Properties

Label 1216.2.s.g.31.1
Level $1216$
Weight $2$
Character 1216.31
Analytic conductor $9.710$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(31,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1216.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.s (of order \(6\), degree \(2\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} - 18 x^{9} + 11 x^{8} - 36 x^{7} + 180 x^{6} - 120 x^{5} + 31 x^{4} + 18 x^{3} + 162 x^{2} - 54 x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(-1.17819 + 1.17819i\) of defining polynomial
Character \(\chi\) \(=\) 1216.31
Dual form 1216.2.s.g.863.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.71294 + 1.56632i) q^{3} +(-1.50000 + 0.866025i) q^{5} +2.00000i q^{7} +(3.40671 - 5.90059i) q^{9} +O(q^{10})\) \(q+(-2.71294 + 1.56632i) q^{3} +(-1.50000 + 0.866025i) q^{5} +2.00000i q^{7} +(3.40671 - 5.90059i) q^{9} +(0.225929 - 0.391320i) q^{13} +(2.71294 - 4.69896i) q^{15} +(-2.90671 - 5.03457i) q^{17} +(-4.16854 + 1.27407i) q^{19} +(-3.13264 - 5.42589i) q^{21} +(-5.70234 - 3.29225i) q^{23} +(-1.00000 + 1.73205i) q^{25} +11.9461i q^{27} +(2.90671 - 5.03457i) q^{29} +8.33708 q^{31} +(-1.73205 - 3.00000i) q^{35} -3.45186 q^{37} +1.41551i q^{39} +(-0.677786 + 0.391320i) q^{41} +(-0.276452 - 0.478830i) q^{43} +11.8012i q^{45} +(8.13883 + 4.69896i) q^{47} +3.00000 q^{49} +(15.7715 + 9.10567i) q^{51} +(4.31342 - 7.47106i) q^{53} +(9.31342 - 9.98575i) q^{57} +(0.506187 - 0.292247i) q^{59} +(10.8979 + 6.29191i) q^{61} +(11.8012 + 6.81342i) q^{63} +0.782640i q^{65} +(-12.5523 - 7.24710i) q^{67} +20.6268 q^{69} +(4.91970 + 8.52117i) q^{71} +(2.50000 + 4.33013i) q^{73} -6.26527i q^{75} +(-3.18765 - 5.52117i) q^{79} +(-8.49120 - 14.7072i) q^{81} +8.33708 q^{83} +(8.72013 + 5.03457i) q^{85} +18.2113i q^{87} +(-9.11804 - 5.26430i) q^{89} +(0.782640 + 0.451857i) q^{91} +(-22.6180 + 13.0585i) q^{93} +(5.14943 - 5.52117i) q^{95} +(-0.677786 + 0.391320i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{5} + 12 q^{9} - 2 q^{13} - 6 q^{17} - 4 q^{21} - 12 q^{25} + 6 q^{29} - 32 q^{37} + 6 q^{41} + 36 q^{49} - 6 q^{53} + 54 q^{57} + 30 q^{61} + 132 q^{69} + 30 q^{73} - 30 q^{81} + 18 q^{85} + 78 q^{89} - 84 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) −2.71294 + 1.56632i −1.56632 + 0.904315i −0.569726 + 0.821835i \(0.692951\pi\)
−0.996593 + 0.0824799i \(0.973716\pi\)
\(4\) 0 0
\(5\) −1.50000 + 0.866025i −0.670820 + 0.387298i −0.796387 0.604787i \(-0.793258\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 3.40671 5.90059i 1.13557 1.96686i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0.225929 0.391320i 0.0626613 0.108533i −0.832993 0.553284i \(-0.813375\pi\)
0.895654 + 0.444751i \(0.146708\pi\)
\(14\) 0 0
\(15\) 2.71294 4.69896i 0.700479 1.21327i
\(16\) 0 0
\(17\) −2.90671 5.03457i −0.704980 1.22106i −0.966699 0.255918i \(-0.917622\pi\)
0.261718 0.965144i \(-0.415711\pi\)
\(18\) 0 0
\(19\) −4.16854 + 1.27407i −0.956329 + 0.292292i
\(20\) 0 0
\(21\) −3.13264 5.42589i −0.683598 1.18403i
\(22\) 0 0
\(23\) −5.70234 3.29225i −1.18902 0.686481i −0.230936 0.972969i \(-0.574179\pi\)
−0.958084 + 0.286488i \(0.907512\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 11.9461i 2.29902i
\(28\) 0 0
\(29\) 2.90671 5.03457i 0.539762 0.934896i −0.459154 0.888357i \(-0.651847\pi\)
0.998916 0.0465391i \(-0.0148192\pi\)
\(30\) 0 0
\(31\) 8.33708 1.49738 0.748692 0.662918i \(-0.230682\pi\)
0.748692 + 0.662918i \(0.230682\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.73205 3.00000i −0.292770 0.507093i
\(36\) 0 0
\(37\) −3.45186 −0.567482 −0.283741 0.958901i \(-0.591576\pi\)
−0.283741 + 0.958901i \(0.591576\pi\)
\(38\) 0 0
\(39\) 1.41551i 0.226662i
\(40\) 0 0
\(41\) −0.677786 + 0.391320i −0.105852 + 0.0611139i −0.551992 0.833850i \(-0.686132\pi\)
0.446139 + 0.894964i \(0.352799\pi\)
\(42\) 0 0
\(43\) −0.276452 0.478830i −0.0421586 0.0730209i 0.844176 0.536066i \(-0.180090\pi\)
−0.886335 + 0.463045i \(0.846757\pi\)
\(44\) 0 0
\(45\) 11.8012i 1.75922i
\(46\) 0 0
\(47\) 8.13883 + 4.69896i 1.18717 + 0.685413i 0.957662 0.287894i \(-0.0929549\pi\)
0.229508 + 0.973307i \(0.426288\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 15.7715 + 9.10567i 2.20845 + 1.27505i
\(52\) 0 0
\(53\) 4.31342 7.47106i 0.592493 1.02623i −0.401402 0.915902i \(-0.631477\pi\)
0.993895 0.110327i \(-0.0351897\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.31342 9.98575i 1.23359 1.32264i
\(58\) 0 0
\(59\) 0.506187 0.292247i 0.0659000 0.0380474i −0.466688 0.884422i \(-0.654553\pi\)
0.532588 + 0.846375i \(0.321220\pi\)
\(60\) 0 0
\(61\) 10.8979 + 6.29191i 1.39533 + 0.805597i 0.993899 0.110291i \(-0.0351781\pi\)
0.401435 + 0.915887i \(0.368511\pi\)
\(62\) 0 0
\(63\) 11.8012 + 6.81342i 1.48681 + 0.858410i
\(64\) 0 0
\(65\) 0.782640i 0.0970745i
\(66\) 0 0
\(67\) −12.5523 7.24710i −1.53351 0.885374i −0.999196 0.0400897i \(-0.987236\pi\)
−0.534317 0.845284i \(-0.679431\pi\)
\(68\) 0 0
\(69\) 20.6268 2.48318
\(70\) 0 0
\(71\) 4.91970 + 8.52117i 0.583861 + 1.01128i 0.995016 + 0.0997116i \(0.0317920\pi\)
−0.411155 + 0.911565i \(0.634875\pi\)
\(72\) 0 0
\(73\) 2.50000 + 4.33013i 0.292603 + 0.506803i 0.974424 0.224716i \(-0.0721453\pi\)
−0.681822 + 0.731519i \(0.738812\pi\)
\(74\) 0 0
\(75\) 6.26527i 0.723452i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.18765 5.52117i −0.358639 0.621180i 0.629095 0.777328i \(-0.283426\pi\)
−0.987734 + 0.156148i \(0.950092\pi\)
\(80\) 0 0
\(81\) −8.49120 14.7072i −0.943467 1.63413i
\(82\) 0 0
\(83\) 8.33708 0.915114 0.457557 0.889180i \(-0.348725\pi\)
0.457557 + 0.889180i \(0.348725\pi\)
\(84\) 0 0
\(85\) 8.72013 + 5.03457i 0.945831 + 0.546076i
\(86\) 0 0
\(87\) 18.2113i 1.95246i
\(88\) 0 0
\(89\) −9.11804 5.26430i −0.966510 0.558015i −0.0683396 0.997662i \(-0.521770\pi\)
−0.898171 + 0.439647i \(0.855103\pi\)
\(90\) 0 0
\(91\) 0.782640 + 0.451857i 0.0820429 + 0.0473675i
\(92\) 0 0
\(93\) −22.6180 + 13.0585i −2.34538 + 1.35411i
\(94\) 0 0
\(95\) 5.14943 5.52117i 0.528321 0.566460i
\(96\) 0 0
\(97\) −0.677786 + 0.391320i −0.0688187 + 0.0397325i −0.534015 0.845475i \(-0.679317\pi\)
0.465196 + 0.885208i \(0.345984\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.7201 + 8.49867i 1.46471 + 0.845649i 0.999223 0.0394071i \(-0.0125469\pi\)
0.465484 + 0.885056i \(0.345880\pi\)
\(102\) 0 0
\(103\) −14.3159 −1.41059 −0.705293 0.708916i \(-0.749184\pi\)
−0.705293 + 0.708916i \(0.749184\pi\)
\(104\) 0 0
\(105\) 9.39791 + 5.42589i 0.917142 + 0.529512i
\(106\) 0 0
\(107\) 11.6268i 1.12401i −0.827134 0.562004i \(-0.810030\pi\)
0.827134 0.562004i \(-0.189970\pi\)
\(108\) 0 0
\(109\) 3.17198 + 5.49404i 0.303821 + 0.526233i 0.976998 0.213248i \(-0.0684042\pi\)
−0.673177 + 0.739481i \(0.735071\pi\)
\(110\) 0 0
\(111\) 9.36469 5.40671i 0.888857 0.513182i
\(112\) 0 0
\(113\) 16.8306i 1.58329i 0.610984 + 0.791643i \(0.290774\pi\)
−0.610984 + 0.791643i \(0.709226\pi\)
\(114\) 0 0
\(115\) 11.4047 1.06349
\(116\) 0 0
\(117\) −1.53935 2.66623i −0.142313 0.246493i
\(118\) 0 0
\(119\) 10.0691 5.81342i 0.923036 0.532915i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 1.22586 2.12326i 0.110532 0.191448i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 5.70234 9.87674i 0.506001 0.876419i −0.493975 0.869476i \(-0.664457\pi\)
0.999976 0.00694308i \(-0.00221007\pi\)
\(128\) 0 0
\(129\) 1.50000 + 0.866025i 0.132068 + 0.0762493i
\(130\) 0 0
\(131\) 0.198253 + 0.343384i 0.0173214 + 0.0300016i 0.874556 0.484924i \(-0.161153\pi\)
−0.857235 + 0.514926i \(0.827819\pi\)
\(132\) 0 0
\(133\) −2.54814 8.33708i −0.220952 0.722917i
\(134\) 0 0
\(135\) −10.3456 17.9191i −0.890406 1.54223i
\(136\) 0 0
\(137\) 3.09329 5.35774i 0.264278 0.457742i −0.703096 0.711095i \(-0.748200\pi\)
0.967374 + 0.253352i \(0.0815331\pi\)
\(138\) 0 0
\(139\) 8.61354 14.9191i 0.730591 1.26542i −0.226040 0.974118i \(-0.572578\pi\)
0.956631 0.291302i \(-0.0940885\pi\)
\(140\) 0 0
\(141\) −29.4403 −2.47932
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 10.0691i 0.836196i
\(146\) 0 0
\(147\) −8.13883 + 4.69896i −0.671279 + 0.387563i
\(148\) 0 0
\(149\) 14.7201 8.49867i 1.20592 0.696238i 0.244054 0.969762i \(-0.421522\pi\)
0.961865 + 0.273523i \(0.0881891\pi\)
\(150\) 0 0
\(151\) −0.459470 −0.0373911 −0.0186956 0.999825i \(-0.505951\pi\)
−0.0186956 + 0.999825i \(0.505951\pi\)
\(152\) 0 0
\(153\) −39.6092 −3.20222
\(154\) 0 0
\(155\) −12.5056 + 7.22013i −1.00448 + 0.579935i
\(156\) 0 0
\(157\) −3.54234 + 2.04517i −0.282710 + 0.163222i −0.634649 0.772800i \(-0.718855\pi\)
0.351940 + 0.936023i \(0.385522\pi\)
\(158\) 0 0
\(159\) 27.0247i 2.14320i
\(160\) 0 0
\(161\) 6.58449 11.4047i 0.518931 0.898815i
\(162\) 0 0
\(163\) −7.38767 −0.578647 −0.289324 0.957231i \(-0.593430\pi\)
−0.289324 + 0.957231i \(0.593430\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.71294 4.69896i 0.209934 0.363616i −0.741760 0.670666i \(-0.766008\pi\)
0.951693 + 0.307050i \(0.0993418\pi\)
\(168\) 0 0
\(169\) 6.39791 + 11.0815i 0.492147 + 0.852424i
\(170\) 0 0
\(171\) −6.68323 + 28.9373i −0.511080 + 2.21289i
\(172\) 0 0
\(173\) 1.50000 + 2.59808i 0.114043 + 0.197528i 0.917397 0.397974i \(-0.130287\pi\)
−0.803354 + 0.595502i \(0.796953\pi\)
\(174\) 0 0
\(175\) −3.46410 2.00000i −0.261861 0.151186i
\(176\) 0 0
\(177\) −0.915505 + 1.58570i −0.0688136 + 0.119189i
\(178\) 0 0
\(179\) 6.79582i 0.507944i −0.967212 0.253972i \(-0.918263\pi\)
0.967212 0.253972i \(-0.0817371\pi\)
\(180\) 0 0
\(181\) −1.95186 + 3.38072i −0.145080 + 0.251287i −0.929403 0.369067i \(-0.879677\pi\)
0.784323 + 0.620353i \(0.213011\pi\)
\(182\) 0 0
\(183\) −39.4206 −2.91405
\(184\) 0 0
\(185\) 5.17779 2.98940i 0.380678 0.219785i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −23.8921 −1.73790
\(190\) 0 0
\(191\) 24.4227i 1.76716i −0.468279 0.883581i \(-0.655126\pi\)
0.468279 0.883581i \(-0.344874\pi\)
\(192\) 0 0
\(193\) 20.2958 11.7178i 1.46093 0.843466i 0.461871 0.886947i \(-0.347178\pi\)
0.999054 + 0.0434813i \(0.0138449\pi\)
\(194\) 0 0
\(195\) −1.22586 2.12326i −0.0877859 0.152050i
\(196\) 0 0
\(197\) 4.87298i 0.347186i −0.984818 0.173593i \(-0.944462\pi\)
0.984818 0.173593i \(-0.0555377\pi\)
\(198\) 0 0
\(199\) 22.6844 + 13.0969i 1.60806 + 0.928412i 0.989804 + 0.142433i \(0.0454926\pi\)
0.618253 + 0.785979i \(0.287841\pi\)
\(200\) 0 0
\(201\) 45.4051 3.20263
\(202\) 0 0
\(203\) 10.0691 + 5.81342i 0.706715 + 0.408022i
\(204\) 0 0
\(205\) 0.677786 1.17396i 0.0473386 0.0819929i
\(206\) 0 0
\(207\) −38.8524 + 22.4315i −2.70043 + 1.55909i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −19.3138 + 11.1508i −1.32962 + 0.767654i −0.985240 0.171178i \(-0.945243\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(212\) 0 0
\(213\) −26.6937 15.4116i −1.82902 1.05599i
\(214\) 0 0
\(215\) 0.829357 + 0.478830i 0.0565617 + 0.0326559i
\(216\) 0 0
\(217\) 16.6742i 1.13192i
\(218\) 0 0
\(219\) −13.5647 7.83159i −0.916619 0.529210i
\(220\) 0 0
\(221\) −2.62684 −0.176700
\(222\) 0 0
\(223\) −3.66235 6.34338i −0.245249 0.424784i 0.716952 0.697122i \(-0.245537\pi\)
−0.962202 + 0.272338i \(0.912203\pi\)
\(224\) 0 0
\(225\) 6.81342 + 11.8012i 0.454228 + 0.786746i
\(226\) 0 0
\(227\) 0.373165i 0.0247678i 0.999923 + 0.0123839i \(0.00394202\pi\)
−0.999923 + 0.0123839i \(0.996058\pi\)
\(228\) 0 0
\(229\) 26.5765i 1.75623i −0.478453 0.878113i \(-0.658802\pi\)
0.478453 0.878113i \(-0.341198\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.677786 + 1.17396i 0.0444032 + 0.0769087i 0.887373 0.461053i \(-0.152528\pi\)
−0.842970 + 0.537961i \(0.819195\pi\)
\(234\) 0 0
\(235\) −16.2777 −1.06184
\(236\) 0 0
\(237\) 17.2958 + 9.98575i 1.12348 + 0.648644i
\(238\) 0 0
\(239\) 13.2713i 0.858447i −0.903198 0.429223i \(-0.858787\pi\)
0.903198 0.429223i \(-0.141213\pi\)
\(240\) 0 0
\(241\) −7.07570 4.08516i −0.455786 0.263148i 0.254485 0.967077i \(-0.418094\pi\)
−0.710271 + 0.703929i \(0.751427\pi\)
\(242\) 0 0
\(243\) 15.0356 + 8.68078i 0.964531 + 0.556872i
\(244\) 0 0
\(245\) −4.50000 + 2.59808i −0.287494 + 0.165985i
\(246\) 0 0
\(247\) −0.443224 + 1.91908i −0.0282016 + 0.122108i
\(248\) 0 0
\(249\) −22.6180 + 13.0585i −1.43336 + 0.827551i
\(250\) 0 0
\(251\) 7.43439 12.8767i 0.469255 0.812773i −0.530128 0.847918i \(-0.677856\pi\)
0.999382 + 0.0351451i \(0.0111893\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −31.5429 −1.97530
\(256\) 0 0
\(257\) −4.10209 2.36834i −0.255881 0.147733i 0.366573 0.930389i \(-0.380531\pi\)
−0.622454 + 0.782656i \(0.713864\pi\)
\(258\) 0 0
\(259\) 6.90371i 0.428976i
\(260\) 0 0
\(261\) −19.8046 34.3026i −1.22588 2.12328i
\(262\) 0 0
\(263\) 2.25347 1.30104i 0.138955 0.0802258i −0.428911 0.903347i \(-0.641103\pi\)
0.567866 + 0.823121i \(0.307769\pi\)
\(264\) 0 0
\(265\) 14.9421i 0.917887i
\(266\) 0 0
\(267\) 32.9823 2.01848
\(268\) 0 0
\(269\) 3.67779 + 6.37011i 0.224239 + 0.388393i 0.956091 0.293071i \(-0.0946772\pi\)
−0.731852 + 0.681463i \(0.761344\pi\)
\(270\) 0 0
\(271\) −18.6045 + 10.7413i −1.13014 + 0.652487i −0.943970 0.330030i \(-0.892941\pi\)
−0.186171 + 0.982517i \(0.559608\pi\)
\(272\) 0 0
\(273\) −2.83101 −0.171341
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.2900i 1.03886i 0.854514 + 0.519429i \(0.173855\pi\)
−0.854514 + 0.519429i \(0.826145\pi\)
\(278\) 0 0
\(279\) 28.4020 49.1937i 1.70038 2.94515i
\(280\) 0 0
\(281\) −23.2958 13.4499i −1.38971 0.802351i −0.396430 0.918065i \(-0.629751\pi\)
−0.993283 + 0.115714i \(0.963084\pi\)
\(282\) 0 0
\(283\) −1.93030 3.34338i −0.114745 0.198744i 0.802933 0.596069i \(-0.203272\pi\)
−0.917678 + 0.397326i \(0.869938\pi\)
\(284\) 0 0
\(285\) −5.32221 + 23.0443i −0.315261 + 1.36503i
\(286\) 0 0
\(287\) −0.782640 1.35557i −0.0461978 0.0800169i
\(288\) 0 0
\(289\) −8.39791 + 14.5456i −0.493995 + 0.855624i
\(290\) 0 0
\(291\) 1.22586 2.12326i 0.0718614 0.124468i
\(292\) 0 0
\(293\) −17.5246 −1.02380 −0.511898 0.859046i \(-0.671057\pi\)
−0.511898 + 0.859046i \(0.671057\pi\)
\(294\) 0 0
\(295\) −0.506187 + 0.876742i −0.0294714 + 0.0510459i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.57664 + 1.48763i −0.149011 + 0.0860316i
\(300\) 0 0
\(301\) 0.957659 0.552905i 0.0551986 0.0318689i
\(302\) 0 0
\(303\) −53.2465 −3.05893
\(304\) 0 0
\(305\) −21.7958 −1.24803
\(306\) 0 0
\(307\) 25.9340 14.9730i 1.48013 0.854556i 0.480387 0.877056i \(-0.340496\pi\)
0.999747 + 0.0225007i \(0.00716279\pi\)
\(308\) 0 0
\(309\) 38.8382 22.4232i 2.20943 1.27561i
\(310\) 0 0
\(311\) 1.27126i 0.0720867i −0.999350 0.0360434i \(-0.988525\pi\)
0.999350 0.0360434i \(-0.0114754\pi\)
\(312\) 0 0
\(313\) 9.07570 15.7196i 0.512989 0.888523i −0.486898 0.873459i \(-0.661872\pi\)
0.999887 0.0150637i \(-0.00479510\pi\)
\(314\) 0 0
\(315\) −23.6024 −1.32984
\(316\) 0 0
\(317\) 12.4912 21.6354i 0.701576 1.21517i −0.266337 0.963880i \(-0.585813\pi\)
0.967913 0.251285i \(-0.0808532\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 18.2113 + 31.5429i 1.01646 + 1.76056i
\(322\) 0 0
\(323\) 18.5311 + 17.2835i 1.03110 + 0.961677i
\(324\) 0 0
\(325\) 0.451857 + 0.782640i 0.0250645 + 0.0434130i
\(326\) 0 0
\(327\) −17.2108 9.93668i −0.951761 0.549499i
\(328\) 0 0
\(329\) −9.39791 + 16.2777i −0.518124 + 0.897417i
\(330\) 0 0
\(331\) 17.7842i 0.977509i −0.872421 0.488755i \(-0.837451\pi\)
0.872421 0.488755i \(-0.162549\pi\)
\(332\) 0 0
\(333\) −11.7595 + 20.3680i −0.644415 + 1.11616i
\(334\) 0 0
\(335\) 25.1047 1.37162
\(336\) 0 0
\(337\) 12.1180 6.99635i 0.660112 0.381116i −0.132208 0.991222i \(-0.542207\pi\)
0.792320 + 0.610106i \(0.208873\pi\)
\(338\) 0 0
\(339\) −26.3620 45.6604i −1.43179 2.47993i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) −30.9403 + 17.8634i −1.66577 + 0.961731i
\(346\) 0 0
\(347\) −4.68997 8.12326i −0.251771 0.436079i 0.712243 0.701933i \(-0.247679\pi\)
−0.964013 + 0.265854i \(0.914346\pi\)
\(348\) 0 0
\(349\) 9.90236i 0.530061i 0.964240 + 0.265031i \(0.0853821\pi\)
−0.964240 + 0.265031i \(0.914618\pi\)
\(350\) 0 0
\(351\) 4.67473 + 2.69896i 0.249519 + 0.144060i
\(352\) 0 0
\(353\) 3.18658 0.169605 0.0848023 0.996398i \(-0.472974\pi\)
0.0848023 + 0.996398i \(0.472974\pi\)
\(354\) 0 0
\(355\) −14.7591 8.52117i −0.783332 0.452257i
\(356\) 0 0
\(357\) −18.2113 + 31.5429i −0.963846 + 1.66943i
\(358\) 0 0
\(359\) 21.0562 12.1568i 1.11130 0.641611i 0.172137 0.985073i \(-0.444933\pi\)
0.939167 + 0.343462i \(0.111600\pi\)
\(360\) 0 0
\(361\) 15.7535 10.6220i 0.829131 0.559055i
\(362\) 0 0
\(363\) 29.8424 17.2295i 1.56632 0.904315i
\(364\) 0 0
\(365\) −7.50000 4.33013i −0.392568 0.226649i
\(366\) 0 0
\(367\) 11.6029 + 6.69896i 0.605668 + 0.349683i 0.771268 0.636510i \(-0.219623\pi\)
−0.165600 + 0.986193i \(0.552956\pi\)
\(368\) 0 0
\(369\) 5.33245i 0.277596i
\(370\) 0 0
\(371\) 14.9421 + 8.62684i 0.775756 + 0.447883i
\(372\) 0 0
\(373\) −11.8921 −0.615750 −0.307875 0.951427i \(-0.599618\pi\)
−0.307875 + 0.951427i \(0.599618\pi\)
\(374\) 0 0
\(375\) 18.9906 + 32.8927i 0.980671 + 1.69857i
\(376\) 0 0
\(377\) −1.31342 2.27491i −0.0676445 0.117164i
\(378\) 0 0
\(379\) 7.80743i 0.401041i −0.979690 0.200520i \(-0.935737\pi\)
0.979690 0.200520i \(-0.0642633\pi\)
\(380\) 0 0
\(381\) 35.7267i 1.83034i
\(382\) 0 0
\(383\) 4.68997 + 8.12326i 0.239646 + 0.415079i 0.960613 0.277891i \(-0.0896353\pi\)
−0.720967 + 0.692970i \(0.756302\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.76717 −0.191496
\(388\) 0 0
\(389\) 15.2535 + 8.80660i 0.773382 + 0.446513i 0.834080 0.551644i \(-0.185999\pi\)
−0.0606975 + 0.998156i \(0.519333\pi\)
\(390\) 0 0
\(391\) 38.2784i 1.93582i
\(392\) 0 0
\(393\) −1.07570 0.621055i −0.0542618 0.0313281i
\(394\) 0 0
\(395\) 9.56295 + 5.52117i 0.481164 + 0.277800i
\(396\) 0 0
\(397\) 4.10209 2.36834i 0.205878 0.118864i −0.393516 0.919318i \(-0.628742\pi\)
0.599394 + 0.800454i \(0.295408\pi\)
\(398\) 0 0
\(399\) 19.9715 + 18.6268i 0.999825 + 0.932508i
\(400\) 0 0
\(401\) −17.1604 + 9.90755i −0.856948 + 0.494759i −0.862989 0.505222i \(-0.831410\pi\)
0.00604076 + 0.999982i \(0.498077\pi\)
\(402\) 0 0
\(403\) 1.88359 3.26247i 0.0938281 0.162515i
\(404\) 0 0
\(405\) 25.4736 + 14.7072i 1.26579 + 0.730806i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 21.5423 + 12.4375i 1.06520 + 0.614994i 0.926866 0.375392i \(-0.122492\pi\)
0.138334 + 0.990386i \(0.455825\pi\)
\(410\) 0 0
\(411\) 19.3803i 0.955961i
\(412\) 0 0
\(413\) 0.584495 + 1.01237i 0.0287611 + 0.0498157i
\(414\) 0 0
\(415\) −12.5056 + 7.22013i −0.613877 + 0.354422i
\(416\) 0 0
\(417\) 53.9662i 2.64273i
\(418\) 0 0
\(419\) 12.4171 0.606613 0.303306 0.952893i \(-0.401909\pi\)
0.303306 + 0.952893i \(0.401909\pi\)
\(420\) 0 0
\(421\) 10.0481 + 17.4039i 0.489717 + 0.848214i 0.999930 0.0118339i \(-0.00376694\pi\)
−0.510213 + 0.860048i \(0.670434\pi\)
\(422\) 0 0
\(423\) 55.4533 32.0160i 2.69623 1.55667i
\(424\) 0 0
\(425\) 11.6268 0.563984
\(426\) 0 0
\(427\) −12.5838 + 21.7958i −0.608974 + 1.05477i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.2186 26.3593i 0.733053 1.26968i −0.222520 0.974928i \(-0.571428\pi\)
0.955572 0.294756i \(-0.0952385\pi\)
\(432\) 0 0
\(433\) −29.6937 17.1437i −1.42699 0.823873i −0.430108 0.902778i \(-0.641524\pi\)
−0.996882 + 0.0789045i \(0.974858\pi\)
\(434\) 0 0
\(435\) −15.7715 27.3170i −0.756184 1.30975i
\(436\) 0 0
\(437\) 27.9650 + 6.45868i 1.33775 + 0.308961i
\(438\) 0 0
\(439\) −17.5035 30.3170i −0.835398 1.44695i −0.893706 0.448653i \(-0.851904\pi\)
0.0583085 0.998299i \(-0.481429\pi\)
\(440\) 0 0
\(441\) 10.2201 17.7018i 0.486673 0.842942i
\(442\) 0 0
\(443\) −18.9124 + 32.7572i −0.898556 + 1.55634i −0.0692140 + 0.997602i \(0.522049\pi\)
−0.829342 + 0.558742i \(0.811284\pi\)
\(444\) 0 0
\(445\) 18.2361 0.864473
\(446\) 0 0
\(447\) −26.6232 + 46.1128i −1.25924 + 2.18106i
\(448\) 0 0
\(449\) 34.9441i 1.64911i −0.565779 0.824557i \(-0.691425\pi\)
0.565779 0.824557i \(-0.308575\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.24652 0.719676i 0.0585664 0.0338133i
\(454\) 0 0
\(455\) −1.56528 −0.0733814
\(456\) 0 0
\(457\) 2.08468 0.0975173 0.0487586 0.998811i \(-0.484473\pi\)
0.0487586 + 0.998811i \(0.484473\pi\)
\(458\) 0 0
\(459\) 60.1432 34.7237i 2.80725 1.62076i
\(460\) 0 0
\(461\) 3.96664 2.29014i 0.184745 0.106663i −0.404775 0.914416i \(-0.632650\pi\)
0.589520 + 0.807754i \(0.299317\pi\)
\(462\) 0 0
\(463\) 0.355572i 0.0165248i −0.999966 0.00826242i \(-0.997370\pi\)
0.999966 0.00826242i \(-0.00263004\pi\)
\(464\) 0 0
\(465\) 22.6180 39.1756i 1.04889 1.81672i
\(466\) 0 0
\(467\) −33.2017 −1.53639 −0.768195 0.640216i \(-0.778845\pi\)
−0.768195 + 0.640216i \(0.778845\pi\)
\(468\) 0 0
\(469\) 14.4942 25.1047i 0.669280 1.15923i
\(470\) 0 0
\(471\) 6.40678 11.0969i 0.295209 0.511317i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.96179 8.49420i 0.0900129 0.389741i
\(476\) 0 0
\(477\) −29.3891 50.9034i −1.34564 2.33071i
\(478\) 0 0
\(479\) −14.0242 8.09687i −0.640782 0.369955i 0.144134 0.989558i \(-0.453960\pi\)
−0.784916 + 0.619603i \(0.787294\pi\)
\(480\) 0 0
\(481\) −0.779874 + 1.35078i −0.0355592 + 0.0615903i
\(482\) 0 0
\(483\) 41.2537i 1.87711i
\(484\) 0 0
\(485\) 0.677786 1.17396i 0.0307767 0.0533068i
\(486\) 0 0
\(487\) −18.5730 −0.841623 −0.420811 0.907148i \(-0.638255\pi\)
−0.420811 + 0.907148i \(0.638255\pi\)
\(488\) 0 0
\(489\) 20.0423 11.5715i 0.906346 0.523279i
\(490\) 0 0
\(491\) 8.53534 + 14.7836i 0.385194 + 0.667176i 0.991796 0.127830i \(-0.0408011\pi\)
−0.606602 + 0.795006i \(0.707468\pi\)
\(492\) 0 0
\(493\) −33.7958 −1.52209
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −17.0423 + 9.83940i −0.764453 + 0.441357i
\(498\) 0 0
\(499\) −0.276452 0.478830i −0.0123757 0.0214354i 0.859771 0.510679i \(-0.170606\pi\)
−0.872147 + 0.489244i \(0.837273\pi\)
\(500\) 0 0
\(501\) 16.9973i 0.759385i
\(502\) 0 0
\(503\) 14.4359 + 8.33459i 0.643666 + 0.371621i 0.786025 0.618194i \(-0.212135\pi\)
−0.142359 + 0.989815i \(0.545469\pi\)
\(504\) 0 0
\(505\) −29.4403 −1.31007
\(506\) 0 0
\(507\) −34.7144 20.0423i −1.54172 0.890112i
\(508\) 0 0
\(509\) 6.11804 10.5968i 0.271177 0.469693i −0.697986 0.716111i \(-0.745920\pi\)
0.969164 + 0.246418i \(0.0792537\pi\)
\(510\) 0 0
\(511\) −8.66025 + 5.00000i −0.383107 + 0.221187i
\(512\) 0 0
\(513\) −15.2201 49.7976i −0.671985 2.19862i
\(514\) 0 0
\(515\) 21.4738 12.3979i 0.946249 0.546317i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −8.13883 4.69896i −0.357255 0.206261i
\(520\) 0 0
\(521\) 3.13056i 0.137152i 0.997646 + 0.0685761i \(0.0218456\pi\)
−0.997646 + 0.0685761i \(0.978154\pi\)
\(522\) 0 0
\(523\) 22.1620 + 12.7952i 0.969077 + 0.559497i 0.898955 0.438042i \(-0.144328\pi\)
0.0701222 + 0.997538i \(0.477661\pi\)
\(524\) 0 0
\(525\) 12.5305 0.546878
\(526\) 0 0
\(527\) −24.2335 41.9736i −1.05563 1.82840i
\(528\) 0 0
\(529\) 10.1778 + 17.6284i 0.442512 + 0.766454i
\(530\) 0 0
\(531\) 3.98241i 0.172822i
\(532\) 0 0
\(533\) 0.353642i 0.0153179i
\(534\) 0 0
\(535\) 10.0691 + 17.4403i 0.435327 + 0.754008i
\(536\) 0 0
\(537\) 10.6444 + 18.4367i 0.459341 + 0.795602i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −24.1180 13.9246i −1.03692 0.598663i −0.117957 0.993019i \(-0.537635\pi\)
−0.918958 + 0.394355i \(0.870968\pi\)
\(542\) 0 0
\(543\) 12.2289i 0.524793i
\(544\) 0 0
\(545\) −9.51595 5.49404i −0.407619 0.235339i
\(546\) 0 0
\(547\) 20.7379 + 11.9730i 0.886688 + 0.511930i 0.872858 0.487975i \(-0.162264\pi\)
0.0138304 + 0.999904i \(0.495597\pi\)
\(548\) 0 0
\(549\) 74.2520 42.8694i 3.16900 1.82962i
\(550\) 0 0
\(551\) −5.70234 + 24.6902i −0.242928 + 1.05184i
\(552\) 0 0
\(553\) 11.0423 6.37530i 0.469568 0.271105i
\(554\) 0 0
\(555\) −9.36469 + 16.2201i −0.397509 + 0.688506i
\(556\) 0 0
\(557\) −29.1340 16.8205i −1.23445 0.712708i −0.266493 0.963837i \(-0.585865\pi\)
−0.967954 + 0.251129i \(0.919198\pi\)
\(558\) 0 0
\(559\) −0.249834 −0.0105669
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.91532i 0.0807211i −0.999185 0.0403605i \(-0.987149\pi\)
0.999185 0.0403605i \(-0.0128506\pi\)
\(564\) 0 0
\(565\) −14.5757 25.2459i −0.613204 1.06210i
\(566\) 0 0
\(567\) 29.4144 16.9824i 1.23529 0.713194i
\(568\) 0 0
\(569\) 44.2306i 1.85424i −0.374762 0.927121i \(-0.622275\pi\)
0.374762 0.927121i \(-0.377725\pi\)
\(570\) 0 0
\(571\) 25.8140 1.08028 0.540141 0.841575i \(-0.318371\pi\)
0.540141 + 0.841575i \(0.318371\pi\)
\(572\) 0 0
\(573\) 38.2537 + 66.2573i 1.59807 + 2.76794i
\(574\) 0 0
\(575\) 11.4047 6.58449i 0.475608 0.274592i
\(576\) 0 0
\(577\) 35.2361 1.46690 0.733449 0.679745i \(-0.237910\pi\)
0.733449 + 0.679745i \(0.237910\pi\)
\(578\) 0 0
\(579\) −36.7076 + 63.5795i −1.52552 + 2.64227i
\(580\) 0 0
\(581\) 16.6742i 0.691761i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.61804 + 2.66623i 0.190932 + 0.110235i
\(586\) 0 0
\(587\) −1.21063 2.09687i −0.0499680 0.0865470i 0.839960 0.542649i \(-0.182579\pi\)
−0.889928 + 0.456102i \(0.849245\pi\)
\(588\) 0 0
\(589\) −34.7535 + 10.6220i −1.43199 + 0.437674i
\(590\) 0 0
\(591\) 7.63264 + 13.2201i 0.313965 + 0.543803i
\(592\) 0 0
\(593\) 2.32221 4.02219i 0.0953619 0.165172i −0.814398 0.580307i \(-0.802933\pi\)
0.909760 + 0.415136i \(0.136266\pi\)
\(594\) 0 0
\(595\) −10.0691 + 17.4403i −0.412794 + 0.714981i
\(596\) 0 0
\(597\) −82.0555 −3.35831
\(598\) 0 0
\(599\) −0.0467176 + 0.0809173i −0.00190883 + 0.00330619i −0.866978 0.498346i \(-0.833941\pi\)
0.865069 + 0.501652i \(0.167274\pi\)
\(600\) 0 0
\(601\) 8.98342i 0.366442i 0.983072 + 0.183221i \(0.0586523\pi\)
−0.983072 + 0.183221i \(0.941348\pi\)
\(602\) 0 0
\(603\) −85.5244 + 49.3775i −3.48282 + 2.01081i
\(604\) 0 0
\(605\) 16.5000 9.52628i 0.670820 0.387298i
\(606\) 0 0
\(607\) 7.84714 0.318506 0.159253 0.987238i \(-0.449091\pi\)
0.159253 + 0.987238i \(0.449091\pi\)
\(608\) 0 0
\(609\) −36.4227 −1.47592
\(610\) 0 0
\(611\) 3.67759 2.12326i 0.148779 0.0858978i
\(612\) 0 0
\(613\) 12.2799 7.08979i 0.495979 0.286354i −0.231072 0.972937i \(-0.574223\pi\)
0.727052 + 0.686583i \(0.240890\pi\)
\(614\) 0 0
\(615\) 4.24652i 0.171236i
\(616\) 0 0
\(617\) 15.1180 26.1852i 0.608629 1.05418i −0.382837 0.923816i \(-0.625053\pi\)
0.991467 0.130361i \(-0.0416137\pi\)
\(618\) 0 0
\(619\) 4.71658 0.189575 0.0947877 0.995498i \(-0.469783\pi\)
0.0947877 + 0.995498i \(0.469783\pi\)
\(620\) 0 0
\(621\) 39.3294 68.1205i 1.57823 2.73358i
\(622\) 0 0
\(623\) 10.5286 18.2361i 0.421820 0.730613i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0335 + 17.3786i 0.400064 + 0.692930i
\(630\) 0 0
\(631\) −17.4150 10.0545i −0.693278 0.400264i 0.111561 0.993758i \(-0.464415\pi\)
−0.804839 + 0.593493i \(0.797748\pi\)
\(632\) 0 0
\(633\) 34.9315 60.5031i 1.38840 2.40478i
\(634\) 0 0
\(635\) 19.7535i 0.783893i
\(636\) 0 0
\(637\) 0.677786 1.17396i 0.0268549 0.0465140i
\(638\) 0 0
\(639\) 67.0399 2.65206
\(640\) 0 0
\(641\) 5.45766 3.15098i 0.215565 0.124456i −0.388330 0.921520i \(-0.626948\pi\)
0.603895 + 0.797064i \(0.293615\pi\)
\(642\) 0 0
\(643\) 15.6780 + 27.1552i 0.618282 + 1.07089i 0.989799 + 0.142469i \(0.0455042\pi\)
−0.371518 + 0.928426i \(0.621162\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) 0 0
\(647\) 13.9648i 0.549014i −0.961585 0.274507i \(-0.911485\pi\)
0.961585 0.274507i \(-0.0885146\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −26.1171 45.2361i −1.02361 1.77294i
\(652\) 0 0
\(653\) 20.1078i 0.786879i −0.919351 0.393439i \(-0.871285\pi\)
0.919351 0.393439i \(-0.128715\pi\)
\(654\) 0 0
\(655\) −0.594759 0.343384i −0.0232392 0.0134171i
\(656\) 0 0
\(657\) 34.0671 1.32908
\(658\) 0 0
\(659\) 7.81566 + 4.51237i 0.304455 + 0.175777i 0.644442 0.764653i \(-0.277090\pi\)
−0.339988 + 0.940430i \(0.610423\pi\)
\(660\) 0 0
\(661\) −12.1720 + 21.0825i −0.473435 + 0.820014i −0.999538 0.0304073i \(-0.990320\pi\)
0.526102 + 0.850421i \(0.323653\pi\)
\(662\) 0 0
\(663\) 7.12646 4.11446i 0.276769 0.159792i
\(664\) 0 0
\(665\) 11.0423 + 10.2989i 0.428204 + 0.399373i
\(666\) 0 0
\(667\) −33.1501 + 19.1392i −1.28358 + 0.741073i
\(668\) 0 0
\(669\) 19.8715 + 11.4728i 0.768277 + 0.443565i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 30.2275i 1.16518i −0.812765 0.582592i \(-0.802038\pi\)
0.812765 0.582592i \(-0.197962\pi\)
\(674\) 0 0
\(675\) −20.6912 11.9461i −0.796404 0.459804i
\(676\) 0 0
\(677\) −39.9824 −1.53665 −0.768324 0.640061i \(-0.778909\pi\)
−0.768324 + 0.640061i \(0.778909\pi\)
\(678\) 0 0
\(679\) −0.782640 1.35557i −0.0300350 0.0520221i
\(680\) 0 0
\(681\) −0.584495 1.01237i −0.0223979 0.0387943i
\(682\) 0 0
\(683\) 16.3556i 0.625829i 0.949781 + 0.312914i \(0.101305\pi\)
−0.949781 + 0.312914i \(0.898695\pi\)
\(684\) 0 0
\(685\) 10.7155i 0.409417i
\(686\) 0 0
\(687\) 41.6273 + 72.1006i 1.58818 + 2.75081i
\(688\) 0 0
\(689\) −1.94905 3.37585i −0.0742529 0.128610i
\(690\) 0 0
\(691\) 20.7846 0.790684 0.395342 0.918534i \(-0.370626\pi\)
0.395342 + 0.918534i \(0.370626\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 29.8382i 1.13183i
\(696\) 0 0
\(697\) 3.94025 + 2.27491i 0.149248 + 0.0861682i
\(698\) 0 0
\(699\) −3.67759 2.12326i −0.139099 0.0803090i
\(700\) 0 0
\(701\) −33.2271 + 19.1837i −1.25497 + 0.724557i −0.972092 0.234599i \(-0.924622\pi\)
−0.282878 + 0.959156i \(0.591289\pi\)
\(702\) 0 0
\(703\) 14.3892 4.39791i 0.542699 0.165870i
\(704\) 0 0
\(705\) 44.1604 25.4960i 1.66318 0.960235i
\(706\) 0 0
\(707\) −16.9973 + 29.4403i −0.639251 + 1.10721i
\(708\) 0 0
\(709\) −9.54234 5.50927i −0.358370 0.206905i 0.309995 0.950738i \(-0.399672\pi\)
−0.668366 + 0.743833i \(0.733006\pi\)
\(710\) 0 0
\(711\) −43.4376 −1.62904
\(712\) 0 0
\(713\) −47.5409 27.4477i −1.78042 1.02793i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.7870 + 36.0042i 0.776306 + 1.34460i
\(718\) 0 0
\(719\) 11.9994 6.92788i 0.447504 0.258366i −0.259272 0.965804i \(-0.583483\pi\)
0.706775 + 0.707438i \(0.250149\pi\)
\(720\) 0 0
\(721\) 28.6318i 1.06630i
\(722\) 0 0
\(723\) 25.5946 0.951874
\(724\) 0 0
\(725\) 5.81342 + 10.0691i 0.215905 + 0.373958i
\(726\) 0 0
\(727\) −22.6111 + 13.0545i −0.838600 + 0.484166i −0.856788 0.515669i \(-0.827543\pi\)
0.0181884 + 0.999835i \(0.494210\pi\)
\(728\) 0 0
\(729\) −3.44025 −0.127417
\(730\) 0 0
\(731\) −1.60713 + 2.78364i −0.0594420 + 0.102957i
\(732\) 0 0
\(733\) 13.2405i 0.489050i 0.969643 + 0.244525i \(0.0786321\pi\)
−0.969643 + 0.244525i \(0.921368\pi\)
\(734\) 0 0
\(735\) 8.13883 14.0969i 0.300205 0.519971i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.1850 34.9614i −0.742517 1.28608i −0.951346 0.308125i \(-0.900299\pi\)
0.208829 0.977952i \(-0.433035\pi\)
\(740\) 0 0
\(741\) −1.80345 5.90059i −0.0662516 0.216764i
\(742\) 0 0
\(743\) 7.67936 + 13.3010i 0.281728 + 0.487968i 0.971811 0.235763i \(-0.0757589\pi\)
−0.690082 + 0.723731i \(0.742426\pi\)
\(744\) 0 0
\(745\) −14.7201 + 25.4960i −0.539304 + 0.934101i
\(746\) 0 0
\(747\) 28.4020 49.1937i 1.03918 1.79990i
\(748\) 0 0
\(749\) 23.2537 0.849671
\(750\) 0 0
\(751\) 8.21703 14.2323i 0.299844 0.519345i −0.676256 0.736666i \(-0.736399\pi\)
0.976100 + 0.217322i \(0.0697321\pi\)
\(752\) 0 0
\(753\) 46.5785i 1.69741i
\(754\) 0 0
\(755\) 0.689205 0.397912i 0.0250827 0.0144815i
\(756\) 0 0
\(757\) −26.0249 + 15.0255i −0.945892 + 0.546111i −0.891803 0.452425i \(-0.850559\pi\)
−0.0540898 + 0.998536i \(0.517226\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.8981 −1.33756 −0.668778 0.743463i \(-0.733182\pi\)
−0.668778 + 0.743463i \(0.733182\pi\)
\(762\) 0 0
\(763\) −10.9881 + 6.34397i −0.397795 + 0.229667i
\(764\) 0 0
\(765\) 59.4139 34.3026i 2.14811 1.24021i
\(766\) 0 0
\(767\) 0.264108i 0.00953640i
\(768\) 0 0
\(769\) −1.45766 + 2.52474i −0.0525645 + 0.0910445i −0.891110 0.453787i \(-0.850073\pi\)
0.838546 + 0.544831i \(0.183406\pi\)
\(770\) 0 0
\(771\) 14.8383 0.534389
\(772\) 0 0
\(773\) 1.87316 3.24442i 0.0673731 0.116694i −0.830371 0.557211i \(-0.811872\pi\)
0.897744 + 0.440517i \(0.145205\pi\)
\(774\) 0 0
\(775\) −8.33708 + 14.4403i −0.299477 + 0.518709i
\(776\) 0 0
\(777\) 10.8134 + 18.7294i 0.387929 + 0.671913i
\(778\) 0 0
\(779\) 2.32681 2.49478i 0.0833666 0.0893848i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 60.1432 + 34.7237i 2.14934 + 1.24092i
\(784\) 0 0
\(785\) 3.54234 6.13551i 0.126432 0.218986i
\(786\) 0 0
\(787\) 0.107890i 0.00384585i 0.999998 + 0.00192293i \(0.000612087\pi\)
−0.999998 + 0.00192293i \(0.999388\pi\)
\(788\) 0 0
\(789\) −4.07570 + 7.05932i −0.145099 + 0.251318i
\(790\) 0 0
\(791\) −33.6611 −1.19685
\(792\) 0 0
\(793\) 4.92430 2.84305i 0.174867 0.100960i
\(794\) 0 0
\(795\) −23.4041 40.5371i −0.830059 1.43770i
\(796\) 0 0
\(797\) 48.8981 1.73206 0.866030 0.499992i \(-0.166664\pi\)
0.866030 + 0.499992i \(0.166664\pi\)
\(798\) 0 0
\(799\) 54.6340i 1.93281i
\(800\) 0 0
\(801\) −62.1250 + 35.8679i −2.19508 + 1.26733i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 22.8094i 0.803924i
\(806\) 0 0
\(807\) −19.9553 11.5212i −0.702458 0.405564i
\(808\) 0 0
\(809\) −3.98241 −0.140014 −0.0700070 0.997547i \(-0.522302\pi\)
−0.0700070 + 0.997547i \(0.522302\pi\)
\(810\) 0 0
\(811\) −12.7392 7.35499i −0.447334 0.258269i 0.259369 0.965778i \(-0.416485\pi\)
−0.706704 + 0.707510i \(0.749819\pi\)
\(812\) 0 0
\(813\) 33.6486 58.2811i 1.18011 2.04401i
\(814\) 0 0
\(815\) 11.0815 6.39791i 0.388168 0.224109i
\(816\) 0 0
\(817\) 1.76247 + 1.64380i 0.0616609 + 0.0575093i
\(818\) 0 0
\(819\) 5.33245 3.07869i 0.186331 0.107578i
\(820\) 0 0
\(821\) −26.0334 15.0304i −0.908570 0.524563i −0.0285995 0.999591i \(-0.509105\pi\)
−0.879971 + 0.475028i \(0.842438\pi\)
\(822\) 0 0
\(823\) 32.3875 + 18.6990i 1.12896 + 0.651805i 0.943673 0.330880i \(-0.107346\pi\)
0.185286 + 0.982685i \(0.440679\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.5117 + 16.4612i 0.991449 + 0.572413i 0.905707 0.423904i \(-0.139341\pi\)
0.0857418 + 0.996317i \(0.472674\pi\)
\(828\) 0 0
\(829\) −7.80743 −0.271163 −0.135582 0.990766i \(-0.543290\pi\)
−0.135582 + 0.990766i \(0.543290\pi\)
\(830\) 0 0
\(831\) −27.0817 46.9069i −0.939454 1.62718i
\(832\) 0 0
\(833\) −8.72013 15.1037i −0.302134 0.523312i
\(834\) 0 0
\(835\) 9.39791i 0.325228i
\(836\) 0 0
\(837\) 99.5953i 3.44252i
\(838\) 0 0
\(839\) −1.60713 2.78364i −0.0554844 0.0961018i 0.836949 0.547281i \(-0.184337\pi\)
−0.892434 + 0.451179i \(0.851004\pi\)
\(840\) 0 0
\(841\) −2.39791 4.15331i −0.0826866 0.143217i
\(842\) 0 0
\(843\) 84.2670 2.90231
\(844\) 0 0
\(845\) −19.1937 11.0815i −0.660285 0.381216i
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) 10.4736 + 6.04694i 0.359453 + 0.207531i
\(850\) 0 0
\(851\) 19.6837 + 11.3644i 0.674747 + 0.389566i
\(852\) 0 0
\(853\) 16.8979 9.75601i 0.578574 0.334040i −0.181993 0.983300i \(-0.558255\pi\)
0.760566 + 0.649260i \(0.224921\pi\)
\(854\) 0 0
\(855\) −15.0356 49.1937i −0.514205 1.68239i
\(856\) 0 0
\(857\) −4.50000 + 2.59808i −0.153717 + 0.0887486i −0.574886 0.818234i \(-0.694953\pi\)
0.421168 + 0.906982i \(0.361620\pi\)
\(858\) 0 0
\(859\) 4.44499 7.69896i 0.151661 0.262685i −0.780177 0.625559i \(-0.784871\pi\)
0.931838 + 0.362874i \(0.118204\pi\)
\(860\) 0 0
\(861\) 4.24652 + 2.45173i 0.144721 + 0.0835546i
\(862\) 0 0
\(863\) 6.43826 0.219161 0.109580 0.993978i \(-0.465049\pi\)
0.109580 + 0.993978i \(0.465049\pi\)
\(864\) 0 0
\(865\) −4.50000 2.59808i −0.153005 0.0883372i
\(866\) 0 0
\(867\) 52.6152i 1.78691i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −5.67187 + 3.27465i −0.192184 + 0.110957i
\(872\) 0 0
\(873\) 5.33245i 0.180476i
\(874\) 0 0
\(875\) 24.2487 0.819756
\(876\) 0 0
\(877\) 27.5699 + 47.7525i 0.930969 + 1.61249i 0.781668 + 0.623694i \(0.214369\pi\)
0.149301 + 0.988792i \(0.452298\pi\)
\(878\) 0 0
\(879\) 47.5431 27.4490i 1.60359 0.925833i
\(880\) 0 0
\(881\) −35.9472 −1.21109 −0.605546 0.795810i \(-0.707045\pi\)
−0.605546 + 0.795810i \(0.707045\pi\)
\(882\) 0 0
\(883\) 2.86448 4.96142i 0.0963974 0.166965i −0.813794 0.581154i \(-0.802601\pi\)
0.910191 + 0.414189i \(0.135935\pi\)
\(884\) 0 0
\(885\) 3.17140i 0.106606i
\(886\) 0 0
\(887\) −7.26276 + 12.5795i −0.243859 + 0.422377i −0.961810 0.273717i \(-0.911747\pi\)
0.717951 + 0.696094i \(0.245080\pi\)
\(888\) 0 0
\(889\) 19.7535 + 11.4047i 0.662511 + 0.382501i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −39.9139 9.21835i −1.33567 0.308480i
\(894\) 0 0
\(895\) 5.88536 + 10.1937i 0.196726 + 0.340739i
\(896\) 0 0
\(897\) 4.66019 8.07169i 0.155599 0.269506i
\(898\) 0 0
\(899\) 24.2335 41.9736i 0.808232 1.39990i
\(900\) 0 0
\(901\) −50.1514 −1.67079
\(902\) 0 0
\(903\) −1.73205 + 3.00000i −0.0576390 + 0.0998337i
\(904\) 0 0
\(905\) 6.76143i 0.224758i
\(906\) 0 0
\(907\) −6.11895 + 3.53277i −0.203176 + 0.117304i −0.598136 0.801394i \(-0.704092\pi\)
0.394960 + 0.918698i \(0.370758\pi\)
\(908\) 0 0
\(909\) 100.294 57.9050i 3.32655 1.92059i
\(910\) 0 0
\(911\) −20.3251 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 59.1308 34.1392i 1.95481 1.12861i
\(916\) 0 0
\(917\) −0.686769 + 0.396506i −0.0226791 + 0.0130938i
\(918\) 0 0
\(919\) 50.7958i 1.67560i 0.545978 + 0.837800i \(0.316158\pi\)
−0.545978 + 0.837800i \(0.683842\pi\)
\(920\) 0 0
\(921\) −46.9051 + 81.2420i −1.54557 + 2.67701i
\(922\) 0 0
\(923\) 4.44600 0.146342
\(924\) 0 0
\(925\) 3.45186 5.97879i 0.113496 0.196581i
\(926\) 0 0
\(927\) −48.7700 + 84.4722i −1.60182 + 2.77443i
\(928\) 0 0
\(929\) 3.72892 + 6.45868i 0.122342 + 0.211903i 0.920691 0.390293i \(-0.127626\pi\)
−0.798349 + 0.602195i \(0.794293\pi\)
\(930\) 0 0
\(931\) −12.5056 + 3.82221i −0.409855 + 0.125268i
\(932\) 0 0
\(933\) 1.99120 + 3.44887i 0.0651891 + 0.112911i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.5847 + 33.9217i −0.639804 + 1.10817i 0.345672 + 0.938355i \(0.387651\pi\)
−0.985476 + 0.169817i \(0.945682\pi\)
\(938\) 0 0
\(939\) 56.8617i 1.85561i
\(940\) 0 0
\(941\) 2.77126 4.79997i 0.0903406 0.156475i −0.817314 0.576193i \(-0.804538\pi\)
0.907654 + 0.419718i \(0.137871\pi\)
\(942\) 0 0
\(943\) 5.15329 0.167814
\(944\) 0 0
\(945\) 35.8382 20.6912i 1.16582 0.673084i
\(946\) 0 0
\(947\) −3.28108 5.68300i −0.106621 0.184673i 0.807778 0.589486i \(-0.200670\pi\)
−0.914399 + 0.404813i \(0.867336\pi\)
\(948\) 0 0
\(949\) 2.25929 0.0733395
\(950\) 0 0
\(951\) 78.2608i 2.53778i
\(952\) 0 0
\(953\) 41.1340 23.7487i 1.33246 0.769297i 0.346784 0.937945i \(-0.387274\pi\)
0.985676 + 0.168648i \(0.0539402\pi\)
\(954\) 0 0
\(955\) 21.1506 + 36.6340i 0.684419 + 1.18545i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.7155 + 6.18658i 0.346021 + 0.199775i
\(960\) 0 0
\(961\) 38.5070 1.24216
\(962\) 0 0
\(963\) −68.6052 39.6092i −2.21077 1.27639i
\(964\) 0 0
\(965\) −20.2958 + 35.1534i −0.653346 + 1.13163i
\(966\) 0 0
\(967\) −19.3394 + 11.1656i −0.621913 + 0.359061i −0.777613 0.628743i \(-0.783570\pi\)
0.155701 + 0.987804i \(0.450236\pi\)
\(968\) 0 0
\(969\) −77.3453 17.8634i −2.48469 0.573854i
\(970\) 0 0
\(971\) −19.4549 + 11.2323i −0.624339 + 0.360462i −0.778556 0.627575i \(-0.784048\pi\)
0.154218 + 0.988037i \(0.450714\pi\)
\(972\) 0 0
\(973\) 29.8382 + 17.2271i 0.956568 + 0.552275i
\(974\) 0 0
\(975\) −2.45173 1.41551i −0.0785181 0.0453324i
\(976\) 0 0
\(977\) 28.3287i 0.906315i −0.891431 0.453157i \(-0.850298\pi\)
0.891431 0.453157i \(-0.149702\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 43.2241 1.38004
\(982\) 0 0
\(983\) −11.6811 20.2323i −0.372570 0.645311i 0.617390 0.786657i \(-0.288190\pi\)
−0.989960 + 0.141347i \(0.954857\pi\)
\(984\) 0 0
\(985\) 4.22013 + 7.30947i 0.134464 + 0.232899i
\(986\) 0 0
\(987\) 58.8805i 1.87419i
\(988\) 0 0
\(989\) 3.64060i 0.115764i
\(990\) 0 0
\(991\) −1.85210 3.20794i −0.0588340 0.101904i 0.835108 0.550086i \(-0.185405\pi\)
−0.893942 + 0.448182i \(0.852072\pi\)
\(992\) 0 0
\(993\) 27.8558 + 48.2476i 0.883976 + 1.53109i
\(994\) 0 0
\(995\) −45.3689 −1.43829
\(996\) 0 0
\(997\) 5.29582 + 3.05755i 0.167720 + 0.0968335i 0.581511 0.813539i \(-0.302462\pi\)
−0.413790 + 0.910372i \(0.635795\pi\)
\(998\) 0 0
\(999\) 41.2361i 1.30465i
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 1216.2.s.g.31.1 12
4.3 odd 2 inner 1216.2.s.g.31.6 yes 12
8.3 odd 2 1216.2.s.h.31.1 yes 12
8.5 even 2 1216.2.s.h.31.6 yes 12
19.8 odd 6 1216.2.s.h.863.1 yes 12
76.27 even 6 1216.2.s.h.863.6 yes 12
152.27 even 6 inner 1216.2.s.g.863.1 yes 12
152.141 odd 6 inner 1216.2.s.g.863.6 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.s.g.31.1 12 1.1 even 1 trivial
1216.2.s.g.31.6 yes 12 4.3 odd 2 inner
1216.2.s.g.863.1 yes 12 152.27 even 6 inner
1216.2.s.g.863.6 yes 12 152.141 odd 6 inner
1216.2.s.h.31.1 yes 12 8.3 odd 2
1216.2.s.h.31.6 yes 12 8.5 even 2
1216.2.s.h.863.1 yes 12 19.8 odd 6
1216.2.s.h.863.6 yes 12 76.27 even 6