Properties

Label 1216.2.s.i.31.4
Level $1216$
Weight $2$
Character 1216.31
Analytic conductor $9.710$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.4
Character \(\chi\) \(=\) 1216.31
Dual form 1216.2.s.i.863.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.36522 + 1.36556i) q^{3} +(-3.63213 + 2.09701i) q^{5} -0.517694i q^{7} +(2.22950 - 3.86160i) q^{9} +O(q^{10})\) \(q+(-2.36522 + 1.36556i) q^{3} +(-3.63213 + 2.09701i) q^{5} -0.517694i q^{7} +(2.22950 - 3.86160i) q^{9} +5.15654 q^{11} +(1.67164 - 2.89536i) q^{13} +(5.72719 - 9.91978i) q^{15} +(-0.818251 - 1.41725i) q^{17} +(3.66388 + 2.36135i) q^{19} +(0.706940 + 1.22446i) q^{21} +(5.72719 + 3.30659i) q^{23} +(6.29493 - 10.9031i) q^{25} +3.98467i q^{27} +(-3.01997 + 5.23074i) q^{29} -0.896671 q^{31} +(-12.1963 + 7.04155i) q^{33} +(1.08561 + 1.88033i) q^{35} -5.65178 q^{37} +9.13088i q^{39} +(-5.18849 + 2.99557i) q^{41} +(-2.89536 - 5.01492i) q^{43} +18.7011i q^{45} +(2.89268 + 1.67009i) q^{47} +6.73199 q^{49} +(3.87068 + 2.23474i) q^{51} +(4.40275 - 7.62579i) q^{53} +(-18.7292 + 10.8133i) q^{55} +(-11.8904 - 0.581873i) q^{57} +(0.109010 - 0.0629372i) q^{59} +(-10.1584 - 5.86494i) q^{61} +(-1.99912 - 1.15420i) q^{63} +14.0218i q^{65} +(10.6302 + 6.13733i) q^{67} -18.0614 q^{69} +(4.44347 + 7.69632i) q^{71} +(2.88369 + 4.99470i) q^{73} +34.3844i q^{75} -2.66951i q^{77} +(-0.712296 - 1.23373i) q^{79} +(1.24719 + 2.16019i) q^{81} +0.864719 q^{83} +(5.94399 + 3.43177i) q^{85} -16.4958i q^{87} +(12.8317 + 7.40840i) q^{89} +(-1.49891 - 0.865397i) q^{91} +(2.12082 - 1.22446i) q^{93} +(-18.2595 - 0.893551i) q^{95} +(0.0530805 - 0.0306461i) q^{97} +(11.4965 - 19.9125i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 24 q^{9} + 20 q^{17} + 44 q^{25} - 60 q^{33} - 24 q^{41} - 64 q^{49} - 36 q^{57} - 64 q^{73} - 24 q^{81} - 12 q^{89} - 48 q^{97}+O(q^{100}) \) 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
\(3\) −2.36522 + 1.36556i −1.36556 + 0.788405i −0.990357 0.138539i \(-0.955759\pi\)
−0.375201 + 0.926944i \(0.622426\pi\)
\(4\) 0 0
\(5\) −3.63213 + 2.09701i −1.62434 + 0.937813i −0.638600 + 0.769539i \(0.720486\pi\)
−0.985740 + 0.168274i \(0.946181\pi\)
\(6\) 0 0
\(7\) 0.517694i 0.195670i −0.995203 0.0978349i \(-0.968808\pi\)
0.995203 0.0978349i \(-0.0311917\pi\)
\(8\) 0 0
\(9\) 2.22950 3.86160i 0.743165 1.28720i
\(10\) 0 0
\(11\) 5.15654 1.55475 0.777377 0.629035i \(-0.216550\pi\)
0.777377 + 0.629035i \(0.216550\pi\)
\(12\) 0 0
\(13\) 1.67164 2.89536i 0.463629 0.803030i −0.535509 0.844529i \(-0.679880\pi\)
0.999139 + 0.0414999i \(0.0132136\pi\)
\(14\) 0 0
\(15\) 5.72719 9.91978i 1.47875 2.56128i
\(16\) 0 0
\(17\) −0.818251 1.41725i −0.198455 0.343734i 0.749573 0.661922i \(-0.230259\pi\)
−0.948028 + 0.318188i \(0.896926\pi\)
\(18\) 0 0
\(19\) 3.66388 + 2.36135i 0.840551 + 0.541732i
\(20\) 0 0
\(21\) 0.706940 + 1.22446i 0.154267 + 0.267198i
\(22\) 0 0
\(23\) 5.72719 + 3.30659i 1.19420 + 0.689472i 0.959257 0.282537i \(-0.0911759\pi\)
0.234944 + 0.972009i \(0.424509\pi\)
\(24\) 0 0
\(25\) 6.29493 10.9031i 1.25899 2.18063i
\(26\) 0 0
\(27\) 3.98467i 0.766850i
\(28\) 0 0
\(29\) −3.01997 + 5.23074i −0.560795 + 0.971325i 0.436633 + 0.899640i \(0.356171\pi\)
−0.997427 + 0.0716849i \(0.977162\pi\)
\(30\) 0 0
\(31\) −0.896671 −0.161047 −0.0805235 0.996753i \(-0.525659\pi\)
−0.0805235 + 0.996753i \(0.525659\pi\)
\(32\) 0 0
\(33\) −12.1963 + 7.04155i −2.12311 + 1.22578i
\(34\) 0 0
\(35\) 1.08561 + 1.88033i 0.183502 + 0.317834i
\(36\) 0 0
\(37\) −5.65178 −0.929148 −0.464574 0.885534i \(-0.653792\pi\)
−0.464574 + 0.885534i \(0.653792\pi\)
\(38\) 0 0
\(39\) 9.13088i 1.46211i
\(40\) 0 0
\(41\) −5.18849 + 2.99557i −0.810305 + 0.467830i −0.847062 0.531494i \(-0.821631\pi\)
0.0367566 + 0.999324i \(0.488297\pi\)
\(42\) 0 0
\(43\) −2.89536 5.01492i −0.441539 0.764768i 0.556265 0.831005i \(-0.312234\pi\)
−0.997804 + 0.0662372i \(0.978901\pi\)
\(44\) 0 0
\(45\) 18.7011i 2.78780i
\(46\) 0 0
\(47\) 2.89268 + 1.67009i 0.421941 + 0.243608i 0.695907 0.718131i \(-0.255002\pi\)
−0.273966 + 0.961739i \(0.588336\pi\)
\(48\) 0 0
\(49\) 6.73199 0.961713
\(50\) 0 0
\(51\) 3.87068 + 2.23474i 0.542003 + 0.312926i
\(52\) 0 0
\(53\) 4.40275 7.62579i 0.604765 1.04748i −0.387324 0.921944i \(-0.626600\pi\)
0.992089 0.125540i \(-0.0400662\pi\)
\(54\) 0 0
\(55\) −18.7292 + 10.8133i −2.52545 + 1.45807i
\(56\) 0 0
\(57\) −11.8904 0.581873i −1.57493 0.0770709i
\(58\) 0 0
\(59\) 0.109010 0.0629372i 0.0141919 0.00819372i −0.492887 0.870093i \(-0.664058\pi\)
0.507079 + 0.861899i \(0.330725\pi\)
\(60\) 0 0
\(61\) −10.1584 5.86494i −1.30065 0.750928i −0.320131 0.947373i \(-0.603727\pi\)
−0.980515 + 0.196445i \(0.937060\pi\)
\(62\) 0 0
\(63\) −1.99912 1.15420i −0.251866 0.145415i
\(64\) 0 0
\(65\) 14.0218i 1.73919i
\(66\) 0 0
\(67\) 10.6302 + 6.13733i 1.29868 + 0.749794i 0.980176 0.198127i \(-0.0634858\pi\)
0.318505 + 0.947921i \(0.396819\pi\)
\(68\) 0 0
\(69\) −18.0614 −2.17433
\(70\) 0 0
\(71\) 4.44347 + 7.69632i 0.527343 + 0.913385i 0.999492 + 0.0318662i \(0.0101451\pi\)
−0.472149 + 0.881519i \(0.656522\pi\)
\(72\) 0 0
\(73\) 2.88369 + 4.99470i 0.337510 + 0.584585i 0.983964 0.178368i \(-0.0570818\pi\)
−0.646453 + 0.762953i \(0.723749\pi\)
\(74\) 0 0
\(75\) 34.3844i 3.97037i
\(76\) 0 0
\(77\) 2.66951i 0.304218i
\(78\) 0 0
\(79\) −0.712296 1.23373i −0.0801396 0.138806i 0.823170 0.567795i \(-0.192203\pi\)
−0.903310 + 0.428989i \(0.858870\pi\)
\(80\) 0 0
\(81\) 1.24719 + 2.16019i 0.138577 + 0.240022i
\(82\) 0 0
\(83\) 0.864719 0.0949153 0.0474576 0.998873i \(-0.484888\pi\)
0.0474576 + 0.998873i \(0.484888\pi\)
\(84\) 0 0
\(85\) 5.94399 + 3.43177i 0.644717 + 0.372227i
\(86\) 0 0
\(87\) 16.4958i 1.76853i
\(88\) 0 0
\(89\) 12.8317 + 7.40840i 1.36016 + 0.785289i 0.989645 0.143538i \(-0.0458479\pi\)
0.370515 + 0.928826i \(0.379181\pi\)
\(90\) 0 0
\(91\) −1.49891 0.865397i −0.157129 0.0907182i
\(92\) 0 0
\(93\) 2.12082 1.22446i 0.219919 0.126970i
\(94\) 0 0
\(95\) −18.2595 0.893551i −1.87338 0.0916764i
\(96\) 0 0
\(97\) 0.0530805 0.0306461i 0.00538951 0.00311164i −0.497303 0.867577i \(-0.665676\pi\)
0.502692 + 0.864465i \(0.332343\pi\)
\(98\) 0 0
\(99\) 11.4965 19.9125i 1.15544 2.00128i
\(100\) 0 0
\(101\) 3.63213 + 2.09701i 0.361411 + 0.208661i 0.669699 0.742632i \(-0.266423\pi\)
−0.308289 + 0.951293i \(0.599756\pi\)
\(102\) 0 0
\(103\) −6.92820 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(104\) 0 0
\(105\) −5.13540 2.96493i −0.501164 0.289347i
\(106\) 0 0
\(107\) 15.7529i 1.52289i −0.648231 0.761443i \(-0.724491\pi\)
0.648231 0.761443i \(-0.275509\pi\)
\(108\) 0 0
\(109\) 5.59263 + 9.68672i 0.535677 + 0.927819i 0.999130 + 0.0416982i \(0.0132768\pi\)
−0.463453 + 0.886121i \(0.653390\pi\)
\(110\) 0 0
\(111\) 13.3677 7.71784i 1.26880 0.732545i
\(112\) 0 0
\(113\) 8.82565i 0.830247i −0.909765 0.415124i \(-0.863738\pi\)
0.909765 0.415124i \(-0.136262\pi\)
\(114\) 0 0
\(115\) −27.7359 −2.58638
\(116\) 0 0
\(117\) −7.45382 12.9104i −0.689106 1.19357i
\(118\) 0 0
\(119\) −0.733702 + 0.423603i −0.0672584 + 0.0388316i
\(120\) 0 0
\(121\) 15.5899 1.41726
\(122\) 0 0
\(123\) 8.18126 14.1704i 0.737679 1.27770i
\(124\) 0 0
\(125\) 31.8321i 2.84715i
\(126\) 0 0
\(127\) 1.01975 1.76627i 0.0904886 0.156731i −0.817228 0.576314i \(-0.804490\pi\)
0.907717 + 0.419583i \(0.137824\pi\)
\(128\) 0 0
\(129\) 13.6963 + 7.90757i 1.20589 + 0.696223i
\(130\) 0 0
\(131\) 6.13314 + 10.6229i 0.535855 + 0.928127i 0.999121 + 0.0419086i \(0.0133438\pi\)
−0.463267 + 0.886219i \(0.653323\pi\)
\(132\) 0 0
\(133\) 1.22246 1.89677i 0.106001 0.164470i
\(134\) 0 0
\(135\) −8.35591 14.4729i −0.719162 1.24563i
\(136\) 0 0
\(137\) −8.11319 + 14.0524i −0.693156 + 1.20058i 0.277642 + 0.960685i \(0.410447\pi\)
−0.970798 + 0.239897i \(0.922886\pi\)
\(138\) 0 0
\(139\) −2.91442 + 5.04793i −0.247198 + 0.428160i −0.962747 0.270403i \(-0.912843\pi\)
0.715549 + 0.698562i \(0.246177\pi\)
\(140\) 0 0
\(141\) −9.12242 −0.768247
\(142\) 0 0
\(143\) 8.61987 14.9301i 0.720830 1.24851i
\(144\) 0 0
\(145\) 25.3317i 2.10368i
\(146\) 0 0
\(147\) −15.9226 + 9.19292i −1.31327 + 0.758220i
\(148\) 0 0
\(149\) 12.5635 7.25355i 1.02924 0.594234i 0.112476 0.993654i \(-0.464122\pi\)
0.916768 + 0.399420i \(0.130789\pi\)
\(150\) 0 0
\(151\) −2.56743 −0.208935 −0.104467 0.994528i \(-0.533314\pi\)
−0.104467 + 0.994528i \(0.533314\pi\)
\(152\) 0 0
\(153\) −7.29714 −0.589939
\(154\) 0 0
\(155\) 3.25683 1.88033i 0.261595 0.151032i
\(156\) 0 0
\(157\) −4.08584 + 2.35896i −0.326086 + 0.188266i −0.654102 0.756406i \(-0.726953\pi\)
0.328016 + 0.944672i \(0.393620\pi\)
\(158\) 0 0
\(159\) 24.0489i 1.90720i
\(160\) 0 0
\(161\) 1.71180 2.96493i 0.134909 0.233669i
\(162\) 0 0
\(163\) 13.7652 1.07817 0.539086 0.842251i \(-0.318770\pi\)
0.539086 + 0.842251i \(0.318770\pi\)
\(164\) 0 0
\(165\) 29.5324 51.1517i 2.29910 3.98215i
\(166\) 0 0
\(167\) −9.63962 + 16.6963i −0.745936 + 1.29200i 0.203819 + 0.979008i \(0.434664\pi\)
−0.949756 + 0.312991i \(0.898669\pi\)
\(168\) 0 0
\(169\) 0.911244 + 1.57832i 0.0700957 + 0.121409i
\(170\) 0 0
\(171\) 17.2872 8.88380i 1.32199 0.679361i
\(172\) 0 0
\(173\) 2.24935 + 3.89599i 0.171015 + 0.296207i 0.938775 0.344531i \(-0.111962\pi\)
−0.767760 + 0.640738i \(0.778629\pi\)
\(174\) 0 0
\(175\) −5.64449 3.25885i −0.426683 0.246346i
\(176\) 0 0
\(177\) −0.171889 + 0.297720i −0.0129199 + 0.0223780i
\(178\) 0 0
\(179\) 18.2232i 1.36206i 0.732253 + 0.681032i \(0.238469\pi\)
−0.732253 + 0.681032i \(0.761531\pi\)
\(180\) 0 0
\(181\) −3.85614 + 6.67903i −0.286625 + 0.496448i −0.973002 0.230797i \(-0.925867\pi\)
0.686377 + 0.727246i \(0.259200\pi\)
\(182\) 0 0
\(183\) 32.0356 2.36814
\(184\) 0 0
\(185\) 20.5280 11.8519i 1.50925 0.871367i
\(186\) 0 0
\(187\) −4.21934 7.30811i −0.308549 0.534422i
\(188\) 0 0
\(189\) 2.06284 0.150049
\(190\) 0 0
\(191\) 1.22637i 0.0887371i −0.999015 0.0443685i \(-0.985872\pi\)
0.999015 0.0443685i \(-0.0141276\pi\)
\(192\) 0 0
\(193\) −6.83172 + 3.94430i −0.491758 + 0.283917i −0.725304 0.688429i \(-0.758301\pi\)
0.233545 + 0.972346i \(0.424967\pi\)
\(194\) 0 0
\(195\) −19.1476 33.1646i −1.37119 2.37496i
\(196\) 0 0
\(197\) 17.5206i 1.24829i 0.781308 + 0.624146i \(0.214553\pi\)
−0.781308 + 0.624146i \(0.785447\pi\)
\(198\) 0 0
\(199\) −2.44435 1.41124i −0.173275 0.100040i 0.410854 0.911701i \(-0.365231\pi\)
−0.584129 + 0.811661i \(0.698564\pi\)
\(200\) 0 0
\(201\) −33.5235 −2.36457
\(202\) 0 0
\(203\) 2.70792 + 1.56342i 0.190059 + 0.109731i
\(204\) 0 0
\(205\) 12.5635 21.7607i 0.877474 1.51983i
\(206\) 0 0
\(207\) 25.5375 14.7441i 1.77498 1.02478i
\(208\) 0 0
\(209\) 18.8929 + 12.1764i 1.30685 + 0.842260i
\(210\) 0 0
\(211\) −9.03972 + 5.21909i −0.622320 + 0.359297i −0.777772 0.628547i \(-0.783650\pi\)
0.155452 + 0.987844i \(0.450317\pi\)
\(212\) 0 0
\(213\) −21.0195 12.1356i −1.44023 0.831520i
\(214\) 0 0
\(215\) 21.0327 + 12.1432i 1.43442 + 0.828162i
\(216\) 0 0
\(217\) 0.464201i 0.0315120i
\(218\) 0 0
\(219\) −13.6411 7.87569i −0.921780 0.532190i
\(220\) 0 0
\(221\) −5.47128 −0.368038
\(222\) 0 0
\(223\) −0.571419 0.989727i −0.0382651 0.0662770i 0.846259 0.532772i \(-0.178850\pi\)
−0.884524 + 0.466495i \(0.845516\pi\)
\(224\) 0 0
\(225\) −28.0691 48.6170i −1.87127 3.24113i
\(226\) 0 0
\(227\) 16.9989i 1.12825i 0.825688 + 0.564127i \(0.190787\pi\)
−0.825688 + 0.564127i \(0.809213\pi\)
\(228\) 0 0
\(229\) 17.6533i 1.16657i −0.812269 0.583283i \(-0.801768\pi\)
0.812269 0.583283i \(-0.198232\pi\)
\(230\) 0 0
\(231\) 3.64536 + 6.31395i 0.239847 + 0.415428i
\(232\) 0 0
\(233\) −7.84830 13.5937i −0.514159 0.890550i −0.999865 0.0164278i \(-0.994771\pi\)
0.485706 0.874122i \(-0.338563\pi\)
\(234\) 0 0
\(235\) −14.0088 −0.913834
\(236\) 0 0
\(237\) 3.36947 + 1.94536i 0.218870 + 0.126365i
\(238\) 0 0
\(239\) 27.1331i 1.75509i 0.479490 + 0.877547i \(0.340822\pi\)
−0.479490 + 0.877547i \(0.659178\pi\)
\(240\) 0 0
\(241\) −20.6511 11.9229i −1.33025 0.768022i −0.344914 0.938634i \(-0.612092\pi\)
−0.985338 + 0.170613i \(0.945425\pi\)
\(242\) 0 0
\(243\) −16.2522 9.38322i −1.04258 0.601934i
\(244\) 0 0
\(245\) −24.4515 + 14.1171i −1.56215 + 0.901907i
\(246\) 0 0
\(247\) 12.9617 6.66093i 0.824731 0.423825i
\(248\) 0 0
\(249\) −2.04525 + 1.18082i −0.129612 + 0.0748317i
\(250\) 0 0
\(251\) 0.232112 0.402030i 0.0146508 0.0253759i −0.858607 0.512634i \(-0.828670\pi\)
0.873258 + 0.487258i \(0.162003\pi\)
\(252\) 0 0
\(253\) 29.5324 + 17.0506i 1.85669 + 1.07196i
\(254\) 0 0
\(255\) −18.7451 −1.17386
\(256\) 0 0
\(257\) −9.74157 5.62430i −0.607662 0.350834i 0.164388 0.986396i \(-0.447435\pi\)
−0.772050 + 0.635562i \(0.780769\pi\)
\(258\) 0 0
\(259\) 2.92589i 0.181806i
\(260\) 0 0
\(261\) 13.4660 + 23.3238i 0.833526 + 1.44371i
\(262\) 0 0
\(263\) 7.49962 4.32991i 0.462447 0.266994i −0.250626 0.968084i \(-0.580636\pi\)
0.713072 + 0.701090i \(0.247303\pi\)
\(264\) 0 0
\(265\) 36.9305i 2.26863i
\(266\) 0 0
\(267\) −40.4664 −2.47650
\(268\) 0 0
\(269\) 3.91647 + 6.78352i 0.238791 + 0.413598i 0.960368 0.278736i \(-0.0899155\pi\)
−0.721577 + 0.692335i \(0.756582\pi\)
\(270\) 0 0
\(271\) 7.19216 4.15240i 0.436893 0.252240i −0.265386 0.964142i \(-0.585499\pi\)
0.702279 + 0.711902i \(0.252166\pi\)
\(272\) 0 0
\(273\) 4.72700 0.286091
\(274\) 0 0
\(275\) 32.4601 56.2225i 1.95742 3.39034i
\(276\) 0 0
\(277\) 24.2241i 1.45549i −0.685850 0.727743i \(-0.740570\pi\)
0.685850 0.727743i \(-0.259430\pi\)
\(278\) 0 0
\(279\) −1.99912 + 3.46259i −0.119684 + 0.207300i
\(280\) 0 0
\(281\) −20.5811 11.8825i −1.22777 0.708851i −0.261205 0.965283i \(-0.584120\pi\)
−0.966562 + 0.256432i \(0.917453\pi\)
\(282\) 0 0
\(283\) −5.48644 9.50279i −0.326135 0.564882i 0.655607 0.755103i \(-0.272413\pi\)
−0.981741 + 0.190221i \(0.939080\pi\)
\(284\) 0 0
\(285\) 44.4078 22.8209i 2.63049 1.35180i
\(286\) 0 0
\(287\) 1.55079 + 2.68605i 0.0915402 + 0.158552i
\(288\) 0 0
\(289\) 7.16093 12.4031i 0.421231 0.729594i
\(290\) 0 0
\(291\) −0.0836979 + 0.144969i −0.00490646 + 0.00849824i
\(292\) 0 0
\(293\) 17.5459 1.02504 0.512520 0.858675i \(-0.328712\pi\)
0.512520 + 0.858675i \(0.328712\pi\)
\(294\) 0 0
\(295\) −0.263960 + 0.457193i −0.0153684 + 0.0266188i
\(296\) 0 0
\(297\) 20.5471i 1.19226i
\(298\) 0 0
\(299\) 19.1476 11.0549i 1.10733 0.639319i
\(300\) 0 0
\(301\) −2.59619 + 1.49891i −0.149642 + 0.0863958i
\(302\) 0 0
\(303\) −11.4544 −0.658037
\(304\) 0 0
\(305\) 49.1954 2.81692
\(306\) 0 0
\(307\) 20.8230 12.0222i 1.18843 0.686142i 0.230482 0.973077i \(-0.425970\pi\)
0.957950 + 0.286935i \(0.0926364\pi\)
\(308\) 0 0
\(309\) 16.3867 9.46086i 0.932206 0.538210i
\(310\) 0 0
\(311\) 14.7795i 0.838066i −0.907971 0.419033i \(-0.862369\pi\)
0.907971 0.419033i \(-0.137631\pi\)
\(312\) 0 0
\(313\) −9.22387 + 15.9762i −0.521364 + 0.903029i 0.478327 + 0.878182i \(0.341243\pi\)
−0.999691 + 0.0248474i \(0.992090\pi\)
\(314\) 0 0
\(315\) 9.68145 0.545488
\(316\) 0 0
\(317\) 11.6670 20.2079i 0.655285 1.13499i −0.326537 0.945184i \(-0.605882\pi\)
0.981822 0.189803i \(-0.0607849\pi\)
\(318\) 0 0
\(319\) −15.5726 + 26.9725i −0.871898 + 1.51017i
\(320\) 0 0
\(321\) 21.5114 + 37.2589i 1.20065 + 2.07959i
\(322\) 0 0
\(323\) 0.348662 7.12482i 0.0194001 0.396436i
\(324\) 0 0
\(325\) −21.0457 36.4523i −1.16741 2.02201i
\(326\) 0 0
\(327\) −26.4555 15.2741i −1.46300 0.844661i
\(328\) 0 0
\(329\) 0.864595 1.49752i 0.0476667 0.0825611i
\(330\) 0 0
\(331\) 22.7128i 1.24841i 0.781261 + 0.624205i \(0.214577\pi\)
−0.781261 + 0.624205i \(0.785423\pi\)
\(332\) 0 0
\(333\) −12.6006 + 21.8249i −0.690510 + 1.19600i
\(334\) 0 0
\(335\) −51.4803 −2.81267
\(336\) 0 0
\(337\) −15.0531 + 8.69090i −0.819994 + 0.473423i −0.850414 0.526114i \(-0.823649\pi\)
0.0304207 + 0.999537i \(0.490315\pi\)
\(338\) 0 0
\(339\) 12.0519 + 20.8746i 0.654571 + 1.13375i
\(340\) 0 0
\(341\) −4.62372 −0.250388
\(342\) 0 0
\(343\) 7.10896i 0.383848i
\(344\) 0 0
\(345\) 65.6013 37.8749i 3.53186 2.03912i
\(346\) 0 0
\(347\) 6.10782 + 10.5791i 0.327885 + 0.567914i 0.982092 0.188402i \(-0.0603308\pi\)
−0.654207 + 0.756316i \(0.726997\pi\)
\(348\) 0 0
\(349\) 13.8033i 0.738874i 0.929256 + 0.369437i \(0.120449\pi\)
−0.929256 + 0.369437i \(0.879551\pi\)
\(350\) 0 0
\(351\) 11.5371 + 6.66093i 0.615803 + 0.355534i
\(352\) 0 0
\(353\) 5.84663 0.311185 0.155592 0.987821i \(-0.450271\pi\)
0.155592 + 0.987821i \(0.450271\pi\)
\(354\) 0 0
\(355\) −32.2786 18.6360i −1.71317 0.989098i
\(356\) 0 0
\(357\) 1.15691 2.00383i 0.0612301 0.106054i
\(358\) 0 0
\(359\) 14.3471 8.28328i 0.757209 0.437175i −0.0710840 0.997470i \(-0.522646\pi\)
0.828293 + 0.560296i \(0.189313\pi\)
\(360\) 0 0
\(361\) 7.84802 + 17.3034i 0.413053 + 0.910707i
\(362\) 0 0
\(363\) −36.8734 + 21.2889i −1.93535 + 1.11738i
\(364\) 0 0
\(365\) −20.9479 12.0943i −1.09646 0.633043i
\(366\) 0 0
\(367\) 26.2106 + 15.1327i 1.36818 + 0.789919i 0.990696 0.136097i \(-0.0434557\pi\)
0.377485 + 0.926016i \(0.376789\pi\)
\(368\) 0 0
\(369\) 26.7145i 1.39070i
\(370\) 0 0
\(371\) −3.94782 2.27928i −0.204961 0.118334i
\(372\) 0 0
\(373\) −11.1762 −0.578682 −0.289341 0.957226i \(-0.593436\pi\)
−0.289341 + 0.957226i \(0.593436\pi\)
\(374\) 0 0
\(375\) −43.4686 75.2898i −2.24471 3.88795i
\(376\) 0 0
\(377\) 10.0966 + 17.4878i 0.520002 + 0.900669i
\(378\) 0 0
\(379\) 0.728987i 0.0374456i −0.999825 0.0187228i \(-0.994040\pi\)
0.999825 0.0187228i \(-0.00595999\pi\)
\(380\) 0 0
\(381\) 5.57014i 0.285367i
\(382\) 0 0
\(383\) −0.531034 0.919778i −0.0271346 0.0469985i 0.852139 0.523315i \(-0.175305\pi\)
−0.879274 + 0.476317i \(0.841972\pi\)
\(384\) 0 0
\(385\) 5.59799 + 9.69600i 0.285300 + 0.494154i
\(386\) 0 0
\(387\) −25.8208 −1.31255
\(388\) 0 0
\(389\) 27.3980 + 15.8183i 1.38914 + 0.802018i 0.993218 0.116268i \(-0.0370930\pi\)
0.395918 + 0.918286i \(0.370426\pi\)
\(390\) 0 0
\(391\) 10.8225i 0.547317i
\(392\) 0 0
\(393\) −29.0124 16.7503i −1.46348 0.844941i
\(394\) 0 0
\(395\) 5.17431 + 2.98739i 0.260348 + 0.150312i
\(396\) 0 0
\(397\) 1.05761 0.610610i 0.0530798 0.0306456i −0.473225 0.880941i \(-0.656910\pi\)
0.526305 + 0.850296i \(0.323577\pi\)
\(398\) 0 0
\(399\) −0.301232 + 6.15560i −0.0150804 + 0.308165i
\(400\) 0 0
\(401\) −24.3396 + 14.0524i −1.21546 + 0.701746i −0.963943 0.266107i \(-0.914262\pi\)
−0.251516 + 0.967853i \(0.580929\pi\)
\(402\) 0 0
\(403\) −1.49891 + 2.59619i −0.0746661 + 0.129325i
\(404\) 0 0
\(405\) −9.05991 5.23074i −0.450191 0.259918i
\(406\) 0 0
\(407\) −29.1436 −1.44460
\(408\) 0 0
\(409\) 20.1185 + 11.6154i 0.994798 + 0.574347i 0.906705 0.421766i \(-0.138589\pi\)
0.0880927 + 0.996112i \(0.471923\pi\)
\(410\) 0 0
\(411\) 44.3161i 2.18595i
\(412\) 0 0
\(413\) −0.0325822 0.0564340i −0.00160326 0.00277693i
\(414\) 0 0
\(415\) −3.14078 + 1.81333i −0.154175 + 0.0890128i
\(416\) 0 0
\(417\) 15.9193i 0.779569i
\(418\) 0 0
\(419\) 2.85347 0.139401 0.0697005 0.997568i \(-0.477796\pi\)
0.0697005 + 0.997568i \(0.477796\pi\)
\(420\) 0 0
\(421\) −6.04634 10.4726i −0.294680 0.510401i 0.680230 0.732999i \(-0.261880\pi\)
−0.974911 + 0.222597i \(0.928547\pi\)
\(422\) 0 0
\(423\) 12.8984 7.44692i 0.627144 0.362082i
\(424\) 0 0
\(425\) −20.6033 −0.999409
\(426\) 0 0
\(427\) −3.03624 + 5.25892i −0.146934 + 0.254497i
\(428\) 0 0
\(429\) 47.0837i 2.27322i
\(430\) 0 0
\(431\) 5.94883 10.3037i 0.286545 0.496311i −0.686438 0.727189i \(-0.740826\pi\)
0.972983 + 0.230878i \(0.0741598\pi\)
\(432\) 0 0
\(433\) −14.3848 8.30507i −0.691290 0.399116i 0.112805 0.993617i \(-0.464016\pi\)
−0.804095 + 0.594501i \(0.797350\pi\)
\(434\) 0 0
\(435\) 34.5919 + 59.9149i 1.65855 + 2.87270i
\(436\) 0 0
\(437\) 13.1757 + 25.6389i 0.630278 + 1.22647i
\(438\) 0 0
\(439\) 19.5028 + 33.7799i 0.930819 + 1.61223i 0.781925 + 0.623373i \(0.214238\pi\)
0.148894 + 0.988853i \(0.452429\pi\)
\(440\) 0 0
\(441\) 15.0089 25.9963i 0.714712 1.23792i
\(442\) 0 0
\(443\) 5.38865 9.33341i 0.256022 0.443444i −0.709150 0.705057i \(-0.750921\pi\)
0.965173 + 0.261613i \(0.0842546\pi\)
\(444\) 0 0
\(445\) −62.1421 −2.94582
\(446\) 0 0
\(447\) −19.8103 + 34.3124i −0.936994 + 1.62292i
\(448\) 0 0
\(449\) 14.4665i 0.682718i 0.939933 + 0.341359i \(0.110887\pi\)
−0.939933 + 0.341359i \(0.889113\pi\)
\(450\) 0 0
\(451\) −26.7546 + 15.4468i −1.25983 + 0.727361i
\(452\) 0 0
\(453\) 6.07252 3.50597i 0.285312 0.164725i
\(454\) 0 0
\(455\) 7.25900 0.340307
\(456\) 0 0
\(457\) 30.5191 1.42762 0.713812 0.700338i \(-0.246967\pi\)
0.713812 + 0.700338i \(0.246967\pi\)
\(458\) 0 0
\(459\) 5.64728 3.26046i 0.263593 0.152185i
\(460\) 0 0
\(461\) 18.5484 10.7089i 0.863883 0.498763i −0.00142759 0.999999i \(-0.500454\pi\)
0.865311 + 0.501236i \(0.167121\pi\)
\(462\) 0 0
\(463\) 23.2533i 1.08067i −0.841449 0.540337i \(-0.818297\pi\)
0.841449 0.540337i \(-0.181703\pi\)
\(464\) 0 0
\(465\) −5.13540 + 8.89478i −0.238149 + 0.412486i
\(466\) 0 0
\(467\) 5.15654 0.238616 0.119308 0.992857i \(-0.461932\pi\)
0.119308 + 0.992857i \(0.461932\pi\)
\(468\) 0 0
\(469\) 3.17726 5.50317i 0.146712 0.254113i
\(470\) 0 0
\(471\) 6.44260 11.1589i 0.296859 0.514175i
\(472\) 0 0
\(473\) −14.9301 25.8596i −0.686484 1.18903i
\(474\) 0 0
\(475\) 48.8101 25.0832i 2.23956 1.15090i
\(476\) 0 0
\(477\) −19.6318 34.0033i −0.898880 1.55691i
\(478\) 0 0
\(479\) −11.3508 6.55337i −0.518630 0.299431i 0.217744 0.976006i \(-0.430130\pi\)
−0.736374 + 0.676575i \(0.763464\pi\)
\(480\) 0 0
\(481\) −9.44774 + 16.3640i −0.430780 + 0.746133i
\(482\) 0 0
\(483\) 9.35025i 0.425451i
\(484\) 0 0
\(485\) −0.128530 + 0.222621i −0.00583627 + 0.0101087i
\(486\) 0 0
\(487\) 18.2172 0.825499 0.412750 0.910845i \(-0.364568\pi\)
0.412750 + 0.910845i \(0.364568\pi\)
\(488\) 0 0
\(489\) −32.5576 + 18.7972i −1.47231 + 0.850037i
\(490\) 0 0
\(491\) 6.46771 + 11.2024i 0.291884 + 0.505557i 0.974255 0.225448i \(-0.0723847\pi\)
−0.682372 + 0.731005i \(0.739051\pi\)
\(492\) 0 0
\(493\) 9.88438 0.445170
\(494\) 0 0
\(495\) 96.4331i 4.33434i
\(496\) 0 0
\(497\) 3.98433 2.30036i 0.178722 0.103185i
\(498\) 0 0
\(499\) 1.73102 + 2.99822i 0.0774912 + 0.134219i 0.902167 0.431387i \(-0.141976\pi\)
−0.824676 + 0.565606i \(0.808642\pi\)
\(500\) 0 0
\(501\) 52.6538i 2.35240i
\(502\) 0 0
\(503\) 1.56977 + 0.906306i 0.0699925 + 0.0404102i 0.534588 0.845113i \(-0.320467\pi\)
−0.464595 + 0.885523i \(0.653800\pi\)
\(504\) 0 0
\(505\) −17.5899 −0.782739
\(506\) 0 0
\(507\) −4.31058 2.48871i −0.191439 0.110528i
\(508\) 0 0
\(509\) −17.7070 + 30.6694i −0.784847 + 1.35940i 0.144243 + 0.989542i \(0.453925\pi\)
−0.929090 + 0.369853i \(0.879408\pi\)
\(510\) 0 0
\(511\) 2.58572 1.49287i 0.114386 0.0660406i
\(512\) 0 0
\(513\) −9.40922 + 14.5993i −0.415427 + 0.644577i
\(514\) 0 0
\(515\) 25.1642 14.5285i 1.10887 0.640204i
\(516\) 0 0
\(517\) 14.9162 + 8.61189i 0.656015 + 0.378750i
\(518\) 0 0
\(519\) −10.6404 6.14324i −0.467062 0.269658i
\(520\) 0 0
\(521\) 22.3979i 0.981271i −0.871365 0.490636i \(-0.836765\pi\)
0.871365 0.490636i \(-0.163235\pi\)
\(522\) 0 0
\(523\) 7.06121 + 4.07679i 0.308765 + 0.178266i 0.646374 0.763021i \(-0.276285\pi\)
−0.337609 + 0.941287i \(0.609618\pi\)
\(524\) 0 0
\(525\) 17.8006 0.776881
\(526\) 0 0
\(527\) 0.733702 + 1.27081i 0.0319606 + 0.0553573i
\(528\) 0 0
\(529\) 10.3671 + 17.9564i 0.450744 + 0.780711i
\(530\) 0 0
\(531\) 0.561272i 0.0243571i
\(532\) 0 0
\(533\) 20.0301i 0.867599i
\(534\) 0 0
\(535\) 33.0340 + 57.2165i 1.42818 + 2.47369i
\(536\) 0 0
\(537\) −24.8848 43.1017i −1.07386 1.85998i
\(538\) 0 0
\(539\) 34.7138 1.49523
\(540\) 0 0
\(541\) 26.9988 + 15.5877i 1.16077 + 0.670169i 0.951488 0.307686i \(-0.0995548\pi\)
0.209280 + 0.977856i \(0.432888\pi\)
\(542\) 0 0
\(543\) 21.0631i 0.903905i
\(544\) 0 0
\(545\) −40.6264 23.4556i −1.74024 1.00473i
\(546\) 0 0
\(547\) −15.5877 8.99959i −0.666484 0.384795i 0.128259 0.991741i \(-0.459061\pi\)
−0.794743 + 0.606946i \(0.792394\pi\)
\(548\) 0 0
\(549\) −45.2961 + 26.1517i −1.93319 + 1.11613i
\(550\) 0 0
\(551\) −23.4164 + 12.0336i −0.997574 + 0.512648i
\(552\) 0 0
\(553\) −0.638695 + 0.368751i −0.0271601 + 0.0156809i
\(554\) 0 0
\(555\) −32.3688 + 56.0644i −1.37398 + 2.37980i
\(556\) 0 0
\(557\) −10.2517 5.91880i −0.434377 0.250788i 0.266833 0.963743i \(-0.414023\pi\)
−0.701209 + 0.712955i \(0.747356\pi\)
\(558\) 0 0
\(559\) −19.3600 −0.818842
\(560\) 0 0
\(561\) 19.9593 + 11.5235i 0.842682 + 0.486523i
\(562\) 0 0
\(563\) 25.7445i 1.08500i 0.840055 + 0.542501i \(0.182522\pi\)
−0.840055 + 0.542501i \(0.817478\pi\)
\(564\) 0 0
\(565\) 18.5075 + 32.0560i 0.778617 + 1.34860i
\(566\) 0 0
\(567\) 1.11832 0.645661i 0.0469650 0.0271152i
\(568\) 0 0
\(569\) 6.17502i 0.258870i −0.991588 0.129435i \(-0.958684\pi\)
0.991588 0.129435i \(-0.0413164\pi\)
\(570\) 0 0
\(571\) −22.7592 −0.952445 −0.476222 0.879325i \(-0.657994\pi\)
−0.476222 + 0.879325i \(0.657994\pi\)
\(572\) 0 0
\(573\) 1.67468 + 2.90063i 0.0699607 + 0.121176i
\(574\) 0 0
\(575\) 72.1045 41.6296i 3.00697 1.73607i
\(576\) 0 0
\(577\) 5.49715 0.228849 0.114425 0.993432i \(-0.463498\pi\)
0.114425 + 0.993432i \(0.463498\pi\)
\(578\) 0 0
\(579\) 10.7723 18.6582i 0.447683 0.775409i
\(580\) 0 0
\(581\) 0.447660i 0.0185721i
\(582\) 0 0
\(583\) 22.7030 39.3227i 0.940261 1.62858i
\(584\) 0 0
\(585\) 54.1466 + 31.2615i 2.23869 + 1.29251i
\(586\) 0 0
\(587\) 3.79820 + 6.57868i 0.156769 + 0.271531i 0.933702 0.358052i \(-0.116559\pi\)
−0.776933 + 0.629583i \(0.783226\pi\)
\(588\) 0 0
\(589\) −3.28530 2.11736i −0.135368 0.0872442i
\(590\) 0 0
\(591\) −23.9254 41.4400i −0.984159 1.70461i
\(592\) 0 0
\(593\) −10.9545 + 18.9737i −0.449846 + 0.779156i −0.998376 0.0569751i \(-0.981854\pi\)
0.548530 + 0.836131i \(0.315188\pi\)
\(594\) 0 0
\(595\) 1.77660 3.07717i 0.0728336 0.126152i
\(596\) 0 0
\(597\) 7.70854 0.315490
\(598\) 0 0
\(599\) −8.88383 + 15.3872i −0.362983 + 0.628706i −0.988450 0.151545i \(-0.951575\pi\)
0.625467 + 0.780251i \(0.284909\pi\)
\(600\) 0 0
\(601\) 20.6279i 0.841428i −0.907193 0.420714i \(-0.861780\pi\)
0.907193 0.420714i \(-0.138220\pi\)
\(602\) 0 0
\(603\) 47.3998 27.3663i 1.93027 1.11444i
\(604\) 0 0
\(605\) −56.6245 + 32.6922i −2.30211 + 1.32913i
\(606\) 0 0
\(607\) 46.1800 1.87439 0.937195 0.348807i \(-0.113413\pi\)
0.937195 + 0.348807i \(0.113413\pi\)
\(608\) 0 0
\(609\) −8.53976 −0.346048
\(610\) 0 0
\(611\) 9.67104 5.58358i 0.391249 0.225887i
\(612\) 0 0
\(613\) −31.0710 + 17.9389i −1.25495 + 0.724544i −0.972088 0.234617i \(-0.924616\pi\)
−0.282859 + 0.959161i \(0.591283\pi\)
\(614\) 0 0
\(615\) 68.6248i 2.76722i
\(616\) 0 0
\(617\) 16.0899 27.8685i 0.647754 1.12194i −0.335904 0.941896i \(-0.609042\pi\)
0.983658 0.180046i \(-0.0576248\pi\)
\(618\) 0 0
\(619\) −2.14561 −0.0862393 −0.0431197 0.999070i \(-0.513730\pi\)
−0.0431197 + 0.999070i \(0.513730\pi\)
\(620\) 0 0
\(621\) −13.1757 + 22.8209i −0.528722 + 0.915773i
\(622\) 0 0
\(623\) 3.83528 6.64290i 0.153657 0.266142i
\(624\) 0 0
\(625\) −35.2777 61.1028i −1.41111 2.44411i
\(626\) 0 0
\(627\) −61.3134 3.00045i −2.44862 0.119826i
\(628\) 0 0
\(629\) 4.62458 + 8.01000i 0.184394 + 0.319380i
\(630\) 0 0
\(631\) 35.7086 + 20.6164i 1.42154 + 0.820725i 0.996430 0.0844176i \(-0.0269030\pi\)
0.425107 + 0.905143i \(0.360236\pi\)
\(632\) 0 0
\(633\) 14.2539 24.6885i 0.566543 0.981281i
\(634\) 0 0
\(635\) 8.55376i 0.339446i
\(636\) 0 0
\(637\) 11.2535 19.4916i 0.445879 0.772284i
\(638\) 0 0
\(639\) 39.6268 1.56761
\(640\) 0 0
\(641\) −18.9918 + 10.9649i −0.750132 + 0.433089i −0.825742 0.564049i \(-0.809243\pi\)
0.0756095 + 0.997138i \(0.475910\pi\)
\(642\) 0 0
\(643\) 19.5674 + 33.8918i 0.771664 + 1.33656i 0.936650 + 0.350266i \(0.113909\pi\)
−0.164986 + 0.986296i \(0.552758\pi\)
\(644\) 0 0
\(645\) −66.3292 −2.61171
\(646\) 0 0
\(647\) 24.2393i 0.952944i −0.879190 0.476472i \(-0.841915\pi\)
0.879190 0.476472i \(-0.158085\pi\)
\(648\) 0 0
\(649\) 0.562116 0.324538i 0.0220650 0.0127392i
\(650\) 0 0
\(651\) −0.633893 1.09794i −0.0248442 0.0430315i
\(652\) 0 0
\(653\) 24.6718i 0.965481i 0.875764 + 0.482740i \(0.160358\pi\)
−0.875764 + 0.482740i \(0.839642\pi\)
\(654\) 0 0
\(655\) −44.5528 25.7225i −1.74082 1.00506i
\(656\) 0 0
\(657\) 25.7167 1.00330
\(658\) 0 0
\(659\) 4.66292 + 2.69214i 0.181642 + 0.104871i 0.588064 0.808815i \(-0.299890\pi\)
−0.406422 + 0.913685i \(0.633224\pi\)
\(660\) 0 0
\(661\) 10.5097 18.2034i 0.408781 0.708030i −0.585972 0.810331i \(-0.699287\pi\)
0.994753 + 0.102301i \(0.0326206\pi\)
\(662\) 0 0
\(663\) 12.9408 7.47135i 0.502577 0.290163i
\(664\) 0 0
\(665\) −0.462585 + 9.45282i −0.0179383 + 0.366565i
\(666\) 0 0
\(667\) −34.5919 + 19.9716i −1.33940 + 0.773305i
\(668\) 0 0
\(669\) 2.70306 + 1.56061i 0.104506 + 0.0603367i
\(670\) 0 0
\(671\) −52.3820 30.2428i −2.02218 1.16751i
\(672\) 0 0
\(673\) 4.54322i 0.175128i 0.996159 + 0.0875641i \(0.0279083\pi\)
−0.996159 + 0.0875641i \(0.972092\pi\)
\(674\) 0 0
\(675\) 43.4454 + 25.0832i 1.67222 + 0.965454i
\(676\) 0 0
\(677\) −4.30681 −0.165524 −0.0827620 0.996569i \(-0.526374\pi\)
−0.0827620 + 0.996569i \(0.526374\pi\)
\(678\) 0 0
\(679\) −0.0158653 0.0274795i −0.000608853 0.00105456i
\(680\) 0 0
\(681\) −23.2129 40.2059i −0.889521 1.54070i
\(682\) 0 0
\(683\) 30.9152i 1.18294i 0.806328 + 0.591469i \(0.201452\pi\)
−0.806328 + 0.591469i \(0.798548\pi\)
\(684\) 0 0
\(685\) 68.0538i 2.60020i
\(686\) 0 0
\(687\) 24.1066 + 41.7539i 0.919726 + 1.59301i
\(688\) 0 0
\(689\) −14.7196 25.4952i −0.560773 0.971288i
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) −10.3086 5.95165i −0.391590 0.226084i
\(694\) 0 0
\(695\) 24.4463i 0.927303i
\(696\) 0 0
\(697\) 8.49096 + 4.90226i 0.321618 + 0.185686i
\(698\) 0 0
\(699\) 37.1259 + 21.4346i 1.40423 + 0.810732i
\(700\) 0 0
\(701\) 8.25585 4.76652i 0.311819 0.180029i −0.335921 0.941890i \(-0.609048\pi\)
0.647740 + 0.761861i \(0.275714\pi\)
\(702\) 0 0
\(703\) −20.7075 13.3459i −0.780996 0.503349i
\(704\) 0 0
\(705\) 33.1339 19.1298i 1.24789 0.720472i
\(706\) 0 0
\(707\) 1.08561 1.88033i 0.0408286 0.0707172i
\(708\) 0 0
\(709\) 3.08068 + 1.77863i 0.115697 + 0.0667979i 0.556732 0.830692i \(-0.312055\pi\)
−0.441035 + 0.897490i \(0.645388\pi\)
\(710\) 0 0
\(711\) −6.35224 −0.238228
\(712\) 0 0
\(713\) −5.13540 2.96493i −0.192322 0.111037i
\(714\) 0 0
\(715\) 72.3039i 2.70401i
\(716\) 0 0
\(717\) −37.0518 64.1756i −1.38373 2.39668i
\(718\) 0 0
\(719\) 44.6518 25.7797i 1.66523 0.961422i 0.695077 0.718935i \(-0.255370\pi\)
0.970155 0.242487i \(-0.0779631\pi\)
\(720\) 0 0
\(721\) 3.58669i 0.133575i
\(722\) 0 0
\(723\) 65.1256 2.42205
\(724\) 0 0
\(725\) 38.0210 + 65.8544i 1.41207 + 2.44577i
\(726\) 0 0
\(727\) −16.0229 + 9.25080i −0.594255 + 0.343093i −0.766778 0.641912i \(-0.778141\pi\)
0.172523 + 0.985005i \(0.444808\pi\)
\(728\) 0 0
\(729\) 43.7702 1.62112
\(730\) 0 0
\(731\) −4.73827 + 8.20692i −0.175251 + 0.303544i
\(732\) 0 0
\(733\) 18.1522i 0.670468i −0.942135 0.335234i \(-0.891185\pi\)
0.942135 0.335234i \(-0.108815\pi\)
\(734\) 0 0
\(735\) 38.5554 66.7799i 1.42214 2.46321i
\(736\) 0 0
\(737\) 54.8149 + 31.6474i 2.01913 + 1.16575i
\(738\) 0 0
\(739\) −19.3465 33.5092i −0.711674 1.23266i −0.964228 0.265073i \(-0.914604\pi\)
0.252554 0.967583i \(-0.418729\pi\)
\(740\) 0 0
\(741\) −21.5612 + 33.4544i −0.792072 + 1.22898i
\(742\) 0 0
\(743\) 11.7183 + 20.2967i 0.429904 + 0.744615i 0.996864 0.0791296i \(-0.0252141\pi\)
−0.566960 + 0.823745i \(0.691881\pi\)
\(744\) 0 0
\(745\) −30.4216 + 52.6917i −1.11456 + 1.93048i
\(746\) 0 0
\(747\) 1.92789 3.33920i 0.0705377 0.122175i
\(748\) 0 0
\(749\) −8.15516 −0.297983
\(750\) 0 0
\(751\) 14.1045 24.4297i 0.514681 0.891453i −0.485174 0.874417i \(-0.661244\pi\)
0.999855 0.0170354i \(-0.00542281\pi\)
\(752\) 0 0
\(753\) 1.26785i 0.0462030i
\(754\) 0 0
\(755\) 9.32525 5.38394i 0.339381 0.195942i
\(756\) 0 0
\(757\) 15.9738 9.22249i 0.580579 0.335197i −0.180785 0.983523i \(-0.557864\pi\)
0.761363 + 0.648325i \(0.224530\pi\)
\(758\) 0 0
\(759\) −93.1341 −3.38055
\(760\) 0 0
\(761\) 2.41232 0.0874467 0.0437234 0.999044i \(-0.486078\pi\)
0.0437234 + 0.999044i \(0.486078\pi\)
\(762\) 0 0
\(763\) 5.01475 2.89527i 0.181546 0.104816i
\(764\) 0 0
\(765\) 26.5042 15.3022i 0.958262 0.553253i
\(766\) 0 0
\(767\) 0.420833i 0.0151954i
\(768\) 0 0
\(769\) −5.86628 + 10.1607i −0.211544 + 0.366404i −0.952198 0.305482i \(-0.901182\pi\)
0.740654 + 0.671886i \(0.234516\pi\)
\(770\) 0 0
\(771\) 30.7212 1.10640
\(772\) 0 0
\(773\) 0.188383 0.326288i 0.00677565 0.0117358i −0.862618 0.505856i \(-0.831177\pi\)
0.869393 + 0.494121i \(0.164510\pi\)
\(774\) 0 0
\(775\) −5.64449 + 9.77654i −0.202756 + 0.351184i
\(776\) 0 0
\(777\) −3.99547 6.92036i −0.143337 0.248267i
\(778\) 0 0
\(779\) −26.0836 1.27643i −0.934542 0.0457329i
\(780\) 0 0
\(781\) 22.9129 + 39.6863i 0.819889 + 1.42009i
\(782\) 0 0
\(783\) −20.8428 12.0336i −0.744860 0.430045i
\(784\) 0 0
\(785\) 9.89355 17.1361i 0.353116 0.611615i
\(786\) 0 0
\(787\) 3.71274i 0.132345i −0.997808 0.0661725i \(-0.978921\pi\)
0.997808 0.0661725i \(-0.0210788\pi\)
\(788\) 0 0
\(789\) −11.8255 + 20.4823i −0.420998 + 0.729190i
\(790\) 0 0
\(791\) −4.56898 −0.162454
\(792\) 0 0
\(793\) −33.9622 + 19.6081i −1.20603 + 0.696304i
\(794\) 0 0
\(795\) −50.4308 87.3487i −1.78860 3.09794i
\(796\) 0 0
\(797\) 1.03774 0.0367587 0.0183794 0.999831i \(-0.494149\pi\)
0.0183794 + 0.999831i \(0.494149\pi\)
\(798\) 0 0
\(799\) 5.46621i 0.193381i
\(800\) 0 0
\(801\) 57.2165 33.0340i 2.02165 1.16720i
\(802\) 0 0
\(803\) 14.8699 + 25.7553i 0.524746 + 0.908886i
\(804\) 0 0
\(805\) 14.3587i 0.506077i
\(806\) 0 0
\(807\) −18.5266 10.6963i −0.652166 0.376528i
\(808\) 0 0
\(809\) 23.8898 0.839921 0.419961 0.907542i \(-0.362044\pi\)
0.419961 + 0.907542i \(0.362044\pi\)
\(810\) 0 0
\(811\) −25.1720 14.5331i −0.883909 0.510325i −0.0119635 0.999928i \(-0.503808\pi\)
−0.871945 + 0.489604i \(0.837142\pi\)
\(812\) 0 0
\(813\) −11.3407 + 19.6426i −0.397735 + 0.688897i
\(814\) 0 0
\(815\) −49.9970 + 28.8658i −1.75132 + 1.01112i
\(816\) 0 0
\(817\) 1.23373 25.2110i 0.0431628 0.882022i
\(818\) 0 0
\(819\) −6.68363 + 3.85880i −0.233545 + 0.134837i
\(820\) 0 0
\(821\) 30.0386 + 17.3428i 1.04835 + 0.605268i 0.922188 0.386741i \(-0.126399\pi\)
0.126166 + 0.992009i \(0.459733\pi\)
\(822\) 0 0
\(823\) −16.7332 9.66093i −0.583284 0.336759i 0.179154 0.983821i \(-0.442664\pi\)
−0.762437 + 0.647062i \(0.775997\pi\)
\(824\) 0 0
\(825\) 177.304i 6.17294i
\(826\) 0 0
\(827\) −31.4175 18.1389i −1.09249 0.630752i −0.158255 0.987398i \(-0.550587\pi\)
−0.934240 + 0.356646i \(0.883920\pi\)
\(828\) 0 0
\(829\) 33.8855 1.17689 0.588446 0.808537i \(-0.299740\pi\)
0.588446 + 0.808537i \(0.299740\pi\)
\(830\) 0 0
\(831\) 33.0794 + 57.2952i 1.14751 + 1.98755i
\(832\) 0 0
\(833\) −5.50846 9.54093i −0.190857 0.330574i
\(834\) 0 0
\(835\) 80.8577i 2.79820i
\(836\) 0 0
\(837\) 3.57294i 0.123499i
\(838\) 0 0
\(839\) 13.0241 + 22.5585i 0.449643 + 0.778805i 0.998363 0.0572016i \(-0.0182178\pi\)
−0.548719 + 0.836007i \(0.684884\pi\)
\(840\) 0 0
\(841\) −3.74045 6.47865i −0.128981 0.223402i
\(842\) 0 0
\(843\) 64.9050 2.23545
\(844\) 0 0
\(845\) −6.61952 3.82178i −0.227719 0.131473i
\(846\) 0 0
\(847\) 8.07077i 0.277315i
\(848\) 0 0
\(849\) 25.9532 + 14.9841i 0.890712 + 0.514253i
\(850\) 0 0
\(851\) −32.3688 18.6881i −1.10959 0.640621i
\(852\) 0 0
\(853\) −36.3566 + 20.9905i −1.24483 + 0.718700i −0.970073 0.242814i \(-0.921929\pi\)
−0.274753 + 0.961515i \(0.588596\pi\)
\(854\) 0 0
\(855\) −44.1600 + 68.5187i −1.51024 + 2.34329i
\(856\) 0 0
\(857\) −34.3601 + 19.8378i −1.17372 + 0.677647i −0.954553 0.298041i \(-0.903667\pi\)
−0.219166 + 0.975688i \(0.570334\pi\)
\(858\) 0 0
\(859\) −3.13531 + 5.43052i −0.106976 + 0.185287i −0.914544 0.404487i \(-0.867450\pi\)
0.807568 + 0.589774i \(0.200783\pi\)
\(860\) 0 0
\(861\) −7.33590 4.23538i −0.250007 0.144341i
\(862\) 0 0
\(863\) 9.41486 0.320486 0.160243 0.987078i \(-0.448772\pi\)
0.160243 + 0.987078i \(0.448772\pi\)
\(864\) 0 0
\(865\) −16.3399 9.43384i −0.555573 0.320760i
\(866\) 0 0
\(867\) 39.1147i 1.32840i
\(868\) 0 0
\(869\) −3.67298 6.36179i −0.124597 0.215809i
\(870\) 0 0
\(871\) 35.5396 20.5188i 1.20421 0.695253i
\(872\) 0 0
\(873\) 0.273301i 0.00924984i
\(874\) 0 0
\(875\) 16.4793 0.557101
\(876\) 0 0
\(877\) −28.2475 48.9261i −0.953850 1.65212i −0.736979 0.675916i \(-0.763748\pi\)
−0.216871 0.976200i \(-0.569585\pi\)
\(878\) 0 0
\(879\) −41.4997 + 23.9599i −1.39975 + 0.808146i
\(880\) 0 0
\(881\) 17.6960 0.596194 0.298097 0.954536i \(-0.403648\pi\)
0.298097 + 0.954536i \(0.403648\pi\)
\(882\) 0 0
\(883\) −18.9929 + 32.8967i −0.639162 + 1.10706i 0.346455 + 0.938067i \(0.387385\pi\)
−0.985617 + 0.168995i \(0.945948\pi\)
\(884\) 0 0
\(885\) 1.44181i 0.0484660i
\(886\) 0 0
\(887\) −6.86641 + 11.8930i −0.230552 + 0.399327i −0.957971 0.286867i \(-0.907386\pi\)
0.727419 + 0.686194i \(0.240720\pi\)
\(888\) 0 0
\(889\) −0.914385 0.527920i −0.0306675 0.0177059i
\(890\) 0 0
\(891\) 6.43117 + 11.1391i 0.215452 + 0.373175i
\(892\) 0 0
\(893\) 6.65476 + 12.9497i 0.222693 + 0.433344i
\(894\) 0 0
\(895\) −38.2143 66.1890i −1.27736 2.21246i
\(896\) 0 0
\(897\) −30.1921 + 52.2942i −1.00808 + 1.74605i
\(898\) 0 0
\(899\) 2.70792 4.69026i 0.0903143 0.156429i
\(900\) 0 0
\(901\) −14.4102 −0.480074
\(902\) 0 0
\(903\) 4.09370 7.09050i 0.136230 0.235957i
\(904\) 0 0
\(905\) 32.3455i 1.07520i
\(906\) 0 0
\(907\) 16.4366 9.48966i 0.545767 0.315099i −0.201646 0.979459i \(-0.564629\pi\)
0.747413 + 0.664360i \(0.231296\pi\)
\(908\) 0 0
\(909\) 16.1957 9.35056i 0.537176 0.310139i
\(910\) 0 0
\(911\) −20.4639 −0.677999 −0.339000 0.940787i \(-0.610089\pi\)
−0.339000 + 0.940787i \(0.610089\pi\)
\(912\) 0 0
\(913\) 4.45896 0.147570
\(914\) 0 0
\(915\) −116.358 + 67.1792i −3.84667 + 2.22087i
\(916\) 0 0
\(917\) 5.49941 3.17509i 0.181606 0.104851i
\(918\) 0 0
\(919\) 8.78362i 0.289745i −0.989450 0.144872i \(-0.953723\pi\)
0.989450 0.144872i \(-0.0462772\pi\)
\(920\) 0 0
\(921\) −32.8339 + 56.8700i −1.08191 + 1.87393i
\(922\) 0 0
\(923\) 29.7115 0.977967
\(924\) 0 0
\(925\) −35.5776 + 61.6222i −1.16978 + 2.02613i
\(926\) 0 0
\(927\) −15.4464 + 26.7539i −0.507326 + 0.878715i
\(928\) 0 0
\(929\) −14.7264 25.5068i −0.483157 0.836852i 0.516656 0.856193i \(-0.327176\pi\)
−0.999813 + 0.0193411i \(0.993843\pi\)
\(930\) 0 0
\(931\) 24.6652 + 15.8966i 0.808370 + 0.520991i
\(932\) 0 0
\(933\) 20.1822 + 34.9566i 0.660735 + 1.14443i
\(934\) 0 0
\(935\) 30.6504 + 17.6960i 1.00238 + 0.578722i
\(936\) 0 0
\(937\) 10.0966 17.4878i 0.329841 0.571302i −0.652639 0.757669i \(-0.726338\pi\)
0.982480 + 0.186367i \(0.0596715\pi\)
\(938\) 0 0
\(939\) 50.3829i 1.64418i
\(940\) 0 0
\(941\) 20.1773 34.9481i 0.657761 1.13927i −0.323433 0.946251i \(-0.604837\pi\)
0.981194 0.193024i \(-0.0618295\pi\)
\(942\) 0 0
\(943\) −39.6206 −1.29022
\(944\) 0 0
\(945\) −7.49250 + 4.32580i −0.243731 + 0.140718i
\(946\) 0 0
\(947\) −8.63893 14.9631i −0.280727 0.486234i 0.690837 0.723011i \(-0.257242\pi\)
−0.971564 + 0.236777i \(0.923909\pi\)
\(948\) 0 0
\(949\) 19.2820 0.625919
\(950\) 0 0
\(951\) 63.7280i 2.06652i
\(952\) 0 0
\(953\) −26.4050 + 15.2449i −0.855342 + 0.493832i −0.862450 0.506143i \(-0.831071\pi\)
0.00710753 + 0.999975i \(0.497738\pi\)
\(954\) 0 0
\(955\) 2.57172 + 4.45434i 0.0832188 + 0.144139i
\(956\) 0 0
\(957\) 85.0611i 2.74963i
\(958\) 0 0
\(959\) 7.27486 + 4.20014i 0.234918 + 0.135630i
\(960\) 0 0
\(961\) −30.1960 −0.974064
\(962\) 0 0
\(963\) −60.8312 35.1209i −1.96026 1.13176i
\(964\) 0 0
\(965\) 16.5425 28.6524i 0.532522 0.922355i
\(966\) 0 0
\(967\) −17.7891 + 10.2705i −0.572058 + 0.330278i −0.757971 0.652289i \(-0.773809\pi\)
0.185913 + 0.982566i \(0.440476\pi\)
\(968\) 0 0
\(969\) 8.90469 + 17.3278i 0.286060 + 0.556651i
\(970\) 0 0
\(971\) 47.6910 27.5344i 1.53048 0.883621i 0.531138 0.847286i \(-0.321765\pi\)
0.999340 0.0363358i \(-0.0115686\pi\)
\(972\) 0 0
\(973\) 2.61328 + 1.50878i 0.0837779 + 0.0483692i
\(974\) 0 0
\(975\) 99.5553 + 57.4783i 3.18832 + 1.84078i
\(976\) 0 0
\(977\) 1.44407i 0.0461999i 0.999733 + 0.0230999i \(0.00735359\pi\)
−0.999733 + 0.0230999i \(0.992646\pi\)
\(978\) 0 0
\(979\) 66.1672 + 38.2017i 2.11471 + 1.22093i
\(980\) 0 0
\(981\) 49.8750 1.59239
\(982\) 0 0
\(983\) −8.04964 13.9424i −0.256743 0.444693i 0.708624 0.705586i \(-0.249316\pi\)
−0.965368 + 0.260893i \(0.915983\pi\)
\(984\) 0 0
\(985\) −36.7409 63.6372i −1.17066 2.02765i
\(986\) 0 0
\(987\) 4.72262i 0.150323i
\(988\) 0 0
\(989\) 38.2952i 1.21772i
\(990\) 0 0
\(991\) 7.11258 + 12.3193i 0.225939 + 0.391337i 0.956601 0.291402i \(-0.0941218\pi\)
−0.730662 + 0.682739i \(0.760788\pi\)
\(992\) 0 0
\(993\) −31.0157 53.7207i −0.984253 1.70478i
\(994\) 0 0
\(995\) 11.8376 0.375277
\(996\) 0 0
\(997\) 28.6169 + 16.5220i 0.906308 + 0.523257i 0.879241 0.476377i \(-0.158050\pi\)
0.0270664 + 0.999634i \(0.491383\pi\)
\(998\) 0 0
\(999\) 22.5205i 0.712517i
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.i.31.4 yes 32
4.3 odd 2 inner 1216.2.s.i.31.14 yes 32
8.3 odd 2 inner 1216.2.s.i.31.3 32
8.5 even 2 inner 1216.2.s.i.31.13 yes 32
19.8 odd 6 inner 1216.2.s.i.863.3 yes 32
76.27 even 6 inner 1216.2.s.i.863.13 yes 32
152.27 even 6 inner 1216.2.s.i.863.4 yes 32
152.141 odd 6 inner 1216.2.s.i.863.14 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.s.i.31.3 32 8.3 odd 2 inner
1216.2.s.i.31.4 yes 32 1.1 even 1 trivial
1216.2.s.i.31.13 yes 32 8.5 even 2 inner
1216.2.s.i.31.14 yes 32 4.3 odd 2 inner
1216.2.s.i.863.3 yes 32 19.8 odd 6 inner
1216.2.s.i.863.4 yes 32 152.27 even 6 inner
1216.2.s.i.863.13 yes 32 76.27 even 6 inner
1216.2.s.i.863.14 yes 32 152.141 odd 6 inner