Properties

Label 2736.2.dc.f.449.8
Level $2736$
Weight $2$
Character 2736.449
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
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] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + \cdots + 414864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.8
Root \(3.16384 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.449
Dual form 2736.2.dc.f.1889.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.76516 - 1.01912i) q^{5} -3.69902 q^{7} +O(q^{10})\) \(q+(1.76516 - 1.01912i) q^{5} -3.69902 q^{7} +0.605557i q^{11} +(2.65015 + 1.53006i) q^{13} +(2.40852 - 1.39056i) q^{17} +(-0.780721 + 4.28841i) q^{19} +(2.23644 + 1.29121i) q^{23} +(-0.422804 + 0.732318i) q^{25} +(-1.70567 + 2.95430i) q^{29} -4.42738i q^{31} +(-6.52937 + 3.76973i) q^{35} +3.34720i q^{37} +(-1.50349 - 2.60412i) q^{41} +(-0.562914 - 0.974996i) q^{43} +(3.05695 + 1.76493i) q^{47} +6.68277 q^{49} +(-0.984893 + 1.70588i) q^{53} +(0.617133 + 1.06891i) q^{55} +(3.48434 + 6.03505i) q^{59} +(4.98180 - 8.62874i) q^{61} +6.23725 q^{65} +(-0.324291 - 0.187230i) q^{67} +(8.03969 + 13.9252i) q^{71} +(8.09850 + 14.0270i) q^{73} -2.23997i q^{77} +(7.44995 - 4.30123i) q^{79} +7.17251i q^{83} +(2.83429 - 4.90913i) q^{85} +(-3.91340 + 6.77822i) q^{89} +(-9.80296 - 5.65974i) q^{91} +(2.99229 + 8.36538i) q^{95} +(8.57756 - 4.95226i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{13} + 12 q^{17} - 4 q^{19} + 14 q^{25} + 12 q^{35} + 8 q^{41} + 2 q^{43} + 36 q^{47} + 32 q^{49} - 8 q^{53} - 12 q^{55} - 8 q^{59} - 2 q^{61} + 8 q^{65} - 30 q^{67} + 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} - 32 q^{89} - 18 q^{91} + 32 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.76516 1.01912i 0.789404 0.455763i −0.0503486 0.998732i \(-0.516033\pi\)
0.839753 + 0.542969i \(0.182700\pi\)
\(6\) 0 0
\(7\) −3.69902 −1.39810 −0.699050 0.715073i \(-0.746393\pi\)
−0.699050 + 0.715073i \(0.746393\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.605557i 0.182582i 0.995824 + 0.0912912i \(0.0290994\pi\)
−0.995824 + 0.0912912i \(0.970901\pi\)
\(12\) 0 0
\(13\) 2.65015 + 1.53006i 0.735019 + 0.424364i 0.820256 0.571997i \(-0.193831\pi\)
−0.0852364 + 0.996361i \(0.527165\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.40852 1.39056i 0.584153 0.337261i −0.178629 0.983916i \(-0.557166\pi\)
0.762782 + 0.646656i \(0.223833\pi\)
\(18\) 0 0
\(19\) −0.780721 + 4.28841i −0.179110 + 0.983829i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.23644 + 1.29121i 0.466330 + 0.269236i 0.714702 0.699429i \(-0.246562\pi\)
−0.248372 + 0.968665i \(0.579896\pi\)
\(24\) 0 0
\(25\) −0.422804 + 0.732318i −0.0845607 + 0.146464i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.70567 + 2.95430i −0.316734 + 0.548600i −0.979805 0.199957i \(-0.935920\pi\)
0.663070 + 0.748557i \(0.269253\pi\)
\(30\) 0 0
\(31\) 4.42738i 0.795181i −0.917563 0.397591i \(-0.869846\pi\)
0.917563 0.397591i \(-0.130154\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.52937 + 3.76973i −1.10367 + 0.637201i
\(36\) 0 0
\(37\) 3.34720i 0.550276i 0.961405 + 0.275138i \(0.0887236\pi\)
−0.961405 + 0.275138i \(0.911276\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.50349 2.60412i −0.234806 0.406696i 0.724410 0.689369i \(-0.242112\pi\)
−0.959216 + 0.282673i \(0.908779\pi\)
\(42\) 0 0
\(43\) −0.562914 0.974996i −0.0858436 0.148686i 0.819907 0.572497i \(-0.194025\pi\)
−0.905750 + 0.423812i \(0.860692\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.05695 + 1.76493i 0.445903 + 0.257442i 0.706098 0.708114i \(-0.250454\pi\)
−0.260196 + 0.965556i \(0.583787\pi\)
\(48\) 0 0
\(49\) 6.68277 0.954681
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.984893 + 1.70588i −0.135285 + 0.234321i −0.925706 0.378243i \(-0.876528\pi\)
0.790421 + 0.612564i \(0.209862\pi\)
\(54\) 0 0
\(55\) 0.617133 + 1.06891i 0.0832142 + 0.144131i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.48434 + 6.03505i 0.453622 + 0.785696i 0.998608 0.0527492i \(-0.0167984\pi\)
−0.544986 + 0.838445i \(0.683465\pi\)
\(60\) 0 0
\(61\) 4.98180 8.62874i 0.637855 1.10480i −0.348048 0.937477i \(-0.613155\pi\)
0.985903 0.167320i \(-0.0535113\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.23725 0.773636
\(66\) 0 0
\(67\) −0.324291 0.187230i −0.0396185 0.0228738i 0.480060 0.877236i \(-0.340615\pi\)
−0.519678 + 0.854362i \(0.673948\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.03969 + 13.9252i 0.954136 + 1.65261i 0.736334 + 0.676618i \(0.236555\pi\)
0.217802 + 0.975993i \(0.430111\pi\)
\(72\) 0 0
\(73\) 8.09850 + 14.0270i 0.947858 + 1.64174i 0.749925 + 0.661523i \(0.230090\pi\)
0.197934 + 0.980215i \(0.436577\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.23997i 0.255268i
\(78\) 0 0
\(79\) 7.44995 4.30123i 0.838185 0.483926i −0.0184618 0.999830i \(-0.505877\pi\)
0.856647 + 0.515903i \(0.172544\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.17251i 0.787285i 0.919264 + 0.393643i \(0.128785\pi\)
−0.919264 + 0.393643i \(0.871215\pi\)
\(84\) 0 0
\(85\) 2.83429 4.90913i 0.307422 0.532470i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.91340 + 6.77822i −0.414820 + 0.718489i −0.995410 0.0957070i \(-0.969489\pi\)
0.580590 + 0.814196i \(0.302822\pi\)
\(90\) 0 0
\(91\) −9.80296 5.65974i −1.02763 0.593302i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.99229 + 8.36538i 0.307003 + 0.858270i
\(96\) 0 0
\(97\) 8.57756 4.95226i 0.870919 0.502826i 0.00326577 0.999995i \(-0.498960\pi\)
0.867654 + 0.497169i \(0.165627\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.3741 + 7.72154i 1.33077 + 0.768322i 0.985418 0.170150i \(-0.0544253\pi\)
0.345355 + 0.938472i \(0.387759\pi\)
\(102\) 0 0
\(103\) 13.3067i 1.31114i −0.755133 0.655572i \(-0.772428\pi\)
0.755133 0.655572i \(-0.227572\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.95982 −0.769505 −0.384753 0.923020i \(-0.625713\pi\)
−0.384753 + 0.923020i \(0.625713\pi\)
\(108\) 0 0
\(109\) 7.06803 4.08073i 0.676995 0.390863i −0.121727 0.992564i \(-0.538843\pi\)
0.798722 + 0.601700i \(0.205510\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.63294 0.341759 0.170879 0.985292i \(-0.445339\pi\)
0.170879 + 0.985292i \(0.445339\pi\)
\(114\) 0 0
\(115\) 5.26357 0.490831
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.90918 + 5.14372i −0.816704 + 0.471524i
\(120\) 0 0
\(121\) 10.6333 0.966664
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.9147i 1.06568i
\(126\) 0 0
\(127\) 5.69581 + 3.28848i 0.505421 + 0.291805i 0.730950 0.682431i \(-0.239077\pi\)
−0.225528 + 0.974237i \(0.572411\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.27444 + 1.89050i −0.286089 + 0.165174i −0.636177 0.771543i \(-0.719485\pi\)
0.350088 + 0.936717i \(0.386152\pi\)
\(132\) 0 0
\(133\) 2.88790 15.8629i 0.250413 1.37549i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.5831 7.26483i −1.07504 0.620676i −0.145488 0.989360i \(-0.546475\pi\)
−0.929555 + 0.368684i \(0.879808\pi\)
\(138\) 0 0
\(139\) −3.79838 + 6.57899i −0.322175 + 0.558023i −0.980936 0.194329i \(-0.937747\pi\)
0.658762 + 0.752352i \(0.271080\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.926541 + 1.60482i −0.0774813 + 0.134202i
\(144\) 0 0
\(145\) 6.95309i 0.577423i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.06792 + 1.77127i −0.251334 + 0.145108i −0.620375 0.784305i \(-0.713020\pi\)
0.369041 + 0.929413i \(0.379686\pi\)
\(150\) 0 0
\(151\) 1.10333i 0.0897877i −0.998992 0.0448938i \(-0.985705\pi\)
0.998992 0.0448938i \(-0.0142950\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.51202 7.81504i −0.362414 0.627719i
\(156\) 0 0
\(157\) −0.691234 1.19725i −0.0551665 0.0955511i 0.837123 0.547014i \(-0.184236\pi\)
−0.892290 + 0.451463i \(0.850902\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.27265 4.77622i −0.651976 0.376419i
\(162\) 0 0
\(163\) 15.3481 1.20216 0.601080 0.799189i \(-0.294737\pi\)
0.601080 + 0.799189i \(0.294737\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.45424 4.25087i 0.189915 0.328942i −0.755307 0.655371i \(-0.772512\pi\)
0.945222 + 0.326429i \(0.105845\pi\)
\(168\) 0 0
\(169\) −1.81781 3.14853i −0.139831 0.242195i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.2798 + 21.2693i 0.933619 + 1.61708i 0.777078 + 0.629404i \(0.216701\pi\)
0.156541 + 0.987671i \(0.449966\pi\)
\(174\) 0 0
\(175\) 1.56396 2.70886i 0.118224 0.204770i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.5239 1.16031 0.580155 0.814506i \(-0.302992\pi\)
0.580155 + 0.814506i \(0.302992\pi\)
\(180\) 0 0
\(181\) −11.6668 6.73585i −0.867189 0.500672i −0.000775855 1.00000i \(-0.500247\pi\)
−0.866413 + 0.499328i \(0.833580\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.41119 + 5.90835i 0.250795 + 0.434390i
\(186\) 0 0
\(187\) 0.842065 + 1.45850i 0.0615779 + 0.106656i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.3381i 1.03747i −0.854935 0.518734i \(-0.826403\pi\)
0.854935 0.518734i \(-0.173597\pi\)
\(192\) 0 0
\(193\) −16.7819 + 9.68904i −1.20799 + 0.697432i −0.962319 0.271922i \(-0.912341\pi\)
−0.245668 + 0.969354i \(0.579007\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.2122i 1.01257i −0.862365 0.506287i \(-0.831018\pi\)
0.862365 0.506287i \(-0.168982\pi\)
\(198\) 0 0
\(199\) −11.3046 + 19.5801i −0.801360 + 1.38800i 0.117362 + 0.993089i \(0.462556\pi\)
−0.918721 + 0.394906i \(0.870777\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.30930 10.9280i 0.442826 0.766997i
\(204\) 0 0
\(205\) −5.30781 3.06447i −0.370714 0.214032i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.59688 0.472771i −0.179630 0.0327022i
\(210\) 0 0
\(211\) 10.1117 5.83798i 0.696116 0.401903i −0.109783 0.993956i \(-0.535016\pi\)
0.805899 + 0.592053i \(0.201682\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.98727 1.14735i −0.135531 0.0782487i
\(216\) 0 0
\(217\) 16.3770i 1.11174i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.51060 0.572485
\(222\) 0 0
\(223\) 14.6238 8.44308i 0.979285 0.565391i 0.0772310 0.997013i \(-0.475392\pi\)
0.902054 + 0.431623i \(0.142059\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.3183 −1.01671 −0.508354 0.861148i \(-0.669746\pi\)
−0.508354 + 0.861148i \(0.669746\pi\)
\(228\) 0 0
\(229\) −17.5798 −1.16171 −0.580854 0.814008i \(-0.697281\pi\)
−0.580854 + 0.814008i \(0.697281\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.4561 + 10.6556i −1.20910 + 0.698074i −0.962563 0.271059i \(-0.912626\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(234\) 0 0
\(235\) 7.19469 0.469330
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.70885i 0.369275i −0.982807 0.184638i \(-0.940889\pi\)
0.982807 0.184638i \(-0.0591111\pi\)
\(240\) 0 0
\(241\) −15.3638 8.87029i −0.989668 0.571385i −0.0844932 0.996424i \(-0.526927\pi\)
−0.905175 + 0.425039i \(0.860260\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.7962 6.81052i 0.753629 0.435108i
\(246\) 0 0
\(247\) −8.63057 + 10.1704i −0.549150 + 0.647126i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.8199 + 14.3298i 1.56662 + 0.904489i 0.996559 + 0.0828875i \(0.0264142\pi\)
0.570062 + 0.821602i \(0.306919\pi\)
\(252\) 0 0
\(253\) −0.781902 + 1.35429i −0.0491577 + 0.0851437i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.27016 + 10.8602i −0.391122 + 0.677443i −0.992598 0.121448i \(-0.961246\pi\)
0.601476 + 0.798891i \(0.294580\pi\)
\(258\) 0 0
\(259\) 12.3814i 0.769341i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.42476 0.822587i 0.0878546 0.0507229i −0.455429 0.890272i \(-0.650514\pi\)
0.543284 + 0.839549i \(0.317181\pi\)
\(264\) 0 0
\(265\) 4.01488i 0.246632i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.25176 9.09632i −0.320206 0.554612i 0.660325 0.750980i \(-0.270419\pi\)
−0.980530 + 0.196368i \(0.937085\pi\)
\(270\) 0 0
\(271\) 6.37145 + 11.0357i 0.387038 + 0.670370i 0.992050 0.125847i \(-0.0401649\pi\)
−0.605012 + 0.796217i \(0.706832\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.443460 0.256032i −0.0267416 0.0154393i
\(276\) 0 0
\(277\) 19.4101 1.16624 0.583119 0.812387i \(-0.301832\pi\)
0.583119 + 0.812387i \(0.301832\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.78424 + 13.4827i −0.464369 + 0.804311i −0.999173 0.0406655i \(-0.987052\pi\)
0.534804 + 0.844976i \(0.320386\pi\)
\(282\) 0 0
\(283\) 5.22827 + 9.05563i 0.310788 + 0.538301i 0.978533 0.206089i \(-0.0660736\pi\)
−0.667745 + 0.744390i \(0.732740\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.56145 + 9.63271i 0.328282 + 0.568601i
\(288\) 0 0
\(289\) −4.63267 + 8.02403i −0.272510 + 0.472002i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.1390 1.64390 0.821949 0.569561i \(-0.192887\pi\)
0.821949 + 0.569561i \(0.192887\pi\)
\(294\) 0 0
\(295\) 12.3008 + 7.10189i 0.716182 + 0.413488i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.95127 + 6.84380i 0.228508 + 0.395787i
\(300\) 0 0
\(301\) 2.08223 + 3.60653i 0.120018 + 0.207877i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.3082i 1.16284i
\(306\) 0 0
\(307\) −11.1659 + 6.44663i −0.637271 + 0.367929i −0.783563 0.621313i \(-0.786600\pi\)
0.146292 + 0.989242i \(0.453266\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.6008i 0.771228i −0.922660 0.385614i \(-0.873990\pi\)
0.922660 0.385614i \(-0.126010\pi\)
\(312\) 0 0
\(313\) 1.12148 1.94246i 0.0633899 0.109794i −0.832589 0.553892i \(-0.813142\pi\)
0.895979 + 0.444097i \(0.146476\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.3488 + 28.3170i −0.918242 + 1.59044i −0.116159 + 0.993231i \(0.537058\pi\)
−0.802084 + 0.597212i \(0.796275\pi\)
\(318\) 0 0
\(319\) −1.78900 1.03288i −0.100165 0.0578301i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.08292 + 11.4144i 0.227180 + 0.635113i
\(324\) 0 0
\(325\) −2.24099 + 1.29383i −0.124308 + 0.0717690i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.3077 6.52853i −0.623416 0.359929i
\(330\) 0 0
\(331\) 18.7734i 1.03188i −0.856625 0.515940i \(-0.827443\pi\)
0.856625 0.515940i \(-0.172557\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.763236 −0.0417000
\(336\) 0 0
\(337\) −8.38982 + 4.84387i −0.457023 + 0.263862i −0.710792 0.703403i \(-0.751663\pi\)
0.253769 + 0.967265i \(0.418330\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.68103 0.145186
\(342\) 0 0
\(343\) 1.17346 0.0633606
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.2531 6.49698i 0.604098 0.348776i −0.166554 0.986032i \(-0.553264\pi\)
0.770652 + 0.637256i \(0.219931\pi\)
\(348\) 0 0
\(349\) 6.98962 0.374146 0.187073 0.982346i \(-0.440100\pi\)
0.187073 + 0.982346i \(0.440100\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.60313i 0.0853258i 0.999090 + 0.0426629i \(0.0135842\pi\)
−0.999090 + 0.0426629i \(0.986416\pi\)
\(354\) 0 0
\(355\) 28.3827 + 16.3868i 1.50640 + 0.869719i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.1453 + 11.0535i −1.01045 + 0.583383i −0.911323 0.411692i \(-0.864938\pi\)
−0.0991256 + 0.995075i \(0.531605\pi\)
\(360\) 0 0
\(361\) −17.7810 6.69610i −0.935840 0.352426i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 28.5903 + 16.5066i 1.49649 + 0.863997i
\(366\) 0 0
\(367\) −4.97781 + 8.62182i −0.259840 + 0.450055i −0.966199 0.257798i \(-0.917003\pi\)
0.706359 + 0.707854i \(0.250336\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.64314 6.31011i 0.189142 0.327604i
\(372\) 0 0
\(373\) 16.4257i 0.850489i −0.905078 0.425245i \(-0.860188\pi\)
0.905078 0.425245i \(-0.139812\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.04054 + 5.21956i −0.465612 + 0.268821i
\(378\) 0 0
\(379\) 25.2429i 1.29664i −0.761367 0.648322i \(-0.775471\pi\)
0.761367 0.648322i \(-0.224529\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.5496 30.3968i −0.896742 1.55320i −0.831633 0.555325i \(-0.812594\pi\)
−0.0651092 0.997878i \(-0.520740\pi\)
\(384\) 0 0
\(385\) −2.28279 3.95391i −0.116342 0.201510i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.8623 + 9.73543i 0.854950 + 0.493606i 0.862318 0.506367i \(-0.169012\pi\)
−0.00736774 + 0.999973i \(0.502345\pi\)
\(390\) 0 0
\(391\) 7.18203 0.363211
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.76691 15.1847i 0.441111 0.764027i
\(396\) 0 0
\(397\) −8.78901 15.2230i −0.441108 0.764021i 0.556664 0.830738i \(-0.312081\pi\)
−0.997772 + 0.0667165i \(0.978748\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.54092 + 14.7933i 0.426513 + 0.738743i 0.996560 0.0828693i \(-0.0264084\pi\)
−0.570047 + 0.821612i \(0.693075\pi\)
\(402\) 0 0
\(403\) 6.77418 11.7332i 0.337446 0.584473i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.02692 −0.100471
\(408\) 0 0
\(409\) 13.3939 + 7.73296i 0.662284 + 0.382370i 0.793147 0.609030i \(-0.208441\pi\)
−0.130862 + 0.991401i \(0.541775\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.8886 22.3238i −0.634208 1.09848i
\(414\) 0 0
\(415\) 7.30962 + 12.6606i 0.358815 + 0.621486i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 25.5092i 1.24621i −0.782140 0.623103i \(-0.785872\pi\)
0.782140 0.623103i \(-0.214128\pi\)
\(420\) 0 0
\(421\) 6.76334 3.90482i 0.329625 0.190309i −0.326050 0.945353i \(-0.605718\pi\)
0.655675 + 0.755044i \(0.272384\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.35174i 0.114076i
\(426\) 0 0
\(427\) −18.4278 + 31.9179i −0.891784 + 1.54461i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.26660 16.0502i 0.446356 0.773112i −0.551789 0.833984i \(-0.686055\pi\)
0.998146 + 0.0608716i \(0.0193880\pi\)
\(432\) 0 0
\(433\) −28.2020 16.2824i −1.35530 0.782484i −0.366316 0.930491i \(-0.619381\pi\)
−0.988986 + 0.148007i \(0.952714\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.28328 + 8.58271i −0.348406 + 0.410567i
\(438\) 0 0
\(439\) 10.5290 6.07891i 0.502521 0.290131i −0.227233 0.973840i \(-0.572968\pi\)
0.729754 + 0.683710i \(0.239635\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.7476 6.20513i −0.510634 0.294815i 0.222460 0.974942i \(-0.428591\pi\)
−0.733094 + 0.680127i \(0.761925\pi\)
\(444\) 0 0
\(445\) 15.9529i 0.756238i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −38.8022 −1.83119 −0.915596 0.402100i \(-0.868280\pi\)
−0.915596 + 0.402100i \(0.868280\pi\)
\(450\) 0 0
\(451\) 1.57695 0.910450i 0.0742555 0.0428714i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −23.0717 −1.08162
\(456\) 0 0
\(457\) −31.4392 −1.47066 −0.735331 0.677708i \(-0.762973\pi\)
−0.735331 + 0.677708i \(0.762973\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.76189 1.59458i 0.128634 0.0742669i −0.434302 0.900767i \(-0.643005\pi\)
0.562936 + 0.826500i \(0.309672\pi\)
\(462\) 0 0
\(463\) −21.6707 −1.00712 −0.503562 0.863959i \(-0.667978\pi\)
−0.503562 + 0.863959i \(0.667978\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.0633i 0.882144i 0.897472 + 0.441072i \(0.145402\pi\)
−0.897472 + 0.441072i \(0.854598\pi\)
\(468\) 0 0
\(469\) 1.19956 + 0.692567i 0.0553906 + 0.0319798i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.590416 0.340877i 0.0271474 0.0156735i
\(474\) 0 0
\(475\) −2.81039 2.38489i −0.128949 0.109426i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7.51906 4.34113i −0.343554 0.198351i 0.318288 0.947994i \(-0.396892\pi\)
−0.661843 + 0.749643i \(0.730225\pi\)
\(480\) 0 0
\(481\) −5.12143 + 8.87058i −0.233517 + 0.404464i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0939 17.4831i 0.458338 0.793865i
\(486\) 0 0
\(487\) 17.7742i 0.805425i 0.915326 + 0.402713i \(0.131933\pi\)
−0.915326 + 0.402713i \(0.868067\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.13906 + 0.657635i −0.0514050 + 0.0296787i −0.525482 0.850805i \(-0.676115\pi\)
0.474077 + 0.880483i \(0.342782\pi\)
\(492\) 0 0
\(493\) 9.48734i 0.427288i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −29.7390 51.5094i −1.33398 2.31051i
\(498\) 0 0
\(499\) 10.7952 + 18.6979i 0.483261 + 0.837033i 0.999815 0.0192214i \(-0.00611874\pi\)
−0.516554 + 0.856255i \(0.672785\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.2076 + 16.2857i 1.25772 + 0.726143i 0.972630 0.232359i \(-0.0746444\pi\)
0.285086 + 0.958502i \(0.407978\pi\)
\(504\) 0 0
\(505\) 31.4766 1.40069
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.4800 19.8839i 0.508842 0.881339i −0.491106 0.871100i \(-0.663407\pi\)
0.999948 0.0102396i \(-0.00325942\pi\)
\(510\) 0 0
\(511\) −29.9565 51.8863i −1.32520 2.29531i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −13.5610 23.4884i −0.597570 1.03502i
\(516\) 0 0
\(517\) −1.06877 + 1.85116i −0.0470044 + 0.0814139i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0841 0.617035 0.308517 0.951219i \(-0.400167\pi\)
0.308517 + 0.951219i \(0.400167\pi\)
\(522\) 0 0
\(523\) −33.1343 19.1301i −1.44886 0.836501i −0.450448 0.892802i \(-0.648736\pi\)
−0.998414 + 0.0563014i \(0.982069\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.15655 10.6635i −0.268183 0.464507i
\(528\) 0 0
\(529\) −8.16555 14.1431i −0.355024 0.614920i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.20176i 0.398572i
\(534\) 0 0
\(535\) −14.0504 + 8.11199i −0.607450 + 0.350712i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.04680i 0.174308i
\(540\) 0 0
\(541\) 2.22232 3.84916i 0.0955448 0.165489i −0.814291 0.580457i \(-0.802874\pi\)
0.909836 + 0.414968i \(0.136207\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.31748 14.4063i 0.356282 0.617098i
\(546\) 0 0
\(547\) −18.2229 10.5210i −0.779156 0.449846i 0.0569750 0.998376i \(-0.481854\pi\)
−0.836131 + 0.548530i \(0.815188\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.3376 9.62108i −0.482998 0.409872i
\(552\) 0 0
\(553\) −27.5575 + 15.9104i −1.17187 + 0.676577i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −21.4094 12.3607i −0.907145 0.523740i −0.0276336 0.999618i \(-0.508797\pi\)
−0.879511 + 0.475878i \(0.842131\pi\)
\(558\) 0 0
\(559\) 3.44518i 0.145716i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20.1514 −0.849281 −0.424640 0.905362i \(-0.639599\pi\)
−0.424640 + 0.905362i \(0.639599\pi\)
\(564\) 0 0
\(565\) 6.41273 3.70239i 0.269786 0.155761i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.91264 0.122104 0.0610520 0.998135i \(-0.480554\pi\)
0.0610520 + 0.998135i \(0.480554\pi\)
\(570\) 0 0
\(571\) −5.30300 −0.221924 −0.110962 0.993825i \(-0.535393\pi\)
−0.110962 + 0.993825i \(0.535393\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.89115 + 1.09186i −0.0788665 + 0.0455336i
\(576\) 0 0
\(577\) 29.0613 1.20984 0.604919 0.796287i \(-0.293205\pi\)
0.604919 + 0.796287i \(0.293205\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 26.5313i 1.10070i
\(582\) 0 0
\(583\) −1.03301 0.596409i −0.0427829 0.0247007i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.60901 3.81571i 0.272783 0.157491i −0.357369 0.933963i \(-0.616326\pi\)
0.630152 + 0.776472i \(0.282993\pi\)
\(588\) 0 0
\(589\) 18.9864 + 3.45655i 0.782322 + 0.142425i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −23.1987 13.3938i −0.952657 0.550017i −0.0587516 0.998273i \(-0.518712\pi\)
−0.893905 + 0.448256i \(0.852045\pi\)
\(594\) 0 0
\(595\) −10.4841 + 18.1590i −0.429806 + 0.744446i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −19.4393 + 33.6699i −0.794270 + 1.37572i 0.129032 + 0.991640i \(0.458813\pi\)
−0.923302 + 0.384075i \(0.874520\pi\)
\(600\) 0 0
\(601\) 20.4550i 0.834377i 0.908820 + 0.417188i \(0.136985\pi\)
−0.908820 + 0.417188i \(0.863015\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.7695 10.8366i 0.763088 0.440569i
\(606\) 0 0
\(607\) 4.65988i 0.189139i 0.995518 + 0.0945694i \(0.0301474\pi\)
−0.995518 + 0.0945694i \(0.969853\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.40092 + 9.35467i 0.218498 + 0.378450i
\(612\) 0 0
\(613\) −5.90269 10.2238i −0.238408 0.412934i 0.721850 0.692050i \(-0.243292\pi\)
−0.960258 + 0.279116i \(0.909959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.1238 24.3202i −1.69584 0.979094i −0.949625 0.313389i \(-0.898536\pi\)
−0.746215 0.665705i \(-0.768131\pi\)
\(618\) 0 0
\(619\) −42.7915 −1.71993 −0.859967 0.510350i \(-0.829516\pi\)
−0.859967 + 0.510350i \(0.829516\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.4758 25.0728i 0.579959 1.00452i
\(624\) 0 0
\(625\) 10.0285 + 17.3698i 0.401138 + 0.694792i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.65449 + 8.06181i 0.185587 + 0.321445i
\(630\) 0 0
\(631\) 18.6092 32.2321i 0.740820 1.28314i −0.211302 0.977421i \(-0.567770\pi\)
0.952122 0.305717i \(-0.0988963\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.4054 0.531976
\(636\) 0 0
\(637\) 17.7103 + 10.2251i 0.701709 + 0.405132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.29142 + 5.70091i 0.130003 + 0.225172i 0.923678 0.383171i \(-0.125168\pi\)
−0.793674 + 0.608343i \(0.791835\pi\)
\(642\) 0 0
\(643\) −6.24623 10.8188i −0.246327 0.426651i 0.716177 0.697919i \(-0.245890\pi\)
−0.962504 + 0.271268i \(0.912557\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.23117i 0.166344i −0.996535 0.0831722i \(-0.973495\pi\)
0.996535 0.0831722i \(-0.0265052\pi\)
\(648\) 0 0
\(649\) −3.65456 + 2.10996i −0.143454 + 0.0828233i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.95004i 0.115444i −0.998333 0.0577219i \(-0.981616\pi\)
0.998333 0.0577219i \(-0.0183837\pi\)
\(654\) 0 0
\(655\) −3.85328 + 6.67408i −0.150560 + 0.260778i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.557166 0.965039i 0.0217041 0.0375926i −0.854969 0.518679i \(-0.826424\pi\)
0.876673 + 0.481086i \(0.159757\pi\)
\(660\) 0 0
\(661\) −21.5103 12.4190i −0.836653 0.483042i 0.0194720 0.999810i \(-0.493801\pi\)
−0.856125 + 0.516768i \(0.827135\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.0686 30.9437i −0.429220 1.19995i
\(666\) 0 0
\(667\) −7.62925 + 4.40475i −0.295406 + 0.170553i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.22519 + 3.01677i 0.201716 + 0.116461i
\(672\) 0 0
\(673\) 4.51767i 0.174144i −0.996202 0.0870718i \(-0.972249\pi\)
0.996202 0.0870718i \(-0.0277509\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.8210 0.454319 0.227159 0.973858i \(-0.427056\pi\)
0.227159 + 0.973858i \(0.427056\pi\)
\(678\) 0 0
\(679\) −31.7286 + 18.3185i −1.21763 + 0.703000i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −21.2681 −0.813803 −0.406901 0.913472i \(-0.633391\pi\)
−0.406901 + 0.913472i \(0.633391\pi\)
\(684\) 0 0
\(685\) −29.6148 −1.13152
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.22023 + 3.01390i −0.198875 + 0.114820i
\(690\) 0 0
\(691\) 34.9630 1.33006 0.665028 0.746819i \(-0.268420\pi\)
0.665028 + 0.746819i \(0.268420\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.4840i 0.587341i
\(696\) 0 0
\(697\) −7.24239 4.18140i −0.274325 0.158382i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.1442 22.0226i 1.44069 0.831781i 0.442791 0.896625i \(-0.353988\pi\)
0.997895 + 0.0648442i \(0.0206550\pi\)
\(702\) 0 0
\(703\) −14.3542 2.61323i −0.541378 0.0985598i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −49.4711 28.5622i −1.86055 1.07419i
\(708\) 0 0
\(709\) 16.9260 29.3166i 0.635668 1.10101i −0.350705 0.936486i \(-0.614058\pi\)
0.986373 0.164523i \(-0.0526085\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.71668 9.90158i 0.214091 0.370817i
\(714\) 0 0
\(715\) 3.77701i 0.141252i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 45.9929 26.5540i 1.71525 0.990298i 0.788144 0.615491i \(-0.211042\pi\)
0.927103 0.374807i \(-0.122291\pi\)
\(720\) 0 0
\(721\) 49.2216i 1.83311i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.44232 2.49818i −0.0535666 0.0927800i
\(726\) 0 0
\(727\) 3.59081 + 6.21946i 0.133176 + 0.230667i 0.924899 0.380213i \(-0.124149\pi\)
−0.791723 + 0.610880i \(0.790816\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.71159 1.56553i −0.100292 0.0579034i
\(732\) 0 0
\(733\) 28.8680 1.06626 0.533132 0.846032i \(-0.321015\pi\)
0.533132 + 0.846032i \(0.321015\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.113378 0.196377i 0.00417634 0.00723364i
\(738\) 0 0
\(739\) −16.1981 28.0559i −0.595855 1.03205i −0.993426 0.114480i \(-0.963480\pi\)
0.397570 0.917572i \(-0.369854\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.2449 + 29.8690i 0.632654 + 1.09579i 0.987007 + 0.160677i \(0.0513678\pi\)
−0.354353 + 0.935112i \(0.615299\pi\)
\(744\) 0 0
\(745\) −3.61025 + 6.25314i −0.132269 + 0.229097i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 29.4436 1.07584
\(750\) 0 0
\(751\) −19.7822 11.4212i −0.721861 0.416767i 0.0935759 0.995612i \(-0.470170\pi\)
−0.815437 + 0.578845i \(0.803504\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.12442 1.94755i −0.0409219 0.0708788i
\(756\) 0 0
\(757\) 2.87620 + 4.98173i 0.104537 + 0.181064i 0.913549 0.406728i \(-0.133330\pi\)
−0.809012 + 0.587792i \(0.799997\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 48.7434i 1.76695i 0.468481 + 0.883474i \(0.344801\pi\)
−0.468481 + 0.883474i \(0.655199\pi\)
\(762\) 0 0
\(763\) −26.1448 + 15.0947i −0.946506 + 0.546465i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.3250i 0.770002i
\(768\) 0 0
\(769\) −3.50200 + 6.06565i −0.126285 + 0.218733i −0.922235 0.386631i \(-0.873639\pi\)
0.795949 + 0.605363i \(0.206972\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.00414 + 8.66743i −0.179987 + 0.311746i −0.941876 0.335962i \(-0.890939\pi\)
0.761889 + 0.647707i \(0.224272\pi\)
\(774\) 0 0
\(775\) 3.24225 + 1.87191i 0.116465 + 0.0672411i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.3414 4.41450i 0.442175 0.158166i
\(780\) 0 0
\(781\) −8.43247 + 4.86849i −0.301738 + 0.174208i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.44028 1.40890i −0.0870973 0.0502856i
\(786\) 0 0
\(787\) 34.2689i 1.22155i 0.791803 + 0.610777i \(0.209143\pi\)
−0.791803 + 0.610777i \(0.790857\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.4383 −0.477812
\(792\) 0 0
\(793\) 26.4050 15.2450i 0.937671 0.541364i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −27.8026 −0.984817 −0.492409 0.870364i \(-0.663883\pi\)
−0.492409 + 0.870364i \(0.663883\pi\)
\(798\) 0 0
\(799\) 9.81700 0.347300
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −8.49416 + 4.90411i −0.299752 + 0.173062i
\(804\) 0 0
\(805\) −19.4701 −0.686230
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38.6452i 1.35869i −0.733817 0.679347i \(-0.762263\pi\)
0.733817 0.679347i \(-0.237737\pi\)
\(810\) 0 0
\(811\) 37.9998 + 21.9392i 1.33435 + 0.770390i 0.985964 0.166960i \(-0.0533951\pi\)
0.348390 + 0.937350i \(0.386728\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 27.0919 15.6415i 0.948989 0.547899i
\(816\) 0 0
\(817\) 4.62066 1.65281i 0.161657 0.0578245i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 45.8864 + 26.4925i 1.60145 + 0.924596i 0.991198 + 0.132387i \(0.0422642\pi\)
0.610250 + 0.792209i \(0.291069\pi\)
\(822\) 0 0
\(823\) −14.0908 + 24.4059i −0.491174 + 0.850737i −0.999948 0.0101621i \(-0.996765\pi\)
0.508775 + 0.860900i \(0.330099\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −17.9263 + 31.0492i −0.623358 + 1.07969i 0.365498 + 0.930812i \(0.380899\pi\)
−0.988856 + 0.148876i \(0.952435\pi\)
\(828\) 0 0
\(829\) 37.7685i 1.31175i −0.754868 0.655877i \(-0.772299\pi\)
0.754868 0.655877i \(-0.227701\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 16.0956 9.29280i 0.557680 0.321976i
\(834\) 0 0
\(835\) 10.0046i 0.346225i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.3542 19.6660i −0.391989 0.678945i 0.600723 0.799457i \(-0.294880\pi\)
−0.992712 + 0.120513i \(0.961546\pi\)
\(840\) 0 0
\(841\) 8.68140 + 15.0366i 0.299359 + 0.518505i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.41744 3.70511i −0.220767 0.127460i
\(846\) 0 0
\(847\) −39.3328 −1.35149
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.32194 + 7.48582i −0.148154 + 0.256611i
\(852\) 0 0
\(853\) 18.3713 + 31.8200i 0.629021 + 1.08950i 0.987749 + 0.156054i \(0.0498772\pi\)
−0.358728 + 0.933442i \(0.616789\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.42058 5.92462i −0.116845 0.202381i 0.801671 0.597766i \(-0.203945\pi\)
−0.918516 + 0.395385i \(0.870611\pi\)
\(858\) 0 0
\(859\) 11.3794 19.7097i 0.388261 0.672488i −0.603955 0.797019i \(-0.706409\pi\)
0.992216 + 0.124531i \(0.0397426\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19.6907 0.670279 0.335139 0.942169i \(-0.391217\pi\)
0.335139 + 0.942169i \(0.391217\pi\)
\(864\) 0 0
\(865\) 43.3518 + 25.0292i 1.47401 + 0.851017i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.60464 + 4.51137i 0.0883564 + 0.153038i
\(870\) 0 0
\(871\) −0.572947 0.992374i −0.0194136 0.0336253i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 44.0728i 1.48993i
\(876\) 0 0
\(877\) 13.0986 7.56250i 0.442309 0.255367i −0.262267 0.964995i \(-0.584470\pi\)
0.704577 + 0.709628i \(0.251137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 41.3230i 1.39221i 0.717942 + 0.696103i \(0.245084\pi\)
−0.717942 + 0.696103i \(0.754916\pi\)
\(882\) 0 0
\(883\) −23.1270 + 40.0572i −0.778286 + 1.34803i 0.154642 + 0.987971i \(0.450578\pi\)
−0.932929 + 0.360061i \(0.882756\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.7167 + 22.0260i −0.426986 + 0.739561i −0.996604 0.0823488i \(-0.973758\pi\)
0.569618 + 0.821910i \(0.307091\pi\)
\(888\) 0 0
\(889\) −21.0689 12.1641i −0.706629 0.407972i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.95539 + 11.7316i −0.333144 + 0.392582i
\(894\) 0 0
\(895\) 27.4021 15.8206i 0.915953 0.528826i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.0798 + 7.55164i 0.436236 + 0.251861i
\(900\) 0 0
\(901\) 5.47822i 0.182506i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −27.4585 −0.912750
\(906\) 0 0
\(907\) −19.5518 + 11.2882i −0.649205 + 0.374819i −0.788152 0.615481i \(-0.788962\pi\)
0.138946 + 0.990300i \(0.455628\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −50.6150 −1.67695 −0.838475 0.544940i \(-0.816552\pi\)
−0.838475 + 0.544940i \(0.816552\pi\)
\(912\) 0 0
\(913\) −4.34336 −0.143744
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.1122 6.99300i 0.399981 0.230929i
\(918\) 0 0
\(919\) 30.3240 1.00030 0.500148 0.865940i \(-0.333279\pi\)
0.500148 + 0.865940i \(0.333279\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 49.2050i 1.61960i
\(924\) 0 0
\(925\) −2.45121 1.41521i −0.0805954 0.0465318i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 25.7976 14.8943i 0.846393 0.488665i −0.0130391 0.999915i \(-0.504151\pi\)
0.859432 + 0.511250i \(0.170817\pi\)
\(930\) 0 0
\(931\) −5.21737 + 28.6585i −0.170993 + 0.939243i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.97276 + 1.71632i 0.0972196 + 0.0561298i
\(936\) 0 0
\(937\) −13.1927 + 22.8504i −0.430985 + 0.746488i −0.996958 0.0779353i \(-0.975167\pi\)
0.565973 + 0.824424i \(0.308501\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.47207 2.54971i 0.0479883 0.0831181i −0.841034 0.540983i \(-0.818052\pi\)
0.889022 + 0.457865i \(0.151386\pi\)
\(942\) 0 0
\(943\) 7.76530i 0.252873i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.65210 + 1.53119i −0.0861815 + 0.0497569i −0.542471 0.840074i \(-0.682511\pi\)
0.456290 + 0.889831i \(0.349178\pi\)
\(948\) 0 0
\(949\) 49.5649i 1.60895i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.84291 + 3.19201i 0.0596977 + 0.103399i 0.894330 0.447409i \(-0.147653\pi\)
−0.834632 + 0.550808i \(0.814320\pi\)
\(954\) 0 0
\(955\) −14.6122 25.3091i −0.472840 0.818982i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 46.5450 + 26.8728i 1.50302 + 0.867767i
\(960\) 0 0
\(961\) 11.3983 0.367687
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19.7485 + 34.2054i −0.635727 + 1.10111i
\(966\) 0 0
\(967\) −16.0399 27.7819i −0.515808 0.893405i −0.999832 0.0183505i \(-0.994159\pi\)
0.484024 0.875055i \(-0.339175\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.26555 12.5843i −0.233163 0.403849i 0.725575 0.688144i \(-0.241574\pi\)
−0.958737 + 0.284294i \(0.908241\pi\)
\(972\) 0 0
\(973\) 14.0503 24.3358i 0.450432 0.780171i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.611504 0.0195637 0.00978187 0.999952i \(-0.496886\pi\)
0.00978187 + 0.999952i \(0.496886\pi\)
\(978\) 0 0
\(979\) −4.10460 2.36979i −0.131183 0.0757388i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.61045 2.78937i −0.0513652 0.0889672i 0.839200 0.543824i \(-0.183024\pi\)
−0.890565 + 0.454856i \(0.849691\pi\)
\(984\) 0 0
\(985\) −14.4838 25.0867i −0.461493 0.799330i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.90736i 0.0924488i
\(990\) 0 0
\(991\) 34.9740 20.1923i 1.11099 0.641428i 0.171902 0.985114i \(-0.445009\pi\)
0.939085 + 0.343686i \(0.111675\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 46.0827i 1.46092i
\(996\) 0 0
\(997\) 27.5964 47.7983i 0.873986 1.51379i 0.0161464 0.999870i \(-0.494860\pi\)
0.857839 0.513918i \(-0.171806\pi\)
\(998\) 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 2736.2.dc.f.449.8 20
3.2 odd 2 2736.2.dc.e.449.3 20
4.3 odd 2 1368.2.cu.b.449.8 yes 20
12.11 even 2 1368.2.cu.a.449.3 20
19.8 odd 6 2736.2.dc.e.1889.3 20
57.8 even 6 inner 2736.2.dc.f.1889.8 20
76.27 even 6 1368.2.cu.a.521.3 yes 20
228.179 odd 6 1368.2.cu.b.521.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.cu.a.449.3 20 12.11 even 2
1368.2.cu.a.521.3 yes 20 76.27 even 6
1368.2.cu.b.449.8 yes 20 4.3 odd 2
1368.2.cu.b.521.8 yes 20 228.179 odd 6
2736.2.dc.e.449.3 20 3.2 odd 2
2736.2.dc.e.1889.3 20 19.8 odd 6
2736.2.dc.f.449.8 20 1.1 even 1 trivial
2736.2.dc.f.1889.8 20 57.8 even 6 inner