Properties

Label 756.2.w.a.521.5
Level $756$
Weight $2$
Character 756.521
Analytic conductor $6.037$
Analytic rank $0$
Dimension $16$
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] = [756,2,Mod(341,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.w (of order \(6\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 521.5
Root \(1.08696 - 1.34852i\) of defining polynomial
Character \(\chi\) \(=\) 756.521
Dual form 756.2.w.a.341.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0382122 + 0.0661855i) q^{5} +(0.232935 + 2.63548i) q^{7} +O(q^{10})\) \(q+(-0.0382122 + 0.0661855i) q^{5} +(0.232935 + 2.63548i) q^{7} +(-4.66300 + 2.69219i) q^{11} +(4.60313 - 2.65762i) q^{13} +(-1.89092 + 3.27516i) q^{17} +(-4.33939 + 2.50535i) q^{19} +(2.02463 + 1.16892i) q^{23} +(2.49708 + 4.32507i) q^{25} +(-8.84430 - 5.10626i) q^{29} +5.74620i q^{31} +(-0.183331 - 0.0852905i) q^{35} +(0.354486 + 0.613988i) q^{37} +(3.29910 + 5.71422i) q^{41} +(0.716520 - 1.24105i) q^{43} +2.92385 q^{47} +(-6.89148 + 1.22779i) q^{49} +(10.4835 + 6.05264i) q^{53} -0.411498i q^{55} +0.579903 q^{59} +2.77868i q^{61} +0.406214i q^{65} +5.27185 q^{67} +3.32103i q^{71} +(-6.17326 - 3.56413i) q^{73} +(-8.18137 - 11.6621i) q^{77} +0.938245 q^{79} +(-6.49790 + 11.2547i) q^{83} +(-0.144512 - 0.250303i) q^{85} +(-1.51794 - 2.62915i) q^{89} +(8.07632 + 11.5124i) q^{91} -0.382940i q^{95} +(-6.18183 - 3.56908i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{7} + 6 q^{11} - 3 q^{13} - 9 q^{17} - 21 q^{23} - 8 q^{25} - 6 q^{29} + 15 q^{35} + q^{37} + 6 q^{41} - 2 q^{43} + 36 q^{47} - 5 q^{49} + 30 q^{59} + 14 q^{67} - 3 q^{77} + 2 q^{79} + 6 q^{85} - 21 q^{89} + 9 q^{91} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.0382122 + 0.0661855i −0.0170890 + 0.0295991i −0.874443 0.485127i \(-0.838773\pi\)
0.857354 + 0.514727i \(0.172107\pi\)
\(6\) 0 0
\(7\) 0.232935 + 2.63548i 0.0880412 + 0.996117i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.66300 + 2.69219i −1.40595 + 0.811725i −0.994994 0.0999316i \(-0.968138\pi\)
−0.410954 + 0.911656i \(0.634804\pi\)
\(12\) 0 0
\(13\) 4.60313 2.65762i 1.27668 0.737091i 0.300442 0.953800i \(-0.402866\pi\)
0.976236 + 0.216709i \(0.0695324\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.89092 + 3.27516i −0.458615 + 0.794344i −0.998888 0.0471458i \(-0.984987\pi\)
0.540273 + 0.841490i \(0.318321\pi\)
\(18\) 0 0
\(19\) −4.33939 + 2.50535i −0.995525 + 0.574767i −0.906921 0.421300i \(-0.861574\pi\)
−0.0886040 + 0.996067i \(0.528241\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.02463 + 1.16892i 0.422164 + 0.243737i 0.696003 0.718039i \(-0.254960\pi\)
−0.273839 + 0.961776i \(0.588293\pi\)
\(24\) 0 0
\(25\) 2.49708 + 4.32507i 0.499416 + 0.865014i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.84430 5.10626i −1.64235 0.948209i −0.979997 0.199013i \(-0.936226\pi\)
−0.662349 0.749196i \(-0.730440\pi\)
\(30\) 0 0
\(31\) 5.74620i 1.03205i 0.856574 + 0.516024i \(0.172589\pi\)
−0.856574 + 0.516024i \(0.827411\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.183331 0.0852905i −0.0309887 0.0144167i
\(36\) 0 0
\(37\) 0.354486 + 0.613988i 0.0582771 + 0.100939i 0.893692 0.448681i \(-0.148106\pi\)
−0.835415 + 0.549620i \(0.814773\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.29910 + 5.71422i 0.515234 + 0.892411i 0.999844 + 0.0176805i \(0.00562816\pi\)
−0.484610 + 0.874730i \(0.661039\pi\)
\(42\) 0 0
\(43\) 0.716520 1.24105i 0.109268 0.189258i −0.806206 0.591635i \(-0.798483\pi\)
0.915474 + 0.402377i \(0.131816\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.92385 0.426487 0.213244 0.976999i \(-0.431597\pi\)
0.213244 + 0.976999i \(0.431597\pi\)
\(48\) 0 0
\(49\) −6.89148 + 1.22779i −0.984497 + 0.175399i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.4835 + 6.05264i 1.44002 + 0.831394i 0.997850 0.0655390i \(-0.0208767\pi\)
0.442167 + 0.896933i \(0.354210\pi\)
\(54\) 0 0
\(55\) 0.411498i 0.0554863i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.579903 0.0754969 0.0377484 0.999287i \(-0.487981\pi\)
0.0377484 + 0.999287i \(0.487981\pi\)
\(60\) 0 0
\(61\) 2.77868i 0.355773i 0.984051 + 0.177887i \(0.0569261\pi\)
−0.984051 + 0.177887i \(0.943074\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.406214i 0.0503847i
\(66\) 0 0
\(67\) 5.27185 0.644059 0.322030 0.946730i \(-0.395635\pi\)
0.322030 + 0.946730i \(0.395635\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.32103i 0.394134i 0.980390 + 0.197067i \(0.0631416\pi\)
−0.980390 + 0.197067i \(0.936858\pi\)
\(72\) 0 0
\(73\) −6.17326 3.56413i −0.722525 0.417150i 0.0931564 0.995651i \(-0.470304\pi\)
−0.815681 + 0.578502i \(0.803638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.18137 11.6621i −0.932354 1.32902i
\(78\) 0 0
\(79\) 0.938245 0.105561 0.0527804 0.998606i \(-0.483192\pi\)
0.0527804 + 0.998606i \(0.483192\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.49790 + 11.2547i −0.713238 + 1.23536i 0.250398 + 0.968143i \(0.419439\pi\)
−0.963635 + 0.267221i \(0.913895\pi\)
\(84\) 0 0
\(85\) −0.144512 0.250303i −0.0156746 0.0271491i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.51794 2.62915i −0.160901 0.278689i 0.774291 0.632830i \(-0.218107\pi\)
−0.935192 + 0.354141i \(0.884773\pi\)
\(90\) 0 0
\(91\) 8.07632 + 11.5124i 0.846629 + 1.20683i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.382940i 0.0392888i
\(96\) 0 0
\(97\) −6.18183 3.56908i −0.627670 0.362385i 0.152179 0.988353i \(-0.451371\pi\)
−0.779849 + 0.625967i \(0.784704\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.08628 7.07765i −0.406600 0.704252i 0.587906 0.808929i \(-0.299952\pi\)
−0.994506 + 0.104677i \(0.966619\pi\)
\(102\) 0 0
\(103\) −6.46599 3.73314i −0.637113 0.367837i 0.146389 0.989227i \(-0.453235\pi\)
−0.783502 + 0.621390i \(0.786568\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.99991 2.30935i 0.386686 0.223253i −0.294037 0.955794i \(-0.594999\pi\)
0.680723 + 0.732541i \(0.261666\pi\)
\(108\) 0 0
\(109\) 5.22792 9.05503i 0.500744 0.867314i −0.499256 0.866455i \(-0.666393\pi\)
1.00000 0.000859385i \(-0.000273551\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.6379 9.60591i 1.56516 0.903648i 0.568445 0.822721i \(-0.307545\pi\)
0.996720 0.0809270i \(-0.0257881\pi\)
\(114\) 0 0
\(115\) −0.154731 + 0.0893340i −0.0144287 + 0.00833044i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.07208 4.22057i −0.831636 0.386899i
\(120\) 0 0
\(121\) 8.99573 15.5811i 0.817793 1.41646i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.763798 −0.0683162
\(126\) 0 0
\(127\) 1.26488 0.112240 0.0561198 0.998424i \(-0.482127\pi\)
0.0561198 + 0.998424i \(0.482127\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.24394 12.5469i 0.632906 1.09623i −0.354049 0.935227i \(-0.615195\pi\)
0.986955 0.160998i \(-0.0514714\pi\)
\(132\) 0 0
\(133\) −7.61359 10.8528i −0.660182 0.941056i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.3414 7.70264i 1.13983 0.658081i 0.193442 0.981112i \(-0.438035\pi\)
0.946389 + 0.323030i \(0.104702\pi\)
\(138\) 0 0
\(139\) 0.374701 0.216333i 0.0317817 0.0183492i −0.484025 0.875054i \(-0.660826\pi\)
0.515807 + 0.856705i \(0.327492\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −14.3096 + 24.7850i −1.19663 + 2.07262i
\(144\) 0 0
\(145\) 0.675921 0.390243i 0.0561322 0.0324079i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.04535 + 2.33558i 0.331408 + 0.191338i 0.656466 0.754356i \(-0.272051\pi\)
−0.325058 + 0.945694i \(0.605384\pi\)
\(150\) 0 0
\(151\) 4.12276 + 7.14083i 0.335506 + 0.581113i 0.983582 0.180463i \(-0.0577595\pi\)
−0.648076 + 0.761575i \(0.724426\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.380316 0.219575i −0.0305477 0.0176367i
\(156\) 0 0
\(157\) 17.5900i 1.40383i 0.712258 + 0.701917i \(0.247672\pi\)
−0.712258 + 0.701917i \(0.752328\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.60905 + 5.60814i −0.205622 + 0.441984i
\(162\) 0 0
\(163\) −5.27097 9.12959i −0.412854 0.715085i 0.582346 0.812941i \(-0.302135\pi\)
−0.995201 + 0.0978563i \(0.968801\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.59146 7.95265i −0.355298 0.615395i 0.631871 0.775074i \(-0.282287\pi\)
−0.987169 + 0.159679i \(0.948954\pi\)
\(168\) 0 0
\(169\) 7.62587 13.2084i 0.586605 1.01603i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.44717 0.186055 0.0930274 0.995664i \(-0.470346\pi\)
0.0930274 + 0.995664i \(0.470346\pi\)
\(174\) 0 0
\(175\) −10.8170 + 7.58846i −0.817686 + 0.573633i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.05509 + 2.91856i 0.377835 + 0.218143i 0.676876 0.736097i \(-0.263333\pi\)
−0.299041 + 0.954240i \(0.596667\pi\)
\(180\) 0 0
\(181\) 16.0704i 1.19451i −0.802053 0.597253i \(-0.796259\pi\)
0.802053 0.597253i \(-0.203741\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.0541828 −0.00398360
\(186\) 0 0
\(187\) 20.3628i 1.48907i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.97223i 0.576850i 0.957502 + 0.288425i \(0.0931316\pi\)
−0.957502 + 0.288425i \(0.906868\pi\)
\(192\) 0 0
\(193\) 0.718054 0.0516867 0.0258433 0.999666i \(-0.491773\pi\)
0.0258433 + 0.999666i \(0.491773\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.5035i 0.962083i −0.876698 0.481042i \(-0.840259\pi\)
0.876698 0.481042i \(-0.159741\pi\)
\(198\) 0 0
\(199\) 21.2568 + 12.2726i 1.50685 + 0.869983i 0.999968 + 0.00796947i \(0.00253679\pi\)
0.506886 + 0.862013i \(0.330797\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 11.3973 24.4984i 0.799932 1.71945i
\(204\) 0 0
\(205\) −0.504265 −0.0352194
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.4897 23.3649i 0.933105 1.61618i
\(210\) 0 0
\(211\) −11.7838 20.4101i −0.811227 1.40509i −0.912005 0.410178i \(-0.865467\pi\)
0.100778 0.994909i \(-0.467867\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.0547597 + 0.0948465i 0.00373458 + 0.00646848i
\(216\) 0 0
\(217\) −15.1440 + 1.33849i −1.02804 + 0.0908628i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.1013i 1.35216i
\(222\) 0 0
\(223\) 6.47489 + 3.73828i 0.433590 + 0.250334i 0.700875 0.713284i \(-0.252793\pi\)
−0.267285 + 0.963618i \(0.586126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.318701 0.552006i −0.0211529 0.0366379i 0.855255 0.518207i \(-0.173400\pi\)
−0.876408 + 0.481569i \(0.840067\pi\)
\(228\) 0 0
\(229\) −1.58351 0.914239i −0.104641 0.0604146i 0.446766 0.894651i \(-0.352576\pi\)
−0.551407 + 0.834236i \(0.685909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.4232 10.0593i 1.14143 0.659007i 0.194649 0.980873i \(-0.437643\pi\)
0.946785 + 0.321866i \(0.104310\pi\)
\(234\) 0 0
\(235\) −0.111727 + 0.193516i −0.00728825 + 0.0126236i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.41455 1.39404i 0.156184 0.0901730i −0.419871 0.907584i \(-0.637925\pi\)
0.576055 + 0.817411i \(0.304591\pi\)
\(240\) 0 0
\(241\) −20.0304 + 11.5645i −1.29027 + 0.744938i −0.978702 0.205286i \(-0.934187\pi\)
−0.311568 + 0.950224i \(0.600854\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.182077 0.503033i 0.0116325 0.0321376i
\(246\) 0 0
\(247\) −13.3165 + 23.0649i −0.847310 + 1.46758i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.6541 1.17743 0.588717 0.808339i \(-0.299633\pi\)
0.588717 + 0.808339i \(0.299633\pi\)
\(252\) 0 0
\(253\) −12.5878 −0.791388
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.43687 + 9.41694i −0.339143 + 0.587413i −0.984272 0.176661i \(-0.943470\pi\)
0.645129 + 0.764074i \(0.276804\pi\)
\(258\) 0 0
\(259\) −1.53558 + 1.07726i −0.0954162 + 0.0669376i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −16.4519 + 9.49852i −1.01447 + 0.585704i −0.912497 0.409083i \(-0.865849\pi\)
−0.101972 + 0.994787i \(0.532515\pi\)
\(264\) 0 0
\(265\) −0.801194 + 0.462570i −0.0492170 + 0.0284154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.29788 + 7.44415i −0.262046 + 0.453878i −0.966786 0.255589i \(-0.917731\pi\)
0.704739 + 0.709467i \(0.251064\pi\)
\(270\) 0 0
\(271\) 1.58706 0.916292i 0.0964073 0.0556608i −0.451021 0.892513i \(-0.648940\pi\)
0.547429 + 0.836852i \(0.315607\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −23.2878 13.4452i −1.40431 0.810776i
\(276\) 0 0
\(277\) −7.90931 13.6993i −0.475224 0.823113i 0.524373 0.851489i \(-0.324300\pi\)
−0.999597 + 0.0283760i \(0.990966\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.95916 + 5.74992i 0.594114 + 0.343012i 0.766722 0.641979i \(-0.221886\pi\)
−0.172609 + 0.984990i \(0.555220\pi\)
\(282\) 0 0
\(283\) 9.92818i 0.590169i −0.955471 0.295085i \(-0.904652\pi\)
0.955471 0.295085i \(-0.0953478\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.2912 + 10.0258i −0.843584 + 0.591802i
\(288\) 0 0
\(289\) 1.34887 + 2.33631i 0.0793454 + 0.137430i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.63598 + 14.9580i 0.504520 + 0.873854i 0.999986 + 0.00522664i \(0.00166370\pi\)
−0.495467 + 0.868627i \(0.665003\pi\)
\(294\) 0 0
\(295\) −0.0221594 + 0.0383812i −0.00129017 + 0.00223464i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.4262 0.718624
\(300\) 0 0
\(301\) 3.43766 + 1.59929i 0.198143 + 0.0921815i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.183908 0.106180i −0.0105306 0.00607982i
\(306\) 0 0
\(307\) 21.6425i 1.23520i 0.786490 + 0.617602i \(0.211896\pi\)
−0.786490 + 0.617602i \(0.788104\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.2032 −1.14562 −0.572808 0.819690i \(-0.694146\pi\)
−0.572808 + 0.819690i \(0.694146\pi\)
\(312\) 0 0
\(313\) 21.8407i 1.23451i 0.786764 + 0.617254i \(0.211755\pi\)
−0.786764 + 0.617254i \(0.788245\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.8594i 1.39624i 0.715981 + 0.698120i \(0.245980\pi\)
−0.715981 + 0.698120i \(0.754020\pi\)
\(318\) 0 0
\(319\) 54.9880 3.07874
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.9496i 1.05439i
\(324\) 0 0
\(325\) 22.9888 + 13.2726i 1.27519 + 0.736230i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.681067 + 7.70574i 0.0375484 + 0.424831i
\(330\) 0 0
\(331\) 16.1444 0.887375 0.443688 0.896181i \(-0.353670\pi\)
0.443688 + 0.896181i \(0.353670\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.201449 + 0.348920i −0.0110063 + 0.0190635i
\(336\) 0 0
\(337\) −7.81522 13.5364i −0.425722 0.737372i 0.570765 0.821113i \(-0.306647\pi\)
−0.996488 + 0.0837408i \(0.973313\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −15.4698 26.7946i −0.837739 1.45101i
\(342\) 0 0
\(343\) −4.84108 17.8764i −0.261394 0.965232i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.3830i 1.73841i 0.494451 + 0.869206i \(0.335369\pi\)
−0.494451 + 0.869206i \(0.664631\pi\)
\(348\) 0 0
\(349\) 26.0421 + 15.0354i 1.39400 + 0.804827i 0.993755 0.111581i \(-0.0355915\pi\)
0.400246 + 0.916408i \(0.368925\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.50607 14.7329i −0.452733 0.784156i 0.545822 0.837901i \(-0.316217\pi\)
−0.998555 + 0.0537453i \(0.982884\pi\)
\(354\) 0 0
\(355\) −0.219804 0.126904i −0.0116660 0.00673537i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.2692 14.5892i 1.33366 0.769987i 0.347798 0.937570i \(-0.386930\pi\)
0.985858 + 0.167583i \(0.0535962\pi\)
\(360\) 0 0
\(361\) 3.05356 5.28892i 0.160714 0.278364i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.471788 0.272387i 0.0246945 0.0142574i
\(366\) 0 0
\(367\) −15.6981 + 9.06329i −0.819433 + 0.473100i −0.850221 0.526426i \(-0.823532\pi\)
0.0307880 + 0.999526i \(0.490198\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.5096 + 29.0388i −0.701385 + 1.50762i
\(372\) 0 0
\(373\) 10.1823 17.6362i 0.527219 0.913170i −0.472278 0.881450i \(-0.656568\pi\)
0.999497 0.0317200i \(-0.0100985\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −54.2820 −2.79566
\(378\) 0 0
\(379\) −21.9961 −1.12986 −0.564931 0.825138i \(-0.691097\pi\)
−0.564931 + 0.825138i \(0.691097\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.3127 + 28.2544i −0.833538 + 1.44373i 0.0616774 + 0.998096i \(0.480355\pi\)
−0.895215 + 0.445634i \(0.852978\pi\)
\(384\) 0 0
\(385\) 1.08449 0.0958523i 0.0552709 0.00488509i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.6400 7.87504i 0.691574 0.399280i −0.112628 0.993637i \(-0.535927\pi\)
0.804201 + 0.594357i \(0.202593\pi\)
\(390\) 0 0
\(391\) −7.65680 + 4.42066i −0.387221 + 0.223562i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.0358524 + 0.0620983i −0.00180393 + 0.00312450i
\(396\) 0 0
\(397\) −2.95864 + 1.70817i −0.148490 + 0.0857308i −0.572404 0.819972i \(-0.693989\pi\)
0.423914 + 0.905702i \(0.360656\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.851348 + 0.491526i 0.0425143 + 0.0245456i 0.521106 0.853492i \(-0.325519\pi\)
−0.478592 + 0.878037i \(0.658853\pi\)
\(402\) 0 0
\(403\) 15.2712 + 26.4505i 0.760713 + 1.31759i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.30594 1.90868i −0.163869 0.0946099i
\(408\) 0 0
\(409\) 28.8900i 1.42852i 0.699880 + 0.714260i \(0.253237\pi\)
−0.699880 + 0.714260i \(0.746763\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.135080 + 1.52832i 0.00664684 + 0.0752037i
\(414\) 0 0
\(415\) −0.496599 0.860135i −0.0243771 0.0422223i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.28926 + 10.8933i 0.307251 + 0.532174i 0.977760 0.209727i \(-0.0672577\pi\)
−0.670509 + 0.741901i \(0.733924\pi\)
\(420\) 0 0
\(421\) −13.0232 + 22.5568i −0.634710 + 1.09935i 0.351866 + 0.936050i \(0.385547\pi\)
−0.986576 + 0.163300i \(0.947786\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.8871 −0.916158
\(426\) 0 0
\(427\) −7.32315 + 0.647252i −0.354392 + 0.0313227i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.28454 + 3.62838i 0.302716 + 0.174773i 0.643662 0.765310i \(-0.277414\pi\)
−0.340947 + 0.940083i \(0.610748\pi\)
\(432\) 0 0
\(433\) 8.29113i 0.398446i 0.979954 + 0.199223i \(0.0638419\pi\)
−0.979954 + 0.199223i \(0.936158\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.7142 −0.560367
\(438\) 0 0
\(439\) 3.27192i 0.156160i 0.996947 + 0.0780802i \(0.0248790\pi\)
−0.996947 + 0.0780802i \(0.975121\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.84907i 0.135363i −0.997707 0.0676817i \(-0.978440\pi\)
0.997707 0.0676817i \(-0.0215602\pi\)
\(444\) 0 0
\(445\) 0.232015 0.0109986
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 19.9802i 0.942925i 0.881886 + 0.471463i \(0.156274\pi\)
−0.881886 + 0.471463i \(0.843726\pi\)
\(450\) 0 0
\(451\) −30.7675 17.7636i −1.44878 0.836455i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.07057 + 0.0946216i −0.0501890 + 0.00443593i
\(456\) 0 0
\(457\) 18.3002 0.856046 0.428023 0.903768i \(-0.359210\pi\)
0.428023 + 0.903768i \(0.359210\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.52954 7.84539i 0.210962 0.365396i −0.741054 0.671445i \(-0.765674\pi\)
0.952016 + 0.306049i \(0.0990071\pi\)
\(462\) 0 0
\(463\) 10.8227 + 18.7455i 0.502974 + 0.871176i 0.999994 + 0.00343694i \(0.00109401\pi\)
−0.497021 + 0.867739i \(0.665573\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13.7761 23.8610i −0.637484 1.10415i −0.985983 0.166845i \(-0.946642\pi\)
0.348500 0.937309i \(-0.386691\pi\)
\(468\) 0 0
\(469\) 1.22800 + 13.8938i 0.0567038 + 0.641558i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.71602i 0.354783i
\(474\) 0 0
\(475\) −21.6716 12.5121i −0.994362 0.574095i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.47325 4.28380i −0.113006 0.195732i 0.803975 0.594663i \(-0.202715\pi\)
−0.916981 + 0.398931i \(0.869381\pi\)
\(480\) 0 0
\(481\) 3.26349 + 1.88418i 0.148802 + 0.0859110i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.472443 0.272765i 0.0214525 0.0123856i
\(486\) 0 0
\(487\) −4.78573 + 8.28913i −0.216862 + 0.375616i −0.953847 0.300293i \(-0.902916\pi\)
0.736985 + 0.675909i \(0.236249\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.0010 19.0531i 1.48931 0.859855i 0.489387 0.872067i \(-0.337221\pi\)
0.999925 + 0.0122119i \(0.00388725\pi\)
\(492\) 0 0
\(493\) 33.4477 19.3110i 1.50641 0.869725i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.75250 + 0.773585i −0.392603 + 0.0347000i
\(498\) 0 0
\(499\) 12.4192 21.5107i 0.555960 0.962951i −0.441868 0.897080i \(-0.645684\pi\)
0.997828 0.0658709i \(-0.0209825\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.2820 1.21645 0.608223 0.793766i \(-0.291883\pi\)
0.608223 + 0.793766i \(0.291883\pi\)
\(504\) 0 0
\(505\) 0.624584 0.0277936
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.8860 + 36.1757i −0.925758 + 1.60346i −0.135420 + 0.990788i \(0.543238\pi\)
−0.790338 + 0.612671i \(0.790095\pi\)
\(510\) 0 0
\(511\) 7.95522 17.0997i 0.351918 0.756446i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.494160 0.285303i 0.0217753 0.0125720i
\(516\) 0 0
\(517\) −13.6339 + 7.87154i −0.599619 + 0.346190i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.02629 + 3.50963i −0.0887732 + 0.153760i −0.906993 0.421146i \(-0.861628\pi\)
0.818220 + 0.574906i \(0.194961\pi\)
\(522\) 0 0
\(523\) 26.2429 15.1514i 1.14752 0.662523i 0.199241 0.979951i \(-0.436152\pi\)
0.948282 + 0.317428i \(0.102819\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.8198 10.8656i −0.819801 0.473312i
\(528\) 0 0
\(529\) −8.76726 15.1853i −0.381185 0.660232i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.3724 + 17.5355i 1.31558 + 0.759548i
\(534\) 0 0
\(535\) 0.352982i 0.0152607i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 28.8296 24.2783i 1.24178 1.04574i
\(540\) 0 0
\(541\) 8.82681 + 15.2885i 0.379494 + 0.657303i 0.990989 0.133946i \(-0.0427647\pi\)
−0.611495 + 0.791249i \(0.709431\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.399541 + 0.692026i 0.0171145 + 0.0296431i
\(546\) 0 0
\(547\) −2.18319 + 3.78140i −0.0933466 + 0.161681i −0.908917 0.416976i \(-0.863090\pi\)
0.815571 + 0.578657i \(0.196423\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 51.1719 2.18000
\(552\) 0 0
\(553\) 0.218550 + 2.47272i 0.00929371 + 0.105151i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.7527 8.51750i −0.625094 0.360898i 0.153756 0.988109i \(-0.450863\pi\)
−0.778849 + 0.627211i \(0.784196\pi\)
\(558\) 0 0
\(559\) 7.61695i 0.322163i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.9198 0.544507 0.272253 0.962226i \(-0.412231\pi\)
0.272253 + 0.962226i \(0.412231\pi\)
\(564\) 0 0
\(565\) 1.46825i 0.0617699i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.7408i 0.911420i 0.890128 + 0.455710i \(0.150615\pi\)
−0.890128 + 0.455710i \(0.849385\pi\)
\(570\) 0 0
\(571\) −33.6508 −1.40824 −0.704122 0.710079i \(-0.748659\pi\)
−0.704122 + 0.710079i \(0.748659\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.6755i 0.486904i
\(576\) 0 0
\(577\) 12.5598 + 7.25141i 0.522871 + 0.301880i 0.738109 0.674682i \(-0.235719\pi\)
−0.215237 + 0.976562i \(0.569052\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −31.1751 14.5035i −1.29336 0.601705i
\(582\) 0 0
\(583\) −65.1793 −2.69945
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.8417 27.4386i 0.653857 1.13251i −0.328322 0.944566i \(-0.606483\pi\)
0.982179 0.187948i \(-0.0601837\pi\)
\(588\) 0 0
\(589\) −14.3963 24.9350i −0.593187 1.02743i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.54101 6.13320i −0.145412 0.251860i 0.784115 0.620616i \(-0.213117\pi\)
−0.929526 + 0.368755i \(0.879784\pi\)
\(594\) 0 0
\(595\) 0.626005 0.439163i 0.0256637 0.0180039i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00650i 0.245419i −0.992443 0.122709i \(-0.960842\pi\)
0.992443 0.122709i \(-0.0391583\pi\)
\(600\) 0 0
\(601\) 0.530083 + 0.306043i 0.0216225 + 0.0124838i 0.510772 0.859716i \(-0.329360\pi\)
−0.489150 + 0.872200i \(0.662693\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.687494 + 1.19077i 0.0279506 + 0.0484119i
\(606\) 0 0
\(607\) 1.77500 + 1.02480i 0.0720450 + 0.0415952i 0.535590 0.844478i \(-0.320089\pi\)
−0.463545 + 0.886073i \(0.653423\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.4588 7.77047i 0.544487 0.314360i
\(612\) 0 0
\(613\) −4.93166 + 8.54189i −0.199188 + 0.345003i −0.948265 0.317479i \(-0.897164\pi\)
0.749077 + 0.662482i \(0.230497\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.2143 + 13.4028i −0.934571 + 0.539575i −0.888254 0.459352i \(-0.848082\pi\)
−0.0463170 + 0.998927i \(0.514748\pi\)
\(618\) 0 0
\(619\) 0.0603011 0.0348148i 0.00242370 0.00139933i −0.498788 0.866724i \(-0.666221\pi\)
0.501211 + 0.865325i \(0.332888\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.57547 4.61291i 0.263441 0.184812i
\(624\) 0 0
\(625\) −12.4562 + 21.5748i −0.498248 + 0.862992i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.68121 −0.106907
\(630\) 0 0
\(631\) 11.8214 0.470603 0.235301 0.971922i \(-0.424392\pi\)
0.235301 + 0.971922i \(0.424392\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.0483338 + 0.0837165i −0.00191807 + 0.00332219i
\(636\) 0 0
\(637\) −28.4594 + 23.9666i −1.12760 + 0.949592i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.7673 10.2580i 0.701766 0.405165i −0.106239 0.994341i \(-0.533881\pi\)
0.808005 + 0.589176i \(0.200547\pi\)
\(642\) 0 0
\(643\) 15.6081 9.01132i 0.615522 0.355372i −0.159602 0.987182i \(-0.551021\pi\)
0.775123 + 0.631810i \(0.217688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.11827 + 15.7933i −0.358476 + 0.620899i −0.987706 0.156320i \(-0.950037\pi\)
0.629230 + 0.777219i \(0.283370\pi\)
\(648\) 0 0
\(649\) −2.70409 + 1.56121i −0.106145 + 0.0612827i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.79559 + 4.50079i 0.305065 + 0.176129i 0.644716 0.764422i \(-0.276976\pi\)
−0.339651 + 0.940552i \(0.610309\pi\)
\(654\) 0 0
\(655\) 0.553614 + 0.958888i 0.0216315 + 0.0374669i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −30.4806 17.5980i −1.18735 0.685519i −0.229650 0.973273i \(-0.573758\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(660\) 0 0
\(661\) 12.5628i 0.488637i −0.969695 0.244318i \(-0.921436\pi\)
0.969695 0.244318i \(-0.0785642\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.00923 0.0892002i 0.0391363 0.00345904i
\(666\) 0 0
\(667\) −11.9376 20.6765i −0.462226 0.800599i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.48072 12.9570i −0.288790 0.500199i
\(672\) 0 0
\(673\) 23.8913 41.3810i 0.920942 1.59512i 0.122982 0.992409i \(-0.460754\pi\)
0.797960 0.602710i \(-0.205913\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 37.0471 1.42384 0.711918 0.702263i \(-0.247827\pi\)
0.711918 + 0.702263i \(0.247827\pi\)
\(678\) 0 0
\(679\) 7.96627 17.1234i 0.305717 0.657137i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −21.6844 12.5195i −0.829732 0.479046i 0.0240289 0.999711i \(-0.492351\pi\)
−0.853761 + 0.520665i \(0.825684\pi\)
\(684\) 0 0
\(685\) 1.17734i 0.0449839i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 64.3424 2.45125
\(690\) 0 0
\(691\) 46.4946i 1.76874i −0.466787 0.884370i \(-0.654589\pi\)
0.466787 0.884370i \(-0.345411\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.0330663i 0.00125428i
\(696\) 0 0
\(697\) −24.9533 −0.945174
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0041i 1.35986i −0.733279 0.679928i \(-0.762011\pi\)
0.733279 0.679928i \(-0.237989\pi\)
\(702\) 0 0
\(703\) −3.07651 1.77622i −0.116033 0.0669915i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.7011 12.4179i 0.665720 0.467025i
\(708\) 0 0
\(709\) 31.8316 1.19546 0.597731 0.801697i \(-0.296069\pi\)
0.597731 + 0.801697i \(0.296069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.71685 + 11.6339i −0.251548 + 0.435694i
\(714\) 0 0
\(715\) −1.09360 1.89418i −0.0408985 0.0708382i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0271 34.6879i −0.746883 1.29364i −0.949310 0.314342i \(-0.898216\pi\)
0.202427 0.979297i \(-0.435117\pi\)
\(720\) 0 0
\(721\) 8.33245 17.9106i 0.310317 0.667024i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 51.0030i 1.89420i
\(726\) 0 0
\(727\) 3.39242 + 1.95862i 0.125818 + 0.0726411i 0.561588 0.827417i \(-0.310191\pi\)
−0.435770 + 0.900058i \(0.643524\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.70976 + 4.69344i 0.100224 + 0.173593i
\(732\) 0 0
\(733\) 20.4239 + 11.7918i 0.754376 + 0.435539i 0.827273 0.561800i \(-0.189891\pi\)
−0.0728971 + 0.997339i \(0.523224\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.5827 + 14.1928i −0.905514 + 0.522798i
\(738\) 0 0
\(739\) 16.8641 29.2094i 0.620355 1.07449i −0.369065 0.929404i \(-0.620322\pi\)
0.989420 0.145083i \(-0.0463448\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29.4003 + 16.9743i −1.07859 + 0.622725i −0.930516 0.366251i \(-0.880641\pi\)
−0.148076 + 0.988976i \(0.547308\pi\)
\(744\) 0 0
\(745\) −0.309164 + 0.178496i −0.0113269 + 0.00653958i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.01796 + 10.0038i 0.256431 + 0.365529i
\(750\) 0 0
\(751\) −1.69831 + 2.94157i −0.0619724 + 0.107339i −0.895347 0.445369i \(-0.853072\pi\)
0.833375 + 0.552709i \(0.186406\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.630160 −0.0229339
\(756\) 0 0
\(757\) 29.1344 1.05891 0.529454 0.848339i \(-0.322397\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.36288 + 14.4849i −0.303154 + 0.525079i −0.976849 0.213931i \(-0.931373\pi\)
0.673694 + 0.739010i \(0.264706\pi\)
\(762\) 0 0
\(763\) 25.0821 + 11.6688i 0.908032 + 0.422440i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.66937 1.54116i 0.0963852 0.0556480i
\(768\) 0 0
\(769\) 24.0816 13.9035i 0.868404 0.501373i 0.00158643 0.999999i \(-0.499495\pi\)
0.866818 + 0.498625i \(0.166162\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.42238 + 11.1239i −0.230997 + 0.400098i −0.958102 0.286428i \(-0.907532\pi\)
0.727105 + 0.686526i \(0.240865\pi\)
\(774\) 0 0
\(775\) −24.8527 + 14.3487i −0.892736 + 0.515421i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28.6322 16.5308i −1.02586 0.592278i
\(780\) 0 0
\(781\) −8.94083 15.4860i −0.319928 0.554132i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.16420 0.672153i −0.0415522 0.0239902i
\(786\) 0 0
\(787\) 7.56610i 0.269702i 0.990866 + 0.134851i \(0.0430556\pi\)
−0.990866 + 0.134851i \(0.956944\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29.1917 + 41.6113i 1.03794 + 1.47953i
\(792\) 0 0
\(793\) 7.38467 + 12.7906i 0.262237 + 0.454208i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.03362 6.98643i −0.142878 0.247472i 0.785701 0.618606i \(-0.212302\pi\)
−0.928579 + 0.371134i \(0.878969\pi\)
\(798\) 0 0
\(799\) −5.52875 + 9.57608i −0.195593 + 0.338777i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 38.3812 1.35444
\(804\) 0 0
\(805\) −0.271480 0.386981i −0.00956842 0.0136393i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.0849492 0.0490454i −0.00298665 0.00172435i 0.498506 0.866886i \(-0.333882\pi\)
−0.501493 + 0.865162i \(0.667216\pi\)
\(810\) 0 0
\(811\) 30.3085i 1.06428i −0.846658 0.532138i \(-0.821389\pi\)
0.846658 0.532138i \(-0.178611\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.805662 0.0282211
\(816\) 0 0
\(817\) 7.18054i 0.251215i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.5742i 0.787846i 0.919143 + 0.393923i \(0.128882\pi\)
−0.919143 + 0.393923i \(0.871118\pi\)
\(822\) 0 0
\(823\) −24.5310 −0.855097 −0.427549 0.903992i \(-0.640623\pi\)
−0.427549 + 0.903992i \(0.640623\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 40.3057i 1.40157i 0.713375 + 0.700783i \(0.247166\pi\)
−0.713375 + 0.700783i \(0.752834\pi\)
\(828\) 0 0
\(829\) −46.8081 27.0247i −1.62571 0.938605i −0.985353 0.170529i \(-0.945452\pi\)
−0.640359 0.768076i \(-0.721214\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.01000 24.8924i 0.312178 0.862470i
\(834\) 0 0
\(835\) 0.701801 0.0242868
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.8650 + 20.5507i −0.409624 + 0.709489i −0.994847 0.101383i \(-0.967673\pi\)
0.585224 + 0.810872i \(0.301007\pi\)
\(840\) 0 0
\(841\) 37.6478 + 65.2079i 1.29820 + 2.24855i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.582803 + 1.00944i 0.0200490 + 0.0347259i
\(846\) 0 0
\(847\) 43.1589 + 20.0787i 1.48296 + 0.689911i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.65746i 0.0568170i
\(852\) 0 0
\(853\) −48.0748 27.7560i −1.64605 0.950347i −0.978621 0.205674i \(-0.934061\pi\)
−0.667429 0.744673i \(-0.732605\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.3048 26.5088i −0.522803 0.905522i −0.999648 0.0265343i \(-0.991553\pi\)
0.476845 0.878988i \(-0.341780\pi\)
\(858\) 0 0
\(859\) −36.4944 21.0700i −1.24517 0.718900i −0.275030 0.961436i \(-0.588688\pi\)
−0.970143 + 0.242535i \(0.922021\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.7782 13.1510i 0.775379 0.447665i −0.0594112 0.998234i \(-0.518922\pi\)
0.834790 + 0.550568i \(0.185589\pi\)
\(864\) 0 0
\(865\) −0.0935118 + 0.161967i −0.00317950 + 0.00550705i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.37504 + 2.52593i −0.148413 + 0.0856863i
\(870\) 0 0
\(871\) 24.2670 14.0106i 0.822256 0.474730i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.177916 2.01297i −0.00601464 0.0680509i
\(876\) 0 0
\(877\) −17.8533 + 30.9228i −0.602863 + 1.04419i 0.389522 + 0.921017i \(0.372640\pi\)
−0.992385 + 0.123172i \(0.960693\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.4482 −0.419392 −0.209696 0.977767i \(-0.567247\pi\)
−0.209696 + 0.977767i \(0.567247\pi\)
\(882\) 0 0
\(883\) 2.35637 0.0792982 0.0396491 0.999214i \(-0.487376\pi\)
0.0396491 + 0.999214i \(0.487376\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.7299 28.9770i 0.561734 0.972952i −0.435611 0.900135i \(-0.643468\pi\)
0.997345 0.0728170i \(-0.0231989\pi\)
\(888\) 0 0
\(889\) 0.294634 + 3.33355i 0.00988172 + 0.111804i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.6877 + 7.32526i −0.424579 + 0.245131i
\(894\) 0 0
\(895\) −0.386333 + 0.223049i −0.0129137 + 0.00745572i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 29.3416 50.8212i 0.978597 1.69498i
\(900\) 0 0
\(901\) −39.6468 + 22.8901i −1.32083 + 0.762579i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.06363 + 0.614087i 0.0353563 + 0.0204130i
\(906\) 0 0
\(907\) 0.467962 + 0.810535i 0.0155384 + 0.0269134i 0.873690 0.486483i \(-0.161720\pi\)
−0.858152 + 0.513396i \(0.828387\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.8739 16.6703i −0.956634 0.552313i −0.0614988 0.998107i \(-0.519588\pi\)
−0.895136 + 0.445794i \(0.852921\pi\)
\(912\) 0 0
\(913\) 69.9743i 2.31581i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 34.7544 + 16.1686i 1.14769 + 0.533935i
\(918\) 0 0
\(919\) −1.73484 3.00483i −0.0572270 0.0991200i 0.835993 0.548741i \(-0.184893\pi\)
−0.893220 + 0.449621i \(0.851559\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.82603 + 15.2871i 0.290512 + 0.503182i
\(924\) 0 0
\(925\) −1.77036 + 3.06635i −0.0582090 + 0.100821i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.1536 −0.497174 −0.248587 0.968610i \(-0.579966\pi\)
−0.248587 + 0.968610i \(0.579966\pi\)
\(930\) 0 0
\(931\) 26.8288 22.5934i 0.879279 0.740470i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.34772 + 0.778108i 0.0440752 + 0.0254468i
\(936\) 0 0
\(937\) 33.6651i 1.09979i 0.835233 + 0.549896i \(0.185333\pi\)
−0.835233 + 0.549896i \(0.814667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 37.7960 1.23212 0.616058 0.787701i \(-0.288729\pi\)
0.616058 + 0.787701i \(0.288729\pi\)
\(942\) 0 0
\(943\) 15.4255i 0.502325i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.9399i 0.355500i −0.984076 0.177750i \(-0.943118\pi\)
0.984076 0.177750i \(-0.0568818\pi\)
\(948\) 0 0
\(949\) −37.8884 −1.22991
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 11.0914i 0.359284i 0.983732 + 0.179642i \(0.0574939\pi\)
−0.983732 + 0.179642i \(0.942506\pi\)
\(954\) 0 0
\(955\) −0.527646 0.304637i −0.0170742 0.00985781i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 23.4078 + 33.3667i 0.755878 + 1.07747i
\(960\) 0 0
\(961\) −2.01885 −0.0651242
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.0274384 + 0.0475248i −0.000883275 + 0.00152988i
\(966\) 0 0
\(967\) 20.1446 + 34.8915i 0.647807 + 1.12203i 0.983646 + 0.180115i \(0.0576470\pi\)
−0.335839 + 0.941920i \(0.609020\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 23.8458 + 41.3021i 0.765248 + 1.32545i 0.940115 + 0.340856i \(0.110717\pi\)
−0.174867 + 0.984592i \(0.555950\pi\)
\(972\) 0 0
\(973\) 0.657423 + 0.937123i 0.0210760 + 0.0300428i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.6901i 0.533963i −0.963702 0.266982i \(-0.913974\pi\)
0.963702 0.266982i \(-0.0860263\pi\)
\(978\) 0 0
\(979\) 14.1563 + 8.17314i 0.452437 + 0.261215i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −16.9255 29.3157i −0.539838 0.935027i −0.998912 0.0466291i \(-0.985152\pi\)
0.459074 0.888398i \(-0.348181\pi\)
\(984\) 0 0
\(985\) 0.893735 + 0.515998i 0.0284768 + 0.0164411i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.90137 1.67511i 0.0922583 0.0532653i
\(990\) 0 0
\(991\) −4.09775 + 7.09751i −0.130169 + 0.225460i −0.923742 0.383016i \(-0.874885\pi\)
0.793572 + 0.608476i \(0.208219\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.62454 + 0.937928i −0.0515014 + 0.0297343i
\(996\) 0 0
\(997\) −18.7391 + 10.8190i −0.593472 + 0.342641i −0.766469 0.642281i \(-0.777988\pi\)
0.172997 + 0.984922i \(0.444655\pi\)
\(998\) 0 0
\(999\) 0 0
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 756.2.w.a.521.5 16
3.2 odd 2 252.2.w.a.101.1 yes 16
4.3 odd 2 3024.2.ca.d.2033.5 16
7.2 even 3 5292.2.bm.a.4625.4 16
7.3 odd 6 5292.2.x.b.4409.4 16
7.4 even 3 5292.2.x.a.4409.5 16
7.5 odd 6 756.2.bm.a.89.5 16
7.6 odd 2 5292.2.w.b.521.4 16
9.2 odd 6 2268.2.t.b.1781.4 16
9.4 even 3 252.2.bm.a.185.3 yes 16
9.5 odd 6 756.2.bm.a.17.5 16
9.7 even 3 2268.2.t.a.1781.5 16
12.11 even 2 1008.2.ca.d.353.8 16
21.2 odd 6 1764.2.bm.a.1685.6 16
21.5 even 6 252.2.bm.a.173.3 yes 16
21.11 odd 6 1764.2.x.a.1469.5 16
21.17 even 6 1764.2.x.b.1469.4 16
21.20 even 2 1764.2.w.b.1109.8 16
28.19 even 6 3024.2.df.d.1601.5 16
36.23 even 6 3024.2.df.d.17.5 16
36.31 odd 6 1008.2.df.d.689.6 16
63.4 even 3 1764.2.x.b.293.4 16
63.5 even 6 inner 756.2.w.a.341.5 16
63.13 odd 6 1764.2.bm.a.1697.6 16
63.23 odd 6 5292.2.w.b.1097.4 16
63.31 odd 6 1764.2.x.a.293.5 16
63.32 odd 6 5292.2.x.b.881.4 16
63.40 odd 6 252.2.w.a.5.1 16
63.41 even 6 5292.2.bm.a.2285.4 16
63.47 even 6 2268.2.t.a.2105.5 16
63.58 even 3 1764.2.w.b.509.8 16
63.59 even 6 5292.2.x.a.881.5 16
63.61 odd 6 2268.2.t.b.2105.4 16
84.47 odd 6 1008.2.df.d.929.6 16
252.103 even 6 1008.2.ca.d.257.8 16
252.131 odd 6 3024.2.ca.d.2609.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.1 16 63.40 odd 6
252.2.w.a.101.1 yes 16 3.2 odd 2
252.2.bm.a.173.3 yes 16 21.5 even 6
252.2.bm.a.185.3 yes 16 9.4 even 3
756.2.w.a.341.5 16 63.5 even 6 inner
756.2.w.a.521.5 16 1.1 even 1 trivial
756.2.bm.a.17.5 16 9.5 odd 6
756.2.bm.a.89.5 16 7.5 odd 6
1008.2.ca.d.257.8 16 252.103 even 6
1008.2.ca.d.353.8 16 12.11 even 2
1008.2.df.d.689.6 16 36.31 odd 6
1008.2.df.d.929.6 16 84.47 odd 6
1764.2.w.b.509.8 16 63.58 even 3
1764.2.w.b.1109.8 16 21.20 even 2
1764.2.x.a.293.5 16 63.31 odd 6
1764.2.x.a.1469.5 16 21.11 odd 6
1764.2.x.b.293.4 16 63.4 even 3
1764.2.x.b.1469.4 16 21.17 even 6
1764.2.bm.a.1685.6 16 21.2 odd 6
1764.2.bm.a.1697.6 16 63.13 odd 6
2268.2.t.a.1781.5 16 9.7 even 3
2268.2.t.a.2105.5 16 63.47 even 6
2268.2.t.b.1781.4 16 9.2 odd 6
2268.2.t.b.2105.4 16 63.61 odd 6
3024.2.ca.d.2033.5 16 4.3 odd 2
3024.2.ca.d.2609.5 16 252.131 odd 6
3024.2.df.d.17.5 16 36.23 even 6
3024.2.df.d.1601.5 16 28.19 even 6
5292.2.w.b.521.4 16 7.6 odd 2
5292.2.w.b.1097.4 16 63.23 odd 6
5292.2.x.a.881.5 16 63.59 even 6
5292.2.x.a.4409.5 16 7.4 even 3
5292.2.x.b.881.4 16 63.32 odd 6
5292.2.x.b.4409.4 16 7.3 odd 6
5292.2.bm.a.2285.4 16 63.41 even 6
5292.2.bm.a.4625.4 16 7.2 even 3