Properties

Label 448.2.m.d.113.4
Level $448$
Weight $2$
Character 448.113
Analytic conductor $3.577$
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] = [448,2,Mod(113,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, 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("448.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.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: \(3.57729801055\)
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 113.4
Root \(-1.12465 + 0.857418i\) of defining polynomial
Character \(\chi\) \(=\) 448.113
Dual form 448.2.m.d.337.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.416854 + 0.416854i) q^{3} +(-1.13169 - 1.13169i) q^{5} -1.00000i q^{7} +2.65247i q^{9} +O(q^{10})\) \(q+(-0.416854 + 0.416854i) q^{3} +(-1.13169 - 1.13169i) q^{5} -1.00000i q^{7} +2.65247i q^{9} +(-3.85718 - 3.85718i) q^{11} +(4.66311 - 4.66311i) q^{13} +0.943500 q^{15} -5.33230 q^{17} +(2.55919 - 2.55919i) q^{19} +(0.416854 + 0.416854i) q^{21} -2.60484i q^{23} -2.43855i q^{25} +(-2.35625 - 2.35625i) q^{27} +(-1.22279 + 1.22279i) q^{29} +0.833708 q^{31} +3.21576 q^{33} +(-1.13169 + 1.13169i) q^{35} +(-4.42967 - 4.42967i) q^{37} +3.88768i q^{39} +0.263382i q^{41} +(-1.25233 - 1.25233i) q^{43} +(3.00177 - 3.00177i) q^{45} +10.7559 q^{47} -1.00000 q^{49} +(2.22279 - 2.22279i) q^{51} +(0.0476221 + 0.0476221i) q^{53} +8.73026i q^{55} +2.13362i q^{57} +(3.60682 + 3.60682i) q^{59} +(4.46399 - 4.46399i) q^{61} +2.65247 q^{63} -10.5544 q^{65} +(-9.50964 + 9.50964i) q^{67} +(1.08584 + 1.08584i) q^{69} +2.05301i q^{71} +5.48268i q^{73} +(1.01652 + 1.01652i) q^{75} +(-3.85718 + 3.85718i) q^{77} -5.21576 q^{79} -5.99297 q^{81} +(-5.84045 + 5.84045i) q^{83} +(6.03452 + 6.03452i) q^{85} -1.01945i q^{87} +6.32651i q^{89} +(-4.66311 - 4.66311i) q^{91} +(-0.347535 + 0.347535i) q^{93} -5.79243 q^{95} +18.8089 q^{97} +(10.2310 - 10.2310i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 4 q^{5}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 −0.817128 0.576457i \(-0.804435\pi\)
0.576457 + 0.817128i \(0.304435\pi\)
\(4\) 0 0
\(5\) −1.13169 1.13169i −0.506108 0.506108i 0.407222 0.913329i \(-0.366498\pi\)
−0.913329 + 0.407222i \(0.866498\pi\)
\(6\) 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 \(-0.942661\pi\)
−0.179163 0.983819i \(-0.557339\pi\)
\(12\) 0 0
\(13\) 4.66311 4.66311i 1.29332 1.29332i 0.360591 0.932724i \(-0.382575\pi\)
0.932724 0.360591i \(-0.117425\pi\)
\(14\) 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 −0.349731 0.936850i \(-0.613727\pi\)
0.936850 + 0.349731i \(0.113727\pi\)
\(20\) 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 −0.811466 0.584399i \(-0.801330\pi\)
0.584399 + 0.811466i \(0.301330\pi\)
\(30\) 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 0.242034 0.970268i \(-0.422185\pi\)
−0.970268 + 0.242034i \(0.922185\pi\)
\(38\) 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 0.605140 0.796119i \(-0.293117\pi\)
−0.796119 + 0.605140i \(0.793117\pi\)
\(44\) 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 0.710370 0.703829i \(-0.248528\pi\)
−0.703829 + 0.710370i \(0.748528\pi\)
\(54\) 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 0.901774 0.432207i \(-0.142265\pi\)
−0.432207 + 0.901774i \(0.642265\pi\)
\(60\) 0 0
\(61\) 4.46399 4.46399i 0.571556 0.571556i −0.361007 0.932563i \(-0.617567\pi\)
0.932563 + 0.361007i \(0.117567\pi\)
\(62\) 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 \(0.556867\pi\)
−0.984084 + 0.177704i \(0.943133\pi\)
\(68\) 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 −0.950819 0.309746i \(-0.899756\pi\)
0.309746 + 0.950819i \(0.399756\pi\)
\(84\) 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 0.548838 0.835929i \(-0.315070\pi\)
−0.835929 + 0.548838i \(0.815070\pi\)
\(102\) 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 0.810886 0.585204i \(-0.198986\pi\)
−0.585204 + 0.810886i \(0.698986\pi\)
\(108\) 0 0
\(109\) −4.92827 + 4.92827i −0.472042 + 0.472042i −0.902575 0.430533i \(-0.858326\pi\)
0.430533 + 0.902575i \(0.358326\pi\)
\(110\) 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 −0.605475 0.795865i \(-0.707017\pi\)
0.795865 + 0.605475i \(0.207017\pi\)
\(132\) 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 0.906924 0.421295i \(-0.138424\pi\)
−0.421295 + 0.906924i \(0.638424\pi\)
\(140\) 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 0.723108 0.690735i \(-0.242713\pi\)
−0.690735 + 0.723108i \(0.742713\pi\)
\(150\) 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 −0.529464 0.848333i \(-0.677607\pi\)
0.848333 + 0.529464i \(0.177607\pi\)
\(158\) 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 −0.826713 0.562623i \(-0.809792\pi\)
0.562623 + 0.826713i \(0.309792\pi\)
\(164\) 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 −0.261302 0.965257i \(-0.584152\pi\)
0.965257 + 0.261302i \(0.0841519\pi\)
\(174\) 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 −0.229274 0.973362i \(-0.573635\pi\)
0.973362 + 0.229274i \(0.0736350\pi\)
\(180\) 0 0
\(181\) −5.08125 5.08125i −0.377687 0.377687i 0.492580 0.870267i \(-0.336054\pi\)
−0.870267 + 0.492580i \(0.836054\pi\)
\(182\) 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 0.714000 0.700145i \(-0.246882\pi\)
−0.700145 + 0.714000i \(0.746882\pi\)
\(198\) 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 −0.921081 0.389370i \(-0.872693\pi\)
0.389370 + 0.921081i \(0.372693\pi\)
\(212\) 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 −0.195911 0.980622i \(-0.562766\pi\)
0.980622 + 0.195911i \(0.0627664\pi\)
\(228\) 0 0
\(229\) −9.37815 9.37815i −0.619726 0.619726i 0.325735 0.945461i \(-0.394388\pi\)
−0.945461 + 0.325735i \(0.894388\pi\)
\(230\) 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 0.229071 0.973410i \(-0.426431\pi\)
−0.973410 + 0.229071i \(0.926431\pi\)
\(252\) 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 \(-0.349513\pi\)
0.890312 0.455352i \(-0.150487\pi\)
\(270\) 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 0.664040 0.747697i \(-0.268841\pi\)
−0.747697 + 0.664040i \(0.768841\pi\)
\(278\) 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 0.999955 0.00953050i \(-0.00303370\pi\)
−0.00953050 + 0.999955i \(0.503034\pi\)
\(284\) 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 1.00000 0.000302098i \(-9.61608e-5\pi\)
−0.000302098 1.00000i \(0.500096\pi\)
\(294\) 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 −0.377998 0.925806i \(-0.623388\pi\)
0.925806 + 0.377998i \(0.123388\pi\)
\(308\) 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 −0.534683 0.845052i \(-0.679569\pi\)
0.845052 + 0.534683i \(0.179569\pi\)
\(318\) 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 0.984756 0.173943i \(-0.0556509\pi\)
−0.173943 + 0.984756i \(0.555651\pi\)
\(332\) 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 0.541312 0.840822i \(-0.317928\pi\)
−0.840822 + 0.541312i \(0.817928\pi\)
\(348\) 0 0
\(349\) −17.9789 + 17.9789i −0.962388 + 0.962388i −0.999318 0.0369299i \(-0.988242\pi\)
0.0369299 + 0.999318i \(0.488242\pi\)
\(350\) 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 0.987698 0.156374i \(-0.0499805\pi\)
−0.156374 + 0.987698i \(0.549981\pi\)
\(374\) 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 0.173622 0.984812i \(-0.444453\pi\)
−0.984812 + 0.173622i \(0.944453\pi\)
\(380\) 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 0.966023 0.258458i \(-0.0832143\pi\)
−0.258458 + 0.966023i \(0.583214\pi\)
\(390\) 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 −0.734850 0.678230i \(-0.762747\pi\)
0.678230 + 0.734850i \(0.262747\pi\)
\(398\) 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 −0.590999 0.806672i \(-0.701266\pi\)
0.806672 + 0.590999i \(0.201266\pi\)
\(420\) 0 0
\(421\) −7.57494 7.57494i −0.369180 0.369180i 0.497998 0.867178i \(-0.334069\pi\)
−0.867178 + 0.497998i \(0.834069\pi\)
\(422\) 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 0.755617 0.655014i \(-0.227337\pi\)
−0.655014 + 0.755617i \(0.727337\pi\)
\(444\) 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 −0.842249 0.539089i \(-0.818769\pi\)
0.539089 + 0.842249i \(0.318769\pi\)
\(462\) 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 −0.976641 0.214879i \(-0.931064\pi\)
0.214879 + 0.976641i \(0.431064\pi\)
\(468\) 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 0.458773 0.888554i \(-0.348289\pi\)
−0.888554 + 0.458773i \(0.848289\pi\)
\(492\) 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 −0.891304 0.453406i \(-0.850209\pi\)
0.453406 + 0.891304i \(0.350209\pi\)
\(500\) 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 −0.710936 0.703257i \(-0.751728\pi\)
0.703257 + 0.710936i \(0.251728\pi\)
\(510\) 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 0.673795 0.738918i \(-0.264663\pi\)
−0.738918 + 0.673795i \(0.764663\pi\)
\(524\) 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 −0.397907 0.917426i \(-0.630263\pi\)
0.917426 + 0.397907i \(0.130263\pi\)
\(542\) 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 −0.918918 0.394449i \(-0.870935\pi\)
0.394449 + 0.918918i \(0.370935\pi\)
\(548\) 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 −0.754689 0.656083i \(-0.772212\pi\)
0.656083 + 0.754689i \(0.272212\pi\)
\(558\) 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 −0.405811 0.913957i \(-0.633011\pi\)
0.913957 + 0.405811i \(0.133011\pi\)
\(564\) 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 0.0743968 0.997229i \(-0.476297\pi\)
−0.997229 + 0.0743968i \(0.976297\pi\)
\(572\) 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 0.974913 0.222587i \(-0.0714502\pi\)
−0.222587 + 0.974913i \(0.571450\pi\)
\(588\) 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 0.283726 0.958905i \(-0.408429\pi\)
−0.958905 + 0.283726i \(0.908429\pi\)
\(614\) 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 \(-0.971887\pi\)
−0.0882048 0.996102i \(-0.528113\pi\)
\(620\) 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 −0.922089 0.386978i \(-0.873519\pi\)
0.386978 + 0.922089i \(0.373519\pi\)
\(644\) 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 −0.646673 0.762767i \(-0.723840\pi\)
0.762767 + 0.646673i \(0.223840\pi\)
\(654\) 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 −0.216391 0.976307i \(-0.569429\pi\)
0.976307 + 0.216391i \(0.0694287\pi\)
\(660\) 0 0
\(661\) 11.9741 + 11.9741i 0.465739 + 0.465739i 0.900531 0.434792i \(-0.143178\pi\)
−0.434792 + 0.900531i \(0.643178\pi\)
\(662\) 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 0.706870 0.707344i \(-0.250107\pi\)
−0.707344 + 0.706870i \(0.750107\pi\)
\(678\) 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 0.923781 0.382921i \(-0.125082\pi\)
−0.382921 + 0.923781i \(0.625082\pi\)
\(684\) 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 \(0.572369\pi\)
−0.974266 + 0.225402i \(0.927631\pi\)
\(692\) 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 −0.949814 0.312816i \(-0.898728\pi\)
0.312816 + 0.949814i \(0.398728\pi\)
\(702\) 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 0.985007 0.172514i \(-0.0551890\pi\)
−0.172514 + 0.985007i \(0.555189\pi\)
\(710\) 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 \(0.503135\pi\)
−0.999951 + 0.00984960i \(0.996865\pi\)
\(734\) 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 \(-0.485931\pi\)
0.999023 0.0441845i \(-0.0140690\pi\)
\(740\) 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 0.0858779 0.996306i \(-0.472631\pi\)
−0.996306 + 0.0858779i \(0.972631\pi\)
\(758\) 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 0.792368 0.610044i \(-0.208848\pi\)
−0.610044 + 0.792368i \(0.708848\pi\)
\(774\) 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 −0.941552 0.336868i \(-0.890632\pi\)
0.336868 + 0.941552i \(0.390632\pi\)
\(788\) 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 \(-0.400218\pi\)
0.951268 0.308366i \(-0.0997820\pi\)
\(798\) 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 0.249117 0.968473i \(-0.419860\pi\)
−0.968473 + 0.249117i \(0.919860\pi\)
\(812\) 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 \(-0.995326\pi\)
−0.0146829 0.999892i \(-0.504674\pi\)
\(822\) 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 0.990122 0.140210i \(-0.0447777\pi\)
−0.140210 + 0.990122i \(0.544778\pi\)
\(828\) 0 0
\(829\) −11.3861 + 11.3861i −0.395455 + 0.395455i −0.876626 0.481172i \(-0.840211\pi\)
0.481172 + 0.876626i \(0.340211\pi\)
\(830\) 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 0.578354 0.815786i \(-0.303695\pi\)
−0.815786 + 0.578354i \(0.803695\pi\)
\(854\) 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 0.734304 0.678821i \(-0.237509\pi\)
−0.678821 + 0.734304i \(0.737509\pi\)
\(860\) 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 −0.991005 0.133827i \(-0.957273\pi\)
0.133827 + 0.991005i \(0.457273\pi\)
\(878\) 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 −0.168325 0.985732i \(-0.553836\pi\)
0.985732 + 0.168325i \(0.0538358\pi\)
\(884\) 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 \(0.130437\pi\)
0.398408 + 0.917208i \(0.369563\pi\)
\(908\) 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 −0.962184 0.272401i \(-0.912182\pi\)
0.272401 + 0.962184i \(0.412182\pi\)
\(942\) 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 −0.872923 0.487857i \(-0.837779\pi\)
0.487857 + 0.872923i \(0.337779\pi\)
\(948\) 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 0.985908 0.167289i \(-0.0535014\pi\)
−0.167289 + 0.985908i \(0.553501\pi\)
\(972\) 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 0.897390 0.441237i \(-0.145460\pi\)
−0.441237 + 0.897390i \(0.645460\pi\)
\(998\) 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 448.2.m.d.113.4 12
4.3 odd 2 112.2.m.d.85.5 yes 12
8.3 odd 2 896.2.m.g.225.4 12
8.5 even 2 896.2.m.h.225.3 12
16.3 odd 4 112.2.m.d.29.5 12
16.5 even 4 896.2.m.h.673.3 12
16.11 odd 4 896.2.m.g.673.4 12
16.13 even 4 inner 448.2.m.d.337.4 12
28.3 even 6 784.2.x.m.373.4 24
28.11 odd 6 784.2.x.l.373.4 24
28.19 even 6 784.2.x.m.165.1 24
28.23 odd 6 784.2.x.l.165.1 24
28.27 even 2 784.2.m.h.197.5 12
32.3 odd 8 7168.2.a.bj.1.7 12
32.13 even 8 7168.2.a.bi.1.7 12
32.19 odd 8 7168.2.a.bj.1.6 12
32.29 even 8 7168.2.a.bi.1.6 12
112.3 even 12 784.2.x.m.765.1 24
112.19 even 12 784.2.x.m.557.4 24
112.51 odd 12 784.2.x.l.557.4 24
112.67 odd 12 784.2.x.l.765.1 24
112.83 even 4 784.2.m.h.589.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.d.29.5 12 16.3 odd 4
112.2.m.d.85.5 yes 12 4.3 odd 2
448.2.m.d.113.4 12 1.1 even 1 trivial
448.2.m.d.337.4 12 16.13 even 4 inner
784.2.m.h.197.5 12 28.27 even 2
784.2.m.h.589.5 12 112.83 even 4
784.2.x.l.165.1 24 28.23 odd 6
784.2.x.l.373.4 24 28.11 odd 6
784.2.x.l.557.4 24 112.51 odd 12
784.2.x.l.765.1 24 112.67 odd 12
784.2.x.m.165.1 24 28.19 even 6
784.2.x.m.373.4 24 28.3 even 6
784.2.x.m.557.4 24 112.19 even 12
784.2.x.m.765.1 24 112.3 even 12
896.2.m.g.225.4 12 8.3 odd 2
896.2.m.g.673.4 12 16.11 odd 4
896.2.m.h.225.3 12 8.5 even 2
896.2.m.h.673.3 12 16.5 even 4
7168.2.a.bi.1.6 12 32.29 even 8
7168.2.a.bi.1.7 12 32.13 even 8
7168.2.a.bj.1.6 12 32.19 odd 8
7168.2.a.bj.1.7 12 32.3 odd 8