Properties

Label 672.2.j.d.239.1
Level $672$
Weight $2$
Character 672.239
Analytic conductor $5.366$
Analytic rank $0$
Dimension $12$
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] = [672,2,Mod(239,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("672.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.1
Root \(-1.19877 - 0.750295i\) of defining polynomial
Character \(\chi\) \(=\) 672.239
Dual form 672.2.j.d.239.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.67298 - 0.448478i) q^{3} -0.896956 q^{5} +1.00000i q^{7} +(2.59774 + 1.50059i) q^{9} +O(q^{10})\) \(q+(-1.67298 - 0.448478i) q^{3} -0.896956 q^{5} +1.00000i q^{7} +(2.59774 + 1.50059i) q^{9} -1.84951i q^{13} +(1.50059 + 0.402265i) q^{15} -4.12397i q^{17} +0.654037 q^{19} +(0.448478 - 1.67298i) q^{21} -7.12515 q^{23} -4.19547 q^{25} +(-3.67298 - 3.67549i) q^{27} -7.79627 q^{29} -0.896956i q^{35} +10.6919i q^{37} +(-0.829463 + 3.09419i) q^{39} -10.1263i q^{41} +1.19547 q^{43} +(-2.33005 - 1.34596i) q^{45} -9.59019 q^{47} -1.00000 q^{49} +(-1.84951 + 6.89932i) q^{51} -8.24793 q^{53} +(-1.09419 - 0.293321i) q^{57} -6.89932i q^{59} -4.54143i q^{61} +(-1.50059 + 2.59774i) q^{63} +1.65893i q^{65} +5.49646 q^{67} +(11.9202 + 3.19547i) q^{69} +4.12397 q^{71} -6.00000 q^{73} +(7.01894 + 1.88158i) q^{75} -13.3839i q^{79} +(4.49646 + 7.79627i) q^{81} -13.3533i q^{83} +3.69901i q^{85} +(13.0430 + 3.49646i) q^{87} +13.7142i q^{89} +1.84951 q^{91} -0.586642 q^{95} -8.99291 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{9} + 48 q^{19} + 4 q^{25} - 24 q^{27} - 40 q^{43} - 12 q^{49} - 8 q^{51} + 40 q^{57} + 40 q^{67} - 72 q^{73} + 24 q^{75} + 28 q^{81} + 8 q^{91} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.67298 0.448478i −0.965896 0.258929i
\(4\) 0 0
\(5\) −0.896956 −0.401131 −0.200565 0.979680i \(-0.564278\pi\)
−0.200565 + 0.979680i \(0.564278\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.59774 + 1.50059i 0.865912 + 0.500197i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.84951i 0.512961i −0.966549 0.256480i \(-0.917437\pi\)
0.966549 0.256480i \(-0.0825629\pi\)
\(14\) 0 0
\(15\) 1.50059 + 0.402265i 0.387451 + 0.103864i
\(16\) 0 0
\(17\) 4.12397i 1.00021i −0.865965 0.500104i \(-0.833295\pi\)
0.865965 0.500104i \(-0.166705\pi\)
\(18\) 0 0
\(19\) 0.654037 0.150046 0.0750232 0.997182i \(-0.476097\pi\)
0.0750232 + 0.997182i \(0.476097\pi\)
\(20\) 0 0
\(21\) 0.448478 1.67298i 0.0978659 0.365075i
\(22\) 0 0
\(23\) −7.12515 −1.48570 −0.742848 0.669460i \(-0.766525\pi\)
−0.742848 + 0.669460i \(0.766525\pi\)
\(24\) 0 0
\(25\) −4.19547 −0.839094
\(26\) 0 0
\(27\) −3.67298 3.67549i −0.706866 0.707348i
\(28\) 0 0
\(29\) −7.79627 −1.44773 −0.723866 0.689941i \(-0.757637\pi\)
−0.723866 + 0.689941i \(0.757637\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.896956i 0.151613i
\(36\) 0 0
\(37\) 10.6919i 1.75774i 0.477060 + 0.878871i \(0.341703\pi\)
−0.477060 + 0.878871i \(0.658297\pi\)
\(38\) 0 0
\(39\) −0.829463 + 3.09419i −0.132820 + 0.495467i
\(40\) 0 0
\(41\) 10.1263i 1.58147i −0.612161 0.790733i \(-0.709699\pi\)
0.612161 0.790733i \(-0.290301\pi\)
\(42\) 0 0
\(43\) 1.19547 0.182308 0.0911538 0.995837i \(-0.470945\pi\)
0.0911538 + 0.995837i \(0.470945\pi\)
\(44\) 0 0
\(45\) −2.33005 1.34596i −0.347344 0.200644i
\(46\) 0 0
\(47\) −9.59019 −1.39887 −0.699436 0.714695i \(-0.746565\pi\)
−0.699436 + 0.714695i \(0.746565\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −1.84951 + 6.89932i −0.258983 + 0.966098i
\(52\) 0 0
\(53\) −8.24793 −1.13294 −0.566470 0.824082i \(-0.691691\pi\)
−0.566470 + 0.824082i \(0.691691\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.09419 0.293321i −0.144929 0.0388513i
\(58\) 0 0
\(59\) 6.89932i 0.898215i −0.893478 0.449107i \(-0.851742\pi\)
0.893478 0.449107i \(-0.148258\pi\)
\(60\) 0 0
\(61\) 4.54143i 0.581471i −0.956803 0.290735i \(-0.906100\pi\)
0.956803 0.290735i \(-0.0939000\pi\)
\(62\) 0 0
\(63\) −1.50059 + 2.59774i −0.189057 + 0.327284i
\(64\) 0 0
\(65\) 1.65893i 0.205764i
\(66\) 0 0
\(67\) 5.49646 0.671499 0.335750 0.941951i \(-0.391010\pi\)
0.335750 + 0.941951i \(0.391010\pi\)
\(68\) 0 0
\(69\) 11.9202 + 3.19547i 1.43503 + 0.384689i
\(70\) 0 0
\(71\) 4.12397 0.489425 0.244712 0.969596i \(-0.421306\pi\)
0.244712 + 0.969596i \(0.421306\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 7.01894 + 1.88158i 0.810478 + 0.217266i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.3839i 1.50580i −0.658134 0.752901i \(-0.728654\pi\)
0.658134 0.752901i \(-0.271346\pi\)
\(80\) 0 0
\(81\) 4.49646 + 7.79627i 0.499606 + 0.866253i
\(82\) 0 0
\(83\) 13.3533i 1.46572i −0.680380 0.732860i \(-0.738185\pi\)
0.680380 0.732860i \(-0.261815\pi\)
\(84\) 0 0
\(85\) 3.69901i 0.401214i
\(86\) 0 0
\(87\) 13.0430 + 3.49646i 1.39836 + 0.374859i
\(88\) 0 0
\(89\) 13.7142i 1.45370i 0.686798 + 0.726849i \(0.259016\pi\)
−0.686798 + 0.726849i \(0.740984\pi\)
\(90\) 0 0
\(91\) 1.84951 0.193881
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.586642 −0.0601882
\(96\) 0 0
\(97\) −8.99291 −0.913092 −0.456546 0.889700i \(-0.650914\pi\)
−0.456546 + 0.889700i \(0.650914\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.35097 0.731449 0.365725 0.930723i \(-0.380821\pi\)
0.365725 + 0.930723i \(0.380821\pi\)
\(102\) 0 0
\(103\) 13.3839i 1.31875i 0.751814 + 0.659375i \(0.229179\pi\)
−0.751814 + 0.659375i \(0.770821\pi\)
\(104\) 0 0
\(105\) −0.402265 + 1.50059i −0.0392570 + 0.146443i
\(106\) 0 0
\(107\) 1.79391i 0.173424i 0.996233 + 0.0867120i \(0.0276360\pi\)
−0.996233 + 0.0867120i \(0.972364\pi\)
\(108\) 0 0
\(109\) 5.30807i 0.508421i −0.967149 0.254211i \(-0.918184\pi\)
0.967149 0.254211i \(-0.0818156\pi\)
\(110\) 0 0
\(111\) 4.79509 17.8874i 0.455130 1.69780i
\(112\) 0 0
\(113\) 3.00118i 0.282327i −0.989986 0.141164i \(-0.954916\pi\)
0.989986 0.141164i \(-0.0450844\pi\)
\(114\) 0 0
\(115\) 6.39094 0.595958
\(116\) 0 0
\(117\) 2.77535 4.80453i 0.256581 0.444179i
\(118\) 0 0
\(119\) 4.12397 0.378043
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −4.54143 + 16.9412i −0.409487 + 1.52753i
\(124\) 0 0
\(125\) 8.24793 0.737717
\(126\) 0 0
\(127\) 14.1884i 1.25902i −0.776994 0.629508i \(-0.783257\pi\)
0.776994 0.629508i \(-0.216743\pi\)
\(128\) 0 0
\(129\) −2.00000 0.536142i −0.176090 0.0472047i
\(130\) 0 0
\(131\) 16.9412i 1.48016i 0.672521 + 0.740078i \(0.265211\pi\)
−0.672521 + 0.740078i \(0.734789\pi\)
\(132\) 0 0
\(133\) 0.654037i 0.0567122i
\(134\) 0 0
\(135\) 3.29450 + 3.29675i 0.283546 + 0.283739i
\(136\) 0 0
\(137\) 11.8358i 1.01120i 0.862769 + 0.505598i \(0.168728\pi\)
−0.862769 + 0.505598i \(0.831272\pi\)
\(138\) 0 0
\(139\) −11.0450 −0.936823 −0.468411 0.883510i \(-0.655173\pi\)
−0.468411 + 0.883510i \(0.655173\pi\)
\(140\) 0 0
\(141\) 16.0442 + 4.30099i 1.35117 + 0.362208i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.99291 0.580730
\(146\) 0 0
\(147\) 1.67298 + 0.448478i 0.137985 + 0.0369898i
\(148\) 0 0
\(149\) 4.66011 0.381771 0.190885 0.981612i \(-0.438864\pi\)
0.190885 + 0.981612i \(0.438864\pi\)
\(150\) 0 0
\(151\) 12.5793i 1.02369i 0.859078 + 0.511845i \(0.171038\pi\)
−0.859078 + 0.511845i \(0.828962\pi\)
\(152\) 0 0
\(153\) 6.18838 10.7130i 0.500301 0.866092i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.5343i 1.55901i −0.626396 0.779505i \(-0.715471\pi\)
0.626396 0.779505i \(-0.284529\pi\)
\(158\) 0 0
\(159\) 13.7986 + 3.69901i 1.09430 + 0.293351i
\(160\) 0 0
\(161\) 7.12515i 0.561540i
\(162\) 0 0
\(163\) −9.19547 −0.720245 −0.360122 0.932905i \(-0.617265\pi\)
−0.360122 + 0.932905i \(0.617265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.8358 0.915878 0.457939 0.888984i \(-0.348588\pi\)
0.457939 + 0.888984i \(0.348588\pi\)
\(168\) 0 0
\(169\) 9.57932 0.736871
\(170\) 0 0
\(171\) 1.69901 + 0.981441i 0.129927 + 0.0750527i
\(172\) 0 0
\(173\) −19.3557 −1.47159 −0.735793 0.677206i \(-0.763190\pi\)
−0.735793 + 0.677206i \(0.763190\pi\)
\(174\) 0 0
\(175\) 4.19547i 0.317148i
\(176\) 0 0
\(177\) −3.09419 + 11.5424i −0.232574 + 0.867582i
\(178\) 0 0
\(179\) 14.7019i 1.09888i 0.835535 + 0.549438i \(0.185158\pi\)
−0.835535 + 0.549438i \(0.814842\pi\)
\(180\) 0 0
\(181\) 4.54143i 0.337562i −0.985654 0.168781i \(-0.946017\pi\)
0.985654 0.168781i \(-0.0539831\pi\)
\(182\) 0 0
\(183\) −2.03673 + 7.59774i −0.150560 + 0.561641i
\(184\) 0 0
\(185\) 9.59019i 0.705084i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.67549 3.67298i 0.267352 0.267170i
\(190\) 0 0
\(191\) −1.70943 −0.123690 −0.0618449 0.998086i \(-0.519698\pi\)
−0.0618449 + 0.998086i \(0.519698\pi\)
\(192\) 0 0
\(193\) 16.1884 1.16527 0.582633 0.812736i \(-0.302023\pi\)
0.582633 + 0.812736i \(0.302023\pi\)
\(194\) 0 0
\(195\) 0.743992 2.77535i 0.0532783 0.198747i
\(196\) 0 0
\(197\) 9.59019 0.683272 0.341636 0.939832i \(-0.389019\pi\)
0.341636 + 0.939832i \(0.389019\pi\)
\(198\) 0 0
\(199\) 14.3909i 1.02015i 0.860131 + 0.510073i \(0.170382\pi\)
−0.860131 + 0.510073i \(0.829618\pi\)
\(200\) 0 0
\(201\) −9.19547 2.46504i −0.648598 0.173870i
\(202\) 0 0
\(203\) 7.79627i 0.547191i
\(204\) 0 0
\(205\) 9.08287i 0.634375i
\(206\) 0 0
\(207\) −18.5092 10.6919i −1.28648 0.743140i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −15.8874 −1.09373 −0.546867 0.837220i \(-0.684180\pi\)
−0.546867 + 0.837220i \(0.684180\pi\)
\(212\) 0 0
\(213\) −6.89932 1.84951i −0.472733 0.126726i
\(214\) 0 0
\(215\) −1.07228 −0.0731292
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.0379 + 2.69087i 0.678298 + 0.181832i
\(220\) 0 0
\(221\) −7.62730 −0.513068
\(222\) 0 0
\(223\) 9.00709i 0.603159i 0.953441 + 0.301580i \(0.0975139\pi\)
−0.953441 + 0.301580i \(0.902486\pi\)
\(224\) 0 0
\(225\) −10.8987 6.29568i −0.726581 0.419712i
\(226\) 0 0
\(227\) 22.9435i 1.52282i −0.648274 0.761408i \(-0.724509\pi\)
0.648274 0.761408i \(-0.275491\pi\)
\(228\) 0 0
\(229\) 10.9324i 0.722432i 0.932482 + 0.361216i \(0.117638\pi\)
−0.932482 + 0.361216i \(0.882362\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.24793i 0.540340i −0.962813 0.270170i \(-0.912920\pi\)
0.962813 0.270170i \(-0.0870799\pi\)
\(234\) 0 0
\(235\) 8.60197 0.561131
\(236\) 0 0
\(237\) −6.00236 + 22.3909i −0.389895 + 1.45445i
\(238\) 0 0
\(239\) −4.87958 −0.315634 −0.157817 0.987468i \(-0.550446\pi\)
−0.157817 + 0.987468i \(0.550446\pi\)
\(240\) 0 0
\(241\) −4.99291 −0.321622 −0.160811 0.986985i \(-0.551411\pi\)
−0.160811 + 0.986985i \(0.551411\pi\)
\(242\) 0 0
\(243\) −4.02603 15.0596i −0.258270 0.966073i
\(244\) 0 0
\(245\) 0.896956 0.0573044
\(246\) 0 0
\(247\) 1.20965i 0.0769679i
\(248\) 0 0
\(249\) −5.98868 + 22.3399i −0.379517 + 1.41573i
\(250\) 0 0
\(251\) 11.1078i 0.701116i 0.936541 + 0.350558i \(0.114008\pi\)
−0.936541 + 0.350558i \(0.885992\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 1.65893 6.18838i 0.103886 0.387532i
\(256\) 0 0
\(257\) 11.4686i 0.715390i 0.933838 + 0.357695i \(0.116437\pi\)
−0.933838 + 0.357695i \(0.883563\pi\)
\(258\) 0 0
\(259\) −10.6919 −0.664364
\(260\) 0 0
\(261\) −20.2527 11.6990i −1.25361 0.724151i
\(262\) 0 0
\(263\) −12.3719 −0.762884 −0.381442 0.924393i \(-0.624572\pi\)
−0.381442 + 0.924393i \(0.624572\pi\)
\(264\) 0 0
\(265\) 7.39803 0.454457
\(266\) 0 0
\(267\) 6.15049 22.9435i 0.376404 1.40412i
\(268\) 0 0
\(269\) 24.7374 1.50827 0.754134 0.656721i \(-0.228057\pi\)
0.754134 + 0.656721i \(0.228057\pi\)
\(270\) 0 0
\(271\) 20.7819i 1.26241i −0.775616 0.631205i \(-0.782561\pi\)
0.775616 0.631205i \(-0.217439\pi\)
\(272\) 0 0
\(273\) −3.09419 0.829463i −0.187269 0.0502014i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.60906i 0.0966790i 0.998831 + 0.0483395i \(0.0153929\pi\)
−0.998831 + 0.0483395i \(0.984607\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.66011i 0.277999i −0.990292 0.138999i \(-0.955611\pi\)
0.990292 0.138999i \(-0.0443886\pi\)
\(282\) 0 0
\(283\) −13.7369 −0.816574 −0.408287 0.912854i \(-0.633874\pi\)
−0.408287 + 0.912854i \(0.633874\pi\)
\(284\) 0 0
\(285\) 0.981441 + 0.263096i 0.0581356 + 0.0155845i
\(286\) 0 0
\(287\) 10.1263 0.597738
\(288\) 0 0
\(289\) −0.00708757 −0.000416916
\(290\) 0 0
\(291\) 15.0450 + 4.03312i 0.881952 + 0.236426i
\(292\) 0 0
\(293\) −19.3557 −1.13077 −0.565386 0.824826i \(-0.691273\pi\)
−0.565386 + 0.824826i \(0.691273\pi\)
\(294\) 0 0
\(295\) 6.18838i 0.360302i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.1780i 0.762104i
\(300\) 0 0
\(301\) 1.19547i 0.0689058i
\(302\) 0 0
\(303\) −12.2980 3.29675i −0.706504 0.189393i
\(304\) 0 0
\(305\) 4.07347i 0.233246i
\(306\) 0 0
\(307\) 30.8198 1.75898 0.879489 0.475920i \(-0.157885\pi\)
0.879489 + 0.475920i \(0.157885\pi\)
\(308\) 0 0
\(309\) 6.00236 22.3909i 0.341462 1.27378i
\(310\) 0 0
\(311\) −6.00236 −0.340363 −0.170181 0.985413i \(-0.554435\pi\)
−0.170181 + 0.985413i \(0.554435\pi\)
\(312\) 0 0
\(313\) −13.7748 −0.778597 −0.389299 0.921112i \(-0.627283\pi\)
−0.389299 + 0.921112i \(0.627283\pi\)
\(314\) 0 0
\(315\) 1.34596 2.33005i 0.0758364 0.131284i
\(316\) 0 0
\(317\) 3.75679 0.211003 0.105501 0.994419i \(-0.466355\pi\)
0.105501 + 0.994419i \(0.466355\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.804530 3.00118i 0.0449045 0.167510i
\(322\) 0 0
\(323\) 2.69722i 0.150078i
\(324\) 0 0
\(325\) 7.75955i 0.430423i
\(326\) 0 0
\(327\) −2.38055 + 8.88031i −0.131645 + 0.491082i
\(328\) 0 0
\(329\) 9.59019i 0.528724i
\(330\) 0 0
\(331\) 6.20256 0.340923 0.170462 0.985364i \(-0.445474\pi\)
0.170462 + 0.985364i \(0.445474\pi\)
\(332\) 0 0
\(333\) −16.0442 + 27.7748i −0.879217 + 1.52205i
\(334\) 0 0
\(335\) −4.93008 −0.269359
\(336\) 0 0
\(337\) −14.5793 −0.794186 −0.397093 0.917778i \(-0.629981\pi\)
−0.397093 + 0.917778i \(0.629981\pi\)
\(338\) 0 0
\(339\) −1.34596 + 5.02092i −0.0731027 + 0.272699i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −10.6919 2.86620i −0.575634 0.154311i
\(346\) 0 0
\(347\) 8.41690i 0.451843i 0.974146 + 0.225921i \(0.0725393\pi\)
−0.974146 + 0.225921i \(0.927461\pi\)
\(348\) 0 0
\(349\) 25.9253i 1.38775i −0.720096 0.693874i \(-0.755902\pi\)
0.720096 0.693874i \(-0.244098\pi\)
\(350\) 0 0
\(351\) −6.79784 + 6.79321i −0.362842 + 0.362594i
\(352\) 0 0
\(353\) 13.7142i 0.729931i −0.931021 0.364965i \(-0.881081\pi\)
0.931021 0.364965i \(-0.118919\pi\)
\(354\) 0 0
\(355\) −3.69901 −0.196323
\(356\) 0 0
\(357\) −6.89932 1.84951i −0.365151 0.0978863i
\(358\) 0 0
\(359\) 26.3055 1.38835 0.694176 0.719805i \(-0.255769\pi\)
0.694176 + 0.719805i \(0.255769\pi\)
\(360\) 0 0
\(361\) −18.5722 −0.977486
\(362\) 0 0
\(363\) −18.4028 4.93326i −0.965896 0.258929i
\(364\) 0 0
\(365\) 5.38173 0.281693
\(366\) 0 0
\(367\) 19.7748i 1.03224i −0.856518 0.516118i \(-0.827377\pi\)
0.856518 0.516118i \(-0.172623\pi\)
\(368\) 0 0
\(369\) 15.1955 26.3055i 0.791045 1.36941i
\(370\) 0 0
\(371\) 8.24793i 0.428211i
\(372\) 0 0
\(373\) 3.77479i 0.195451i −0.995213 0.0977257i \(-0.968843\pi\)
0.995213 0.0977257i \(-0.0311568\pi\)
\(374\) 0 0
\(375\) −13.7986 3.69901i −0.712558 0.191016i
\(376\) 0 0
\(377\) 14.4193i 0.742630i
\(378\) 0 0
\(379\) 8.59350 0.441418 0.220709 0.975340i \(-0.429163\pi\)
0.220709 + 0.975340i \(0.429163\pi\)
\(380\) 0 0
\(381\) −6.36318 + 23.7369i −0.325995 + 1.21608i
\(382\) 0 0
\(383\) −17.8381 −0.911485 −0.455743 0.890112i \(-0.650626\pi\)
−0.455743 + 0.890112i \(0.650626\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.10552 + 1.79391i 0.157862 + 0.0911896i
\(388\) 0 0
\(389\) −7.79627 −0.395287 −0.197643 0.980274i \(-0.563329\pi\)
−0.197643 + 0.980274i \(0.563329\pi\)
\(390\) 0 0
\(391\) 29.3839i 1.48601i
\(392\) 0 0
\(393\) 7.59774 28.3422i 0.383255 1.42968i
\(394\) 0 0
\(395\) 12.0047i 0.604023i
\(396\) 0 0
\(397\) 22.2263i 1.11550i 0.830007 + 0.557752i \(0.188336\pi\)
−0.830007 + 0.557752i \(0.811664\pi\)
\(398\) 0 0
\(399\) 0.293321 1.09419i 0.0146844 0.0547781i
\(400\) 0 0
\(401\) 5.24675i 0.262010i −0.991382 0.131005i \(-0.958180\pi\)
0.991382 0.131005i \(-0.0418204\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4.03312 6.99291i −0.200407 0.347481i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 28.1657 1.39271 0.696353 0.717699i \(-0.254805\pi\)
0.696353 + 0.717699i \(0.254805\pi\)
\(410\) 0 0
\(411\) 5.30807 19.8010i 0.261828 0.976711i
\(412\) 0 0
\(413\) 6.89932 0.339493
\(414\) 0 0
\(415\) 11.9774i 0.587945i
\(416\) 0 0
\(417\) 18.4780 + 4.95343i 0.904874 + 0.242570i
\(418\) 0 0
\(419\) 2.23921i 0.109393i −0.998503 0.0546963i \(-0.982581\pi\)
0.998503 0.0546963i \(-0.0174191\pi\)
\(420\) 0 0
\(421\) 6.99291i 0.340814i 0.985374 + 0.170407i \(0.0545082\pi\)
−0.985374 + 0.170407i \(0.945492\pi\)
\(422\) 0 0
\(423\) −24.9128 14.3909i −1.21130 0.699711i
\(424\) 0 0
\(425\) 17.3020i 0.839269i
\(426\) 0 0
\(427\) 4.54143 0.219775
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.80966 0.472515 0.236257 0.971691i \(-0.424079\pi\)
0.236257 + 0.971691i \(0.424079\pi\)
\(432\) 0 0
\(433\) −23.3839 −1.12376 −0.561878 0.827220i \(-0.689921\pi\)
−0.561878 + 0.827220i \(0.689921\pi\)
\(434\) 0 0
\(435\) −11.6990 3.13617i −0.560925 0.150368i
\(436\) 0 0
\(437\) −4.66011 −0.222923
\(438\) 0 0
\(439\) 20.3768i 0.972530i −0.873811 0.486265i \(-0.838359\pi\)
0.873811 0.486265i \(-0.161641\pi\)
\(440\) 0 0
\(441\) −2.59774 1.50059i −0.123702 0.0714567i
\(442\) 0 0
\(443\) 36.5668i 1.73734i −0.495389 0.868671i \(-0.664974\pi\)
0.495389 0.868671i \(-0.335026\pi\)
\(444\) 0 0
\(445\) 12.3010i 0.583123i
\(446\) 0 0
\(447\) −7.79627 2.08995i −0.368751 0.0988515i
\(448\) 0 0
\(449\) 19.1804i 0.905178i 0.891719 + 0.452589i \(0.149499\pi\)
−0.891719 + 0.452589i \(0.850501\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.64155 21.0450i 0.265063 0.988779i
\(454\) 0 0
\(455\) −1.65893 −0.0777717
\(456\) 0 0
\(457\) −20.5935 −0.963323 −0.481662 0.876357i \(-0.659967\pi\)
−0.481662 + 0.876357i \(0.659967\pi\)
\(458\) 0 0
\(459\) −15.1576 + 15.1472i −0.707495 + 0.707013i
\(460\) 0 0
\(461\) 2.69087 0.125326 0.0626631 0.998035i \(-0.480041\pi\)
0.0626631 + 0.998035i \(0.480041\pi\)
\(462\) 0 0
\(463\) 4.78188i 0.222233i −0.993807 0.111116i \(-0.964557\pi\)
0.993807 0.111116i \(-0.0354427\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.8341i 1.10291i 0.834204 + 0.551456i \(0.185927\pi\)
−0.834204 + 0.551456i \(0.814073\pi\)
\(468\) 0 0
\(469\) 5.49646i 0.253803i
\(470\) 0 0
\(471\) −8.76072 + 32.6806i −0.403673 + 1.50584i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.74399 −0.125903
\(476\) 0 0
\(477\) −21.4259 12.3768i −0.981026 0.566693i
\(478\) 0 0
\(479\) 15.5925 0.712442 0.356221 0.934402i \(-0.384065\pi\)
0.356221 + 0.934402i \(0.384065\pi\)
\(480\) 0 0
\(481\) 19.7748 0.901653
\(482\) 0 0
\(483\) −3.19547 + 11.9202i −0.145399 + 0.542390i
\(484\) 0 0
\(485\) 8.06624 0.366269
\(486\) 0 0
\(487\) 16.2026i 0.734208i 0.930180 + 0.367104i \(0.119651\pi\)
−0.930180 + 0.367104i \(0.880349\pi\)
\(488\) 0 0
\(489\) 15.3839 + 4.12397i 0.695682 + 0.186492i
\(490\) 0 0
\(491\) 6.62299i 0.298891i −0.988770 0.149446i \(-0.952251\pi\)
0.988770 0.149446i \(-0.0477489\pi\)
\(492\) 0 0
\(493\) 32.1516i 1.44803i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.12397i 0.184985i
\(498\) 0 0
\(499\) −21.2713 −0.952232 −0.476116 0.879383i \(-0.657956\pi\)
−0.476116 + 0.879383i \(0.657956\pi\)
\(500\) 0 0
\(501\) −19.8010 5.30807i −0.884643 0.237147i
\(502\) 0 0
\(503\) −1.07228 −0.0478108 −0.0239054 0.999714i \(-0.507610\pi\)
−0.0239054 + 0.999714i \(0.507610\pi\)
\(504\) 0 0
\(505\) −6.59350 −0.293407
\(506\) 0 0
\(507\) −16.0260 4.29611i −0.711741 0.190797i
\(508\) 0 0
\(509\) −1.78755 −0.0792320 −0.0396160 0.999215i \(-0.512613\pi\)
−0.0396160 + 0.999215i \(0.512613\pi\)
\(510\) 0 0
\(511\) 6.00000i 0.265424i
\(512\) 0 0
\(513\) −2.40226 2.40390i −0.106063 0.106135i
\(514\) 0 0
\(515\) 12.0047i 0.528991i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 32.3817 + 8.68060i 1.42140 + 0.381036i
\(520\) 0 0
\(521\) 8.95304i 0.392240i 0.980580 + 0.196120i \(0.0628342\pi\)
−0.980580 + 0.196120i \(0.937166\pi\)
\(522\) 0 0
\(523\) −1.25601 −0.0549214 −0.0274607 0.999623i \(-0.508742\pi\)
−0.0274607 + 0.999623i \(0.508742\pi\)
\(524\) 0 0
\(525\) −1.88158 + 7.01894i −0.0821187 + 0.306332i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 27.7677 1.20729
\(530\) 0 0
\(531\) 10.3531 17.9226i 0.449284 0.777775i
\(532\) 0 0
\(533\) −18.7287 −0.811231
\(534\) 0 0
\(535\) 1.60906i 0.0695657i
\(536\) 0 0
\(537\) 6.59350 24.5961i 0.284530 1.06140i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 44.3768i 1.90791i −0.299956 0.953953i \(-0.596972\pi\)
0.299956 0.953953i \(-0.403028\pi\)
\(542\) 0 0
\(543\) −2.03673 + 7.59774i −0.0874046 + 0.326050i
\(544\) 0 0
\(545\) 4.76111i 0.203943i
\(546\) 0 0
\(547\) 6.80453 0.290941 0.145470 0.989363i \(-0.453530\pi\)
0.145470 + 0.989363i \(0.453530\pi\)
\(548\) 0 0
\(549\) 6.81483 11.7974i 0.290850 0.503503i
\(550\) 0 0
\(551\) −5.09905 −0.217227
\(552\) 0 0
\(553\) 13.3839 0.569139
\(554\) 0 0
\(555\) −4.30099 + 16.0442i −0.182567 + 0.681039i
\(556\) 0 0
\(557\) 46.6087 1.97487 0.987436 0.158017i \(-0.0505101\pi\)
0.987436 + 0.158017i \(0.0505101\pi\)
\(558\) 0 0
\(559\) 2.21103i 0.0935166i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.89932i 0.290772i 0.989375 + 0.145386i \(0.0464423\pi\)
−0.989375 + 0.145386i \(0.953558\pi\)
\(564\) 0 0
\(565\) 2.69193i 0.113250i
\(566\) 0 0
\(567\) −7.79627 + 4.49646i −0.327413 + 0.188833i
\(568\) 0 0
\(569\) 6.75798i 0.283309i −0.989916 0.141655i \(-0.954758\pi\)
0.989916 0.141655i \(-0.0452422\pi\)
\(570\) 0 0
\(571\) −17.1955 −0.719608 −0.359804 0.933028i \(-0.617156\pi\)
−0.359804 + 0.933028i \(0.617156\pi\)
\(572\) 0 0
\(573\) 2.85984 + 0.766640i 0.119471 + 0.0320268i
\(574\) 0 0
\(575\) 29.8933 1.24664
\(576\) 0 0
\(577\) −27.1586 −1.13063 −0.565315 0.824875i \(-0.691245\pi\)
−0.565315 + 0.824875i \(0.691245\pi\)
\(578\) 0 0
\(579\) −27.0829 7.26013i −1.12553 0.301721i
\(580\) 0 0
\(581\) 13.3533 0.553990
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.48937 + 4.30945i −0.102923 + 0.178174i
\(586\) 0 0
\(587\) 26.5313i 1.09507i 0.836784 + 0.547533i \(0.184433\pi\)
−0.836784 + 0.547533i \(0.815567\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −16.0442 4.30099i −0.659970 0.176919i
\(592\) 0 0
\(593\) 27.9644i 1.14836i 0.818728 + 0.574181i \(0.194679\pi\)
−0.818728 + 0.574181i \(0.805321\pi\)
\(594\) 0 0
\(595\) −3.69901 −0.151645
\(596\) 0 0
\(597\) 6.45402 24.0758i 0.264145 0.985356i
\(598\) 0 0
\(599\) 35.0391 1.43166 0.715829 0.698275i \(-0.246049\pi\)
0.715829 + 0.698275i \(0.246049\pi\)
\(600\) 0 0
\(601\) 6.37677 0.260114 0.130057 0.991507i \(-0.458484\pi\)
0.130057 + 0.991507i \(0.458484\pi\)
\(602\) 0 0
\(603\) 14.2783 + 8.24793i 0.581459 + 0.335882i
\(604\) 0 0
\(605\) −9.86651 −0.401131
\(606\) 0 0
\(607\) 8.60197i 0.349143i −0.984644 0.174572i \(-0.944146\pi\)
0.984644 0.174572i \(-0.0558541\pi\)
\(608\) 0 0
\(609\) −3.49646 + 13.0430i −0.141684 + 0.528530i
\(610\) 0 0
\(611\) 17.7371i 0.717567i
\(612\) 0 0
\(613\) 19.2939i 0.779273i 0.920969 + 0.389637i \(0.127399\pi\)
−0.920969 + 0.389637i \(0.872601\pi\)
\(614\) 0 0
\(615\) 4.07347 15.1955i 0.164258 0.612741i
\(616\) 0 0
\(617\) 10.0758i 0.405638i 0.979216 + 0.202819i \(0.0650102\pi\)
−0.979216 + 0.202819i \(0.934990\pi\)
\(618\) 0 0
\(619\) 16.7298 0.672428 0.336214 0.941786i \(-0.390853\pi\)
0.336214 + 0.941786i \(0.390853\pi\)
\(620\) 0 0
\(621\) 26.1705 + 26.1884i 1.05019 + 1.05090i
\(622\) 0 0
\(623\) −13.7142 −0.549446
\(624\) 0 0
\(625\) 13.5793 0.543173
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 44.0931 1.75811
\(630\) 0 0
\(631\) 20.7819i 0.827314i −0.910433 0.413657i \(-0.864251\pi\)
0.910433 0.413657i \(-0.135749\pi\)
\(632\) 0 0
\(633\) 26.5793 + 7.12515i 1.05643 + 0.283199i
\(634\) 0 0
\(635\) 12.7264i 0.505030i
\(636\) 0 0
\(637\) 1.84951i 0.0732801i
\(638\) 0 0
\(639\) 10.7130 + 6.18838i 0.423799 + 0.244809i
\(640\) 0 0
\(641\) 37.6051i 1.48531i −0.669672 0.742657i \(-0.733565\pi\)
0.669672 0.742657i \(-0.266435\pi\)
\(642\) 0 0
\(643\) 25.7369 1.01496 0.507482 0.861662i \(-0.330576\pi\)
0.507482 + 0.861662i \(0.330576\pi\)
\(644\) 0 0
\(645\) 1.79391 + 0.480896i 0.0706352 + 0.0189352i
\(646\) 0 0
\(647\) 41.8476 1.64520 0.822599 0.568622i \(-0.192523\pi\)
0.822599 + 0.568622i \(0.192523\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.0418 0.392968 0.196484 0.980507i \(-0.437048\pi\)
0.196484 + 0.980507i \(0.437048\pi\)
\(654\) 0 0
\(655\) 15.1955i 0.593736i
\(656\) 0 0
\(657\) −15.5864 9.00354i −0.608084 0.351262i
\(658\) 0 0
\(659\) 48.5843i 1.89257i 0.323327 + 0.946287i \(0.395199\pi\)
−0.323327 + 0.946287i \(0.604801\pi\)
\(660\) 0 0
\(661\) 0.164668i 0.00640485i 0.999995 + 0.00320242i \(0.00101936\pi\)
−0.999995 + 0.00320242i \(0.998981\pi\)
\(662\) 0 0
\(663\) 12.7603 + 3.42068i 0.495570 + 0.132848i
\(664\) 0 0
\(665\) 0.586642i 0.0227490i
\(666\) 0 0
\(667\) 55.5496 2.15089
\(668\) 0 0
\(669\) 4.03948 15.0687i 0.156175 0.582589i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 7.00709 0.270103 0.135052 0.990839i \(-0.456880\pi\)
0.135052 + 0.990839i \(0.456880\pi\)
\(674\) 0 0
\(675\) 15.4099 + 15.4204i 0.593127 + 0.593531i
\(676\) 0 0
\(677\) 6.27869 0.241310 0.120655 0.992695i \(-0.461501\pi\)
0.120655 + 0.992695i \(0.461501\pi\)
\(678\) 0 0
\(679\) 8.99291i 0.345116i
\(680\) 0 0
\(681\) −10.2897 + 38.3841i −0.394301 + 1.47088i
\(682\) 0 0
\(683\) 30.2945i 1.15919i −0.814906 0.579593i \(-0.803211\pi\)
0.814906 0.579593i \(-0.196789\pi\)
\(684\) 0 0
\(685\) 10.6161i 0.405622i
\(686\) 0 0
\(687\) 4.90293 18.2897i 0.187058 0.697794i
\(688\) 0 0
\(689\) 15.2546i 0.581154i
\(690\) 0 0
\(691\) 15.6469 0.595238 0.297619 0.954685i \(-0.403807\pi\)
0.297619 + 0.954685i \(0.403807\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.90686 0.375788
\(696\) 0 0
\(697\) −41.7606 −1.58180
\(698\) 0 0
\(699\) −3.69901 + 13.7986i −0.139910 + 0.521912i
\(700\) 0 0
\(701\) −6.45402 −0.243765 −0.121882 0.992545i \(-0.538893\pi\)
−0.121882 + 0.992545i \(0.538893\pi\)
\(702\) 0 0
\(703\) 6.99291i 0.263743i
\(704\) 0 0
\(705\) −14.3909 3.85779i −0.541994 0.145293i
\(706\) 0 0
\(707\) 7.35097i 0.276462i
\(708\) 0 0
\(709\) 18.0900i 0.679383i 0.940537 + 0.339691i \(0.110323\pi\)
−0.940537 + 0.339691i \(0.889677\pi\)
\(710\) 0 0
\(711\) 20.0837 34.7677i 0.753197 1.30389i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.16344 + 2.18838i 0.304869 + 0.0817267i
\(718\) 0 0
\(719\) 5.83339 0.217549 0.108774 0.994066i \(-0.465307\pi\)
0.108774 + 0.994066i \(0.465307\pi\)
\(720\) 0 0
\(721\) −13.3839 −0.498441
\(722\) 0 0
\(723\) 8.35305 + 2.23921i 0.310653 + 0.0832771i
\(724\) 0 0
\(725\) 32.7090 1.21478
\(726\) 0 0
\(727\) 4.22521i 0.156704i −0.996926 0.0783521i \(-0.975034\pi\)
0.996926 0.0783521i \(-0.0249658\pi\)
\(728\) 0 0
\(729\) −0.0184116 + 27.0000i −0.000681912 + 1.00000i
\(730\) 0 0
\(731\) 4.93008i 0.182346i
\(732\) 0 0
\(733\) 43.6101i 1.61078i −0.592747 0.805388i \(-0.701957\pi\)
0.592747 0.805388i \(-0.298043\pi\)
\(734\) 0 0
\(735\) −1.50059 0.402265i −0.0553501 0.0148378i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.97735 0.219880 0.109940 0.993938i \(-0.464934\pi\)
0.109940 + 0.993938i \(0.464934\pi\)
\(740\) 0 0
\(741\) −0.542499 + 2.02371i −0.0199292 + 0.0743430i
\(742\) 0 0
\(743\) −49.9770 −1.83348 −0.916740 0.399485i \(-0.869189\pi\)
−0.916740 + 0.399485i \(0.869189\pi\)
\(744\) 0 0
\(745\) −4.17991 −0.153140
\(746\) 0 0
\(747\) 20.0379 34.6884i 0.733148 1.26918i
\(748\) 0 0
\(749\) −1.79391 −0.0655481
\(750\) 0 0
\(751\) 31.7974i 1.16031i 0.814508 + 0.580153i \(0.197007\pi\)
−0.814508 + 0.580153i \(0.802993\pi\)
\(752\) 0 0
\(753\) 4.98159 18.5831i 0.181539 0.677206i
\(754\) 0 0
\(755\) 11.2831i 0.410634i
\(756\) 0 0
\(757\) 34.0900i 1.23902i 0.784989 + 0.619510i \(0.212669\pi\)
−0.784989 + 0.619510i \(0.787331\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.806113i 0.0292216i 0.999893 + 0.0146108i \(0.00465092\pi\)
−0.999893 + 0.0146108i \(0.995349\pi\)
\(762\) 0 0
\(763\) 5.30807 0.192165
\(764\) 0 0
\(765\) −5.55071 + 9.60906i −0.200686 + 0.347416i
\(766\) 0 0
\(767\) −12.7603 −0.460749
\(768\) 0 0
\(769\) −17.7748 −0.640975 −0.320488 0.947253i \(-0.603847\pi\)
−0.320488 + 0.947253i \(0.603847\pi\)
\(770\) 0 0
\(771\) 5.14341 19.1867i 0.185235 0.690993i
\(772\) 0 0
\(773\) −17.3928 −0.625576 −0.312788 0.949823i \(-0.601263\pi\)
−0.312788 + 0.949823i \(0.601263\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 17.8874 + 4.79509i 0.641707 + 0.172023i
\(778\) 0 0
\(779\) 6.62299i 0.237293i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 28.6356 + 28.6551i 1.02335 + 1.02405i
\(784\) 0 0
\(785\) 17.5214i 0.625367i
\(786\) 0 0
\(787\) −19.0450 −0.678880 −0.339440 0.940628i \(-0.610238\pi\)
−0.339440 + 0.940628i \(0.610238\pi\)
\(788\) 0 0
\(789\) 20.6980 + 5.54852i 0.736867 + 0.197533i
\(790\) 0 0
\(791\) 3.00118 0.106710
\(792\) 0 0
\(793\) −8.39941 −0.298272
\(794\) 0 0
\(795\) −12.3768 3.31785i −0.438959 0.117672i
\(796\) 0 0
\(797\) −41.2333 −1.46056 −0.730279 0.683149i \(-0.760610\pi\)
−0.730279 + 0.683149i \(0.760610\pi\)
\(798\) 0 0
\(799\) 39.5496i 1.39916i
\(800\) 0 0
\(801\) −20.5793 + 35.6257i −0.727135 + 1.25877i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 6.39094i 0.225251i
\(806\) 0 0
\(807\) −41.3853 11.0942i −1.45683 0.390534i
\(808\) 0 0
\(809\) 37.5041i 1.31857i −0.751891 0.659287i \(-0.770858\pi\)
0.751891 0.659287i \(-0.229142\pi\)
\(810\) 0 0
\(811\) −52.9097 −1.85791 −0.928956 0.370190i \(-0.879292\pi\)
−0.928956 + 0.370190i \(0.879292\pi\)
\(812\) 0 0
\(813\) −9.32021 + 34.7677i −0.326874 + 1.21936i
\(814\) 0 0
\(815\) 8.24793 0.288912
\(816\) 0 0
\(817\) 0.781882 0.0273546
\(818\) 0 0
\(819\) 4.80453 + 2.77535i 0.167884 + 0.0969787i
\(820\) 0 0
\(821\) −31.0161 −1.08247 −0.541235 0.840871i \(-0.682043\pi\)
−0.541235 + 0.840871i \(0.682043\pi\)
\(822\) 0 0
\(823\) 42.1657i 1.46981i −0.678173 0.734903i \(-0.737228\pi\)
0.678173 0.734903i \(-0.262772\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.7067i 0.928682i 0.885656 + 0.464341i \(0.153709\pi\)
−0.885656 + 0.464341i \(0.846291\pi\)
\(828\) 0 0
\(829\) 49.3233i 1.71307i −0.516089 0.856535i \(-0.672613\pi\)
0.516089 0.856535i \(-0.327387\pi\)
\(830\) 0 0
\(831\) 0.721627 2.69193i 0.0250330 0.0933819i
\(832\) 0 0
\(833\) 4.12397i 0.142887i
\(834\) 0 0
\(835\) −10.6161 −0.367387
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −46.3387 −1.59979 −0.799895 0.600140i \(-0.795111\pi\)
−0.799895 + 0.600140i \(0.795111\pi\)
\(840\) 0 0
\(841\) 31.7819 1.09593
\(842\) 0 0
\(843\) −2.08995 + 7.79627i −0.0719819 + 0.268518i
\(844\) 0 0
\(845\) −8.59223 −0.295582
\(846\) 0 0
\(847\) 11.0000i 0.377964i
\(848\) 0 0
\(849\) 22.9816 + 6.16070i 0.788726 + 0.211435i
\(850\) 0 0
\(851\) 76.1815i 2.61147i
\(852\) 0 0
\(853\) 1.84951i 0.0633259i −0.999499 0.0316630i \(-0.989920\pi\)
0.999499 0.0316630i \(-0.0100803\pi\)
\(854\) 0 0
\(855\) −1.52394 0.880309i −0.0521177 0.0301059i
\(856\) 0 0
\(857\) 55.5617i 1.89795i −0.315348 0.948976i \(-0.602121\pi\)
0.315348 0.948976i \(-0.397879\pi\)
\(858\) 0 0
\(859\) −21.9621 −0.749338 −0.374669 0.927159i \(-0.622244\pi\)
−0.374669 + 0.927159i \(0.622244\pi\)
\(860\) 0 0
\(861\) −16.9412 4.54143i −0.577353 0.154772i
\(862\) 0 0
\(863\) 23.0344 0.784099 0.392049 0.919944i \(-0.371766\pi\)
0.392049 + 0.919944i \(0.371766\pi\)
\(864\) 0 0
\(865\) 17.3612 0.590299
\(866\) 0 0
\(867\) 0.0118574 + 0.00317862i 0.000402698 + 0.000107952i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 10.1657i 0.344453i
\(872\) 0 0
\(873\) −23.3612 13.4947i −0.790657 0.456726i
\(874\) 0 0
\(875\) 8.24793i 0.278831i
\(876\) 0 0
\(877\) 24.6778i 0.833308i 0.909065 + 0.416654i \(0.136797\pi\)
−0.909065 + 0.416654i \(0.863203\pi\)
\(878\) 0 0
\(879\) 32.3817 + 8.68060i 1.09221 + 0.292789i
\(880\) 0 0
\(881\) 16.3987i 0.552485i 0.961088 + 0.276242i \(0.0890893\pi\)
−0.961088 + 0.276242i \(0.910911\pi\)
\(882\) 0 0
\(883\) −18.2783 −0.615115 −0.307558 0.951529i \(-0.599512\pi\)
−0.307558 + 0.951529i \(0.599512\pi\)
\(884\) 0 0
\(885\) 2.77535 10.3531i 0.0932925 0.348014i
\(886\) 0 0
\(887\) −28.6696 −0.962629 −0.481315 0.876548i \(-0.659841\pi\)
−0.481315 + 0.876548i \(0.659841\pi\)
\(888\) 0 0
\(889\) 14.1884 0.475863
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.27233 −0.209896
\(894\) 0 0
\(895\) 13.1870i 0.440793i
\(896\) 0 0
\(897\) 5.91005 22.0466i 0.197331 0.736113i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 34.0142i 1.13318i
\(902\) 0 0
\(903\) 0.536142 2.00000i 0.0178417 0.0665558i
\(904\) 0 0
\(905\) 4.07347i 0.135407i
\(906\) 0 0
\(907\) −5.87322 −0.195017 −0.0975086 0.995235i \(-0.531087\pi\)
−0.0975086 + 0.995235i \(0.531087\pi\)
\(908\) 0 0
\(909\) 19.0959 + 11.0308i 0.633371 + 0.365869i
\(910\) 0 0
\(911\) −40.4548 −1.34033 −0.670164 0.742213i \(-0.733776\pi\)
−0.670164 + 0.742213i \(0.733776\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1.82686 6.81483i 0.0603941 0.225291i
\(916\) 0 0
\(917\) −16.9412 −0.559446
\(918\) 0 0
\(919\) 27.3697i 0.902842i −0.892311 0.451421i \(-0.850917\pi\)
0.892311 0.451421i \(-0.149083\pi\)
\(920\) 0 0
\(921\) −51.5609 13.8220i −1.69899 0.455450i
\(922\) 0 0
\(923\) 7.62730i 0.251056i
\(924\) 0 0
\(925\) 44.8577i 1.47491i
\(926\) 0 0
\(927\) −20.0837 + 34.7677i −0.659635 + 1.14192i
\(928\) 0 0
\(929\) 23.3043i 0.764590i −0.924040 0.382295i \(-0.875134\pi\)
0.924040 0.382295i \(-0.124866\pi\)
\(930\) 0 0
\(931\) −0.654037 −0.0214352
\(932\) 0 0
\(933\) 10.0418 + 2.69193i 0.328755 + 0.0881297i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 46.7535 1.52737 0.763686 0.645588i \(-0.223388\pi\)
0.763686 + 0.645588i \(0.223388\pi\)
\(938\) 0 0
\(939\) 23.0450 + 6.17769i 0.752044 + 0.201601i
\(940\) 0 0
\(941\) −51.4441 −1.67703 −0.838515 0.544879i \(-0.816576\pi\)
−0.838515 + 0.544879i \(0.816576\pi\)
\(942\) 0 0
\(943\) 72.1516i 2.34958i
\(944\) 0 0
\(945\) −3.29675 + 3.29450i −0.107243 + 0.107170i
\(946\) 0 0
\(947\) 16.4959i 0.536043i −0.963413 0.268022i \(-0.913630\pi\)
0.963413 0.268022i \(-0.0863699\pi\)
\(948\) 0 0
\(949\) 11.0970i 0.360225i
\(950\) 0 0
\(951\) −6.28505 1.68484i −0.203807 0.0546346i
\(952\) 0 0
\(953\) 21.3249i 0.690783i 0.938459 + 0.345391i \(0.112254\pi\)
−0.938459 + 0.345391i \(0.887746\pi\)
\(954\) 0 0
\(955\) 1.53328 0.0496158
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.8358 −0.382196
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) −2.69193 + 4.66011i −0.0867461 + 0.150170i
\(964\) 0 0
\(965\) −14.5203 −0.467424
\(966\) 0 0
\(967\) 16.2026i 0.521039i 0.965469 + 0.260520i \(0.0838939\pi\)
−0.965469 + 0.260520i \(0.916106\pi\)
\(968\) 0 0
\(969\) −1.20965 + 4.51241i −0.0388594 + 0.144959i
\(970\) 0 0
\(971\) 0.175328i 0.00562655i 0.999996 + 0.00281328i \(0.000895495\pi\)
−0.999996 + 0.00281328i \(0.999105\pi\)
\(972\) 0 0
\(973\) 11.0450i 0.354086i
\(974\) 0 0
\(975\) 3.47999 12.9816i 0.111449 0.415744i
\(976\) 0 0
\(977\) 46.6087i 1.49114i −0.666425 0.745572i \(-0.732176\pi\)
0.666425 0.745572i \(-0.267824\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 7.96524 13.7890i 0.254311 0.440248i
\(982\) 0 0
\(983\) −40.4373 −1.28975 −0.644875 0.764288i \(-0.723091\pi\)
−0.644875 + 0.764288i \(0.723091\pi\)
\(984\) 0 0
\(985\) −8.60197 −0.274082
\(986\) 0 0
\(987\) −4.30099 + 16.0442i −0.136902 + 0.510693i
\(988\) 0 0
\(989\) −8.51790 −0.270853
\(990\) 0 0
\(991\) 16.6020i 0.527379i 0.964608 + 0.263690i \(0.0849394\pi\)
−0.964608 + 0.263690i \(0.915061\pi\)
\(992\) 0 0
\(993\) −10.3768 2.78171i −0.329297 0.0882749i
\(994\) 0 0
\(995\) 12.9080i 0.409212i
\(996\) 0 0
\(997\) 31.3091i 0.991570i 0.868445 + 0.495785i \(0.165120\pi\)
−0.868445 + 0.495785i \(0.834880\pi\)
\(998\) 0 0
\(999\) 39.2980 39.2713i 1.24333 1.24249i
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 672.2.j.d.239.1 12
3.2 odd 2 inner 672.2.j.d.239.4 12
4.3 odd 2 168.2.j.d.155.10 yes 12
8.3 odd 2 inner 672.2.j.d.239.2 12
8.5 even 2 168.2.j.d.155.4 yes 12
12.11 even 2 168.2.j.d.155.3 12
24.5 odd 2 168.2.j.d.155.9 yes 12
24.11 even 2 inner 672.2.j.d.239.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.d.155.3 12 12.11 even 2
168.2.j.d.155.4 yes 12 8.5 even 2
168.2.j.d.155.9 yes 12 24.5 odd 2
168.2.j.d.155.10 yes 12 4.3 odd 2
672.2.j.d.239.1 12 1.1 even 1 trivial
672.2.j.d.239.2 12 8.3 odd 2 inner
672.2.j.d.239.3 12 24.11 even 2 inner
672.2.j.d.239.4 12 3.2 odd 2 inner