Properties

Label 1216.2.s.i.31.1
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 31.1
Character \(\chi\) \(=\) 1216.31
Dual form 1216.2.s.i.863.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.52536 + 1.45802i) q^{3} +(1.80656 - 1.04302i) q^{5} +3.63975i q^{7} +(2.75164 - 4.76598i) q^{9} +O(q^{10})\) \(q+(-2.52536 + 1.45802i) q^{3} +(1.80656 - 1.04302i) q^{5} +3.63975i q^{7} +(2.75164 - 4.76598i) q^{9} -3.65398 q^{11} +(-0.883346 + 1.53000i) q^{13} +(-3.04148 + 5.26799i) q^{15} +(2.68776 + 4.65534i) q^{17} +(1.96933 - 3.88867i) q^{19} +(-5.30682 - 9.19168i) q^{21} +(-3.04148 - 1.75600i) q^{23} +(-0.324233 + 0.561588i) q^{25} +7.29965i q^{27} +(-2.87617 + 4.98168i) q^{29} +6.30423 q^{31} +(9.22761 - 5.32756i) q^{33} +(3.79632 + 6.57542i) q^{35} -4.58389 q^{37} -5.15174i q^{39} +(-6.75491 + 3.89995i) q^{41} +(1.53000 + 2.65004i) q^{43} -11.4800i q^{45} +(6.26920 + 3.61952i) q^{47} -6.24777 q^{49} +(-13.5751 - 7.83761i) q^{51} +(2.03269 - 3.52073i) q^{53} +(-6.60112 + 3.81116i) q^{55} +(0.696478 + 12.6916i) q^{57} +(-9.00169 + 5.19713i) q^{59} +(-9.65731 - 5.57565i) q^{61} +(17.3470 + 10.0153i) q^{63} +3.68538i q^{65} +(-7.01096 - 4.04778i) q^{67} +10.2411 q^{69} +(-7.92564 - 13.7276i) q^{71} +(-7.76363 - 13.4470i) q^{73} -1.89095i q^{75} -13.2995i q^{77} +(-7.68926 - 13.3182i) q^{79} +(-2.38811 - 4.13633i) q^{81} +4.97484 q^{83} +(9.71119 + 5.60676i) q^{85} -16.7741i q^{87} +(5.44655 + 3.14456i) q^{89} +(-5.56882 - 3.21516i) q^{91} +(-15.9205 + 9.19168i) q^{93} +(-0.498237 - 9.07915i) q^{95} +(-12.4192 + 7.17025i) q^{97} +(-10.0544 + 17.4148i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 24 q^{9}+O(q^{10}) \) 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
\(3\) −2.52536 + 1.45802i −1.45802 + 0.841788i −0.998914 0.0465944i \(-0.985163\pi\)
−0.459105 + 0.888382i \(0.651830\pi\)
\(4\) 0 0
\(5\) 1.80656 1.04302i 0.807917 0.466451i −0.0383149 0.999266i \(-0.512199\pi\)
0.846232 + 0.532815i \(0.178866\pi\)
\(6\) 0 0
\(7\) 3.63975i 1.37570i 0.725855 + 0.687848i \(0.241444\pi\)
−0.725855 + 0.687848i \(0.758556\pi\)
\(8\) 0 0
\(9\) 2.75164 4.76598i 0.917213 1.58866i
\(10\) 0 0
\(11\) −3.65398 −1.10171 −0.550857 0.834599i \(-0.685699\pi\)
−0.550857 + 0.834599i \(0.685699\pi\)
\(12\) 0 0
\(13\) −0.883346 + 1.53000i −0.244996 + 0.424346i −0.962131 0.272589i \(-0.912120\pi\)
0.717134 + 0.696935i \(0.245453\pi\)
\(14\) 0 0
\(15\) −3.04148 + 5.26799i −0.785306 + 1.36019i
\(16\) 0 0
\(17\) 2.68776 + 4.65534i 0.651878 + 1.12909i 0.982667 + 0.185381i \(0.0593518\pi\)
−0.330789 + 0.943705i \(0.607315\pi\)
\(18\) 0 0
\(19\) 1.96933 3.88867i 0.451795 0.892122i
\(20\) 0 0
\(21\) −5.30682 9.19168i −1.15804 2.00579i
\(22\) 0 0
\(23\) −3.04148 1.75600i −0.634192 0.366151i 0.148182 0.988960i \(-0.452658\pi\)
−0.782374 + 0.622809i \(0.785991\pi\)
\(24\) 0 0
\(25\) −0.324233 + 0.561588i −0.0648466 + 0.112318i
\(26\) 0 0
\(27\) 7.29965i 1.40482i
\(28\) 0 0
\(29\) −2.87617 + 4.98168i −0.534092 + 0.925074i 0.465115 + 0.885250i \(0.346013\pi\)
−0.999207 + 0.0398238i \(0.987320\pi\)
\(30\) 0 0
\(31\) 6.30423 1.13227 0.566136 0.824312i \(-0.308438\pi\)
0.566136 + 0.824312i \(0.308438\pi\)
\(32\) 0 0
\(33\) 9.22761 5.32756i 1.60632 0.927410i
\(34\) 0 0
\(35\) 3.79632 + 6.57542i 0.641695 + 1.11145i
\(36\) 0 0
\(37\) −4.58389 −0.753587 −0.376794 0.926297i \(-0.622973\pi\)
−0.376794 + 0.926297i \(0.622973\pi\)
\(38\) 0 0
\(39\) 5.15174i 0.824939i
\(40\) 0 0
\(41\) −6.75491 + 3.89995i −1.05494 + 0.609070i −0.924028 0.382324i \(-0.875124\pi\)
−0.130912 + 0.991394i \(0.541791\pi\)
\(42\) 0 0
\(43\) 1.53000 + 2.65004i 0.233323 + 0.404127i 0.958784 0.284136i \(-0.0917068\pi\)
−0.725461 + 0.688263i \(0.758373\pi\)
\(44\) 0 0
\(45\) 11.4800i 1.71134i
\(46\) 0 0
\(47\) 6.26920 + 3.61952i 0.914457 + 0.527962i 0.881862 0.471507i \(-0.156290\pi\)
0.0325945 + 0.999469i \(0.489623\pi\)
\(48\) 0 0
\(49\) −6.24777 −0.892538
\(50\) 0 0
\(51\) −13.5751 7.83761i −1.90090 1.09749i
\(52\) 0 0
\(53\) 2.03269 3.52073i 0.279212 0.483609i −0.691977 0.721919i \(-0.743260\pi\)
0.971189 + 0.238310i \(0.0765936\pi\)
\(54\) 0 0
\(55\) −6.60112 + 3.81116i −0.890094 + 0.513896i
\(56\) 0 0
\(57\) 0.696478 + 12.6916i 0.0922508 + 1.68105i
\(58\) 0 0
\(59\) −9.00169 + 5.19713i −1.17192 + 0.676608i −0.954132 0.299388i \(-0.903218\pi\)
−0.217788 + 0.975996i \(0.569884\pi\)
\(60\) 0 0
\(61\) −9.65731 5.57565i −1.23649 0.713889i −0.268117 0.963386i \(-0.586401\pi\)
−0.968375 + 0.249498i \(0.919735\pi\)
\(62\) 0 0
\(63\) 17.3470 + 10.0153i 2.18551 + 1.26181i
\(64\) 0 0
\(65\) 3.68538i 0.457115i
\(66\) 0 0
\(67\) −7.01096 4.04778i −0.856525 0.494515i 0.00632174 0.999980i \(-0.497988\pi\)
−0.862847 + 0.505465i \(0.831321\pi\)
\(68\) 0 0
\(69\) 10.2411 1.23288
\(70\) 0 0
\(71\) −7.92564 13.7276i −0.940601 1.62917i −0.764329 0.644826i \(-0.776930\pi\)
−0.176271 0.984342i \(-0.556404\pi\)
\(72\) 0 0
\(73\) −7.76363 13.4470i −0.908664 1.57385i −0.815921 0.578163i \(-0.803770\pi\)
−0.0927429 0.995690i \(-0.529563\pi\)
\(74\) 0 0
\(75\) 1.89095i 0.218348i
\(76\) 0 0
\(77\) 13.2995i 1.51562i
\(78\) 0 0
\(79\) −7.68926 13.3182i −0.865110 1.49841i −0.866938 0.498416i \(-0.833915\pi\)
0.00182823 0.999998i \(-0.499418\pi\)
\(80\) 0 0
\(81\) −2.38811 4.13633i −0.265346 0.459592i
\(82\) 0 0
\(83\) 4.97484 0.546060 0.273030 0.962006i \(-0.411974\pi\)
0.273030 + 0.962006i \(0.411974\pi\)
\(84\) 0 0
\(85\) 9.71119 + 5.60676i 1.05333 + 0.608138i
\(86\) 0 0
\(87\) 16.7741i 1.79837i
\(88\) 0 0
\(89\) 5.44655 + 3.14456i 0.577333 + 0.333323i 0.760073 0.649838i \(-0.225163\pi\)
−0.182740 + 0.983161i \(0.558497\pi\)
\(90\) 0 0
\(91\) −5.56882 3.21516i −0.583771 0.337040i
\(92\) 0 0
\(93\) −15.9205 + 9.19168i −1.65088 + 0.953133i
\(94\) 0 0
\(95\) −0.498237 9.07915i −0.0511180 0.931501i
\(96\) 0 0
\(97\) −12.4192 + 7.17025i −1.26098 + 0.728029i −0.973265 0.229686i \(-0.926230\pi\)
−0.287718 + 0.957715i \(0.592897\pi\)
\(98\) 0 0
\(99\) −10.0544 + 17.4148i −1.01051 + 1.75025i
\(100\) 0 0
\(101\) −1.80656 1.04302i −0.179759 0.103784i 0.407420 0.913241i \(-0.366428\pi\)
−0.587180 + 0.809457i \(0.699762\pi\)
\(102\) 0 0
\(103\) −6.92820 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(104\) 0 0
\(105\) −19.1742 11.0702i −1.87121 1.08034i
\(106\) 0 0
\(107\) 16.5917i 1.60398i 0.597338 + 0.801990i \(0.296225\pi\)
−0.597338 + 0.801990i \(0.703775\pi\)
\(108\) 0 0
\(109\) −2.72977 4.72810i −0.261464 0.452870i 0.705167 0.709041i \(-0.250872\pi\)
−0.966631 + 0.256172i \(0.917539\pi\)
\(110\) 0 0
\(111\) 11.5760 6.68340i 1.09874 0.634360i
\(112\) 0 0
\(113\) 1.51077i 0.142122i 0.997472 + 0.0710608i \(0.0226384\pi\)
−0.997472 + 0.0710608i \(0.977362\pi\)
\(114\) 0 0
\(115\) −7.32614 −0.683166
\(116\) 0 0
\(117\) 4.86130 + 8.42002i 0.449427 + 0.778431i
\(118\) 0 0
\(119\) −16.9443 + 9.78277i −1.55328 + 0.896785i
\(120\) 0 0
\(121\) 2.35153 0.213776
\(122\) 0 0
\(123\) 11.3724 19.6976i 1.02542 1.77607i
\(124\) 0 0
\(125\) 11.7829i 1.05389i
\(126\) 0 0
\(127\) −5.95721 + 10.3182i −0.528617 + 0.915592i 0.470826 + 0.882226i \(0.343956\pi\)
−0.999443 + 0.0333658i \(0.989377\pi\)
\(128\) 0 0
\(129\) −7.72761 4.46154i −0.680378 0.392817i
\(130\) 0 0
\(131\) 9.14403 + 15.8379i 0.798918 + 1.38377i 0.920322 + 0.391163i \(0.127927\pi\)
−0.121404 + 0.992603i \(0.538740\pi\)
\(132\) 0 0
\(133\) 14.1538 + 7.16787i 1.22729 + 0.621533i
\(134\) 0 0
\(135\) 7.61365 + 13.1872i 0.655279 + 1.13498i
\(136\) 0 0
\(137\) 2.01199 3.48488i 0.171896 0.297733i −0.767187 0.641424i \(-0.778344\pi\)
0.939083 + 0.343691i \(0.111677\pi\)
\(138\) 0 0
\(139\) 0.815078 1.41176i 0.0691340 0.119744i −0.829386 0.558676i \(-0.811310\pi\)
0.898520 + 0.438932i \(0.144643\pi\)
\(140\) 0 0
\(141\) −21.1093 −1.77773
\(142\) 0 0
\(143\) 3.22773 5.59058i 0.269916 0.467508i
\(144\) 0 0
\(145\) 11.9996i 0.996511i
\(146\) 0 0
\(147\) 15.7779 9.10936i 1.30134 0.751328i
\(148\) 0 0
\(149\) −8.13543 + 4.69699i −0.666480 + 0.384793i −0.794742 0.606948i \(-0.792394\pi\)
0.128261 + 0.991740i \(0.459060\pi\)
\(150\) 0 0
\(151\) −9.76833 −0.794935 −0.397468 0.917616i \(-0.630111\pi\)
−0.397468 + 0.917616i \(0.630111\pi\)
\(152\) 0 0
\(153\) 29.5830 2.39164
\(154\) 0 0
\(155\) 11.3890 6.57542i 0.914783 0.528150i
\(156\) 0 0
\(157\) 15.0113 8.66676i 1.19803 0.691683i 0.237914 0.971286i \(-0.423536\pi\)
0.960116 + 0.279604i \(0.0902030\pi\)
\(158\) 0 0
\(159\) 11.8548i 0.940148i
\(160\) 0 0
\(161\) 6.39139 11.0702i 0.503712 0.872454i
\(162\) 0 0
\(163\) 23.8569 1.86861 0.934307 0.356468i \(-0.116019\pi\)
0.934307 + 0.356468i \(0.116019\pi\)
\(164\) 0 0
\(165\) 11.1135 19.2491i 0.865183 1.49854i
\(166\) 0 0
\(167\) 2.72949 4.72761i 0.211214 0.365834i −0.740881 0.671637i \(-0.765592\pi\)
0.952095 + 0.305803i \(0.0989249\pi\)
\(168\) 0 0
\(169\) 4.93940 + 8.55529i 0.379954 + 0.658099i
\(170\) 0 0
\(171\) −13.1144 20.0860i −1.00288 1.53601i
\(172\) 0 0
\(173\) −0.963077 1.66810i −0.0732214 0.126823i 0.827090 0.562069i \(-0.189995\pi\)
−0.900311 + 0.435246i \(0.856661\pi\)
\(174\) 0 0
\(175\) −2.04404 1.18013i −0.154515 0.0892091i
\(176\) 0 0
\(177\) 15.1550 26.2493i 1.13912 1.97302i
\(178\) 0 0
\(179\) 3.44804i 0.257718i 0.991663 + 0.128859i \(0.0411315\pi\)
−0.991663 + 0.128859i \(0.958868\pi\)
\(180\) 0 0
\(181\) 7.65774 13.2636i 0.569195 0.985875i −0.427450 0.904039i \(-0.640588\pi\)
0.996646 0.0818365i \(-0.0260785\pi\)
\(182\) 0 0
\(183\) 32.5176 2.40377
\(184\) 0 0
\(185\) −8.28107 + 4.78108i −0.608836 + 0.351512i
\(186\) 0 0
\(187\) −9.82101 17.0105i −0.718183 1.24393i
\(188\) 0 0
\(189\) −26.5689 −1.93260
\(190\) 0 0
\(191\) 19.0240i 1.37653i 0.725461 + 0.688264i \(0.241627\pi\)
−0.725461 + 0.688264i \(0.758373\pi\)
\(192\) 0 0
\(193\) 0.553454 0.319537i 0.0398385 0.0230008i −0.479948 0.877297i \(-0.659345\pi\)
0.519787 + 0.854296i \(0.326011\pi\)
\(194\) 0 0
\(195\) −5.37335 9.30692i −0.384794 0.666482i
\(196\) 0 0
\(197\) 8.09130i 0.576481i 0.957558 + 0.288241i \(0.0930703\pi\)
−0.957558 + 0.288241i \(0.906930\pi\)
\(198\) 0 0
\(199\) −9.42132 5.43940i −0.667859 0.385589i 0.127406 0.991851i \(-0.459335\pi\)
−0.795265 + 0.606262i \(0.792668\pi\)
\(200\) 0 0
\(201\) 23.6070 1.66511
\(202\) 0 0
\(203\) −18.1320 10.4685i −1.27262 0.734748i
\(204\) 0 0
\(205\) −8.13543 + 14.0910i −0.568203 + 0.984156i
\(206\) 0 0
\(207\) −16.7381 + 9.66374i −1.16338 + 0.671676i
\(208\) 0 0
\(209\) −7.19588 + 14.2091i −0.497750 + 0.982864i
\(210\) 0 0
\(211\) −5.15513 + 2.97631i −0.354894 + 0.204898i −0.666838 0.745202i \(-0.732353\pi\)
0.311945 + 0.950100i \(0.399020\pi\)
\(212\) 0 0
\(213\) 40.0302 + 23.1115i 2.74283 + 1.58357i
\(214\) 0 0
\(215\) 5.52807 + 3.19163i 0.377011 + 0.217667i
\(216\) 0 0
\(217\) 22.9458i 1.55766i
\(218\) 0 0
\(219\) 39.2120 + 22.6390i 2.64970 + 1.52980i
\(220\) 0 0
\(221\) −9.49689 −0.638830
\(222\) 0 0
\(223\) 2.80510 + 4.85857i 0.187843 + 0.325354i 0.944531 0.328422i \(-0.106517\pi\)
−0.756688 + 0.653777i \(0.773184\pi\)
\(224\) 0 0
\(225\) 1.78434 + 3.09057i 0.118956 + 0.206038i
\(226\) 0 0
\(227\) 12.8135i 0.850462i 0.905085 + 0.425231i \(0.139807\pi\)
−0.905085 + 0.425231i \(0.860193\pi\)
\(228\) 0 0
\(229\) 19.0150i 1.25654i −0.777994 0.628272i \(-0.783763\pi\)
0.777994 0.628272i \(-0.216237\pi\)
\(230\) 0 0
\(231\) 19.3910 + 33.5862i 1.27583 + 2.20981i
\(232\) 0 0
\(233\) −5.51586 9.55376i −0.361356 0.625887i 0.626828 0.779158i \(-0.284353\pi\)
−0.988184 + 0.153270i \(0.951020\pi\)
\(234\) 0 0
\(235\) 15.1009 0.985074
\(236\) 0 0
\(237\) 38.8364 + 22.4222i 2.52269 + 1.45648i
\(238\) 0 0
\(239\) 5.56988i 0.360285i −0.983640 0.180143i \(-0.942344\pi\)
0.983640 0.180143i \(-0.0576559\pi\)
\(240\) 0 0
\(241\) 11.2909 + 6.51880i 0.727310 + 0.419913i 0.817437 0.576017i \(-0.195394\pi\)
−0.0901270 + 0.995930i \(0.528727\pi\)
\(242\) 0 0
\(243\) −6.90335 3.98565i −0.442850 0.255680i
\(244\) 0 0
\(245\) −11.2870 + 6.51652i −0.721097 + 0.416325i
\(246\) 0 0
\(247\) 4.21006 + 6.44812i 0.267880 + 0.410284i
\(248\) 0 0
\(249\) −12.5633 + 7.25341i −0.796166 + 0.459666i
\(250\) 0 0
\(251\) −3.63743 + 6.30021i −0.229592 + 0.397666i −0.957687 0.287811i \(-0.907073\pi\)
0.728095 + 0.685476i \(0.240406\pi\)
\(252\) 0 0
\(253\) 11.1135 + 6.41637i 0.698698 + 0.403394i
\(254\) 0 0
\(255\) −32.6990 −2.04769
\(256\) 0 0
\(257\) 1.16433 + 0.672226i 0.0726289 + 0.0419323i 0.535875 0.844298i \(-0.319982\pi\)
−0.463246 + 0.886230i \(0.653315\pi\)
\(258\) 0 0
\(259\) 16.6842i 1.03671i
\(260\) 0 0
\(261\) 15.8284 + 27.4155i 0.979752 + 1.69698i
\(262\) 0 0
\(263\) 4.12310 2.38048i 0.254241 0.146786i −0.367463 0.930038i \(-0.619774\pi\)
0.621705 + 0.783252i \(0.286440\pi\)
\(264\) 0 0
\(265\) 8.48052i 0.520954i
\(266\) 0 0
\(267\) −18.3393 −1.12235
\(268\) 0 0
\(269\) −3.67883 6.37192i −0.224302 0.388503i 0.731808 0.681511i \(-0.238677\pi\)
−0.956110 + 0.293008i \(0.905344\pi\)
\(270\) 0 0
\(271\) 17.7696 10.2593i 1.07943 0.623206i 0.148684 0.988885i \(-0.452496\pi\)
0.930741 + 0.365678i \(0.119163\pi\)
\(272\) 0 0
\(273\) 18.7510 1.13486
\(274\) 0 0
\(275\) 1.18474 2.05203i 0.0714424 0.123742i
\(276\) 0 0
\(277\) 15.8661i 0.953299i 0.879093 + 0.476650i \(0.158149\pi\)
−0.879093 + 0.476650i \(0.841851\pi\)
\(278\) 0 0
\(279\) 17.3470 30.0458i 1.03854 1.79880i
\(280\) 0 0
\(281\) 20.7003 + 11.9513i 1.23488 + 0.712957i 0.968043 0.250786i \(-0.0806890\pi\)
0.266834 + 0.963742i \(0.414022\pi\)
\(282\) 0 0
\(283\) 13.8342 + 23.9615i 0.822357 + 1.42436i 0.903922 + 0.427696i \(0.140675\pi\)
−0.0815654 + 0.996668i \(0.525992\pi\)
\(284\) 0 0
\(285\) 14.4958 + 22.2017i 0.858657 + 1.31512i
\(286\) 0 0
\(287\) −14.1948 24.5862i −0.837895 1.45128i
\(288\) 0 0
\(289\) −5.94812 + 10.3024i −0.349889 + 0.606026i
\(290\) 0 0
\(291\) 20.9087 36.2150i 1.22569 2.12296i
\(292\) 0 0
\(293\) −29.7017 −1.73519 −0.867596 0.497270i \(-0.834336\pi\)
−0.867596 + 0.497270i \(0.834336\pi\)
\(294\) 0 0
\(295\) −10.8414 + 18.7778i −0.631209 + 1.09329i
\(296\) 0 0
\(297\) 26.6727i 1.54771i
\(298\) 0 0
\(299\) 5.37335 3.10231i 0.310749 0.179411i
\(300\) 0 0
\(301\) −9.64547 + 5.56882i −0.555956 + 0.320981i
\(302\) 0 0
\(303\) 6.08295 0.349456
\(304\) 0 0
\(305\) −23.2620 −1.33198
\(306\) 0 0
\(307\) −2.85832 + 1.65025i −0.163133 + 0.0941849i −0.579344 0.815083i \(-0.696691\pi\)
0.416211 + 0.909268i \(0.363358\pi\)
\(308\) 0 0
\(309\) 17.4962 10.1015i 0.995326 0.574651i
\(310\) 0 0
\(311\) 17.9432i 1.01747i 0.860924 + 0.508734i \(0.169886\pi\)
−0.860924 + 0.508734i \(0.830114\pi\)
\(312\) 0 0
\(313\) −2.47542 + 4.28755i −0.139919 + 0.242347i −0.927466 0.373908i \(-0.878018\pi\)
0.787547 + 0.616255i \(0.211351\pi\)
\(314\) 0 0
\(315\) 41.7844 2.35428
\(316\) 0 0
\(317\) −1.58042 + 2.73737i −0.0887654 + 0.153746i −0.906990 0.421153i \(-0.861625\pi\)
0.818224 + 0.574899i \(0.194959\pi\)
\(318\) 0 0
\(319\) 10.5095 18.2029i 0.588417 1.01917i
\(320\) 0 0
\(321\) −24.1910 41.9000i −1.35021 2.33863i
\(322\) 0 0
\(323\) 23.3962 1.28391i 1.30180 0.0714387i
\(324\) 0 0
\(325\) −0.572820 0.992153i −0.0317743 0.0550347i
\(326\) 0 0
\(327\) 13.7873 + 7.96011i 0.762440 + 0.440195i
\(328\) 0 0
\(329\) −13.1742 + 22.8183i −0.726315 + 1.25801i
\(330\) 0 0
\(331\) 2.14294i 0.117786i −0.998264 0.0588932i \(-0.981243\pi\)
0.998264 0.0588932i \(-0.0187571\pi\)
\(332\) 0 0
\(333\) −12.6132 + 21.8467i −0.691200 + 1.19719i
\(334\) 0 0
\(335\) −16.8876 −0.922669
\(336\) 0 0
\(337\) −2.58076 + 1.49000i −0.140583 + 0.0811655i −0.568642 0.822585i \(-0.692531\pi\)
0.428059 + 0.903751i \(0.359198\pi\)
\(338\) 0 0
\(339\) −2.20274 3.81525i −0.119636 0.207216i
\(340\) 0 0
\(341\) −23.0355 −1.24744
\(342\) 0 0
\(343\) 2.73794i 0.147835i
\(344\) 0 0
\(345\) 18.5011 10.6816i 0.996068 0.575080i
\(346\) 0 0
\(347\) −2.76301 4.78568i −0.148326 0.256909i 0.782283 0.622924i \(-0.214055\pi\)
−0.930609 + 0.366015i \(0.880722\pi\)
\(348\) 0 0
\(349\) 25.2867i 1.35357i 0.736182 + 0.676784i \(0.236627\pi\)
−0.736182 + 0.676784i \(0.763373\pi\)
\(350\) 0 0
\(351\) −11.1685 6.44812i −0.596129 0.344175i
\(352\) 0 0
\(353\) −8.95314 −0.476528 −0.238264 0.971200i \(-0.576578\pi\)
−0.238264 + 0.971200i \(0.576578\pi\)
\(354\) 0 0
\(355\) −28.6363 16.5331i −1.51985 0.877488i
\(356\) 0 0
\(357\) 28.5269 49.4101i 1.50981 2.61506i
\(358\) 0 0
\(359\) 0.186249 0.107531i 0.00982987 0.00567528i −0.495077 0.868849i \(-0.664860\pi\)
0.504907 + 0.863174i \(0.331527\pi\)
\(360\) 0 0
\(361\) −11.2435 15.3161i −0.591762 0.806113i
\(362\) 0 0
\(363\) −5.93848 + 3.42858i −0.311689 + 0.179954i
\(364\) 0 0
\(365\) −28.0509 16.1952i −1.46825 0.847695i
\(366\) 0 0
\(367\) −28.4512 16.4263i −1.48514 0.857445i −0.485281 0.874358i \(-0.661283\pi\)
−0.999857 + 0.0169129i \(0.994616\pi\)
\(368\) 0 0
\(369\) 42.9250i 2.23459i
\(370\) 0 0
\(371\) 12.8146 + 7.39849i 0.665298 + 0.384110i
\(372\) 0 0
\(373\) 9.12436 0.472441 0.236221 0.971699i \(-0.424091\pi\)
0.236221 + 0.971699i \(0.424091\pi\)
\(374\) 0 0
\(375\) −17.1797 29.7561i −0.887154 1.53660i
\(376\) 0 0
\(377\) −5.08131 8.80109i −0.261701 0.453279i
\(378\) 0 0
\(379\) 18.7418i 0.962702i 0.876528 + 0.481351i \(0.159854\pi\)
−0.876528 + 0.481351i \(0.840146\pi\)
\(380\) 0 0
\(381\) 34.7429i 1.77993i
\(382\) 0 0
\(383\) 8.23763 + 14.2680i 0.420923 + 0.729060i 0.996030 0.0890180i \(-0.0283729\pi\)
−0.575107 + 0.818078i \(0.695040\pi\)
\(384\) 0 0
\(385\) −13.8717 24.0264i −0.706965 1.22450i
\(386\) 0 0
\(387\) 16.8400 0.856027
\(388\) 0 0
\(389\) 21.5367 + 12.4342i 1.09195 + 0.630440i 0.934096 0.357021i \(-0.116208\pi\)
0.157859 + 0.987462i \(0.449541\pi\)
\(390\) 0 0
\(391\) 18.8788i 0.954742i
\(392\) 0 0
\(393\) −46.1840 26.6643i −2.32967 1.34504i
\(394\) 0 0
\(395\) −27.7822 16.0401i −1.39787 0.807063i
\(396\) 0 0
\(397\) 27.3186 15.7724i 1.37108 0.791595i 0.380018 0.924979i \(-0.375918\pi\)
0.991064 + 0.133385i \(0.0425846\pi\)
\(398\) 0 0
\(399\) −46.1943 + 2.53501i −2.31261 + 0.126909i
\(400\) 0 0
\(401\) 6.03598 3.48488i 0.301423 0.174026i −0.341659 0.939824i \(-0.610989\pi\)
0.643082 + 0.765798i \(0.277656\pi\)
\(402\) 0 0
\(403\) −5.56882 + 9.64547i −0.277403 + 0.480475i
\(404\) 0 0
\(405\) −8.62852 4.98168i −0.428754 0.247542i
\(406\) 0 0
\(407\) 16.7494 0.830238
\(408\) 0 0
\(409\) 12.3455 + 7.12768i 0.610446 + 0.352441i 0.773140 0.634236i \(-0.218685\pi\)
−0.162694 + 0.986677i \(0.552018\pi\)
\(410\) 0 0
\(411\) 11.7341i 0.578800i
\(412\) 0 0
\(413\) −18.9162 32.7639i −0.930807 1.61220i
\(414\) 0 0
\(415\) 8.98734 5.18884i 0.441171 0.254710i
\(416\) 0 0
\(417\) 4.75360i 0.232785i
\(418\) 0 0
\(419\) −17.6856 −0.863998 −0.431999 0.901874i \(-0.642192\pi\)
−0.431999 + 0.901874i \(0.642192\pi\)
\(420\) 0 0
\(421\) 15.9345 + 27.5993i 0.776599 + 1.34511i 0.933891 + 0.357557i \(0.116390\pi\)
−0.157292 + 0.987552i \(0.550277\pi\)
\(422\) 0 0
\(423\) 34.5011 19.9192i 1.67750 0.968507i
\(424\) 0 0
\(425\) −3.48584 −0.169088
\(426\) 0 0
\(427\) 20.2940 35.1502i 0.982094 1.70104i
\(428\) 0 0
\(429\) 18.8243i 0.908848i
\(430\) 0 0
\(431\) 18.3179 31.7276i 0.882344 1.52827i 0.0336170 0.999435i \(-0.489297\pi\)
0.848727 0.528831i \(-0.177369\pi\)
\(432\) 0 0
\(433\) 5.47270 + 3.15966i 0.263001 + 0.151844i 0.625703 0.780062i \(-0.284812\pi\)
−0.362702 + 0.931905i \(0.618146\pi\)
\(434\) 0 0
\(435\) −17.4956 30.3033i −0.838851 1.45293i
\(436\) 0 0
\(437\) −12.8182 + 8.36915i −0.613176 + 0.400351i
\(438\) 0 0
\(439\) −0.0501271 0.0868226i −0.00239243 0.00414382i 0.864827 0.502070i \(-0.167428\pi\)
−0.867219 + 0.497927i \(0.834095\pi\)
\(440\) 0 0
\(441\) −17.1916 + 29.7767i −0.818647 + 1.41794i
\(442\) 0 0
\(443\) −7.29140 + 12.6291i −0.346425 + 0.600026i −0.985612 0.169026i \(-0.945938\pi\)
0.639187 + 0.769052i \(0.279271\pi\)
\(444\) 0 0
\(445\) 13.1193 0.621916
\(446\) 0 0
\(447\) 13.6966 23.7232i 0.647827 1.12207i
\(448\) 0 0
\(449\) 1.42562i 0.0672791i 0.999434 + 0.0336395i \(0.0107098\pi\)
−0.999434 + 0.0336395i \(0.989290\pi\)
\(450\) 0 0
\(451\) 24.6823 14.2503i 1.16224 0.671022i
\(452\) 0 0
\(453\) 24.6686 14.2424i 1.15903 0.669167i
\(454\) 0 0
\(455\) −13.4139 −0.628851
\(456\) 0 0
\(457\) 33.9105 1.58627 0.793134 0.609047i \(-0.208448\pi\)
0.793134 + 0.609047i \(0.208448\pi\)
\(458\) 0 0
\(459\) −33.9823 + 19.6197i −1.58616 + 0.915770i
\(460\) 0 0
\(461\) −24.7141 + 14.2687i −1.15105 + 0.664558i −0.949143 0.314847i \(-0.898047\pi\)
−0.201906 + 0.979405i \(0.564714\pi\)
\(462\) 0 0
\(463\) 4.75537i 0.221001i 0.993876 + 0.110500i \(0.0352454\pi\)
−0.993876 + 0.110500i \(0.964755\pi\)
\(464\) 0 0
\(465\) −19.1742 + 33.2106i −0.889180 + 1.54011i
\(466\) 0 0
\(467\) −3.65398 −0.169086 −0.0845429 0.996420i \(-0.526943\pi\)
−0.0845429 + 0.996420i \(0.526943\pi\)
\(468\) 0 0
\(469\) 14.7329 25.5181i 0.680302 1.17832i
\(470\) 0 0
\(471\) −25.2726 + 43.7734i −1.16450 + 2.01697i
\(472\) 0 0
\(473\) −5.59058 9.68318i −0.257055 0.445233i
\(474\) 0 0
\(475\) 1.54531 + 2.36679i 0.0709035 + 0.108596i
\(476\) 0 0
\(477\) −11.1865 19.3755i −0.512193 0.887144i
\(478\) 0 0
\(479\) −18.7756 10.8401i −0.857879 0.495297i 0.00542242 0.999985i \(-0.498274\pi\)
−0.863301 + 0.504689i \(0.831607\pi\)
\(480\) 0 0
\(481\) 4.04916 7.01336i 0.184626 0.319782i
\(482\) 0 0
\(483\) 37.2750i 1.69607i
\(484\) 0 0
\(485\) −14.9574 + 25.9070i −0.679180 + 1.17637i
\(486\) 0 0
\(487\) 11.0163 0.499195 0.249598 0.968350i \(-0.419702\pi\)
0.249598 + 0.968350i \(0.419702\pi\)
\(488\) 0 0
\(489\) −60.2473 + 34.7838i −2.72448 + 1.57298i
\(490\) 0 0
\(491\) 18.1742 + 31.4787i 0.820192 + 1.42061i 0.905539 + 0.424263i \(0.139467\pi\)
−0.0853474 + 0.996351i \(0.527200\pi\)
\(492\) 0 0
\(493\) −30.9219 −1.39265
\(494\) 0 0
\(495\) 41.9477i 1.88541i
\(496\) 0 0
\(497\) 49.9651 28.8473i 2.24124 1.29398i
\(498\) 0 0
\(499\) 1.93139 + 3.34526i 0.0864608 + 0.149755i 0.906013 0.423250i \(-0.139111\pi\)
−0.819552 + 0.573005i \(0.805778\pi\)
\(500\) 0 0
\(501\) 15.9186i 0.711190i
\(502\) 0 0
\(503\) −18.0173 10.4023i −0.803352 0.463816i 0.0412899 0.999147i \(-0.486853\pi\)
−0.844642 + 0.535332i \(0.820187\pi\)
\(504\) 0 0
\(505\) −4.35153 −0.193641
\(506\) 0 0
\(507\) −24.9475 14.4035i −1.10796 0.639681i
\(508\) 0 0
\(509\) −4.17192 + 7.22598i −0.184917 + 0.320286i −0.943549 0.331234i \(-0.892535\pi\)
0.758631 + 0.651520i \(0.225868\pi\)
\(510\) 0 0
\(511\) 48.9437 28.2577i 2.16514 1.25005i
\(512\) 0 0
\(513\) 28.3859 + 14.3754i 1.25327 + 0.634690i
\(514\) 0 0
\(515\) −12.5162 + 7.22623i −0.551530 + 0.318426i
\(516\) 0 0
\(517\) −22.9075 13.2257i −1.00747 0.581664i
\(518\) 0 0
\(519\) 4.86424 + 2.80837i 0.213516 + 0.123274i
\(520\) 0 0
\(521\) 19.2965i 0.845397i −0.906270 0.422698i \(-0.861083\pi\)
0.906270 0.422698i \(-0.138917\pi\)
\(522\) 0 0
\(523\) −32.3479 18.6761i −1.41448 0.816648i −0.418670 0.908138i \(-0.637504\pi\)
−0.995806 + 0.0914902i \(0.970837\pi\)
\(524\) 0 0
\(525\) 6.88258 0.300381
\(526\) 0 0
\(527\) 16.9443 + 29.3483i 0.738104 + 1.27843i
\(528\) 0 0
\(529\) −5.33295 9.23694i −0.231867 0.401606i
\(530\) 0 0
\(531\) 57.2024i 2.48237i
\(532\) 0 0
\(533\) 13.7800i 0.596879i
\(534\) 0 0
\(535\) 17.3054 + 29.9738i 0.748178 + 1.29588i
\(536\) 0 0
\(537\) −5.02730 8.70754i −0.216944 0.375758i
\(538\) 0 0
\(539\) 22.8292 0.983323
\(540\) 0 0
\(541\) 13.9488 + 8.05336i 0.599707 + 0.346241i 0.768926 0.639337i \(-0.220791\pi\)
−0.169219 + 0.985578i \(0.554125\pi\)
\(542\) 0 0
\(543\) 44.6605i 1.91657i
\(544\) 0 0
\(545\) −9.86297 5.69439i −0.422483 0.243921i
\(546\) 0 0
\(547\) −8.05336 4.64961i −0.344337 0.198803i 0.317851 0.948141i \(-0.397039\pi\)
−0.662188 + 0.749338i \(0.730372\pi\)
\(548\) 0 0
\(549\) −53.1468 + 30.6843i −2.26825 + 1.30958i
\(550\) 0 0
\(551\) 13.7080 + 20.9950i 0.583978 + 0.894419i
\(552\) 0 0
\(553\) 48.4749 27.9870i 2.06136 1.19013i
\(554\) 0 0
\(555\) 13.9418 24.1479i 0.591796 1.02502i
\(556\) 0 0
\(557\) 19.6532 + 11.3468i 0.832732 + 0.480778i 0.854787 0.518979i \(-0.173688\pi\)
−0.0220553 + 0.999757i \(0.507021\pi\)
\(558\) 0 0
\(559\) −5.40608 −0.228653
\(560\) 0 0
\(561\) 49.6032 + 28.6384i 2.09425 + 1.20912i
\(562\) 0 0
\(563\) 29.7497i 1.25380i 0.779100 + 0.626900i \(0.215677\pi\)
−0.779100 + 0.626900i \(0.784323\pi\)
\(564\) 0 0
\(565\) 1.57576 + 2.72930i 0.0662928 + 0.114822i
\(566\) 0 0
\(567\) 15.0552 8.69212i 0.632259 0.365035i
\(568\) 0 0
\(569\) 35.2216i 1.47657i 0.674490 + 0.738284i \(0.264363\pi\)
−0.674490 + 0.738284i \(0.735637\pi\)
\(570\) 0 0
\(571\) 8.45311 0.353752 0.176876 0.984233i \(-0.443401\pi\)
0.176876 + 0.984233i \(0.443401\pi\)
\(572\) 0 0
\(573\) −27.7373 48.0425i −1.15874 2.00700i
\(574\) 0 0
\(575\) 1.97229 1.13870i 0.0822503 0.0474872i
\(576\) 0 0
\(577\) −10.3569 −0.431163 −0.215582 0.976486i \(-0.569165\pi\)
−0.215582 + 0.976486i \(0.569165\pi\)
\(578\) 0 0
\(579\) −0.931781 + 1.61389i −0.0387235 + 0.0670711i
\(580\) 0 0
\(581\) 18.1072i 0.751212i
\(582\) 0 0
\(583\) −7.42740 + 12.8646i −0.307612 + 0.532799i
\(584\) 0 0
\(585\) 17.5644 + 10.1408i 0.726200 + 0.419272i
\(586\) 0 0
\(587\) 4.87469 + 8.44321i 0.201200 + 0.348488i 0.948915 0.315531i \(-0.102183\pi\)
−0.747715 + 0.664019i \(0.768849\pi\)
\(588\) 0 0
\(589\) 12.4151 24.5151i 0.511556 1.01013i
\(590\) 0 0
\(591\) −11.7973 20.4335i −0.485275 0.840521i
\(592\) 0 0
\(593\) 16.3226 28.2716i 0.670290 1.16098i −0.307532 0.951538i \(-0.599503\pi\)
0.977822 0.209438i \(-0.0671636\pi\)
\(594\) 0 0
\(595\) −20.4072 + 35.3463i −0.836613 + 1.44906i
\(596\) 0 0
\(597\) 31.7230 1.29834
\(598\) 0 0
\(599\) −14.0691 + 24.3684i −0.574848 + 0.995666i 0.421210 + 0.906963i \(0.361605\pi\)
−0.996058 + 0.0887029i \(0.971728\pi\)
\(600\) 0 0
\(601\) 15.8077i 0.644808i 0.946602 + 0.322404i \(0.104491\pi\)
−0.946602 + 0.322404i \(0.895509\pi\)
\(602\) 0 0
\(603\) −38.5833 + 22.2761i −1.57123 + 0.907151i
\(604\) 0 0
\(605\) 4.24818 2.45269i 0.172713 0.0997160i
\(606\) 0 0
\(607\) 7.52198 0.305308 0.152654 0.988280i \(-0.451218\pi\)
0.152654 + 0.988280i \(0.451218\pi\)
\(608\) 0 0
\(609\) 61.0533 2.47401
\(610\) 0 0
\(611\) −11.0757 + 6.39459i −0.448077 + 0.258697i
\(612\) 0 0
\(613\) 6.55967 3.78723i 0.264942 0.152965i −0.361645 0.932316i \(-0.617785\pi\)
0.626587 + 0.779351i \(0.284451\pi\)
\(614\) 0 0
\(615\) 47.4464i 1.91322i
\(616\) 0 0
\(617\) 2.85153 4.93900i 0.114798 0.198837i −0.802901 0.596113i \(-0.796711\pi\)
0.917699 + 0.397276i \(0.130044\pi\)
\(618\) 0 0
\(619\) −28.5470 −1.14740 −0.573701 0.819065i \(-0.694493\pi\)
−0.573701 + 0.819065i \(0.694493\pi\)
\(620\) 0 0
\(621\) 12.8182 22.2017i 0.514375 0.890924i
\(622\) 0 0
\(623\) −11.4454 + 19.8241i −0.458551 + 0.794234i
\(624\) 0 0
\(625\) 10.6686 + 18.4785i 0.426743 + 0.739141i
\(626\) 0 0
\(627\) −2.54491 46.3749i −0.101634 1.85203i
\(628\) 0 0
\(629\) −12.3204 21.3396i −0.491247 0.850864i
\(630\) 0 0
\(631\) 1.32283 + 0.763739i 0.0526612 + 0.0304040i 0.526099 0.850423i \(-0.323654\pi\)
−0.473438 + 0.880827i \(0.656987\pi\)
\(632\) 0 0
\(633\) 8.67904 15.0325i 0.344961 0.597490i
\(634\) 0 0
\(635\) 24.8539i 0.986297i
\(636\) 0 0
\(637\) 5.51894 9.55909i 0.218668 0.378745i
\(638\) 0 0
\(639\) −87.2340 −3.45092
\(640\) 0 0
\(641\) −24.4668 + 14.1259i −0.966382 + 0.557941i −0.898131 0.439727i \(-0.855075\pi\)
−0.0682507 + 0.997668i \(0.521742\pi\)
\(642\) 0 0
\(643\) −14.5235 25.1554i −0.572750 0.992032i −0.996282 0.0861507i \(-0.972543\pi\)
0.423532 0.905881i \(-0.360790\pi\)
\(644\) 0 0
\(645\) −18.6138 −0.732919
\(646\) 0 0
\(647\) 33.4084i 1.31342i 0.754144 + 0.656709i \(0.228052\pi\)
−0.754144 + 0.656709i \(0.771948\pi\)
\(648\) 0 0
\(649\) 32.8919 18.9902i 1.29112 0.745429i
\(650\) 0 0
\(651\) −33.4554 57.9465i −1.31122 2.27110i
\(652\) 0 0
\(653\) 33.9732i 1.32948i −0.747077 0.664738i \(-0.768543\pi\)
0.747077 0.664738i \(-0.231457\pi\)
\(654\) 0 0
\(655\) 33.0384 + 19.0748i 1.29092 + 0.745312i
\(656\) 0 0
\(657\) −85.4508 −3.33375
\(658\) 0 0
\(659\) 9.84953 + 5.68663i 0.383683 + 0.221520i 0.679420 0.733750i \(-0.262232\pi\)
−0.295736 + 0.955270i \(0.595565\pi\)
\(660\) 0 0
\(661\) 22.0983 38.2753i 0.859523 1.48874i −0.0128614 0.999917i \(-0.504094\pi\)
0.872384 0.488820i \(-0.162573\pi\)
\(662\) 0 0
\(663\) 23.9831 13.8466i 0.931426 0.537759i
\(664\) 0 0
\(665\) 33.0458 1.81346i 1.28146 0.0703228i
\(666\) 0 0
\(667\) 17.4956 10.1011i 0.677433 0.391116i
\(668\) 0 0
\(669\) −14.1678 8.17978i −0.547758 0.316248i
\(670\) 0 0
\(671\) 35.2876 + 20.3733i 1.36226 + 0.786502i
\(672\) 0 0
\(673\) 0.883049i 0.0340390i −0.999855 0.0170195i \(-0.994582\pi\)
0.999855 0.0170195i \(-0.00541774\pi\)
\(674\) 0 0
\(675\) −4.09939 2.36679i −0.157786 0.0910976i
\(676\) 0 0
\(677\) −5.99154 −0.230273 −0.115137 0.993350i \(-0.536731\pi\)
−0.115137 + 0.993350i \(0.536731\pi\)
\(678\) 0 0
\(679\) −26.0979 45.2029i −1.00155 1.73473i
\(680\) 0 0
\(681\) −18.6823 32.3587i −0.715908 1.23999i
\(682\) 0 0
\(683\) 10.2700i 0.392970i 0.980507 + 0.196485i \(0.0629528\pi\)
−0.980507 + 0.196485i \(0.937047\pi\)
\(684\) 0 0
\(685\) 8.39417i 0.320725i
\(686\) 0 0
\(687\) 27.7242 + 48.0197i 1.05774 + 1.83206i
\(688\) 0 0
\(689\) 3.59114 + 6.22004i 0.136812 + 0.236965i
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) −63.3853 36.5955i −2.40781 1.39015i
\(694\) 0 0
\(695\) 3.40056i 0.128991i
\(696\) 0 0
\(697\) −36.3112 20.9643i −1.37538 0.794079i
\(698\) 0 0
\(699\) 27.8591 + 16.0845i 1.05373 + 0.608371i
\(700\) 0 0
\(701\) 21.2289 12.2565i 0.801806 0.462923i −0.0422963 0.999105i \(-0.513467\pi\)
0.844102 + 0.536182i \(0.180134\pi\)
\(702\) 0 0
\(703\) −9.02720 + 17.8252i −0.340467 + 0.672292i
\(704\) 0 0
\(705\) −38.1352 + 22.0174i −1.43626 + 0.829223i
\(706\) 0 0
\(707\) 3.79632 6.57542i 0.142775 0.247294i
\(708\) 0 0
\(709\) −45.3505 26.1832i −1.70318 0.983329i −0.942506 0.334190i \(-0.891537\pi\)
−0.760670 0.649139i \(-0.775129\pi\)
\(710\) 0 0
\(711\) −84.6323 −3.17396
\(712\) 0 0
\(713\) −19.1742 11.0702i −0.718078 0.414582i
\(714\) 0 0
\(715\) 13.4663i 0.503610i
\(716\) 0 0
\(717\) 8.12098 + 14.0660i 0.303284 + 0.525303i
\(718\) 0 0
\(719\) 17.0020 9.81611i 0.634068 0.366079i −0.148258 0.988949i \(-0.547367\pi\)
0.782326 + 0.622870i \(0.214033\pi\)
\(720\) 0 0
\(721\) 25.2169i 0.939127i
\(722\) 0 0
\(723\) −38.0181 −1.41391
\(724\) 0 0
\(725\) −1.86510 3.23045i −0.0692680 0.119976i
\(726\) 0 0
\(727\) 29.6121 17.0966i 1.09825 0.634077i 0.162492 0.986710i \(-0.448047\pi\)
0.935762 + 0.352633i \(0.114713\pi\)
\(728\) 0 0
\(729\) 37.5733 1.39160
\(730\) 0 0
\(731\) −8.22455 + 14.2453i −0.304196 + 0.526883i
\(732\) 0 0
\(733\) 41.0010i 1.51441i 0.653179 + 0.757204i \(0.273435\pi\)
−0.653179 + 0.757204i \(0.726565\pi\)
\(734\) 0 0
\(735\) 19.0024 32.9132i 0.700915 1.21402i
\(736\) 0 0
\(737\) 25.6179 + 14.7905i 0.943647 + 0.544815i
\(738\) 0 0
\(739\) 13.3449 + 23.1141i 0.490902 + 0.850267i 0.999945 0.0104741i \(-0.00333406\pi\)
−0.509043 + 0.860741i \(0.670001\pi\)
\(740\) 0 0
\(741\) −20.0334 10.1455i −0.735946 0.372704i
\(742\) 0 0
\(743\) 4.75843 + 8.24184i 0.174570 + 0.302364i 0.940012 0.341141i \(-0.110813\pi\)
−0.765443 + 0.643504i \(0.777480\pi\)
\(744\) 0 0
\(745\) −9.79808 + 16.9708i −0.358974 + 0.621761i
\(746\) 0 0
\(747\) 13.6890 23.7100i 0.500853 0.867503i
\(748\) 0 0
\(749\) −60.3896 −2.20659
\(750\) 0 0
\(751\) −1.40013 + 2.42510i −0.0510916 + 0.0884933i −0.890440 0.455100i \(-0.849603\pi\)
0.839349 + 0.543594i \(0.182937\pi\)
\(752\) 0 0
\(753\) 21.2138i 0.773072i
\(754\) 0 0
\(755\) −17.6471 + 10.1885i −0.642242 + 0.370799i
\(756\) 0 0
\(757\) 4.41111 2.54676i 0.160325 0.0925635i −0.417691 0.908589i \(-0.637161\pi\)
0.578016 + 0.816026i \(0.303827\pi\)
\(758\) 0 0
\(759\) −37.4207 −1.35829
\(760\) 0 0
\(761\) −26.4509 −0.958844 −0.479422 0.877584i \(-0.659154\pi\)
−0.479422 + 0.877584i \(0.659154\pi\)
\(762\) 0 0
\(763\) 17.2091 9.93567i 0.623011 0.359695i
\(764\) 0 0
\(765\) 53.4434 30.8555i 1.93225 1.11558i
\(766\) 0 0
\(767\) 18.3634i 0.663066i
\(768\) 0 0
\(769\) −16.1355 + 27.9474i −0.581860 + 1.00781i 0.413399 + 0.910550i \(0.364341\pi\)
−0.995259 + 0.0972605i \(0.968992\pi\)
\(770\) 0 0
\(771\) −3.92048 −0.141192
\(772\) 0 0
\(773\) 2.08657 3.61405i 0.0750487 0.129988i −0.826059 0.563584i \(-0.809422\pi\)
0.901107 + 0.433596i \(0.142755\pi\)
\(774\) 0 0
\(775\) −2.04404 + 3.54038i −0.0734240 + 0.127174i
\(776\) 0 0
\(777\) 24.3259 + 42.1337i 0.872687 + 1.51154i
\(778\) 0 0
\(779\) 1.86296 + 33.9479i 0.0667475 + 1.21631i
\(780\) 0 0
\(781\) 28.9601 + 50.1604i 1.03627 + 1.79488i
\(782\) 0 0
\(783\) −36.3645 20.9950i −1.29956 0.750302i
\(784\) 0 0
\(785\) 18.0791 31.3140i 0.645272 1.11764i
\(786\) 0 0
\(787\) 18.0039i 0.641770i −0.947118 0.320885i \(-0.896020\pi\)
0.947118 0.320885i \(-0.103980\pi\)
\(788\) 0 0
\(789\) −6.94156 + 12.0231i −0.247126 + 0.428035i
\(790\) 0 0
\(791\) −5.49883 −0.195516
\(792\) 0 0
\(793\) 17.0615 9.85046i 0.605872 0.349800i
\(794\) 0 0
\(795\) 12.3648 + 21.4164i 0.438533 + 0.759561i
\(796\) 0 0
\(797\) 49.2555 1.74472 0.872360 0.488864i \(-0.162589\pi\)
0.872360 + 0.488864i \(0.162589\pi\)
\(798\) 0 0
\(799\) 38.9137i 1.37667i
\(800\) 0 0
\(801\) 29.9738 17.3054i 1.05907 0.611457i
\(802\) 0 0
\(803\) 28.3681 + 49.1350i 1.00109 + 1.73394i
\(804\) 0 0
\(805\) 26.6653i 0.939828i
\(806\) 0 0
\(807\) 18.5808 + 10.7276i 0.654074 + 0.377630i
\(808\) 0 0
\(809\) −11.1309 −0.391341 −0.195671 0.980670i \(-0.562688\pi\)
−0.195671 + 0.980670i \(0.562688\pi\)
\(810\) 0 0
\(811\) 9.34237 + 5.39382i 0.328055 + 0.189403i 0.654977 0.755649i \(-0.272678\pi\)
−0.326922 + 0.945051i \(0.606012\pi\)
\(812\) 0 0
\(813\) −29.9164 + 51.8168i −1.04921 + 1.81729i
\(814\) 0 0
\(815\) 43.0988 24.8831i 1.50969 0.871618i
\(816\) 0 0
\(817\) 13.3182 0.730863i 0.465945 0.0255697i
\(818\) 0 0
\(819\) −30.6467 + 17.6939i −1.07088 + 0.618275i
\(820\) 0 0
\(821\) −5.11190 2.95135i −0.178406 0.103003i 0.408137 0.912921i \(-0.366178\pi\)
−0.586544 + 0.809918i \(0.699512\pi\)
\(822\) 0 0
\(823\) 5.97231 + 3.44812i 0.208182 + 0.120194i 0.600466 0.799650i \(-0.294982\pi\)
−0.392284 + 0.919844i \(0.628315\pi\)
\(824\) 0 0
\(825\) 6.90949i 0.240557i
\(826\) 0 0
\(827\) 14.8328 + 8.56370i 0.515786 + 0.297789i 0.735209 0.677841i \(-0.237084\pi\)
−0.219423 + 0.975630i \(0.570417\pi\)
\(828\) 0 0
\(829\) −42.7083 −1.48332 −0.741660 0.670776i \(-0.765961\pi\)
−0.741660 + 0.670776i \(0.765961\pi\)
\(830\) 0 0
\(831\) −23.1330 40.0676i −0.802476 1.38993i
\(832\) 0 0
\(833\) −16.7925 29.0855i −0.581826 1.00775i
\(834\) 0 0
\(835\) 11.3876i 0.394084i
\(836\) 0 0
\(837\) 46.0187i 1.59064i
\(838\) 0 0
\(839\) −24.1003 41.7429i −0.832033 1.44112i −0.896423 0.443199i \(-0.853843\pi\)
0.0643898 0.997925i \(-0.479490\pi\)
\(840\) 0 0
\(841\) −2.04474 3.54159i −0.0705082 0.122124i
\(842\) 0 0
\(843\) −69.7011 −2.40063
\(844\) 0 0
\(845\) 17.8466 + 10.3037i 0.613942 + 0.354460i
\(846\) 0 0
\(847\) 8.55899i 0.294090i
\(848\) 0 0
\(849\) −69.8727 40.3410i −2.39802 1.38450i
\(850\) 0 0
\(851\) 13.9418 + 8.04930i 0.477919 + 0.275926i
\(852\) 0 0
\(853\) −26.2023 + 15.1279i −0.897150 + 0.517970i −0.876274 0.481813i \(-0.839979\pi\)
−0.0208753 + 0.999782i \(0.506645\pi\)
\(854\) 0 0
\(855\) −44.6420 22.6079i −1.52672 0.773175i
\(856\) 0 0
\(857\) −15.6812 + 9.05353i −0.535658 + 0.309263i −0.743318 0.668939i \(-0.766749\pi\)
0.207659 + 0.978201i \(0.433416\pi\)
\(858\) 0 0
\(859\) 1.99360 3.45302i 0.0680209 0.117816i −0.830009 0.557750i \(-0.811665\pi\)
0.898030 + 0.439934i \(0.144998\pi\)
\(860\) 0 0
\(861\) 71.6943 + 41.3927i 2.44333 + 1.41066i
\(862\) 0 0
\(863\) 5.83148 0.198506 0.0992529 0.995062i \(-0.468355\pi\)
0.0992529 + 0.995062i \(0.468355\pi\)
\(864\) 0 0
\(865\) −3.47971 2.00901i −0.118314 0.0683084i
\(866\) 0 0
\(867\) 34.6899i 1.17813i
\(868\) 0 0
\(869\) 28.0964 + 48.6644i 0.953104 + 1.65083i
\(870\) 0 0
\(871\) 12.3862 7.15119i 0.419691 0.242309i
\(872\) 0 0
\(873\) 78.9198i 2.67103i
\(874\) 0 0
\(875\) −42.8867 −1.44984
\(876\) 0 0
\(877\) 15.5211 + 26.8834i 0.524111 + 0.907787i 0.999606 + 0.0280684i \(0.00893564\pi\)
−0.475495 + 0.879718i \(0.657731\pi\)
\(878\) 0 0
\(879\) 75.0076 43.3056i 2.52994 1.46066i
\(880\) 0 0
\(881\) −20.4870 −0.690223 −0.345112 0.938562i \(-0.612159\pi\)
−0.345112 + 0.938562i \(0.612159\pi\)
\(882\) 0 0
\(883\) 23.0901 39.9932i 0.777043 1.34588i −0.156596 0.987663i \(-0.550052\pi\)
0.933639 0.358215i \(-0.116615\pi\)
\(884\) 0 0
\(885\) 63.2277i 2.12538i
\(886\) 0 0
\(887\) 7.75932 13.4395i 0.260532 0.451255i −0.705851 0.708360i \(-0.749435\pi\)
0.966384 + 0.257105i \(0.0827686\pi\)
\(888\) 0 0
\(889\) −37.5556 21.6828i −1.25958 0.727216i
\(890\) 0 0
\(891\) 8.72610 + 15.1140i 0.292335 + 0.506339i
\(892\) 0 0
\(893\) 26.4213 17.2508i 0.884154 0.577276i
\(894\) 0 0
\(895\) 3.59636 + 6.22908i 0.120213 + 0.208215i
\(896\) 0 0
\(897\) −9.04644 + 15.6689i −0.302052 + 0.523169i
\(898\) 0 0
\(899\) −18.1320 + 31.4056i −0.604738 + 1.04744i
\(900\) 0 0
\(901\) 21.8536 0.728047
\(902\) 0 0
\(903\) 16.2389 28.1266i 0.540396 0.935993i
\(904\) 0 0
\(905\) 31.9486i 1.06201i
\(906\) 0 0
\(907\) −3.72331 + 2.14966i −0.123631 + 0.0713782i −0.560540 0.828127i \(-0.689406\pi\)
0.436909 + 0.899506i \(0.356073\pi\)
\(908\) 0 0
\(909\) −9.94199 + 5.74001i −0.329755 + 0.190384i
\(910\) 0 0
\(911\) 50.6153 1.67696 0.838479 0.544934i \(-0.183445\pi\)
0.838479 + 0.544934i \(0.183445\pi\)
\(912\) 0 0
\(913\) −18.1780 −0.601602
\(914\) 0 0
\(915\) 58.7450 33.9164i 1.94205 1.12124i
\(916\) 0 0
\(917\) −57.6461 + 33.2820i −1.90364 + 1.09907i
\(918\) 0 0
\(919\) 24.9736i 0.823804i 0.911228 + 0.411902i \(0.135135\pi\)
−0.911228 + 0.411902i \(0.864865\pi\)
\(920\) 0 0
\(921\) 4.81220 8.33498i 0.158567 0.274647i
\(922\) 0 0
\(923\) 28.0043 0.921774
\(924\) 0 0
\(925\) 1.48625 2.57426i 0.0488676 0.0846411i
\(926\) 0 0
\(927\) −19.0639 + 33.0197i −0.626141 + 1.08451i
\(928\) 0 0
\(929\) 5.52399 + 9.56783i 0.181236 + 0.313910i 0.942302 0.334765i \(-0.108657\pi\)
−0.761066 + 0.648675i \(0.775324\pi\)
\(930\) 0 0
\(931\) −12.3039 + 24.2955i −0.403245 + 0.796253i
\(932\) 0 0
\(933\) −26.1616 45.3132i −0.856491 1.48349i
\(934\) 0 0
\(935\) −35.4844 20.4870i −1.16047 0.669995i
\(936\) 0 0
\(937\) −16.9219 + 29.3096i −0.552815 + 0.957504i 0.445255 + 0.895404i \(0.353113\pi\)
−0.998070 + 0.0621001i \(0.980220\pi\)
\(938\) 0 0
\(939\) 14.4368i 0.471128i
\(940\) 0 0
\(941\) 24.2116 41.9358i 0.789277 1.36707i −0.137133 0.990553i \(-0.543789\pi\)
0.926410 0.376516i \(-0.122878\pi\)
\(942\) 0 0
\(943\) 27.3932 0.892046
\(944\) 0 0
\(945\) −47.9982 + 27.7118i −1.56138 + 0.901464i
\(946\) 0 0
\(947\) −25.7669 44.6295i −0.837311 1.45026i −0.892135 0.451769i \(-0.850793\pi\)
0.0548242 0.998496i \(-0.482540\pi\)
\(948\) 0 0
\(949\) 27.4319 0.890477
\(950\) 0 0
\(951\) 9.21715i 0.298887i
\(952\) 0 0
\(953\) −0.728762 + 0.420751i −0.0236069 + 0.0136295i −0.511757 0.859130i \(-0.671005\pi\)
0.488150 + 0.872760i \(0.337672\pi\)
\(954\) 0 0
\(955\) 19.8423 + 34.3679i 0.642083 + 1.11212i
\(956\) 0 0
\(957\) 61.2920i 1.98129i
\(958\) 0 0
\(959\) 12.6841 + 7.32315i 0.409590 + 0.236477i
\(960\) 0 0
\(961\) 8.74330 0.282042
\(962\) 0 0
\(963\) 79.0756 + 45.6543i 2.54818 + 1.47119i
\(964\) 0 0
\(965\) 0.666564 1.15452i 0.0214575 0.0371654i
\(966\) 0 0
\(967\) −37.3932 + 21.5890i −1.20248 + 0.694255i −0.961107 0.276177i \(-0.910933\pi\)
−0.241377 + 0.970431i \(0.577599\pi\)
\(968\) 0 0
\(969\) −57.2118 + 37.3544i −1.83791 + 1.20000i
\(970\) 0 0
\(971\) 15.5296 8.96600i 0.498367 0.287733i −0.229672 0.973268i \(-0.573765\pi\)
0.728039 + 0.685536i \(0.240432\pi\)
\(972\) 0 0
\(973\) 5.13844 + 2.96668i 0.164731 + 0.0951074i
\(974\) 0 0
\(975\) 2.89316 + 1.67036i 0.0926551 + 0.0534945i
\(976\) 0 0
\(977\) 40.5841i 1.29840i 0.760618 + 0.649200i \(0.224896\pi\)
−0.760618 + 0.649200i \(0.775104\pi\)
\(978\) 0 0
\(979\) −19.9015 11.4902i −0.636056 0.367227i
\(980\) 0 0
\(981\) −30.0453 −0.959274
\(982\) 0 0
\(983\) 7.93699 + 13.7473i 0.253151 + 0.438470i 0.964392 0.264479i \(-0.0851998\pi\)
−0.711241 + 0.702948i \(0.751866\pi\)
\(984\) 0 0
\(985\) 8.43936 + 14.6174i 0.268900 + 0.465749i
\(986\) 0 0
\(987\) 76.8327i 2.44561i
\(988\) 0 0
\(989\) 10.7467i 0.341725i
\(990\) 0 0
\(991\) −7.06529 12.2374i −0.224436 0.388735i 0.731714 0.681612i \(-0.238721\pi\)
−0.956150 + 0.292877i \(0.905387\pi\)
\(992\) 0 0
\(993\) 3.12444 + 5.41169i 0.0991511 + 0.171735i
\(994\) 0 0
\(995\) −22.6935 −0.719433
\(996\) 0 0
\(997\) 4.24947 + 2.45344i 0.134582 + 0.0777011i 0.565779 0.824557i \(-0.308575\pi\)
−0.431197 + 0.902258i \(0.641909\pi\)
\(998\) 0 0
\(999\) 33.4608i 1.05865i
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.31.1 32
4.3 odd 2 inner 1216.2.s.i.31.15 yes 32
8.3 odd 2 inner 1216.2.s.i.31.2 yes 32
8.5 even 2 inner 1216.2.s.i.31.16 yes 32
19.8 odd 6 inner 1216.2.s.i.863.2 yes 32
76.27 even 6 inner 1216.2.s.i.863.16 yes 32
152.27 even 6 inner 1216.2.s.i.863.1 yes 32
152.141 odd 6 inner 1216.2.s.i.863.15 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.s.i.31.1 32 1.1 even 1 trivial
1216.2.s.i.31.2 yes 32 8.3 odd 2 inner
1216.2.s.i.31.15 yes 32 4.3 odd 2 inner
1216.2.s.i.31.16 yes 32 8.5 even 2 inner
1216.2.s.i.863.1 yes 32 152.27 even 6 inner
1216.2.s.i.863.2 yes 32 19.8 odd 6 inner
1216.2.s.i.863.15 yes 32 152.141 odd 6 inner
1216.2.s.i.863.16 yes 32 76.27 even 6 inner