\) −8.62749 + 4.98108i −0.402697 + 0.232497i
\(460\) 0 0
\(461\) −11.7449 + 6.78091i −0.547014 + 0.315819i −0.747917 0.663793i \(-0.768946\pi\)
0.200903 + 0.979611i \(0.435612\pi\)
\(462\) 0 0
\(463\) −16.0544 −0.746112 −0.373056 0.927809i \(-0.621690\pi\)
−0.373056 + 0.927809i \(0.621690\pi\)
\(464\) 0 0
\(465\) 4.80101 8.31559i 0.222641 0.385626i
\(466\) 0 0
\(467\) 36.4819i 1.68818i −0.536201 0.844091i \(-0.680141\pi\)
0.536201 0.844091i \(-0.319859\pi\)
\(468\) 0 0
\(469\) 18.1806 + 10.4966i 0.839502 + 0.484686i
\(470\) 0 0
\(471\) −5.05297 + 8.75200i −0.232828 + 0.403271i
\(472\) 0 0
\(473\) −3.56884 6.18141i −0.164095 0.284221i
\(474\) 0 0
\(475\) −36.2054 21.3964i −1.66122 0.981734i
\(476\) 0 0
\(477\) −3.70936 + 2.14160i −0.169840 + 0.0980570i
\(478\) 0 0
\(479\) −6.04549 + 10.4711i −0.276225 + 0.478437i −0.970444 0.241328i \(-0.922417\pi\)
0.694218 + 0.719765i \(0.255750\pi\)
\(480\) 0 0
\(481\) 5.52740 9.57373i 0.252028 0.436525i
\(482\) 0 0
\(483\) 12.1876i 0.554554i
\(484\) 0 0
\(485\) 44.2370 + 25.5402i 2.00870 + 1.15972i
\(486\) 0 0
\(487\) 33.9780 1.53969 0.769845 0.638231i \(-0.220334\pi\)
0.769845 + 0.638231i \(0.220334\pi\)
\(488\) 0 0
\(489\) 5.36928 + 9.29987i 0.242807 + 0.420555i
\(490\) 0 0
\(491\) −8.85946 + 5.11501i −0.399822 + 0.230837i −0.686407 0.727218i \(-0.740813\pi\)
0.286585 + 0.958055i \(0.407480\pi\)
\(492\) 0 0
\(493\) 28.5467i 1.28568i
\(494\) 0 0
\(495\) 26.8191 1.20543
\(496\) 0 0
\(497\) −8.64377 14.9715i −0.387726 0.671562i
\(498\) 0 0
\(499\) −24.9924 + 14.4294i −1.11881 + 0.645947i −0.941098 0.338135i \(-0.890204\pi\)
−0.177716 + 0.984082i \(0.556871\pi\)
\(500\) 0 0
\(501\) 2.72143i 0.121584i
\(502\) 0 0
\(503\) 13.5515 23.4718i 0.604229 1.04656i −0.387943 0.921683i \(-0.626814\pi\)
0.992173 0.124873i \(-0.0398523\pi\)
\(504\) 0 0
\(505\) −6.96794 −0.310069
\(506\) 0 0
\(507\) −5.20737 3.00648i −0.231267 0.133522i
\(508\) 0 0
\(509\) 20.2327 + 11.6814i 0.896798 + 0.517767i 0.876160 0.482020i \(-0.160097\pi\)
0.0206383 + 0.999787i \(0.493430\pi\)
\(510\) 0 0
\(511\) 4.30336 + 7.45364i 0.190370 + 0.329730i
\(512\) 0 0
\(513\) 14.1431 + 0.143655i 0.624433 + 0.00634252i
\(514\) 0 0
\(515\) −32.9650 + 19.0323i −1.45261 + 0.838665i
\(516\) 0 0
\(517\) 5.24678 + 3.02923i 0.230753 + 0.133225i
\(518\) 0 0
\(519\) 4.77158 8.26463i 0.209449 0.362777i
\(520\) 0 0
\(521\) 2.49127 0.109144 0.0545722 0.998510i \(-0.482621\pi\)
0.0545722 + 0.998510i \(0.482621\pi\)
\(522\) 0 0
\(523\) 24.6860 + 14.2525i 1.07944 + 0.623217i 0.930746 0.365666i \(-0.119159\pi\)
0.148697 + 0.988883i \(0.452492\pi\)
\(524\) 0 0
\(525\) 13.1313i 0.573095i
\(526\) 0 0
\(527\) 6.73309 + 11.6621i 0.293298 + 0.508007i
\(528\) 0 0
\(529\) −28.5937 49.5257i −1.24320 2.15329i
\(530\) 0 0
\(531\) 32.2188i 1.39818i
\(532\) 0 0
\(533\) 19.6230i 0.849968i
\(534\) 0 0
\(535\) 25.3713 + 43.9443i 1.09690 + 1.89988i
\(536\) 0 0
\(537\) 4.93707 + 8.55125i 0.213050 + 0.369014i
\(538\) 0 0
\(539\) 3.50862i 0.151127i
\(540\) 0 0
\(541\) 5.26285 + 3.03851i 0.226267 + 0.130636i 0.608849 0.793286i \(-0.291632\pi\)
−0.382581 + 0.923922i \(0.624965\pi\)
\(542\) 0 0
\(543\) −0.236620 −0.0101544
\(544\) 0 0
\(545\) 14.4671 25.0577i 0.619702 1.07336i
\(546\) 0 0
\(547\) 16.7508 + 9.67108i 0.716212 + 0.413505i 0.813357 0.581765i \(-0.197638\pi\)
−0.0971447 + 0.995270i \(0.530971\pi\)
\(548\) 0 0
\(549\) −1.27560 + 0.736466i −0.0544412 + 0.0314316i
\(550\) 0 0
\(551\) 20.6201 34.8918i 0.878447 1.48644i
\(552\) 0 0
\(553\) −4.74602 8.22034i −0.201821 0.349564i
\(554\) 0 0
\(555\) −13.2881 7.67188i −0.564047 0.325653i
\(556\) 0 0
\(557\) 19.3179 + 11.1532i 0.818525 + 0.472576i 0.849908 0.526932i \(-0.176658\pi\)
−0.0313823 + 0.999507i \(0.509991\pi\)
\(558\) 0 0
\(559\) 4.29412 0.181622
\(560\) 0 0
\(561\) 2.30201 3.98720i 0.0971911 0.168340i
\(562\) 0 0
\(563\) 25.2448i 1.06394i 0.846763 + 0.531970i \(0.178548\pi\)
−0.846763 + 0.531970i \(0.821452\pi\)
\(564\) 0 0
\(565\) 23.4586 13.5439i 0.986913 0.569794i
\(566\) 0 0
\(567\) 7.33158 + 12.6987i 0.307897 + 0.533294i
\(568\) 0 0
\(569\) −17.1784 −0.720155 −0.360078 0.932922i \(-0.617250\pi\)
−0.360078 + 0.932922i \(0.617250\pi\)
\(570\) 0 0
\(571\) 8.64144i 0.361633i 0.983517 + 0.180816i \(0.0578740\pi\)
−0.983517 + 0.180816i \(0.942126\pi\)
\(572\) 0 0
\(573\) −12.3233 + 7.11484i −0.514812 + 0.297227i
\(574\) 0 0
\(575\) −43.1981 74.8214i −1.80149 3.12027i
\(576\) 0 0
\(577\) 24.5806 1.02331 0.511653 0.859192i \(-0.329034\pi\)
0.511653 + 0.859192i \(0.329034\pi\)
\(578\) 0 0
\(579\) −4.71389 2.72157i −0.195903 0.113104i
\(580\) 0 0
\(581\) 19.6369i 0.814678i
\(582\) 0 0
\(583\) 2.10064 3.63841i 0.0869995 0.150688i
\(584\) 0 0
\(585\) −8.06737 + 13.9731i −0.333545 + 0.577716i
\(586\) 0 0
\(587\) −33.7781 + 19.5018i −1.39417 + 0.804926i −0.993774 0.111415i \(-0.964462\pi\)
−0.400399 + 0.916341i \(0.631128\pi\)
\(588\) 0 0
\(589\) 0.194183 19.1177i 0.00800117 0.787730i
\(590\) 0 0
\(591\) 1.67352 + 2.89862i 0.0688394 + 0.119233i
\(592\) 0 0
\(593\) −3.40127 + 5.89117i −0.139673 + 0.241921i −0.927373 0.374138i \(-0.877939\pi\)
0.787700 + 0.616059i \(0.211272\pi\)
\(594\) 0 0
\(595\) −24.2135 13.9797i −0.992659 0.573112i
\(596\) 0 0
\(597\) 9.60418i 0.393073i
\(598\) 0 0
\(599\) −18.1953 + 31.5152i −0.743439 + 1.28767i 0.207481 + 0.978239i \(0.433473\pi\)
−0.950921 + 0.309435i \(0.899860\pi\)
\(600\) 0 0
\(601\) −0.372849 −0.0152088 −0.00760441 0.999971i \(-0.502421\pi\)
−0.00760441 + 0.999971i \(0.502421\pi\)
\(602\) 0 0
\(603\) 20.4223 11.7908i 0.831661 0.480159i
\(604\) 0 0
\(605\) 13.6780 7.89698i 0.556089 0.321058i
\(606\) 0 0
\(607\) 31.5367 1.28003 0.640017 0.768361i \(-0.278927\pi\)
0.640017 + 0.768361i \(0.278927\pi\)
\(608\) 0 0
\(609\) −12.6549 −0.512801
\(610\) 0 0
\(611\) −3.15653 + 1.82242i −0.127700 + 0.0737274i
\(612\) 0 0
\(613\) −16.9009 + 9.75775i −0.682621 + 0.394112i −0.800842 0.598876i \(-0.795614\pi\)
0.118221 + 0.992987i \(0.462281\pi\)
\(614\) 0 0
\(615\) −27.2362 −1.09827
\(616\) 0 0
\(617\) 18.5156 32.0700i 0.745410 1.29109i −0.204592 0.978847i \(-0.565587\pi\)
0.950003 0.312241i \(-0.101080\pi\)
\(618\) 0 0
\(619\) 8.86666i 0.356381i 0.983996 + 0.178190i \(0.0570243\pi\)
−0.983996 + 0.178190i \(0.942976\pi\)
\(620\) 0 0
\(621\) 25.1637 + 14.5282i 1.00978 + 0.582998i
\(622\) 0 0
\(623\) 11.4951 19.9101i 0.460542 0.797682i
\(624\) 0 0
\(625\) −9.92273 17.1867i −0.396909 0.687467i
\(626\) 0 0
\(627\) −5.69375 + 3.21063i −0.227387 + 0.128220i
\(628\) 0 0
\(629\) 18.6356 10.7593i 0.743051 0.429001i
\(630\) 0 0
\(631\) −2.13344 + 3.69522i −0.0849307 + 0.147104i −0.905362 0.424641i \(-0.860400\pi\)
0.820431 + 0.571746i \(0.193734\pi\)
\(632\) 0 0
\(633\) −1.04252 + 1.80570i −0.0414364 + 0.0717700i
\(634\) 0 0
\(635\) 22.3442i 0.886702i
\(636\) 0 0
\(637\) −1.82803 1.05542i −0.0724294 0.0418171i
\(638\) 0 0
\(639\) −19.4192 −0.768210
\(640\) 0 0
\(641\) −11.7085 20.2797i −0.462458 0.801000i 0.536625 0.843821i \(-0.319699\pi\)
−0.999083 + 0.0428208i \(0.986366\pi\)
\(642\) 0 0
\(643\) 18.6718 10.7802i 0.736344 0.425128i −0.0843944 0.996432i \(-0.526896\pi\)
0.820739 + 0.571304i \(0.193562\pi\)
\(644\) 0 0
\(645\) 5.96012i 0.234679i
\(646\) 0 0
\(647\) 27.5504 1.08312 0.541559 0.840663i \(-0.317834\pi\)
0.541559 + 0.840663i \(0.317834\pi\)
\(648\) 0 0
\(649\) 15.8013 + 27.3686i 0.620255 + 1.07431i
\(650\) 0 0
\(651\) −5.16982 + 2.98480i −0.202621 + 0.116984i
\(652\) 0 0
\(653\) 11.9425i 0.467348i 0.972315 + 0.233674i \(0.0750748\pi\)
−0.972315 + 0.233674i \(0.924925\pi\)
\(654\) 0 0
\(655\) 27.0976 46.9345i 1.05879 1.83388i
\(656\) 0 0
\(657\) 9.66796 0.377183
\(658\) 0 0
\(659\) −30.7336 17.7441i −1.19721 0.691211i −0.237279 0.971442i \(-0.576256\pi\)
−0.959933 + 0.280231i \(0.909589\pi\)
\(660\) 0 0
\(661\) −15.5565 8.98158i −0.605080 0.349343i 0.165958 0.986133i \(-0.446928\pi\)
−0.771037 + 0.636790i \(0.780262\pi\)
\(662\) 0 0
\(663\) 1.38492 + 2.39876i 0.0537859 + 0.0931599i
\(664\) 0 0
\(665\) 19.4976 + 34.5771i 0.756083 + 1.34084i
\(666\) 0 0
\(667\) 72.1069 41.6309i 2.79199 1.61196i
\(668\) 0 0
\(669\) 8.58268 + 4.95521i 0.331826 + 0.191580i
\(670\) 0 0
\(671\) 0.722381 1.25120i 0.0278872 0.0483020i
\(672\) 0 0
\(673\) −28.4280 −1.09582 −0.547909 0.836538i \(-0.684576\pi\)
−0.547909 + 0.836538i \(0.684576\pi\)
\(674\) 0 0
\(675\) 27.1121 + 15.6532i 1.04354 + 0.602491i
\(676\) 0 0
\(677\) 42.7219i 1.64194i 0.570974 + 0.820968i \(0.306566\pi\)
−0.570974 + 0.820968i \(0.693434\pi\)
\(678\) 0 0
\(679\) −15.8784 27.5023i −0.609358 1.05544i
\(680\) 0 0
\(681\) 4.31569 + 7.47499i 0.165377 + 0.286442i
\(682\) 0 0
\(683\) 5.28834i 0.202353i 0.994869 + 0.101176i \(0.0322606\pi\)
−0.994869 + 0.101176i \(0.967739\pi\)
\(684\) 0 0
\(685\) 4.71630i 0.180201i
\(686\) 0 0
\(687\) 2.58609 + 4.47923i 0.0986654 + 0.170893i
\(688\) 0 0
\(689\) 1.26377 + 2.18892i 0.0481458 + 0.0833911i
\(690\) 0 0
\(691\) 2.46172i 0.0936482i −0.998903 0.0468241i \(-0.985090\pi\)
0.998903 0.0468241i \(-0.0149100\pi\)
\(692\) 0 0
\(693\) −14.4397 8.33676i −0.548518 0.316687i
\(694\) 0 0
\(695\) 23.0146 0.872992
\(696\) 0 0
\(697\) 19.0985 33.0796i 0.723407 1.25298i
\(698\) 0 0
\(699\) −5.59563 3.23064i −0.211646 0.122194i
\(700\) 0 0
\(701\) 31.3878 18.1217i 1.18550 0.684449i 0.228220 0.973610i \(-0.426710\pi\)
0.957280 + 0.289161i \(0.0933763\pi\)
\(702\) 0 0
\(703\) −30.5495 0.310299i −1.15220 0.0117031i
\(704\) 0 0
\(705\) 2.52947 + 4.38118i 0.0952655 + 0.165005i
\(706\) 0 0
\(707\) 3.75161 + 2.16599i 0.141094 + 0.0814605i
\(708\) 0 0
\(709\) 11.9138 + 6.87845i 0.447433 + 0.258325i 0.706745 0.707468i \(-0.250163\pi\)
−0.259313 + 0.965793i \(0.583496\pi\)
\(710\) 0 0
\(711\) −10.6624 −0.399872
\(712\) 0 0
\(713\) 19.6383 34.0145i 0.735460 1.27385i
\(714\) 0 0
\(715\) 15.8261i 0.591864i
\(716\) 0 0
\(717\) 3.20523 1.85054i 0.119701 0.0691097i
\(718\) 0 0
\(719\) −21.5301 37.2913i −0.802938 1.39073i −0.917674 0.397334i \(-0.869935\pi\)
0.114736 0.993396i \(-0.463398\pi\)
\(720\) 0 0
\(721\) 23.6649 0.881328
\(722\) 0 0
\(723\) 11.6062i 0.431640i
\(724\) 0 0
\(725\) 77.6901 44.8544i 2.88534 1.66585i
\(726\) 0 0
\(727\) 10.5087 + 18.2017i 0.389747 + 0.675062i 0.992415 0.122930i \(-0.0392290\pi\)
−0.602668 + 0.797992i \(0.705896\pi\)
\(728\) 0 0
\(729\) 10.9031 0.403819
\(730\) 0 0
\(731\) 7.23881 + 4.17933i 0.267737 + 0.154578i
\(732\) 0 0
\(733\) 39.1556i 1.44624i −0.690720 0.723122i \(-0.742706\pi\)
0.690720 0.723122i \(-0.257294\pi\)
\(734\) 0 0
\(735\) −1.46489 + 2.53726i −0.0540332 + 0.0935883i
\(736\) 0 0
\(737\) −11.5653 + 20.0317i −0.426014 + 0.737877i
\(738\) 0 0
\(739\) −13.9285 + 8.04160i −0.512367 + 0.295815i −0.733806 0.679359i \(-0.762258\pi\)
0.221439 + 0.975174i \(0.428925\pi\)
\(740\) 0 0
\(741\) 0.0399413 3.93230i 0.00146728 0.144457i
\(742\) 0 0
\(743\) −17.2042 29.7985i −0.631161 1.09320i −0.987315 0.158775i \(-0.949245\pi\)
0.356154 0.934427i \(-0.384088\pi\)
\(744\) 0 0
\(745\) 1.10288 1.91024i 0.0404063 0.0699858i
\(746\) 0 0
\(747\) −19.1030 11.0291i −0.698942 0.403534i
\(748\) 0 0
\(749\) 31.5468i 1.15269i
\(750\) 0 0
\(751\) 16.1770 28.0194i 0.590307 1.02244i −0.403884 0.914810i \(-0.632340\pi\)
0.994191 0.107631i \(-0.0343266\pi\)
\(752\) 0 0
\(753\) 6.48346 0.236270
\(754\) 0 0
\(755\) 43.8820 25.3353i 1.59703 0.922046i
\(756\) 0 0
\(757\) −32.1757 + 18.5767i −1.16945 + 0.675180i −0.953550 0.301236i \(-0.902601\pi\)
−0.215896 + 0.976416i \(0.569267\pi\)
\(758\) 0 0
\(759\) −13.4285 −0.487423
\(760\) 0 0
\(761\) 43.8152 1.58830 0.794150 0.607721i \(-0.207916\pi\)
0.794150 + 0.607721i \(0.207916\pi\)
\(762\) 0 0
\(763\) −15.5785 + 8.99423i −0.563978 + 0.325613i
\(764\) 0 0
\(765\) −27.1991 + 15.7034i −0.983387 + 0.567759i
\(766\) 0 0
\(767\) −19.0125 −0.686503
\(768\) 0 0
\(769\) −5.81793 + 10.0770i −0.209800 + 0.363384i −0.951651 0.307180i \(-0.900615\pi\)
0.741851 + 0.670564i \(0.233948\pi\)
\(770\) 0 0
\(771\) 2.67325i 0.0962749i
\(772\) 0 0
\(773\) −20.2046 11.6652i −0.726711 0.419567i 0.0905070 0.995896i \(-0.471151\pi\)
−0.817218 + 0.576329i \(0.804485\pi\)
\(774\) 0 0
\(775\) 21.1589 36.6483i 0.760050 1.31645i
\(776\) 0 0
\(777\) 4.76963 + 8.26123i 0.171109 + 0.296370i
\(778\) 0 0
\(779\) −47.2378 + 26.6368i −1.69247 + 0.954361i
\(780\) 0 0
\(781\) 16.4958 9.52387i 0.590267 0.340791i
\(782\) 0 0
\(783\) −15.0853 + 26.1285i −0.539104 + 0.933755i
\(784\) 0 0
\(785\) −33.8101 + 58.5608i −1.20673 + 2.09012i
\(786\) 0 0
\(787\) 47.9991i 1.71098i −0.517815 0.855492i \(-0.673255\pi\)
0.517815 0.855492i \(-0.326745\pi\)
\(788\) 0 0
\(789\) 5.46514 + 3.15530i 0.194564 + 0.112332i
\(790\) 0 0
\(791\) −16.8405 −0.598779
\(792\) 0 0
\(793\) 0.434594 + 0.752739i 0.0154329 + 0.0267305i
\(794\) 0 0
\(795\) 3.03815 1.75408i 0.107752 0.0622108i
\(796\) 0 0
\(797\) 15.8257i 0.560576i 0.959916 + 0.280288i \(0.0904299\pi\)
−0.959916 + 0.280288i \(0.909570\pi\)
\(798\) 0 0
\(799\) −7.09483 −0.250997
\(800\) 0 0
\(801\) −12.9125 22.3651i −0.456240 0.790231i
\(802\) 0 0
\(803\) −8.21256 + 4.74152i −0.289815 + 0.167325i
\(804\) 0 0
\(805\) 81.5487i 2.87421i
\(806\) 0 0
\(807\) 4.28931 7.42931i 0.150991 0.261524i
\(808\) 0 0
\(809\) −15.5885 −0.548063 −0.274031 0.961721i \(-0.588357\pi\)
−0.274031 + 0.961721i \(0.588357\pi\)
\(810\) 0 0
\(811\) 30.1683 + 17.4177i 1.05935 + 0.611618i 0.925253 0.379350i \(-0.123852\pi\)
0.134100 + 0.990968i \(0.457186\pi\)
\(812\) 0 0
\(813\) −8.25572 4.76644i −0.289541 0.167166i
\(814\) 0 0
\(815\) 35.9266 + 62.2267i 1.25845 + 2.17971i
\(816\) 0 0
\(817\) −5.82893 10.3371i −0.203929 0.361648i
\(818\) 0 0
\(819\) 8.68711 5.01551i 0.303552 0.175256i
\(820\) 0 0
\(821\) 35.5478 + 20.5235i 1.24063 + 0.716276i 0.969222 0.246190i \(-0.0791786\pi\)
0.271404 + 0.962465i \(0.412512\pi\)
\(822\) 0 0
\(823\) 3.83395 6.64060i 0.133643 0.231477i −0.791435 0.611253i \(-0.790666\pi\)
0.925078 + 0.379776i \(0.123999\pi\)
\(824\) 0 0
\(825\) −14.4683 −0.503720
\(826\) 0 0
\(827\) 10.6903 + 6.17207i 0.371740 + 0.214624i 0.674218 0.738532i \(-0.264481\pi\)
−0.302478 + 0.953156i \(0.597814\pi\)
\(828\) 0 0
\(829\) 27.6662i 0.960887i 0.877026 + 0.480444i \(0.159524\pi\)
−0.877026 + 0.480444i \(0.840476\pi\)
\(830\) 0 0
\(831\) 4.47546 + 7.75172i 0.155252 + 0.268904i
\(832\) 0 0
\(833\) −2.05441 3.55834i −0.0711810 0.123289i
\(834\) 0 0
\(835\) 18.2095i 0.630164i
\(836\) 0 0
\(837\) 14.2322i 0.491936i
\(838\) 0 0
\(839\) 7.03223 + 12.1802i 0.242779 + 0.420506i 0.961505 0.274788i \(-0.0886076\pi\)
−0.718726 + 0.695294i \(0.755274\pi\)
\(840\) 0 0
\(841\) 28.7271 + 49.7568i 0.990590 + 1.71575i
\(842\) 0 0
\(843\) 4.00446i 0.137921i
\(844\) 0 0
\(845\) −34.8432 20.1167i −1.19864 0.692037i
\(846\) 0 0
\(847\) −9.81915 −0.337390
\(848\) 0 0
\(849\) 0.570510 0.988153i 0.0195799 0.0339133i
\(850\) 0 0
\(851\) −54.3542 31.3814i −1.86324 1.07574i
\(852\) 0 0
\(853\) 7.50368 4.33225i 0.256921 0.148333i −0.366008 0.930612i \(-0.619276\pi\)
0.622929 + 0.782278i \(0.285942\pi\)
\(854\) 0 0
\(855\) 44.5877 + 0.452889i 1.52487 + 0.0154885i
\(856\) 0 0
\(857\) −6.95703 12.0499i −0.237648 0.411618i 0.722391 0.691485i \(-0.243043\pi\)
−0.960039 + 0.279867i \(0.909710\pi\)
\(858\) 0 0
\(859\) −12.5401 7.24001i −0.427861 0.247026i 0.270574 0.962699i \(-0.412787\pi\)
−0.698435 + 0.715673i \(0.746120\pi\)
\(860\) 0 0
\(861\) 14.6643 + 8.46642i 0.499757 + 0.288535i
\(862\) 0 0
\(863\) 26.5065 0.902291 0.451145 0.892450i \(-0.351016\pi\)
0.451145 + 0.892450i \(0.351016\pi\)
\(864\) 0 0
\(865\) 31.9273 55.2997i 1.08556 1.88025i
\(866\) 0 0
\(867\) 4.33228i 0.147132i
\(868\) 0 0
\(869\) 9.05732 5.22925i 0.307249 0.177390i
\(870\) 0 0
\(871\) −6.95785 12.0513i −0.235758 0.408344i
\(872\) 0 0
\(873\) −35.6726 −1.20733
\(874\) 0 0
\(875\) 42.3292i 1.43099i
\(876\) 0 0
\(877\) 15.4900 8.94317i 0.523061 0.301989i −0.215125 0.976586i \(-0.569016\pi\)
0.738186 + 0.674597i \(0.235683\pi\)
\(878\) 0 0
\(879\) −5.12971 8.88491i −0.173021 0.299681i
\(880\) 0 0
\(881\) −16.0705 −0.541427 −0.270714 0.962660i \(-0.587260\pi\)
−0.270714 + 0.962660i \(0.587260\pi\)
\(882\) 0 0
\(883\) −14.4924 8.36718i −0.487708 0.281578i 0.235915 0.971774i \(-0.424191\pi\)
−0.723623 + 0.690196i \(0.757525\pi\)
\(884\) 0 0
\(885\) 26.3889i 0.887052i
\(886\) 0 0
\(887\) −23.8735 + 41.3502i −0.801595 + 1.38840i 0.116971 + 0.993135i \(0.462681\pi\)
−0.918566 + 0.395268i \(0.870652\pi\)
\(888\) 0 0
\(889\) −6.94572 + 12.0303i −0.232952 + 0.403485i
\(890\) 0 0
\(891\) −13.9916 + 8.07807i −0.468737 + 0.270625i
\(892\) 0 0
\(893\) 8.67180 + 5.12480i 0.290191 + 0.171495i
\(894\) 0 0
\(895\) 33.0346 + 57.2176i 1.10422 + 1.91257i
\(896\) 0 0
\(897\) 4.03938 6.99641i 0.134871 0.233603i
\(898\) 0 0
\(899\) 35.3187 + 20.3913i 1.17794 + 0.680086i
\(900\) 0 0
\(901\) 4.91995i 0.163907i
\(902\) 0 0
\(903\) −1.85271 + 3.20899i −0.0616543 + 0.106788i
\(904\) 0 0
\(905\) −1.58326 −0.0526293
\(906\) 0 0
\(907\) 21.5036 12.4151i 0.714016 0.412237i −0.0985303 0.995134i \(-0.531414\pi\)
0.812546 + 0.582897i \(0.198081\pi\)
\(908\) 0 0
\(909\) 4.21419 2.43306i 0.139776 0.0806997i
\(910\) 0 0
\(911\) 12.9835 0.430161 0.215081 0.976596i \(-0.430999\pi\)
0.215081 + 0.976596i \(0.430999\pi\)
\(912\) 0 0
\(913\) 21.6363 0.716058
\(914\) 0 0
\(915\) 1.04478 0.603204i 0.0345394 0.0199413i
\(916\) 0 0
\(917\) −29.1793 + 16.8467i −0.963585 + 0.556326i
\(918\) 0 0
\(919\) −49.4768 −1.63209 −0.816045 0.577988i \(-0.803838\pi\)
−0.816045 + 0.577988i \(0.803838\pi\)
\(920\) 0 0
\(921\) 8.43535 14.6105i 0.277954 0.481431i
\(922\) 0 0
\(923\) 11.4594i 0.377190i
\(924\) 0 0
\(925\) −58.5629 33.8113i −1.92554 1.11171i
\(926\) 0 0
\(927\) 13.2914 23.0214i 0.436548 0.756123i
\(928\) 0 0
\(929\) −9.58486 16.6015i −0.314469 0.544676i 0.664856 0.746972i \(-0.268493\pi\)
−0.979324 + 0.202296i \(0.935160\pi\)
\(930\) 0 0
\(931\) −0.0592493 + 5.83321i −0.00194182 + 0.191176i
\(932\) 0 0
\(933\) −9.02894 + 5.21286i −0.295594 + 0.170661i
\(934\) 0 0
\(935\) 15.4031 26.6789i 0.503735 0.872494i
\(936\) 0 0
\(937\) −21.9199 + 37.9663i −0.716091 + 1.24031i 0.246447 + 0.969156i \(0.420737\pi\)
−0.962537 + 0.271149i \(0.912596\pi\)
\(938\) 0 0
\(939\) 17.0698i 0.557050i
\(940\) 0 0
\(941\) −25.4920 14.7178i −0.831017 0.479788i 0.0231840 0.999731i \(-0.492620\pi\)
−0.854201 + 0.519944i \(0.825953\pi\)
\(942\) 0 0
\(943\) −111.409 −3.62796
\(944\) 0 0
\(945\) −14.7749 25.5909i −0.480627 0.832471i
\(946\) 0 0
\(947\) −1.17630 + 0.679140i −0.0382248 + 0.0220691i −0.518991 0.854780i \(-0.673692\pi\)
0.480766 + 0.876849i \(0.340359\pi\)
\(948\) 0 0
\(949\) 5.70513i 0.185196i
\(950\) 0 0
\(951\) 3.65327 0.118466
\(952\) 0 0
\(953\) 19.0604 + 33.0136i 0.617428 + 1.06942i 0.989953 + 0.141395i \(0.0451587\pi\)
−0.372525 + 0.928022i \(0.621508\pi\)
\(954\) 0 0
\(955\) −82.4567 + 47.6064i −2.66824 + 1.54051i
\(956\) 0 0
\(957\) 13.9434i 0.450725i
\(958\) 0 0
\(959\) −1.46607 + 2.53930i −0.0473418 + 0.0819984i
\(960\) 0 0
\(961\) −11.7619 −0.379417
\(962\) 0 0
\(963\) −30.6890 17.7183i −0.988938 0.570964i
\(964\) 0 0
\(965\) −31.5413 18.2104i −1.01535 0.586213i
\(966\) 0 0
\(967\) 9.12156 + 15.7990i 0.293330 + 0.508062i 0.974595 0.223975i \(-0.0719033\pi\)
−0.681265 + 0.732037i \(0.738570\pi\)
\(968\) 0 0
\(969\) 3.89451 6.59000i 0.125110 0.211701i
\(970\) 0 0
\(971\) 33.6284 19.4153i 1.07919 0.623068i 0.148509 0.988911i \(-0.452553\pi\)
0.930677 + 0.365843i \(0.119219\pi\)
\(972\) 0 0
\(973\) −12.3913 7.15411i −0.397246 0.229350i
\(974\) 0 0
\(975\) 4.35215 7.53815i 0.139380 0.241414i
\(976\) 0 0
\(977\) −4.02782 −0.128861 −0.0644306 0.997922i \(-0.520523\pi\)
−0.0644306 + 0.997922i \(0.520523\pi\)
\(978\) 0 0
\(979\) 21.9373 + 12.6655i 0.701120 + 0.404792i
\(980\) 0 0
\(981\) 20.2065i 0.645143i
\(982\) 0 0
\(983\) 21.2147 + 36.7449i 0.676643 + 1.17198i 0.975986 + 0.217835i \(0.0698995\pi\)
−0.299342 + 0.954146i \(0.596767\pi\)
\(984\) 0 0
\(985\) 11.1977 + 19.3951i 0.356790 + 0.617978i
\(986\) 0 0
\(987\) 3.14516i 0.100112i
\(988\) 0 0
\(989\) 24.3796i 0.775225i
\(990\) 0 0
\(991\) 29.4383 + 50.9886i 0.935138 + 1.61971i 0.774387 + 0.632712i \(0.218058\pi\)
0.160751 + 0.986995i \(0.448608\pi\)
\(992\) 0 0
\(993\) 1.77950 + 3.08218i 0.0564707 + 0.0978101i
\(994\) 0 0
\(995\) 64.2629i 2.03727i
\(996\) 0 0
\(997\) −42.1664 24.3448i −1.33542 0.771007i −0.349298 0.937012i \(-0.613580\pi\)
−0.986125 + 0.166005i \(0.946913\pi\)
\(998\) 0 0
\(999\) 22.7426 0.719544
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1216.2.t.d.1185.7 yes 24
4.3 odd 2 inner 1216.2.t.d.1185.5 yes 24
8.3 odd 2 1216.2.t.e.1185.7 yes 24
8.5 even 2 1216.2.t.e.1185.5 yes 24
19.11 even 3 1216.2.t.e.353.5 yes 24
76.11 odd 6 1216.2.t.e.353.7 yes 24
152.11 odd 6 inner 1216.2.t.d.353.5 24
152.125 even 6 inner 1216.2.t.d.353.7 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.t.d.353.5 24 152.11 odd 6 inner
1216.2.t.d.353.7 yes 24 152.125 even 6 inner
1216.2.t.d.1185.5 yes 24 4.3 odd 2 inner
1216.2.t.d.1185.7 yes 24 1.1 even 1 trivial
1216.2.t.e.353.5 yes 24 19.11 even 3
1216.2.t.e.353.7 yes 24 76.11 odd 6
1216.2.t.e.1185.5 yes 24 8.5 even 2
1216.2.t.e.1185.7 yes 24 8.3 odd 2