Properties

Label 896.2.m.h.225.3
Level $896$
Weight $2$
Character 896.225
Analytic conductor $7.155$
Analytic rank $0$
Dimension $12$
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] = [896,2,Mod(225,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("896.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.20138089353117696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) 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 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 225.3
Root \(-1.12465 + 0.857418i\) of defining polynomial
Character \(\chi\) \(=\) 896.225
Dual form 896.2.m.h.673.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.416854 - 0.416854i) q^{3} +(1.13169 + 1.13169i) q^{5} -1.00000i q^{7} +2.65247i q^{9} +O(q^{10})\) \(q+(0.416854 - 0.416854i) q^{3} +(1.13169 + 1.13169i) q^{5} -1.00000i q^{7} +2.65247i q^{9} +(3.85718 + 3.85718i) q^{11} +(-4.66311 + 4.66311i) q^{13} +0.943500 q^{15} -5.33230 q^{17} +(-2.55919 + 2.55919i) q^{19} +(-0.416854 - 0.416854i) q^{21} -2.60484i q^{23} -2.43855i q^{25} +(2.35625 + 2.35625i) q^{27} +(1.22279 - 1.22279i) q^{29} +0.833708 q^{31} +3.21576 q^{33} +(1.13169 - 1.13169i) q^{35} +(4.42967 + 4.42967i) q^{37} +3.88768i q^{39} +0.263382i q^{41} +(1.25233 + 1.25233i) q^{43} +(-3.00177 + 3.00177i) q^{45} +10.7559 q^{47} -1.00000 q^{49} +(-2.22279 + 2.22279i) q^{51} +(-0.0476221 - 0.0476221i) q^{53} +8.73026i q^{55} +2.13362i q^{57} +(-3.60682 - 3.60682i) q^{59} +(-4.46399 + 4.46399i) q^{61} +2.65247 q^{63} -10.5544 q^{65} +(9.50964 - 9.50964i) q^{67} +(-1.08584 - 1.08584i) q^{69} +2.05301i q^{71} +5.48268i q^{73} +(-1.01652 - 1.01652i) q^{75} +(3.85718 - 3.85718i) q^{77} -5.21576 q^{79} -5.99297 q^{81} +(5.84045 - 5.84045i) q^{83} +(-6.03452 - 6.03452i) q^{85} -1.01945i q^{87} +6.32651i q^{89} +(4.66311 + 4.66311i) q^{91} +(0.347535 - 0.347535i) q^{93} -5.79243 q^{95} +18.8089 q^{97} +(-10.2310 + 10.2310i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} - 4 q^{5} + 24 q^{15} - 8 q^{17} - 4 q^{21} + 4 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{35} + 20 q^{37} + 16 q^{43} - 40 q^{45} - 16 q^{47} - 12 q^{49} - 16 q^{51} - 4 q^{53} - 16 q^{59} + 20 q^{61} - 12 q^{63} + 32 q^{65} + 24 q^{67} + 4 q^{69} - 40 q^{75} - 24 q^{79} - 44 q^{81} - 20 q^{83} + 8 q^{85} + 48 q^{93} + 48 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.416854 0.416854i 0.240671 0.240671i −0.576457 0.817128i \(-0.695565\pi\)
0.817128 + 0.576457i \(0.195565\pi\)
\(4\) 0 0
\(5\) 1.13169 + 1.13169i 0.506108 + 0.506108i 0.913329 0.407222i \(-0.133502\pi\)
−0.407222 + 0.913329i \(0.633502\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.65247i 0.884155i
\(10\) 0 0
\(11\) 3.85718 + 3.85718i 1.16298 + 1.16298i 0.983819 + 0.179163i \(0.0573390\pi\)
0.179163 + 0.983819i \(0.442661\pi\)
\(12\) 0 0
\(13\) −4.66311 + 4.66311i −1.29332 + 1.29332i −0.360591 + 0.932724i \(0.617425\pi\)
−0.932724 + 0.360591i \(0.882575\pi\)
\(14\) 0 0
\(15\) 0.943500 0.243611
\(16\) 0 0
\(17\) −5.33230 −1.29327 −0.646637 0.762798i \(-0.723825\pi\)
−0.646637 + 0.762798i \(0.723825\pi\)
\(18\) 0 0
\(19\) −2.55919 + 2.55919i −0.587119 + 0.587119i −0.936850 0.349731i \(-0.886273\pi\)
0.349731 + 0.936850i \(0.386273\pi\)
\(20\) 0 0
\(21\) −0.416854 0.416854i −0.0909650 0.0909650i
\(22\) 0 0
\(23\) 2.60484i 0.543147i −0.962418 0.271574i \(-0.912456\pi\)
0.962418 0.271574i \(-0.0875441\pi\)
\(24\) 0 0
\(25\) 2.43855i 0.487710i
\(26\) 0 0
\(27\) 2.35625 + 2.35625i 0.453461 + 0.453461i
\(28\) 0 0
\(29\) 1.22279 1.22279i 0.227067 0.227067i −0.584399 0.811466i \(-0.698670\pi\)
0.811466 + 0.584399i \(0.198670\pi\)
\(30\) 0 0
\(31\) 0.833708 0.149738 0.0748692 0.997193i \(-0.476146\pi\)
0.0748692 + 0.997193i \(0.476146\pi\)
\(32\) 0 0
\(33\) 3.21576 0.559792
\(34\) 0 0
\(35\) 1.13169 1.13169i 0.191291 0.191291i
\(36\) 0 0
\(37\) 4.42967 + 4.42967i 0.728234 + 0.728234i 0.970268 0.242034i \(-0.0778145\pi\)
−0.242034 + 0.970268i \(0.577815\pi\)
\(38\) 0 0
\(39\) 3.88768i 0.622526i
\(40\) 0 0
\(41\) 0.263382i 0.0411333i 0.999788 + 0.0205667i \(0.00654703\pi\)
−0.999788 + 0.0205667i \(0.993453\pi\)
\(42\) 0 0
\(43\) 1.25233 + 1.25233i 0.190979 + 0.190979i 0.796119 0.605140i \(-0.206883\pi\)
−0.605140 + 0.796119i \(0.706883\pi\)
\(44\) 0 0
\(45\) −3.00177 + 3.00177i −0.447478 + 0.447478i
\(46\) 0 0
\(47\) 10.7559 1.56891 0.784455 0.620186i \(-0.212943\pi\)
0.784455 + 0.620186i \(0.212943\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.22279 + 2.22279i −0.311253 + 0.311253i
\(52\) 0 0
\(53\) −0.0476221 0.0476221i −0.00654139 0.00654139i 0.703829 0.710370i \(-0.251472\pi\)
−0.710370 + 0.703829i \(0.751472\pi\)
\(54\) 0 0
\(55\) 8.73026i 1.17719i
\(56\) 0 0
\(57\) 2.13362i 0.282605i
\(58\) 0 0
\(59\) −3.60682 3.60682i −0.469567 0.469567i 0.432207 0.901774i \(-0.357735\pi\)
−0.901774 + 0.432207i \(0.857735\pi\)
\(60\) 0 0
\(61\) −4.46399 + 4.46399i −0.571556 + 0.571556i −0.932563 0.361007i \(-0.882433\pi\)
0.361007 + 0.932563i \(0.382433\pi\)
\(62\) 0 0
\(63\) 2.65247 0.334179
\(64\) 0 0
\(65\) −10.5544 −1.30911
\(66\) 0 0
\(67\) 9.50964 9.50964i 1.16179 1.16179i 0.177704 0.984084i \(-0.443133\pi\)
0.984084 0.177704i \(-0.0568668\pi\)
\(68\) 0 0
\(69\) −1.08584 1.08584i −0.130720 0.130720i
\(70\) 0 0
\(71\) 2.05301i 0.243647i 0.992552 + 0.121824i \(0.0388743\pi\)
−0.992552 + 0.121824i \(0.961126\pi\)
\(72\) 0 0
\(73\) 5.48268i 0.641700i 0.947130 + 0.320850i \(0.103968\pi\)
−0.947130 + 0.320850i \(0.896032\pi\)
\(74\) 0 0
\(75\) −1.01652 1.01652i −0.117378 0.117378i
\(76\) 0 0
\(77\) 3.85718 3.85718i 0.439566 0.439566i
\(78\) 0 0
\(79\) −5.21576 −0.586819 −0.293409 0.955987i \(-0.594790\pi\)
−0.293409 + 0.955987i \(0.594790\pi\)
\(80\) 0 0
\(81\) −5.99297 −0.665885
\(82\) 0 0
\(83\) 5.84045 5.84045i 0.641073 0.641073i −0.309746 0.950819i \(-0.600244\pi\)
0.950819 + 0.309746i \(0.100244\pi\)
\(84\) 0 0
\(85\) −6.03452 6.03452i −0.654535 0.654535i
\(86\) 0 0
\(87\) 1.01945i 0.109297i
\(88\) 0 0
\(89\) 6.32651i 0.670609i 0.942110 + 0.335304i \(0.108839\pi\)
−0.942110 + 0.335304i \(0.891161\pi\)
\(90\) 0 0
\(91\) 4.66311 + 4.66311i 0.488827 + 0.488827i
\(92\) 0 0
\(93\) 0.347535 0.347535i 0.0360377 0.0360377i
\(94\) 0 0
\(95\) −5.79243 −0.594291
\(96\) 0 0
\(97\) 18.8089 1.90976 0.954878 0.296999i \(-0.0959858\pi\)
0.954878 + 0.296999i \(0.0959858\pi\)
\(98\) 0 0
\(99\) −10.2310 + 10.2310i −1.02826 + 1.02826i
\(100\) 0 0
\(101\) 2.88523 + 2.88523i 0.287091 + 0.287091i 0.835929 0.548838i \(-0.184930\pi\)
−0.548838 + 0.835929i \(0.684930\pi\)
\(102\) 0 0
\(103\) 7.74040i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(104\) 0 0
\(105\) 0.943500i 0.0920762i
\(106\) 0 0
\(107\) −2.33447 2.33447i −0.225682 0.225682i 0.585204 0.810886i \(-0.301014\pi\)
−0.810886 + 0.585204i \(0.801014\pi\)
\(108\) 0 0
\(109\) 4.92827 4.92827i 0.472042 0.472042i −0.430533 0.902575i \(-0.641674\pi\)
0.902575 + 0.430533i \(0.141674\pi\)
\(110\) 0 0
\(111\) 3.69306 0.350529
\(112\) 0 0
\(113\) 5.24381 0.493296 0.246648 0.969105i \(-0.420671\pi\)
0.246648 + 0.969105i \(0.420671\pi\)
\(114\) 0 0
\(115\) 2.94788 2.94788i 0.274891 0.274891i
\(116\) 0 0
\(117\) −12.3687 12.3687i −1.14349 1.14349i
\(118\) 0 0
\(119\) 5.33230i 0.488811i
\(120\) 0 0
\(121\) 18.7556i 1.70506i
\(122\) 0 0
\(123\) 0.109792 + 0.109792i 0.00989959 + 0.00989959i
\(124\) 0 0
\(125\) 8.41814 8.41814i 0.752941 0.752941i
\(126\) 0 0
\(127\) 17.6789 1.56875 0.784373 0.620290i \(-0.212985\pi\)
0.784373 + 0.620290i \(0.212985\pi\)
\(128\) 0 0
\(129\) 1.04408 0.0919262
\(130\) 0 0
\(131\) −2.17911 + 2.17911i −0.190390 + 0.190390i −0.795865 0.605475i \(-0.792983\pi\)
0.605475 + 0.795865i \(0.292983\pi\)
\(132\) 0 0
\(133\) 2.55919 + 2.55919i 0.221910 + 0.221910i
\(134\) 0 0
\(135\) 5.33310i 0.459000i
\(136\) 0 0
\(137\) 8.54650i 0.730177i −0.930973 0.365088i \(-0.881039\pi\)
0.930973 0.365088i \(-0.118961\pi\)
\(138\) 0 0
\(139\) −5.72549 5.72549i −0.485629 0.485629i 0.421295 0.906924i \(-0.361576\pi\)
−0.906924 + 0.421295i \(0.861576\pi\)
\(140\) 0 0
\(141\) 4.48364 4.48364i 0.377591 0.377591i
\(142\) 0 0
\(143\) −35.9729 −3.00821
\(144\) 0 0
\(145\) 2.76764 0.229840
\(146\) 0 0
\(147\) −0.416854 + 0.416854i −0.0343815 + 0.0343815i
\(148\) 0 0
\(149\) −0.395157 0.395157i −0.0323725 0.0323725i 0.690735 0.723108i \(-0.257287\pi\)
−0.723108 + 0.690735i \(0.757287\pi\)
\(150\) 0 0
\(151\) 3.71559i 0.302371i −0.988505 0.151185i \(-0.951691\pi\)
0.988505 0.151185i \(-0.0483090\pi\)
\(152\) 0 0
\(153\) 14.1437i 1.14345i
\(154\) 0 0
\(155\) 0.943500 + 0.943500i 0.0757837 + 0.0757837i
\(156\) 0 0
\(157\) −3.99542 + 3.99542i −0.318869 + 0.318869i −0.848333 0.529464i \(-0.822393\pi\)
0.529464 + 0.848333i \(0.322393\pi\)
\(158\) 0 0
\(159\) −0.0397029 −0.00314864
\(160\) 0 0
\(161\) −2.60484 −0.205290
\(162\) 0 0
\(163\) 3.37168 3.37168i 0.264090 0.264090i −0.562623 0.826713i \(-0.690208\pi\)
0.826713 + 0.562623i \(0.190208\pi\)
\(164\) 0 0
\(165\) 3.63925 + 3.63925i 0.283315 + 0.283315i
\(166\) 0 0
\(167\) 12.4233i 0.961345i −0.876900 0.480673i \(-0.840393\pi\)
0.876900 0.480673i \(-0.159607\pi\)
\(168\) 0 0
\(169\) 30.4893i 2.34533i
\(170\) 0 0
\(171\) −6.78817 6.78817i −0.519105 0.519105i
\(172\) 0 0
\(173\) −9.25908 + 9.25908i −0.703955 + 0.703955i −0.965257 0.261302i \(-0.915848\pi\)
0.261302 + 0.965257i \(0.415848\pi\)
\(174\) 0 0
\(175\) −2.43855 −0.184337
\(176\) 0 0
\(177\) −3.00703 −0.226022
\(178\) 0 0
\(179\) −9.95523 + 9.95523i −0.744088 + 0.744088i −0.973362 0.229274i \(-0.926365\pi\)
0.229274 + 0.973362i \(0.426365\pi\)
\(180\) 0 0
\(181\) 5.08125 + 5.08125i 0.377687 + 0.377687i 0.870267 0.492580i \(-0.163946\pi\)
−0.492580 + 0.870267i \(0.663946\pi\)
\(182\) 0 0
\(183\) 3.72167i 0.275114i
\(184\) 0 0
\(185\) 10.0260i 0.737129i
\(186\) 0 0
\(187\) −20.5676 20.5676i −1.50405 1.50405i
\(188\) 0 0
\(189\) 2.35625 2.35625i 0.171392 0.171392i
\(190\) 0 0
\(191\) −20.7927 −1.50451 −0.752254 0.658873i \(-0.771033\pi\)
−0.752254 + 0.658873i \(0.771033\pi\)
\(192\) 0 0
\(193\) 13.3447 0.960574 0.480287 0.877111i \(-0.340533\pi\)
0.480287 + 0.877111i \(0.340533\pi\)
\(194\) 0 0
\(195\) −4.39965 + 4.39965i −0.315065 + 0.315065i
\(196\) 0 0
\(197\) −0.194462 0.194462i −0.0138549 0.0138549i 0.700145 0.714000i \(-0.253118\pi\)
−0.714000 + 0.700145i \(0.753118\pi\)
\(198\) 0 0
\(199\) 14.4003i 1.02081i −0.859935 0.510403i \(-0.829496\pi\)
0.859935 0.510403i \(-0.170504\pi\)
\(200\) 0 0
\(201\) 7.92827i 0.559217i
\(202\) 0 0
\(203\) −1.22279 1.22279i −0.0858232 0.0858232i
\(204\) 0 0
\(205\) −0.298067 + 0.298067i −0.0208179 + 0.0208179i
\(206\) 0 0
\(207\) 6.90926 0.480227
\(208\) 0 0
\(209\) −19.7425 −1.36562
\(210\) 0 0
\(211\) 7.72356 7.72356i 0.531711 0.531711i −0.389370 0.921081i \(-0.627307\pi\)
0.921081 + 0.389370i \(0.127307\pi\)
\(212\) 0 0
\(213\) 0.855805 + 0.855805i 0.0586388 + 0.0586388i
\(214\) 0 0
\(215\) 2.83451i 0.193312i
\(216\) 0 0
\(217\) 0.833708i 0.0565958i
\(218\) 0 0
\(219\) 2.28548 + 2.28548i 0.154438 + 0.154438i
\(220\) 0 0
\(221\) 24.8651 24.8651i 1.67261 1.67261i
\(222\) 0 0
\(223\) 7.06285 0.472963 0.236482 0.971636i \(-0.424006\pi\)
0.236482 + 0.971636i \(0.424006\pi\)
\(224\) 0 0
\(225\) 6.46817 0.431212
\(226\) 0 0
\(227\) −11.8229 + 11.8229i −0.784711 + 0.784711i −0.980622 0.195911i \(-0.937234\pi\)
0.195911 + 0.980622i \(0.437234\pi\)
\(228\) 0 0
\(229\) 9.37815 + 9.37815i 0.619726 + 0.619726i 0.945461 0.325735i \(-0.105612\pi\)
−0.325735 + 0.945461i \(0.605612\pi\)
\(230\) 0 0
\(231\) 3.21576i 0.211581i
\(232\) 0 0
\(233\) 24.4385i 1.60102i −0.599318 0.800511i \(-0.704561\pi\)
0.599318 0.800511i \(-0.295439\pi\)
\(234\) 0 0
\(235\) 12.1724 + 12.1724i 0.794037 + 0.794037i
\(236\) 0 0
\(237\) −2.17421 + 2.17421i −0.141230 + 0.141230i
\(238\) 0 0
\(239\) 6.27660 0.406000 0.203000 0.979179i \(-0.434931\pi\)
0.203000 + 0.979179i \(0.434931\pi\)
\(240\) 0 0
\(241\) −15.0124 −0.967034 −0.483517 0.875335i \(-0.660641\pi\)
−0.483517 + 0.875335i \(0.660641\pi\)
\(242\) 0 0
\(243\) −9.56695 + 9.56695i −0.613720 + 0.613720i
\(244\) 0 0
\(245\) −1.13169 1.13169i −0.0723011 0.0723011i
\(246\) 0 0
\(247\) 23.8676i 1.51866i
\(248\) 0 0
\(249\) 4.86923i 0.308575i
\(250\) 0 0
\(251\) 11.7926 + 11.7926i 0.744339 + 0.744339i 0.973410 0.229071i \(-0.0735687\pi\)
−0.229071 + 0.973410i \(0.573569\pi\)
\(252\) 0 0
\(253\) 10.0473 10.0473i 0.631671 0.631671i
\(254\) 0 0
\(255\) −5.03103 −0.315055
\(256\) 0 0
\(257\) 20.3977 1.27237 0.636186 0.771536i \(-0.280511\pi\)
0.636186 + 0.771536i \(0.280511\pi\)
\(258\) 0 0
\(259\) 4.42967 4.42967i 0.275247 0.275247i
\(260\) 0 0
\(261\) 3.24341 + 3.24341i 0.200762 + 0.200762i
\(262\) 0 0
\(263\) 23.3452i 1.43953i 0.694219 + 0.719764i \(0.255750\pi\)
−0.694219 + 0.719764i \(0.744250\pi\)
\(264\) 0 0
\(265\) 0.107787i 0.00662130i
\(266\) 0 0
\(267\) 2.63723 + 2.63723i 0.161396 + 0.161396i
\(268\) 0 0
\(269\) −22.0705 + 22.0705i −1.34566 + 1.34566i −0.455352 + 0.890312i \(0.650487\pi\)
−0.890312 + 0.455352i \(0.849513\pi\)
\(270\) 0 0
\(271\) −1.46392 −0.0889266 −0.0444633 0.999011i \(-0.514158\pi\)
−0.0444633 + 0.999011i \(0.514158\pi\)
\(272\) 0 0
\(273\) 3.88768 0.235293
\(274\) 0 0
\(275\) 9.40592 9.40592i 0.567199 0.567199i
\(276\) 0 0
\(277\) 1.39234 + 1.39234i 0.0836578 + 0.0836578i 0.747697 0.664040i \(-0.231159\pi\)
−0.664040 + 0.747697i \(0.731159\pi\)
\(278\) 0 0
\(279\) 2.21138i 0.132392i
\(280\) 0 0
\(281\) 5.66742i 0.338090i 0.985608 + 0.169045i \(0.0540683\pi\)
−0.985608 + 0.169045i \(0.945932\pi\)
\(282\) 0 0
\(283\) −16.6615 16.6615i −0.990424 0.990424i 0.00953050 0.999955i \(-0.496966\pi\)
−0.999955 + 0.00953050i \(0.996966\pi\)
\(284\) 0 0
\(285\) −2.41460 + 2.41460i −0.143029 + 0.143029i
\(286\) 0 0
\(287\) 0.263382 0.0155469
\(288\) 0 0
\(289\) 11.4334 0.672555
\(290\) 0 0
\(291\) 7.84057 7.84057i 0.459622 0.459622i
\(292\) 0 0
\(293\) −17.1121 17.1121i −0.999698 0.999698i 0.000302098 1.00000i \(-0.499904\pi\)
−1.00000 0.000302098i \(0.999904\pi\)
\(294\) 0 0
\(295\) 8.16360i 0.475303i
\(296\) 0 0
\(297\) 18.1770i 1.05473i
\(298\) 0 0
\(299\) 12.1467 + 12.1467i 0.702461 + 0.702461i
\(300\) 0 0
\(301\) 1.25233 1.25233i 0.0721833 0.0721833i
\(302\) 0 0
\(303\) 2.40544 0.138189
\(304\) 0 0
\(305\) −10.1037 −0.578537
\(306\) 0 0
\(307\) −9.59837 + 9.59837i −0.547808 + 0.547808i −0.925806 0.377998i \(-0.876612\pi\)
0.377998 + 0.925806i \(0.376612\pi\)
\(308\) 0 0
\(309\) −3.22662 3.22662i −0.183556 0.183556i
\(310\) 0 0
\(311\) 19.1866i 1.08797i −0.839094 0.543987i \(-0.816914\pi\)
0.839094 0.543987i \(-0.183086\pi\)
\(312\) 0 0
\(313\) 5.03963i 0.284857i 0.989805 + 0.142428i \(0.0454911\pi\)
−0.989805 + 0.142428i \(0.954509\pi\)
\(314\) 0 0
\(315\) 3.00177 + 3.00177i 0.169131 + 0.169131i
\(316\) 0 0
\(317\) −5.52596 + 5.52596i −0.310369 + 0.310369i −0.845052 0.534683i \(-0.820431\pi\)
0.534683 + 0.845052i \(0.320431\pi\)
\(318\) 0 0
\(319\) 9.43305 0.528149
\(320\) 0 0
\(321\) −1.94627 −0.108630
\(322\) 0 0
\(323\) 13.6464 13.6464i 0.759306 0.759306i
\(324\) 0 0
\(325\) 11.3712 + 11.3712i 0.630763 + 0.630763i
\(326\) 0 0
\(327\) 4.10874i 0.227214i
\(328\) 0 0
\(329\) 10.7559i 0.592992i
\(330\) 0 0
\(331\) −14.7514 14.7514i −0.810813 0.810813i 0.173943 0.984756i \(-0.444349\pi\)
−0.984756 + 0.173943i \(0.944349\pi\)
\(332\) 0 0
\(333\) −11.7496 + 11.7496i −0.643872 + 0.643872i
\(334\) 0 0
\(335\) 21.5239 1.17598
\(336\) 0 0
\(337\) 16.7111 0.910311 0.455156 0.890412i \(-0.349584\pi\)
0.455156 + 0.890412i \(0.349584\pi\)
\(338\) 0 0
\(339\) 2.18590 2.18590i 0.118722 0.118722i
\(340\) 0 0
\(341\) 3.21576 + 3.21576i 0.174143 + 0.174143i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 2.45767i 0.132316i
\(346\) 0 0
\(347\) 5.57925 + 5.57925i 0.299510 + 0.299510i 0.840822 0.541312i \(-0.182072\pi\)
−0.541312 + 0.840822i \(0.682072\pi\)
\(348\) 0 0
\(349\) 17.9789 17.9789i 0.962388 0.962388i −0.0369299 0.999318i \(-0.511758\pi\)
0.999318 + 0.0369299i \(0.0117578\pi\)
\(350\) 0 0
\(351\) −21.9750 −1.17294
\(352\) 0 0
\(353\) −25.0318 −1.33231 −0.666155 0.745814i \(-0.732061\pi\)
−0.666155 + 0.745814i \(0.732061\pi\)
\(354\) 0 0
\(355\) −2.32337 + 2.32337i −0.123312 + 0.123312i
\(356\) 0 0
\(357\) 2.22279 + 2.22279i 0.117643 + 0.117643i
\(358\) 0 0
\(359\) 17.9910i 0.949526i 0.880114 + 0.474763i \(0.157466\pi\)
−0.880114 + 0.474763i \(0.842534\pi\)
\(360\) 0 0
\(361\) 5.90105i 0.310582i
\(362\) 0 0
\(363\) 7.81836 + 7.81836i 0.410357 + 0.410357i
\(364\) 0 0
\(365\) −6.20470 + 6.20470i −0.324769 + 0.324769i
\(366\) 0 0
\(367\) −10.3077 −0.538060 −0.269030 0.963132i \(-0.586703\pi\)
−0.269030 + 0.963132i \(0.586703\pi\)
\(368\) 0 0
\(369\) −0.698611 −0.0363682
\(370\) 0 0
\(371\) −0.0476221 + 0.0476221i −0.00247241 + 0.00247241i
\(372\) 0 0
\(373\) −16.0555 16.0555i −0.831324 0.831324i 0.156374 0.987698i \(-0.450019\pi\)
−0.987698 + 0.156374i \(0.950019\pi\)
\(374\) 0 0
\(375\) 7.01827i 0.362422i
\(376\) 0 0
\(377\) 11.4040i 0.587338i
\(378\) 0 0
\(379\) 15.7922 + 15.7922i 0.811190 + 0.811190i 0.984812 0.173622i \(-0.0555471\pi\)
−0.173622 + 0.984812i \(0.555547\pi\)
\(380\) 0 0
\(381\) 7.36951 7.36951i 0.377551 0.377551i
\(382\) 0 0
\(383\) −18.3633 −0.938319 −0.469160 0.883113i \(-0.655443\pi\)
−0.469160 + 0.883113i \(0.655443\pi\)
\(384\) 0 0
\(385\) 8.73026 0.444935
\(386\) 0 0
\(387\) −3.32177 + 3.32177i −0.168855 + 0.168855i
\(388\) 0 0
\(389\) −13.9554 13.9554i −0.707565 0.707565i 0.258458 0.966023i \(-0.416786\pi\)
−0.966023 + 0.258458i \(0.916786\pi\)
\(390\) 0 0
\(391\) 13.8898i 0.702438i
\(392\) 0 0
\(393\) 1.81674i 0.0916426i
\(394\) 0 0
\(395\) −5.90263 5.90263i −0.296993 0.296993i
\(396\) 0 0
\(397\) 1.12815 1.12815i 0.0566202 0.0566202i −0.678230 0.734850i \(-0.737253\pi\)
0.734850 + 0.678230i \(0.237253\pi\)
\(398\) 0 0
\(399\) 2.13362 0.106815
\(400\) 0 0
\(401\) −14.3470 −0.716454 −0.358227 0.933635i \(-0.616619\pi\)
−0.358227 + 0.933635i \(0.616619\pi\)
\(402\) 0 0
\(403\) −3.88768 + 3.88768i −0.193659 + 0.193659i
\(404\) 0 0
\(405\) −6.78219 6.78219i −0.337010 0.337010i
\(406\) 0 0
\(407\) 34.1721i 1.69385i
\(408\) 0 0
\(409\) 39.2518i 1.94088i 0.241350 + 0.970438i \(0.422410\pi\)
−0.241350 + 0.970438i \(0.577590\pi\)
\(410\) 0 0
\(411\) −3.56264 3.56264i −0.175732 0.175732i
\(412\) 0 0
\(413\) −3.60682 + 3.60682i −0.177480 + 0.177480i
\(414\) 0 0
\(415\) 13.2192 0.648904
\(416\) 0 0
\(417\) −4.77338 −0.233754
\(418\) 0 0
\(419\) −4.41473 + 4.41473i −0.215673 + 0.215673i −0.806672 0.590999i \(-0.798734\pi\)
0.590999 + 0.806672i \(0.298734\pi\)
\(420\) 0 0
\(421\) 7.57494 + 7.57494i 0.369180 + 0.369180i 0.867178 0.497998i \(-0.165931\pi\)
−0.497998 + 0.867178i \(0.665931\pi\)
\(422\) 0 0
\(423\) 28.5297i 1.38716i
\(424\) 0 0
\(425\) 13.0031i 0.630743i
\(426\) 0 0
\(427\) 4.46399 + 4.46399i 0.216028 + 0.216028i
\(428\) 0 0
\(429\) −14.9955 + 14.9955i −0.723987 + 0.723987i
\(430\) 0 0
\(431\) 18.5396 0.893020 0.446510 0.894779i \(-0.352667\pi\)
0.446510 + 0.894779i \(0.352667\pi\)
\(432\) 0 0
\(433\) 7.21190 0.346582 0.173291 0.984871i \(-0.444560\pi\)
0.173291 + 0.984871i \(0.444560\pi\)
\(434\) 0 0
\(435\) 1.15370 1.15370i 0.0553159 0.0553159i
\(436\) 0 0
\(437\) 6.66630 + 6.66630i 0.318892 + 0.318892i
\(438\) 0 0
\(439\) 15.1615i 0.723617i 0.932252 + 0.361809i \(0.117841\pi\)
−0.932252 + 0.361809i \(0.882159\pi\)
\(440\) 0 0
\(441\) 2.65247i 0.126308i
\(442\) 0 0
\(443\) −2.11746 2.11746i −0.100603 0.100603i 0.655014 0.755617i \(-0.272663\pi\)
−0.755617 + 0.655014i \(0.772663\pi\)
\(444\) 0 0
\(445\) −7.15965 + 7.15965i −0.339400 + 0.339400i
\(446\) 0 0
\(447\) −0.329445 −0.0155822
\(448\) 0 0
\(449\) 4.29509 0.202698 0.101349 0.994851i \(-0.467684\pi\)
0.101349 + 0.994851i \(0.467684\pi\)
\(450\) 0 0
\(451\) −1.01591 + 1.01591i −0.0478373 + 0.0478373i
\(452\) 0 0
\(453\) −1.54886 1.54886i −0.0727718 0.0727718i
\(454\) 0 0
\(455\) 10.5544i 0.494798i
\(456\) 0 0
\(457\) 27.7833i 1.29965i 0.760085 + 0.649823i \(0.225157\pi\)
−0.760085 + 0.649823i \(0.774843\pi\)
\(458\) 0 0
\(459\) −12.5643 12.5643i −0.586449 0.586449i
\(460\) 0 0
\(461\) 6.50912 6.50912i 0.303160 0.303160i −0.539089 0.842249i \(-0.681231\pi\)
0.842249 + 0.539089i \(0.181231\pi\)
\(462\) 0 0
\(463\) 39.1018 1.81722 0.908608 0.417650i \(-0.137146\pi\)
0.908608 + 0.417650i \(0.137146\pi\)
\(464\) 0 0
\(465\) 0.786604 0.0364779
\(466\) 0 0
\(467\) 16.4618 16.4618i 0.761762 0.761762i −0.214879 0.976641i \(-0.568936\pi\)
0.976641 + 0.214879i \(0.0689356\pi\)
\(468\) 0 0
\(469\) −9.50964 9.50964i −0.439114 0.439114i
\(470\) 0 0
\(471\) 3.33101i 0.153485i
\(472\) 0 0
\(473\) 9.66094i 0.444210i
\(474\) 0 0
\(475\) 6.24073 + 6.24073i 0.286344 + 0.286344i
\(476\) 0 0
\(477\) 0.126316 0.126316i 0.00578361 0.00578361i
\(478\) 0 0
\(479\) 37.2565 1.70229 0.851145 0.524930i \(-0.175908\pi\)
0.851145 + 0.524930i \(0.175908\pi\)
\(480\) 0 0
\(481\) −41.3121 −1.88367
\(482\) 0 0
\(483\) −1.08584 + 1.08584i −0.0494074 + 0.0494074i
\(484\) 0 0
\(485\) 21.2859 + 21.2859i 0.966542 + 0.966542i
\(486\) 0 0
\(487\) 38.4219i 1.74106i −0.492113 0.870531i \(-0.663775\pi\)
0.492113 0.870531i \(-0.336225\pi\)
\(488\) 0 0
\(489\) 2.81100i 0.127118i
\(490\) 0 0
\(491\) 9.52330 + 9.52330i 0.429781 + 0.429781i 0.888554 0.458773i \(-0.151711\pi\)
−0.458773 + 0.888554i \(0.651711\pi\)
\(492\) 0 0
\(493\) −6.52029 + 6.52029i −0.293659 + 0.293659i
\(494\) 0 0
\(495\) −23.1567 −1.04082
\(496\) 0 0
\(497\) 2.05301 0.0920901
\(498\) 0 0
\(499\) 9.78190 9.78190i 0.437898 0.437898i −0.453406 0.891304i \(-0.649791\pi\)
0.891304 + 0.453406i \(0.149791\pi\)
\(500\) 0 0
\(501\) −5.17871 5.17871i −0.231368 0.231368i
\(502\) 0 0
\(503\) 11.0554i 0.492938i −0.969151 0.246469i \(-0.920730\pi\)
0.969151 0.246469i \(-0.0792703\pi\)
\(504\) 0 0
\(505\) 6.53038i 0.290598i
\(506\) 0 0
\(507\) −12.7096 12.7096i −0.564452 0.564452i
\(508\) 0 0
\(509\) 0.173240 0.173240i 0.00767871 0.00767871i −0.703257 0.710936i \(-0.748272\pi\)
0.710936 + 0.703257i \(0.248272\pi\)
\(510\) 0 0
\(511\) 5.48268 0.242540
\(512\) 0 0
\(513\) −12.0602 −0.532472
\(514\) 0 0
\(515\) 8.75973 8.75973i 0.386000 0.386000i
\(516\) 0 0
\(517\) 41.4874 + 41.4874i 1.82461 + 1.82461i
\(518\) 0 0
\(519\) 7.71937i 0.338843i
\(520\) 0 0
\(521\) 29.9861i 1.31371i −0.754015 0.656857i \(-0.771885\pi\)
0.754015 0.656857i \(-0.228115\pi\)
\(522\) 0 0
\(523\) 1.48931 + 1.48931i 0.0651228 + 0.0651228i 0.738918 0.673795i \(-0.235337\pi\)
−0.673795 + 0.738918i \(0.735337\pi\)
\(524\) 0 0
\(525\) −1.01652 + 1.01652i −0.0443646 + 0.0443646i
\(526\) 0 0
\(527\) −4.44558 −0.193653
\(528\) 0 0
\(529\) 16.2148 0.704991
\(530\) 0 0
\(531\) 9.56695 9.56695i 0.415170 0.415170i
\(532\) 0 0
\(533\) −1.22818 1.22818i −0.0531983 0.0531983i
\(534\) 0 0
\(535\) 5.28380i 0.228439i
\(536\) 0 0
\(537\) 8.29975i 0.358161i
\(538\) 0 0
\(539\) −3.85718 3.85718i −0.166140 0.166140i
\(540\) 0 0
\(541\) −12.0837 + 12.0837i −0.519519 + 0.519519i −0.917426 0.397907i \(-0.869737\pi\)
0.397907 + 0.917426i \(0.369737\pi\)
\(542\) 0 0
\(543\) 4.23628 0.181796
\(544\) 0 0
\(545\) 11.1545 0.477808
\(546\) 0 0
\(547\) 12.2663 12.2663i 0.524468 0.524468i −0.394449 0.918918i \(-0.629065\pi\)
0.918918 + 0.394449i \(0.129065\pi\)
\(548\) 0 0
\(549\) −11.8406 11.8406i −0.505344 0.505344i
\(550\) 0 0
\(551\) 6.25872i 0.266631i
\(552\) 0 0
\(553\) 5.21576i 0.221797i
\(554\) 0 0
\(555\) 4.17940 + 4.17940i 0.177406 + 0.177406i
\(556\) 0 0
\(557\) 2.32720 2.32720i 0.0986065 0.0986065i −0.656083 0.754689i \(-0.727788\pi\)
0.754689 + 0.656083i \(0.227788\pi\)
\(558\) 0 0
\(559\) −11.6795 −0.493992
\(560\) 0 0
\(561\) −17.1474 −0.723964
\(562\) 0 0
\(563\) −12.0571 + 12.0571i −0.508146 + 0.508146i −0.913957 0.405811i \(-0.866989\pi\)
0.405811 + 0.913957i \(0.366989\pi\)
\(564\) 0 0
\(565\) 5.93437 + 5.93437i 0.249661 + 0.249661i
\(566\) 0 0
\(567\) 5.99297i 0.251681i
\(568\) 0 0
\(569\) 7.99770i 0.335281i 0.985848 + 0.167641i \(0.0536148\pi\)
−0.985848 + 0.167641i \(0.946385\pi\)
\(570\) 0 0
\(571\) 22.0516 + 22.0516i 0.922832 + 0.922832i 0.997229 0.0743968i \(-0.0237031\pi\)
−0.0743968 + 0.997229i \(0.523703\pi\)
\(572\) 0 0
\(573\) −8.66753 + 8.66753i −0.362091 + 0.362091i
\(574\) 0 0
\(575\) −6.35204 −0.264899
\(576\) 0 0
\(577\) 43.4199 1.80760 0.903798 0.427960i \(-0.140768\pi\)
0.903798 + 0.427960i \(0.140768\pi\)
\(578\) 0 0
\(579\) 5.56280 5.56280i 0.231182 0.231182i
\(580\) 0 0
\(581\) −5.84045 5.84045i −0.242303 0.242303i
\(582\) 0 0
\(583\) 0.367373i 0.0152150i
\(584\) 0 0
\(585\) 27.9952i 1.15746i
\(586\) 0 0
\(587\) −18.2274 18.2274i −0.752326 0.752326i 0.222587 0.974913i \(-0.428550\pi\)
−0.974913 + 0.222587i \(0.928550\pi\)
\(588\) 0 0
\(589\) −2.13362 + 2.13362i −0.0879143 + 0.0879143i
\(590\) 0 0
\(591\) −0.162125 −0.00666892
\(592\) 0 0
\(593\) −39.7514 −1.63239 −0.816197 0.577773i \(-0.803922\pi\)
−0.816197 + 0.577773i \(0.803922\pi\)
\(594\) 0 0
\(595\) −6.03452 + 6.03452i −0.247391 + 0.247391i
\(596\) 0 0
\(597\) −6.00280 6.00280i −0.245678 0.245678i
\(598\) 0 0
\(599\) 37.5296i 1.53342i 0.641996 + 0.766708i \(0.278107\pi\)
−0.641996 + 0.766708i \(0.721893\pi\)
\(600\) 0 0
\(601\) 3.99899i 0.163122i −0.996668 0.0815611i \(-0.974009\pi\)
0.996668 0.0815611i \(-0.0259906\pi\)
\(602\) 0 0
\(603\) 25.2240 + 25.2240i 1.02720 + 1.02720i
\(604\) 0 0
\(605\) −21.2256 + 21.2256i −0.862942 + 0.862942i
\(606\) 0 0
\(607\) 24.3672 0.989035 0.494517 0.869168i \(-0.335345\pi\)
0.494517 + 0.869168i \(0.335345\pi\)
\(608\) 0 0
\(609\) −1.01945 −0.0413103
\(610\) 0 0
\(611\) −50.1560 + 50.1560i −2.02909 + 2.02909i
\(612\) 0 0
\(613\) 16.7167 + 16.7167i 0.675179 + 0.675179i 0.958905 0.283726i \(-0.0915706\pi\)
−0.283726 + 0.958905i \(0.591571\pi\)
\(614\) 0 0
\(615\) 0.248501i 0.0100205i
\(616\) 0 0
\(617\) 2.64202i 0.106364i −0.998585 0.0531819i \(-0.983064\pi\)
0.998585 0.0531819i \(-0.0169363\pi\)
\(618\) 0 0
\(619\) 26.9772 + 26.9772i 1.08431 + 1.08431i 0.996102 + 0.0882048i \(0.0281130\pi\)
0.0882048 + 0.996102i \(0.471887\pi\)
\(620\) 0 0
\(621\) 6.13767 6.13767i 0.246296 0.246296i
\(622\) 0 0
\(623\) 6.32651 0.253466
\(624\) 0 0
\(625\) 6.86071 0.274428
\(626\) 0 0
\(627\) −8.22975 + 8.22975i −0.328665 + 0.328665i
\(628\) 0 0
\(629\) −23.6204 23.6204i −0.941805 0.941805i
\(630\) 0 0
\(631\) 18.9710i 0.755223i −0.925964 0.377611i \(-0.876746\pi\)
0.925964 0.377611i \(-0.123254\pi\)
\(632\) 0 0
\(633\) 6.43919i 0.255935i
\(634\) 0 0
\(635\) 20.0070 + 20.0070i 0.793954 + 0.793954i
\(636\) 0 0
\(637\) 4.66311 4.66311i 0.184759 0.184759i
\(638\) 0 0
\(639\) −5.44554 −0.215422
\(640\) 0 0
\(641\) 31.3762 1.23929 0.619643 0.784884i \(-0.287277\pi\)
0.619643 + 0.784884i \(0.287277\pi\)
\(642\) 0 0
\(643\) 13.5690 13.5690i 0.535110 0.535110i −0.386978 0.922089i \(-0.626481\pi\)
0.922089 + 0.386978i \(0.126481\pi\)
\(644\) 0 0
\(645\) 1.18158 + 1.18158i 0.0465245 + 0.0465245i
\(646\) 0 0
\(647\) 39.6587i 1.55915i −0.626312 0.779573i \(-0.715436\pi\)
0.626312 0.779573i \(-0.284564\pi\)
\(648\) 0 0
\(649\) 27.8242i 1.09220i
\(650\) 0 0
\(651\) −0.347535 0.347535i −0.0136210 0.0136210i
\(652\) 0 0
\(653\) −2.96664 + 2.96664i −0.116094 + 0.116094i −0.762767 0.646673i \(-0.776160\pi\)
0.646673 + 0.762767i \(0.276160\pi\)
\(654\) 0 0
\(655\) −4.93216 −0.192715
\(656\) 0 0
\(657\) −14.5426 −0.567362
\(658\) 0 0
\(659\) −19.5078 + 19.5078i −0.759915 + 0.759915i −0.976307 0.216391i \(-0.930571\pi\)
0.216391 + 0.976307i \(0.430571\pi\)
\(660\) 0 0
\(661\) −11.9741 11.9741i −0.465739 0.465739i 0.434792 0.900531i \(-0.356822\pi\)
−0.900531 + 0.434792i \(0.856822\pi\)
\(662\) 0 0
\(663\) 20.7303i 0.805097i
\(664\) 0 0
\(665\) 5.79243i 0.224621i
\(666\) 0 0
\(667\) −3.18518 3.18518i −0.123331 0.123331i
\(668\) 0 0
\(669\) 2.94418 2.94418i 0.113828 0.113828i
\(670\) 0 0
\(671\) −34.4368 −1.32942
\(672\) 0 0
\(673\) −30.9400 −1.19265 −0.596324 0.802744i \(-0.703373\pi\)
−0.596324 + 0.802744i \(0.703373\pi\)
\(674\) 0 0
\(675\) 5.74585 5.74585i 0.221158 0.221158i
\(676\) 0 0
\(677\) 0.0123354 + 0.0123354i 0.000474089 + 0.000474089i 0.707344 0.706870i \(-0.249893\pi\)
−0.706870 + 0.707344i \(0.749893\pi\)
\(678\) 0 0
\(679\) 18.8089i 0.721820i
\(680\) 0 0
\(681\) 9.85681i 0.377714i
\(682\) 0 0
\(683\) −14.1350 14.1350i −0.540860 0.540860i 0.382921 0.923781i \(-0.374918\pi\)
−0.923781 + 0.382921i \(0.874918\pi\)
\(684\) 0 0
\(685\) 9.67199 9.67199i 0.369548 0.369548i
\(686\) 0 0
\(687\) 7.81864 0.298300
\(688\) 0 0
\(689\) 0.444134 0.0169202
\(690\) 0 0
\(691\) 31.5355 31.5355i 1.19967 1.19967i 0.225402 0.974266i \(-0.427631\pi\)
0.974266 0.225402i \(-0.0723695\pi\)
\(692\) 0 0
\(693\) 10.2310 + 10.2310i 0.388645 + 0.388645i
\(694\) 0 0
\(695\) 12.9590i 0.491561i
\(696\) 0 0
\(697\) 1.40443i 0.0531966i
\(698\) 0 0
\(699\) −10.1873 10.1873i −0.385319 0.385319i
\(700\) 0 0
\(701\) 16.8654 16.8654i 0.636998 0.636998i −0.312816 0.949814i \(-0.601272\pi\)
0.949814 + 0.312816i \(0.101272\pi\)
\(702\) 0 0
\(703\) −22.6728 −0.855120
\(704\) 0 0
\(705\) 10.1482 0.382203
\(706\) 0 0
\(707\) 2.88523 2.88523i 0.108510 0.108510i
\(708\) 0 0
\(709\) −21.6343 21.6343i −0.812493 0.812493i 0.172514 0.985007i \(-0.444811\pi\)
−0.985007 + 0.172514i \(0.944811\pi\)
\(710\) 0 0
\(711\) 13.8346i 0.518839i
\(712\) 0 0
\(713\) 2.17168i 0.0813300i
\(714\) 0 0
\(715\) −40.7102 40.7102i −1.52248 1.52248i
\(716\) 0 0
\(717\) 2.61643 2.61643i 0.0977123 0.0977123i
\(718\) 0 0
\(719\) −40.3698 −1.50554 −0.752769 0.658284i \(-0.771283\pi\)
−0.752769 + 0.658284i \(0.771283\pi\)
\(720\) 0 0
\(721\) −7.74040 −0.288267
\(722\) 0 0
\(723\) −6.25799 + 6.25799i −0.232737 + 0.232737i
\(724\) 0 0
\(725\) −2.98184 2.98184i −0.110743 0.110743i
\(726\) 0 0
\(727\) 3.43634i 0.127447i −0.997968 0.0637234i \(-0.979702\pi\)
0.997968 0.0637234i \(-0.0202976\pi\)
\(728\) 0 0
\(729\) 10.0029i 0.370476i
\(730\) 0 0
\(731\) −6.67782 6.67782i −0.246988 0.246988i
\(732\) 0 0
\(733\) 27.3393 27.3393i 1.00980 1.00980i 0.00984960 0.999951i \(-0.496865\pi\)
0.999951 0.00984960i \(-0.00313528\pi\)
\(734\) 0 0
\(735\) −0.943500 −0.0348015
\(736\) 0 0
\(737\) 73.3607 2.70228
\(738\) 0 0
\(739\) −28.3591 + 28.3591i −1.04321 + 1.04321i −0.0441845 + 0.999023i \(0.514069\pi\)
−0.999023 + 0.0441845i \(0.985931\pi\)
\(740\) 0 0
\(741\) −9.94932 9.94932i −0.365497 0.365497i
\(742\) 0 0
\(743\) 34.1733i 1.25370i −0.779141 0.626848i \(-0.784344\pi\)
0.779141 0.626848i \(-0.215656\pi\)
\(744\) 0 0
\(745\) 0.894391i 0.0327679i
\(746\) 0 0
\(747\) 15.4916 + 15.4916i 0.566808 + 0.566808i
\(748\) 0 0
\(749\) −2.33447 + 2.33447i −0.0852997 + 0.0852997i
\(750\) 0 0
\(751\) −8.55791 −0.312282 −0.156141 0.987735i \(-0.549905\pi\)
−0.156141 + 0.987735i \(0.549905\pi\)
\(752\) 0 0
\(753\) 9.83155 0.358281
\(754\) 0 0
\(755\) 4.20490 4.20490i 0.153032 0.153032i
\(756\) 0 0
\(757\) 25.0492 + 25.0492i 0.910428 + 0.910428i 0.996306 0.0858779i \(-0.0273695\pi\)
−0.0858779 + 0.996306i \(0.527369\pi\)
\(758\) 0 0
\(759\) 8.37655i 0.304050i
\(760\) 0 0
\(761\) 28.4224i 1.03031i −0.857097 0.515155i \(-0.827734\pi\)
0.857097 0.515155i \(-0.172266\pi\)
\(762\) 0 0
\(763\) −4.92827 4.92827i −0.178415 0.178415i
\(764\) 0 0
\(765\) 16.0063 16.0063i 0.578711 0.578711i
\(766\) 0 0
\(767\) 33.6380 1.21460
\(768\) 0 0
\(769\) −12.0189 −0.433413 −0.216707 0.976237i \(-0.569531\pi\)
−0.216707 + 0.976237i \(0.569531\pi\)
\(770\) 0 0
\(771\) 8.50286 8.50286i 0.306223 0.306223i
\(772\) 0 0
\(773\) −5.06913 5.06913i −0.182324 0.182324i 0.610044 0.792368i \(-0.291152\pi\)
−0.792368 + 0.610044i \(0.791152\pi\)
\(774\) 0 0
\(775\) 2.03304i 0.0730290i
\(776\) 0 0
\(777\) 3.69306i 0.132488i
\(778\) 0 0
\(779\) −0.674045 0.674045i −0.0241502 0.0241502i
\(780\) 0 0
\(781\) −7.91882 + 7.91882i −0.283358 + 0.283358i
\(782\) 0 0
\(783\) 5.76241 0.205932
\(784\) 0 0
\(785\) −9.04315 −0.322764
\(786\) 0 0
\(787\) 16.9635 16.9635i 0.604684 0.604684i −0.336868 0.941552i \(-0.609368\pi\)
0.941552 + 0.336868i \(0.109368\pi\)
\(788\) 0 0
\(789\) 9.73155 + 9.73155i 0.346453 + 0.346453i
\(790\) 0 0
\(791\) 5.24381i 0.186448i
\(792\) 0 0
\(793\) 41.6322i 1.47840i
\(794\) 0 0
\(795\) −0.0449314 0.0449314i −0.00159355 0.00159355i
\(796\) 0 0
\(797\) −35.5609 + 35.5609i −1.25963 + 1.25963i −0.308366 + 0.951268i \(0.599782\pi\)
−0.951268 + 0.308366i \(0.900218\pi\)
\(798\) 0 0
\(799\) −57.3537 −2.02903
\(800\) 0 0
\(801\) −16.7808 −0.592922
\(802\) 0 0
\(803\) −21.1477 + 21.1477i −0.746285 + 0.746285i
\(804\) 0 0
\(805\) −2.94788 2.94788i −0.103899 0.103899i
\(806\) 0 0
\(807\) 18.4004i 0.647724i
\(808\) 0 0
\(809\) 11.8621i 0.417050i −0.978017 0.208525i \(-0.933134\pi\)
0.978017 0.208525i \(-0.0668663\pi\)
\(810\) 0 0
\(811\) 20.4859 + 20.4859i 0.719357 + 0.719357i 0.968473 0.249117i \(-0.0801403\pi\)
−0.249117 + 0.968473i \(0.580140\pi\)
\(812\) 0 0
\(813\) −0.610240 + 0.610240i −0.0214020 + 0.0214020i
\(814\) 0 0
\(815\) 7.63140 0.267316
\(816\) 0 0
\(817\) −6.40993 −0.224255
\(818\) 0 0
\(819\) −12.3687 + 12.3687i −0.432199 + 0.432199i
\(820\) 0 0
\(821\) 29.0707 + 29.0707i 1.01458 + 1.01458i 0.999892 + 0.0146829i \(0.00467389\pi\)
0.0146829 + 0.999892i \(0.495326\pi\)
\(822\) 0 0
\(823\) 51.3595i 1.79028i 0.445785 + 0.895140i \(0.352925\pi\)
−0.445785 + 0.895140i \(0.647075\pi\)
\(824\) 0 0
\(825\) 7.84180i 0.273016i
\(826\) 0 0
\(827\) −24.4414 24.4414i −0.849912 0.849912i 0.140210 0.990122i \(-0.455222\pi\)
−0.990122 + 0.140210i \(0.955222\pi\)
\(828\) 0 0
\(829\) 11.3861 11.3861i 0.395455 0.395455i −0.481172 0.876626i \(-0.659789\pi\)
0.876626 + 0.481172i \(0.159789\pi\)
\(830\) 0 0
\(831\) 1.16081 0.0402680
\(832\) 0 0
\(833\) 5.33230 0.184753
\(834\) 0 0
\(835\) 14.0594 14.0594i 0.486544 0.486544i
\(836\) 0 0
\(837\) 1.96443 + 1.96443i 0.0679006 + 0.0679006i
\(838\) 0 0
\(839\) 35.2906i 1.21837i 0.793029 + 0.609184i \(0.208503\pi\)
−0.793029 + 0.609184i \(0.791497\pi\)
\(840\) 0 0
\(841\) 26.0096i 0.896881i
\(842\) 0 0
\(843\) 2.36249 + 2.36249i 0.0813683 + 0.0813683i
\(844\) 0 0
\(845\) 34.5044 34.5044i 1.18699 1.18699i
\(846\) 0 0
\(847\) 18.7556 0.644451
\(848\) 0 0
\(849\) −13.8908 −0.476732
\(850\) 0 0
\(851\) 11.5386 11.5386i 0.395538 0.395538i
\(852\) 0 0
\(853\) 6.93449 + 6.93449i 0.237432 + 0.237432i 0.815786 0.578354i \(-0.196305\pi\)
−0.578354 + 0.815786i \(0.696305\pi\)
\(854\) 0 0
\(855\) 15.3642i 0.525445i
\(856\) 0 0
\(857\) 29.1791i 0.996737i −0.866965 0.498369i \(-0.833933\pi\)
0.866965 0.498369i \(-0.166067\pi\)
\(858\) 0 0
\(859\) −1.62614 1.62614i −0.0554833 0.0554833i 0.678821 0.734304i \(-0.262491\pi\)
−0.734304 + 0.678821i \(0.762491\pi\)
\(860\) 0 0
\(861\) 0.109792 0.109792i 0.00374169 0.00374169i
\(862\) 0 0
\(863\) 33.7059 1.14736 0.573681 0.819079i \(-0.305515\pi\)
0.573681 + 0.819079i \(0.305515\pi\)
\(864\) 0 0
\(865\) −20.9568 −0.712554
\(866\) 0 0
\(867\) 4.76608 4.76608i 0.161864 0.161864i
\(868\) 0 0
\(869\) −20.1181 20.1181i −0.682460 0.682460i
\(870\) 0 0
\(871\) 88.6891i 3.00512i
\(872\) 0 0
\(873\) 49.8900i 1.68852i
\(874\) 0 0
\(875\) −8.41814 8.41814i −0.284585 0.284585i
\(876\) 0 0
\(877\) 25.3846 25.3846i 0.857178 0.857178i −0.133827 0.991005i \(-0.542727\pi\)
0.991005 + 0.133827i \(0.0427266\pi\)
\(878\) 0 0
\(879\) −14.2665 −0.481196
\(880\) 0 0
\(881\) 13.0482 0.439606 0.219803 0.975544i \(-0.429459\pi\)
0.219803 + 0.975544i \(0.429459\pi\)
\(882\) 0 0
\(883\) −24.2895 + 24.2895i −0.817407 + 0.817407i −0.985732 0.168325i \(-0.946164\pi\)
0.168325 + 0.985732i \(0.446164\pi\)
\(884\) 0 0
\(885\) −3.40303 3.40303i −0.114392 0.114392i
\(886\) 0 0
\(887\) 19.8489i 0.666461i −0.942845 0.333230i \(-0.891861\pi\)
0.942845 0.333230i \(-0.108139\pi\)
\(888\) 0 0
\(889\) 17.6789i 0.592930i
\(890\) 0 0
\(891\) −23.1159 23.1159i −0.774413 0.774413i
\(892\) 0 0
\(893\) −27.5264 + 27.5264i −0.921137 + 0.921137i
\(894\) 0 0
\(895\) −22.5325 −0.753178
\(896\) 0 0
\(897\) 10.1268 0.338124
\(898\) 0 0
\(899\) 1.01945 1.01945i 0.0340006 0.0340006i
\(900\) 0 0
\(901\) 0.253935 + 0.253935i 0.00845981 + 0.00845981i
\(902\) 0 0
\(903\) 1.04408i 0.0347448i
\(904\) 0 0
\(905\) 11.5008i 0.382300i
\(906\) 0 0
\(907\) −39.6217 39.6217i −1.31562 1.31562i −0.917208 0.398408i \(-0.869563\pi\)
−0.398408 0.917208i \(-0.630437\pi\)
\(908\) 0 0
\(909\) −7.65297 + 7.65297i −0.253833 + 0.253833i
\(910\) 0 0
\(911\) −11.6260 −0.385188 −0.192594 0.981279i \(-0.561690\pi\)
−0.192594 + 0.981279i \(0.561690\pi\)
\(912\) 0 0
\(913\) 45.0553 1.49111
\(914\) 0 0
\(915\) −4.21178 + 4.21178i −0.139237 + 0.139237i
\(916\) 0 0
\(917\) 2.17911 + 2.17911i 0.0719606 + 0.0719606i
\(918\) 0 0
\(919\) 42.1056i 1.38894i −0.719523 0.694468i \(-0.755640\pi\)
0.719523 0.694468i \(-0.244360\pi\)
\(920\) 0 0
\(921\) 8.00224i 0.263683i
\(922\) 0 0
\(923\) −9.57342 9.57342i −0.315113 0.315113i
\(924\) 0 0
\(925\) 10.8020 10.8020i 0.355167 0.355167i
\(926\) 0 0
\(927\) 20.5311 0.674331
\(928\) 0 0
\(929\) 30.3239 0.994894 0.497447 0.867494i \(-0.334271\pi\)
0.497447 + 0.867494i \(0.334271\pi\)
\(930\) 0 0
\(931\) 2.55919 2.55919i 0.0838742 0.0838742i
\(932\) 0 0
\(933\) −7.99802 7.99802i −0.261843 0.261843i
\(934\) 0 0
\(935\) 46.5524i 1.52243i
\(936\) 0 0
\(937\) 17.7772i 0.580757i −0.956912 0.290378i \(-0.906219\pi\)
0.956912 0.290378i \(-0.0937812\pi\)
\(938\) 0 0
\(939\) 2.10079 + 2.10079i 0.0685567 + 0.0685567i
\(940\) 0 0
\(941\) 21.1596 21.1596i 0.689783 0.689783i −0.272401 0.962184i \(-0.587818\pi\)
0.962184 + 0.272401i \(0.0878178\pi\)
\(942\) 0 0
\(943\) 0.686068 0.0223415
\(944\) 0 0
\(945\) 5.33310 0.173486
\(946\) 0 0
\(947\) 11.8498 11.8498i 0.385066 0.385066i −0.487857 0.872923i \(-0.662221\pi\)
0.872923 + 0.487857i \(0.162221\pi\)
\(948\) 0 0
\(949\) −25.5664 25.5664i −0.829920 0.829920i
\(950\) 0 0
\(951\) 4.60704i 0.149394i
\(952\) 0 0
\(953\) 14.1855i 0.459513i 0.973248 + 0.229757i \(0.0737930\pi\)
−0.973248 + 0.229757i \(0.926207\pi\)
\(954\) 0 0
\(955\) −23.5309 23.5309i −0.761443 0.761443i
\(956\) 0 0
\(957\) 3.93220 3.93220i 0.127110 0.127110i
\(958\) 0 0
\(959\) −8.54650 −0.275981
\(960\) 0 0
\(961\) −30.3049 −0.977578
\(962\) 0 0
\(963\) 6.19211 6.19211i 0.199538 0.199538i
\(964\) 0 0
\(965\) 15.1021 + 15.1021i 0.486154 + 0.486154i
\(966\) 0 0
\(967\) 8.54873i 0.274909i 0.990508 + 0.137454i \(0.0438920\pi\)
−0.990508 + 0.137454i \(0.956108\pi\)
\(968\) 0 0
\(969\) 11.3771i 0.365485i
\(970\) 0 0
\(971\) −25.5089 25.5089i −0.818619 0.818619i 0.167289 0.985908i \(-0.446499\pi\)
−0.985908 + 0.167289i \(0.946499\pi\)
\(972\) 0 0
\(973\) −5.72549 + 5.72549i −0.183551 + 0.183551i
\(974\) 0 0
\(975\) 9.48030 0.303613
\(976\) 0 0
\(977\) 13.2375 0.423506 0.211753 0.977323i \(-0.432083\pi\)
0.211753 + 0.977323i \(0.432083\pi\)
\(978\) 0 0
\(979\) −24.4025 + 24.4025i −0.779906 + 0.779906i
\(980\) 0 0
\(981\) 13.0721 + 13.0721i 0.417359 + 0.417359i
\(982\) 0 0
\(983\) 19.9233i 0.635455i −0.948182 0.317727i \(-0.897080\pi\)
0.948182 0.317727i \(-0.102920\pi\)
\(984\) 0 0
\(985\) 0.440142i 0.0140241i
\(986\) 0 0
\(987\) −4.48364 4.48364i −0.142716 0.142716i
\(988\) 0 0
\(989\) 3.26213 3.26213i 0.103730 0.103730i
\(990\) 0 0
\(991\) −54.3594 −1.72678 −0.863392 0.504533i \(-0.831665\pi\)
−0.863392 + 0.504533i \(0.831665\pi\)
\(992\) 0 0
\(993\) −12.2984 −0.390278
\(994\) 0 0
\(995\) 16.2966 16.2966i 0.516638 0.516638i
\(996\) 0 0
\(997\) −14.4032 14.4032i −0.456153 0.456153i 0.441237 0.897390i \(-0.354540\pi\)
−0.897390 + 0.441237i \(0.854540\pi\)
\(998\) 0 0
\(999\) 20.8749i 0.660452i
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 896.2.m.h.225.3 12
4.3 odd 2 896.2.m.g.225.4 12
8.3 odd 2 112.2.m.d.85.5 yes 12
8.5 even 2 448.2.m.d.113.4 12
16.3 odd 4 896.2.m.g.673.4 12
16.5 even 4 448.2.m.d.337.4 12
16.11 odd 4 112.2.m.d.29.5 12
16.13 even 4 inner 896.2.m.h.673.3 12
32.3 odd 8 7168.2.a.bj.1.6 12
32.13 even 8 7168.2.a.bi.1.6 12
32.19 odd 8 7168.2.a.bj.1.7 12
32.29 even 8 7168.2.a.bi.1.7 12
56.3 even 6 784.2.x.m.373.4 24
56.11 odd 6 784.2.x.l.373.4 24
56.19 even 6 784.2.x.m.165.1 24
56.27 even 2 784.2.m.h.197.5 12
56.51 odd 6 784.2.x.l.165.1 24
112.11 odd 12 784.2.x.l.765.1 24
112.27 even 4 784.2.m.h.589.5 12
112.59 even 12 784.2.x.m.765.1 24
112.75 even 12 784.2.x.m.557.4 24
112.107 odd 12 784.2.x.l.557.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.d.29.5 12 16.11 odd 4
112.2.m.d.85.5 yes 12 8.3 odd 2
448.2.m.d.113.4 12 8.5 even 2
448.2.m.d.337.4 12 16.5 even 4
784.2.m.h.197.5 12 56.27 even 2
784.2.m.h.589.5 12 112.27 even 4
784.2.x.l.165.1 24 56.51 odd 6
784.2.x.l.373.4 24 56.11 odd 6
784.2.x.l.557.4 24 112.107 odd 12
784.2.x.l.765.1 24 112.11 odd 12
784.2.x.m.165.1 24 56.19 even 6
784.2.x.m.373.4 24 56.3 even 6
784.2.x.m.557.4 24 112.75 even 12
784.2.x.m.765.1 24 112.59 even 12
896.2.m.g.225.4 12 4.3 odd 2
896.2.m.g.673.4 12 16.3 odd 4
896.2.m.h.225.3 12 1.1 even 1 trivial
896.2.m.h.673.3 12 16.13 even 4 inner
7168.2.a.bi.1.6 12 32.13 even 8
7168.2.a.bi.1.7 12 32.29 even 8
7168.2.a.bj.1.6 12 32.3 odd 8
7168.2.a.bj.1.7 12 32.19 odd 8