Properties

Label 240.2.v.c.113.2
Level $240$
Weight $2$
Character 240.113
Analytic conductor $1.916$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,2,Mod(17,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 113.2
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 240.113
Dual form 240.2.v.c.17.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 - 1.00000i) q^{3} +(-1.00000 + 2.00000i) q^{5} +(2.41421 - 2.41421i) q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\) \(q+(1.41421 - 1.00000i) q^{3} +(-1.00000 + 2.00000i) q^{5} +(2.41421 - 2.41421i) q^{7} +(1.00000 - 2.82843i) q^{9} -0.828427i q^{11} +(3.82843 + 3.82843i) q^{13} +(0.585786 + 3.82843i) q^{15} +(-1.82843 - 1.82843i) q^{17} -0.828427i q^{19} +(1.00000 - 5.82843i) q^{21} +(-4.41421 + 4.41421i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(-1.41421 - 5.00000i) q^{27} +3.65685 q^{29} -5.65685 q^{31} +(-0.828427 - 1.17157i) q^{33} +(2.41421 + 7.24264i) q^{35} +(-5.82843 + 5.82843i) q^{37} +(9.24264 + 1.58579i) q^{39} +5.65685i q^{41} +(0.414214 + 0.414214i) q^{43} +(4.65685 + 4.82843i) q^{45} +(-3.58579 - 3.58579i) q^{47} -4.65685i q^{49} +(-4.41421 - 0.757359i) q^{51} +(-3.00000 + 3.00000i) q^{53} +(1.65685 + 0.828427i) q^{55} +(-0.828427 - 1.17157i) q^{57} +4.00000 q^{59} +0.343146 q^{61} +(-4.41421 - 9.24264i) q^{63} +(-11.4853 + 3.82843i) q^{65} +(-10.0711 + 10.0711i) q^{67} +(-1.82843 + 10.6569i) q^{69} -10.4853i q^{71} +(-4.65685 - 4.65685i) q^{73} +(-8.24264 - 2.65685i) q^{75} +(-2.00000 - 2.00000i) q^{77} +0.828427i q^{79} +(-7.00000 - 5.65685i) q^{81} +(-3.24264 + 3.24264i) q^{83} +(5.48528 - 1.82843i) q^{85} +(5.17157 - 3.65685i) q^{87} +15.6569 q^{89} +18.4853 q^{91} +(-8.00000 + 5.65685i) q^{93} +(1.65685 + 0.828427i) q^{95} +(1.00000 - 1.00000i) q^{97} +(-2.34315 - 0.828427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{13} + 8 q^{15} + 4 q^{17} + 4 q^{21} - 12 q^{23} - 12 q^{25} - 8 q^{29} + 8 q^{33} + 4 q^{35} - 12 q^{37} + 20 q^{39} - 4 q^{43} - 4 q^{45} - 20 q^{47} - 12 q^{51} - 12 q^{53} - 16 q^{55} + 8 q^{57} + 16 q^{59} + 24 q^{61} - 12 q^{63} - 12 q^{65} - 12 q^{67} + 4 q^{69} + 4 q^{73} - 16 q^{75} - 8 q^{77} - 28 q^{81} + 4 q^{83} - 12 q^{85} + 32 q^{87} + 40 q^{89} + 40 q^{91} - 32 q^{93} - 16 q^{95} + 4 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.41421 1.00000i 0.816497 0.577350i
\(4\) 0 0
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 0 0
\(7\) 2.41421 2.41421i 0.912487 0.912487i −0.0839804 0.996467i \(-0.526763\pi\)
0.996467 + 0.0839804i \(0.0267633\pi\)
\(8\) 0 0
\(9\) 1.00000 2.82843i 0.333333 0.942809i
\(10\) 0 0
\(11\) 0.828427i 0.249780i −0.992171 0.124890i \(-0.960142\pi\)
0.992171 0.124890i \(-0.0398578\pi\)
\(12\) 0 0
\(13\) 3.82843 + 3.82843i 1.06181 + 1.06181i 0.997959 + 0.0638555i \(0.0203397\pi\)
0.0638555 + 0.997959i \(0.479660\pi\)
\(14\) 0 0
\(15\) 0.585786 + 3.82843i 0.151249 + 0.988496i
\(16\) 0 0
\(17\) −1.82843 1.82843i −0.443459 0.443459i 0.449714 0.893173i \(-0.351526\pi\)
−0.893173 + 0.449714i \(0.851526\pi\)
\(18\) 0 0
\(19\) 0.828427i 0.190054i −0.995475 0.0950271i \(-0.969706\pi\)
0.995475 0.0950271i \(-0.0302938\pi\)
\(20\) 0 0
\(21\) 1.00000 5.82843i 0.218218 1.27187i
\(22\) 0 0
\(23\) −4.41421 + 4.41421i −0.920427 + 0.920427i −0.997059 0.0766323i \(-0.975583\pi\)
0.0766323 + 0.997059i \(0.475583\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) −1.41421 5.00000i −0.272166 0.962250i
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) −0.828427 1.17157i −0.144211 0.203945i
\(34\) 0 0
\(35\) 2.41421 + 7.24264i 0.408077 + 1.22423i
\(36\) 0 0
\(37\) −5.82843 + 5.82843i −0.958188 + 0.958188i −0.999160 0.0409727i \(-0.986954\pi\)
0.0409727 + 0.999160i \(0.486954\pi\)
\(38\) 0 0
\(39\) 9.24264 + 1.58579i 1.48001 + 0.253929i
\(40\) 0 0
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) 0.414214 + 0.414214i 0.0631670 + 0.0631670i 0.737985 0.674818i \(-0.235778\pi\)
−0.674818 + 0.737985i \(0.735778\pi\)
\(44\) 0 0
\(45\) 4.65685 + 4.82843i 0.694203 + 0.719779i
\(46\) 0 0
\(47\) −3.58579 3.58579i −0.523041 0.523041i 0.395448 0.918488i \(-0.370589\pi\)
−0.918488 + 0.395448i \(0.870589\pi\)
\(48\) 0 0
\(49\) 4.65685i 0.665265i
\(50\) 0 0
\(51\) −4.41421 0.757359i −0.618114 0.106052i
\(52\) 0 0
\(53\) −3.00000 + 3.00000i −0.412082 + 0.412082i −0.882463 0.470381i \(-0.844116\pi\)
0.470381 + 0.882463i \(0.344116\pi\)
\(54\) 0 0
\(55\) 1.65685 + 0.828427i 0.223410 + 0.111705i
\(56\) 0 0
\(57\) −0.828427 1.17157i −0.109728 0.155179i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 0.343146 0.0439353 0.0219677 0.999759i \(-0.493007\pi\)
0.0219677 + 0.999759i \(0.493007\pi\)
\(62\) 0 0
\(63\) −4.41421 9.24264i −0.556139 1.16446i
\(64\) 0 0
\(65\) −11.4853 + 3.82843i −1.42457 + 0.474858i
\(66\) 0 0
\(67\) −10.0711 + 10.0711i −1.23038 + 1.23038i −0.266558 + 0.963819i \(0.585886\pi\)
−0.963819 + 0.266558i \(0.914114\pi\)
\(68\) 0 0
\(69\) −1.82843 + 10.6569i −0.220117 + 1.28293i
\(70\) 0 0
\(71\) 10.4853i 1.24437i −0.782869 0.622187i \(-0.786244\pi\)
0.782869 0.622187i \(-0.213756\pi\)
\(72\) 0 0
\(73\) −4.65685 4.65685i −0.545044 0.545044i 0.379960 0.925003i \(-0.375938\pi\)
−0.925003 + 0.379960i \(0.875938\pi\)
\(74\) 0 0
\(75\) −8.24264 2.65685i −0.951778 0.306787i
\(76\) 0 0
\(77\) −2.00000 2.00000i −0.227921 0.227921i
\(78\) 0 0
\(79\) 0.828427i 0.0932053i 0.998914 + 0.0466027i \(0.0148395\pi\)
−0.998914 + 0.0466027i \(0.985161\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) −3.24264 + 3.24264i −0.355926 + 0.355926i −0.862309 0.506383i \(-0.830982\pi\)
0.506383 + 0.862309i \(0.330982\pi\)
\(84\) 0 0
\(85\) 5.48528 1.82843i 0.594962 0.198321i
\(86\) 0 0
\(87\) 5.17157 3.65685i 0.554451 0.392056i
\(88\) 0 0
\(89\) 15.6569 1.65962 0.829812 0.558044i \(-0.188448\pi\)
0.829812 + 0.558044i \(0.188448\pi\)
\(90\) 0 0
\(91\) 18.4853 1.93778
\(92\) 0 0
\(93\) −8.00000 + 5.65685i −0.829561 + 0.586588i
\(94\) 0 0
\(95\) 1.65685 + 0.828427i 0.169990 + 0.0849948i
\(96\) 0 0
\(97\) 1.00000 1.00000i 0.101535 0.101535i −0.654515 0.756049i \(-0.727127\pi\)
0.756049 + 0.654515i \(0.227127\pi\)
\(98\) 0 0
\(99\) −2.34315 0.828427i −0.235495 0.0832601i
\(100\) 0 0
\(101\) 9.65685i 0.960893i −0.877024 0.480446i \(-0.840475\pi\)
0.877024 0.480446i \(-0.159525\pi\)
\(102\) 0 0
\(103\) 5.58579 + 5.58579i 0.550384 + 0.550384i 0.926552 0.376168i \(-0.122758\pi\)
−0.376168 + 0.926552i \(0.622758\pi\)
\(104\) 0 0
\(105\) 10.6569 + 7.82843i 1.04000 + 0.763976i
\(106\) 0 0
\(107\) 9.58579 + 9.58579i 0.926693 + 0.926693i 0.997491 0.0707977i \(-0.0225545\pi\)
−0.0707977 + 0.997491i \(0.522554\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i −0.981480 0.191565i \(-0.938644\pi\)
0.981480 0.191565i \(-0.0613564\pi\)
\(110\) 0 0
\(111\) −2.41421 + 14.0711i −0.229147 + 1.33557i
\(112\) 0 0
\(113\) 9.48528 9.48528i 0.892300 0.892300i −0.102439 0.994739i \(-0.532665\pi\)
0.994739 + 0.102439i \(0.0326647\pi\)
\(114\) 0 0
\(115\) −4.41421 13.2426i −0.411628 1.23488i
\(116\) 0 0
\(117\) 14.6569 7.00000i 1.35503 0.647150i
\(118\) 0 0
\(119\) −8.82843 −0.809301
\(120\) 0 0
\(121\) 10.3137 0.937610
\(122\) 0 0
\(123\) 5.65685 + 8.00000i 0.510061 + 0.721336i
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) −5.58579 + 5.58579i −0.495658 + 0.495658i −0.910083 0.414425i \(-0.863983\pi\)
0.414425 + 0.910083i \(0.363983\pi\)
\(128\) 0 0
\(129\) 1.00000 + 0.171573i 0.0880451 + 0.0151061i
\(130\) 0 0
\(131\) 8.82843i 0.771343i −0.922636 0.385672i \(-0.873970\pi\)
0.922636 0.385672i \(-0.126030\pi\)
\(132\) 0 0
\(133\) −2.00000 2.00000i −0.173422 0.173422i
\(134\) 0 0
\(135\) 11.4142 + 2.17157i 0.982379 + 0.186899i
\(136\) 0 0
\(137\) −9.82843 9.82843i −0.839699 0.839699i 0.149120 0.988819i \(-0.452356\pi\)
−0.988819 + 0.149120i \(0.952356\pi\)
\(138\) 0 0
\(139\) 8.82843i 0.748817i −0.927264 0.374409i \(-0.877846\pi\)
0.927264 0.374409i \(-0.122154\pi\)
\(140\) 0 0
\(141\) −8.65685 1.48528i −0.729039 0.125083i
\(142\) 0 0
\(143\) 3.17157 3.17157i 0.265220 0.265220i
\(144\) 0 0
\(145\) −3.65685 + 7.31371i −0.303685 + 0.607370i
\(146\) 0 0
\(147\) −4.65685 6.58579i −0.384091 0.543187i
\(148\) 0 0
\(149\) −13.3137 −1.09070 −0.545351 0.838208i \(-0.683604\pi\)
−0.545351 + 0.838208i \(0.683604\pi\)
\(150\) 0 0
\(151\) 13.6569 1.11138 0.555690 0.831390i \(-0.312454\pi\)
0.555690 + 0.831390i \(0.312454\pi\)
\(152\) 0 0
\(153\) −7.00000 + 3.34315i −0.565916 + 0.270277i
\(154\) 0 0
\(155\) 5.65685 11.3137i 0.454369 0.908739i
\(156\) 0 0
\(157\) 5.48528 5.48528i 0.437773 0.437773i −0.453489 0.891262i \(-0.649821\pi\)
0.891262 + 0.453489i \(0.149821\pi\)
\(158\) 0 0
\(159\) −1.24264 + 7.24264i −0.0985478 + 0.574379i
\(160\) 0 0
\(161\) 21.3137i 1.67976i
\(162\) 0 0
\(163\) 0.414214 + 0.414214i 0.0324437 + 0.0324437i 0.723143 0.690699i \(-0.242697\pi\)
−0.690699 + 0.723143i \(0.742697\pi\)
\(164\) 0 0
\(165\) 3.17157 0.485281i 0.246907 0.0377791i
\(166\) 0 0
\(167\) −9.24264 9.24264i −0.715217 0.715217i 0.252405 0.967622i \(-0.418779\pi\)
−0.967622 + 0.252405i \(0.918779\pi\)
\(168\) 0 0
\(169\) 16.3137i 1.25490i
\(170\) 0 0
\(171\) −2.34315 0.828427i −0.179185 0.0633514i
\(172\) 0 0
\(173\) −0.656854 + 0.656854i −0.0499397 + 0.0499397i −0.731636 0.681696i \(-0.761243\pi\)
0.681696 + 0.731636i \(0.261243\pi\)
\(174\) 0 0
\(175\) −16.8995 2.41421i −1.27748 0.182497i
\(176\) 0 0
\(177\) 5.65685 4.00000i 0.425195 0.300658i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −5.31371 −0.394965 −0.197482 0.980306i \(-0.563277\pi\)
−0.197482 + 0.980306i \(0.563277\pi\)
\(182\) 0 0
\(183\) 0.485281 0.343146i 0.0358730 0.0253661i
\(184\) 0 0
\(185\) −5.82843 17.4853i −0.428514 1.28554i
\(186\) 0 0
\(187\) −1.51472 + 1.51472i −0.110767 + 0.110767i
\(188\) 0 0
\(189\) −15.4853 8.65685i −1.12639 0.629693i
\(190\) 0 0
\(191\) 4.14214i 0.299714i 0.988708 + 0.149857i \(0.0478814\pi\)
−0.988708 + 0.149857i \(0.952119\pi\)
\(192\) 0 0
\(193\) 14.6569 + 14.6569i 1.05502 + 1.05502i 0.998395 + 0.0566281i \(0.0180349\pi\)
0.0566281 + 0.998395i \(0.481965\pi\)
\(194\) 0 0
\(195\) −12.4142 + 16.8995i −0.889000 + 1.21020i
\(196\) 0 0
\(197\) 14.6569 + 14.6569i 1.04426 + 1.04426i 0.998974 + 0.0452834i \(0.0144191\pi\)
0.0452834 + 0.998974i \(0.485581\pi\)
\(198\) 0 0
\(199\) 18.4853i 1.31039i −0.755461 0.655193i \(-0.772587\pi\)
0.755461 0.655193i \(-0.227413\pi\)
\(200\) 0 0
\(201\) −4.17157 + 24.3137i −0.294240 + 1.71496i
\(202\) 0 0
\(203\) 8.82843 8.82843i 0.619634 0.619634i
\(204\) 0 0
\(205\) −11.3137 5.65685i −0.790184 0.395092i
\(206\) 0 0
\(207\) 8.07107 + 16.8995i 0.560978 + 1.17460i
\(208\) 0 0
\(209\) −0.686292 −0.0474718
\(210\) 0 0
\(211\) −20.9706 −1.44367 −0.721837 0.692064i \(-0.756702\pi\)
−0.721837 + 0.692064i \(0.756702\pi\)
\(212\) 0 0
\(213\) −10.4853 14.8284i −0.718440 1.01603i
\(214\) 0 0
\(215\) −1.24264 + 0.414214i −0.0847474 + 0.0282491i
\(216\) 0 0
\(217\) −13.6569 + 13.6569i −0.927088 + 0.927088i
\(218\) 0 0
\(219\) −11.2426 1.92893i −0.759707 0.130345i
\(220\) 0 0
\(221\) 14.0000i 0.941742i
\(222\) 0 0
\(223\) 5.58579 + 5.58579i 0.374052 + 0.374052i 0.868951 0.494899i \(-0.164795\pi\)
−0.494899 + 0.868951i \(0.664795\pi\)
\(224\) 0 0
\(225\) −14.3137 + 4.48528i −0.954247 + 0.299019i
\(226\) 0 0
\(227\) 4.89949 + 4.89949i 0.325191 + 0.325191i 0.850754 0.525563i \(-0.176145\pi\)
−0.525563 + 0.850754i \(0.676145\pi\)
\(228\) 0 0
\(229\) 14.3431i 0.947822i −0.880573 0.473911i \(-0.842842\pi\)
0.880573 0.473911i \(-0.157158\pi\)
\(230\) 0 0
\(231\) −4.82843 0.828427i −0.317687 0.0545065i
\(232\) 0 0
\(233\) 11.8284 11.8284i 0.774906 0.774906i −0.204054 0.978960i \(-0.565412\pi\)
0.978960 + 0.204054i \(0.0654117\pi\)
\(234\) 0 0
\(235\) 10.7574 3.58579i 0.701733 0.233911i
\(236\) 0 0
\(237\) 0.828427 + 1.17157i 0.0538121 + 0.0761018i
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −0.343146 −0.0221040 −0.0110520 0.999939i \(-0.503518\pi\)
−0.0110520 + 0.999939i \(0.503518\pi\)
\(242\) 0 0
\(243\) −15.5563 1.00000i −0.997940 0.0641500i
\(244\) 0 0
\(245\) 9.31371 + 4.65685i 0.595031 + 0.297516i
\(246\) 0 0
\(247\) 3.17157 3.17157i 0.201802 0.201802i
\(248\) 0 0
\(249\) −1.34315 + 7.82843i −0.0851184 + 0.496106i
\(250\) 0 0
\(251\) 26.4853i 1.67174i 0.548930 + 0.835868i \(0.315035\pi\)
−0.548930 + 0.835868i \(0.684965\pi\)
\(252\) 0 0
\(253\) 3.65685 + 3.65685i 0.229904 + 0.229904i
\(254\) 0 0
\(255\) 5.92893 8.07107i 0.371284 0.505430i
\(256\) 0 0
\(257\) 9.48528 + 9.48528i 0.591676 + 0.591676i 0.938084 0.346408i \(-0.112599\pi\)
−0.346408 + 0.938084i \(0.612599\pi\)
\(258\) 0 0
\(259\) 28.1421i 1.74867i
\(260\) 0 0
\(261\) 3.65685 10.3431i 0.226354 0.640225i
\(262\) 0 0
\(263\) 6.89949 6.89949i 0.425441 0.425441i −0.461631 0.887072i \(-0.652736\pi\)
0.887072 + 0.461631i \(0.152736\pi\)
\(264\) 0 0
\(265\) −3.00000 9.00000i −0.184289 0.552866i
\(266\) 0 0
\(267\) 22.1421 15.6569i 1.35508 0.958184i
\(268\) 0 0
\(269\) −16.6274 −1.01379 −0.506896 0.862007i \(-0.669207\pi\)
−0.506896 + 0.862007i \(0.669207\pi\)
\(270\) 0 0
\(271\) −10.3431 −0.628301 −0.314151 0.949373i \(-0.601720\pi\)
−0.314151 + 0.949373i \(0.601720\pi\)
\(272\) 0 0
\(273\) 26.1421 18.4853i 1.58219 1.11878i
\(274\) 0 0
\(275\) −3.31371 + 2.48528i −0.199824 + 0.149868i
\(276\) 0 0
\(277\) −8.17157 + 8.17157i −0.490982 + 0.490982i −0.908616 0.417633i \(-0.862860\pi\)
0.417633 + 0.908616i \(0.362860\pi\)
\(278\) 0 0
\(279\) −5.65685 + 16.0000i −0.338667 + 0.957895i
\(280\) 0 0
\(281\) 5.65685i 0.337460i −0.985662 0.168730i \(-0.946033\pi\)
0.985662 0.168730i \(-0.0539665\pi\)
\(282\) 0 0
\(283\) −5.24264 5.24264i −0.311643 0.311643i 0.533903 0.845546i \(-0.320725\pi\)
−0.845546 + 0.533903i \(0.820725\pi\)
\(284\) 0 0
\(285\) 3.17157 0.485281i 0.187868 0.0287456i
\(286\) 0 0
\(287\) 13.6569 + 13.6569i 0.806139 + 0.806139i
\(288\) 0 0
\(289\) 10.3137i 0.606689i
\(290\) 0 0
\(291\) 0.414214 2.41421i 0.0242816 0.141524i
\(292\) 0 0
\(293\) −16.6569 + 16.6569i −0.973104 + 0.973104i −0.999648 0.0265438i \(-0.991550\pi\)
0.0265438 + 0.999648i \(0.491550\pi\)
\(294\) 0 0
\(295\) −4.00000 + 8.00000i −0.232889 + 0.465778i
\(296\) 0 0
\(297\) −4.14214 + 1.17157i −0.240351 + 0.0679816i
\(298\) 0 0
\(299\) −33.7990 −1.95465
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) −9.65685 13.6569i −0.554772 0.784566i
\(304\) 0 0
\(305\) −0.343146 + 0.686292i −0.0196485 + 0.0392969i
\(306\) 0 0
\(307\) 6.89949 6.89949i 0.393775 0.393775i −0.482256 0.876031i \(-0.660182\pi\)
0.876031 + 0.482256i \(0.160182\pi\)
\(308\) 0 0
\(309\) 13.4853 + 2.31371i 0.767151 + 0.131622i
\(310\) 0 0
\(311\) 5.51472i 0.312711i 0.987701 + 0.156356i \(0.0499746\pi\)
−0.987701 + 0.156356i \(0.950025\pi\)
\(312\) 0 0
\(313\) −2.31371 2.31371i −0.130779 0.130779i 0.638688 0.769466i \(-0.279478\pi\)
−0.769466 + 0.638688i \(0.779478\pi\)
\(314\) 0 0
\(315\) 22.8995 + 0.414214i 1.29024 + 0.0233383i
\(316\) 0 0
\(317\) −4.65685 4.65685i −0.261555 0.261555i 0.564131 0.825686i \(-0.309211\pi\)
−0.825686 + 0.564131i \(0.809211\pi\)
\(318\) 0 0
\(319\) 3.02944i 0.169616i
\(320\) 0 0
\(321\) 23.1421 + 3.97056i 1.29167 + 0.221615i
\(322\) 0 0
\(323\) −1.51472 + 1.51472i −0.0842812 + 0.0842812i
\(324\) 0 0
\(325\) 3.82843 26.7990i 0.212363 1.48654i
\(326\) 0 0
\(327\) −4.00000 5.65685i −0.221201 0.312825i
\(328\) 0 0
\(329\) −17.3137 −0.954536
\(330\) 0 0
\(331\) 9.65685 0.530789 0.265394 0.964140i \(-0.414498\pi\)
0.265394 + 0.964140i \(0.414498\pi\)
\(332\) 0 0
\(333\) 10.6569 + 22.3137i 0.583992 + 1.22278i
\(334\) 0 0
\(335\) −10.0711 30.2132i −0.550241 1.65072i
\(336\) 0 0
\(337\) 1.00000 1.00000i 0.0544735 0.0544735i −0.679345 0.733819i \(-0.737736\pi\)
0.733819 + 0.679345i \(0.237736\pi\)
\(338\) 0 0
\(339\) 3.92893 22.8995i 0.213390 1.24373i
\(340\) 0 0
\(341\) 4.68629i 0.253777i
\(342\) 0 0
\(343\) 5.65685 + 5.65685i 0.305441 + 0.305441i
\(344\) 0 0
\(345\) −19.4853 14.3137i −1.04905 0.770624i
\(346\) 0 0
\(347\) −12.0711 12.0711i −0.648009 0.648009i 0.304503 0.952511i \(-0.401510\pi\)
−0.952511 + 0.304503i \(0.901510\pi\)
\(348\) 0 0
\(349\) 9.65685i 0.516920i 0.966022 + 0.258460i \(0.0832149\pi\)
−0.966022 + 0.258460i \(0.916785\pi\)
\(350\) 0 0
\(351\) 13.7279 24.5563i 0.732742 1.31072i
\(352\) 0 0
\(353\) −15.4853 + 15.4853i −0.824198 + 0.824198i −0.986707 0.162509i \(-0.948041\pi\)
0.162509 + 0.986707i \(0.448041\pi\)
\(354\) 0 0
\(355\) 20.9706 + 10.4853i 1.11300 + 0.556501i
\(356\) 0 0
\(357\) −12.4853 + 8.82843i −0.660791 + 0.467250i
\(358\) 0 0
\(359\) 35.3137 1.86379 0.931893 0.362733i \(-0.118156\pi\)
0.931893 + 0.362733i \(0.118156\pi\)
\(360\) 0 0
\(361\) 18.3137 0.963879
\(362\) 0 0
\(363\) 14.5858 10.3137i 0.765555 0.541329i
\(364\) 0 0
\(365\) 13.9706 4.65685i 0.731253 0.243751i
\(366\) 0 0
\(367\) 4.75736 4.75736i 0.248332 0.248332i −0.571954 0.820286i \(-0.693814\pi\)
0.820286 + 0.571954i \(0.193814\pi\)
\(368\) 0 0
\(369\) 16.0000 + 5.65685i 0.832927 + 0.294484i
\(370\) 0 0
\(371\) 14.4853i 0.752038i
\(372\) 0 0
\(373\) 0.514719 + 0.514719i 0.0266511 + 0.0266511i 0.720307 0.693656i \(-0.244001\pi\)
−0.693656 + 0.720307i \(0.744001\pi\)
\(374\) 0 0
\(375\) 13.5563 13.8284i 0.700047 0.714097i
\(376\) 0 0
\(377\) 14.0000 + 14.0000i 0.721037 + 0.721037i
\(378\) 0 0
\(379\) 29.7990i 1.53067i 0.643631 + 0.765336i \(0.277427\pi\)
−0.643631 + 0.765336i \(0.722573\pi\)
\(380\) 0 0
\(381\) −2.31371 + 13.4853i −0.118535 + 0.690872i
\(382\) 0 0
\(383\) −12.4142 + 12.4142i −0.634337 + 0.634337i −0.949153 0.314816i \(-0.898057\pi\)
0.314816 + 0.949153i \(0.398057\pi\)
\(384\) 0 0
\(385\) 6.00000 2.00000i 0.305788 0.101929i
\(386\) 0 0
\(387\) 1.58579 0.757359i 0.0806101 0.0384987i
\(388\) 0 0
\(389\) −6.68629 −0.339008 −0.169504 0.985529i \(-0.554217\pi\)
−0.169504 + 0.985529i \(0.554217\pi\)
\(390\) 0 0
\(391\) 16.1421 0.816343
\(392\) 0 0
\(393\) −8.82843 12.4853i −0.445335 0.629799i
\(394\) 0 0
\(395\) −1.65685 0.828427i −0.0833654 0.0416827i
\(396\) 0 0
\(397\) 7.82843 7.82843i 0.392897 0.392897i −0.482821 0.875719i \(-0.660388\pi\)
0.875719 + 0.482821i \(0.160388\pi\)
\(398\) 0 0
\(399\) −4.82843 0.828427i −0.241724 0.0414732i
\(400\) 0 0
\(401\) 16.0000i 0.799002i 0.916733 + 0.399501i \(0.130817\pi\)
−0.916733 + 0.399501i \(0.869183\pi\)
\(402\) 0 0
\(403\) −21.6569 21.6569i −1.07880 1.07880i
\(404\) 0 0
\(405\) 18.3137 8.34315i 0.910015 0.414574i
\(406\) 0 0
\(407\) 4.82843 + 4.82843i 0.239336 + 0.239336i
\(408\) 0 0
\(409\) 21.6569i 1.07086i −0.844579 0.535431i \(-0.820149\pi\)
0.844579 0.535431i \(-0.179851\pi\)
\(410\) 0 0
\(411\) −23.7279 4.07107i −1.17041 0.200811i
\(412\) 0 0
\(413\) 9.65685 9.65685i 0.475183 0.475183i
\(414\) 0 0
\(415\) −3.24264 9.72792i −0.159175 0.477525i
\(416\) 0 0
\(417\) −8.82843 12.4853i −0.432330 0.611407i
\(418\) 0 0
\(419\) 29.9411 1.46272 0.731360 0.681992i \(-0.238886\pi\)
0.731360 + 0.681992i \(0.238886\pi\)
\(420\) 0 0
\(421\) 30.9706 1.50941 0.754706 0.656063i \(-0.227779\pi\)
0.754706 + 0.656063i \(0.227779\pi\)
\(422\) 0 0
\(423\) −13.7279 + 6.55635i −0.667474 + 0.318781i
\(424\) 0 0
\(425\) −1.82843 + 12.7990i −0.0886917 + 0.620842i
\(426\) 0 0
\(427\) 0.828427 0.828427i 0.0400904 0.0400904i
\(428\) 0 0
\(429\) 1.31371 7.65685i 0.0634264 0.369676i
\(430\) 0 0
\(431\) 20.1421i 0.970213i 0.874455 + 0.485106i \(0.161219\pi\)
−0.874455 + 0.485106i \(0.838781\pi\)
\(432\) 0 0
\(433\) −15.0000 15.0000i −0.720854 0.720854i 0.247925 0.968779i \(-0.420251\pi\)
−0.968779 + 0.247925i \(0.920251\pi\)
\(434\) 0 0
\(435\) 2.14214 + 14.0000i 0.102708 + 0.671249i
\(436\) 0 0
\(437\) 3.65685 + 3.65685i 0.174931 + 0.174931i
\(438\) 0 0
\(439\) 13.7990i 0.658590i −0.944227 0.329295i \(-0.893189\pi\)
0.944227 0.329295i \(-0.106811\pi\)
\(440\) 0 0
\(441\) −13.1716 4.65685i −0.627218 0.221755i
\(442\) 0 0
\(443\) 0.0710678 0.0710678i 0.00337653 0.00337653i −0.705416 0.708793i \(-0.749240\pi\)
0.708793 + 0.705416i \(0.249240\pi\)
\(444\) 0 0
\(445\) −15.6569 + 31.3137i −0.742206 + 1.48441i
\(446\) 0 0
\(447\) −18.8284 + 13.3137i −0.890554 + 0.629717i
\(448\) 0 0
\(449\) −1.31371 −0.0619977 −0.0309989 0.999519i \(-0.509869\pi\)
−0.0309989 + 0.999519i \(0.509869\pi\)
\(450\) 0 0
\(451\) 4.68629 0.220669
\(452\) 0 0
\(453\) 19.3137 13.6569i 0.907437 0.641655i
\(454\) 0 0
\(455\) −18.4853 + 36.9706i −0.866603 + 1.73321i
\(456\) 0 0
\(457\) 1.00000 1.00000i 0.0467780 0.0467780i −0.683331 0.730109i \(-0.739469\pi\)
0.730109 + 0.683331i \(0.239469\pi\)
\(458\) 0 0
\(459\) −6.55635 + 11.7279i −0.306024 + 0.547413i
\(460\) 0 0
\(461\) 28.9706i 1.34929i −0.738141 0.674647i \(-0.764296\pi\)
0.738141 0.674647i \(-0.235704\pi\)
\(462\) 0 0
\(463\) 21.5858 + 21.5858i 1.00318 + 1.00318i 0.999995 + 0.00318163i \(0.00101275\pi\)
0.00318163 + 0.999995i \(0.498987\pi\)
\(464\) 0 0
\(465\) −3.31371 21.6569i −0.153670 1.00431i
\(466\) 0 0
\(467\) −23.3848 23.3848i −1.08212 1.08212i −0.996312 0.0858066i \(-0.972653\pi\)
−0.0858066 0.996312i \(-0.527347\pi\)
\(468\) 0 0
\(469\) 48.6274i 2.24541i
\(470\) 0 0
\(471\) 2.27208 13.2426i 0.104692 0.610189i
\(472\) 0 0
\(473\) 0.343146 0.343146i 0.0157779 0.0157779i
\(474\) 0 0
\(475\) −3.31371 + 2.48528i −0.152043 + 0.114033i
\(476\) 0 0
\(477\) 5.48528 + 11.4853i 0.251154 + 0.525875i
\(478\) 0 0
\(479\) −22.6274 −1.03387 −0.516937 0.856024i \(-0.672928\pi\)
−0.516937 + 0.856024i \(0.672928\pi\)
\(480\) 0 0
\(481\) −44.6274 −2.03484
\(482\) 0 0
\(483\) 21.3137 + 30.1421i 0.969807 + 1.37151i
\(484\) 0 0
\(485\) 1.00000 + 3.00000i 0.0454077 + 0.136223i
\(486\) 0 0
\(487\) −14.5563 + 14.5563i −0.659611 + 0.659611i −0.955288 0.295677i \(-0.904455\pi\)
0.295677 + 0.955288i \(0.404455\pi\)
\(488\) 0 0
\(489\) 1.00000 + 0.171573i 0.0452216 + 0.00775879i
\(490\) 0 0
\(491\) 21.5147i 0.970946i −0.874252 0.485473i \(-0.838647\pi\)
0.874252 0.485473i \(-0.161353\pi\)
\(492\) 0 0
\(493\) −6.68629 6.68629i −0.301135 0.301135i
\(494\) 0 0
\(495\) 4.00000 3.85786i 0.179787 0.173398i
\(496\) 0 0
\(497\) −25.3137 25.3137i −1.13548 1.13548i
\(498\) 0 0
\(499\) 34.7696i 1.55650i −0.627955 0.778249i \(-0.716108\pi\)
0.627955 0.778249i \(-0.283892\pi\)
\(500\) 0 0
\(501\) −22.3137 3.82843i −0.996903 0.171042i
\(502\) 0 0
\(503\) 5.92893 5.92893i 0.264358 0.264358i −0.562464 0.826822i \(-0.690146\pi\)
0.826822 + 0.562464i \(0.190146\pi\)
\(504\) 0 0
\(505\) 19.3137 + 9.65685i 0.859449 + 0.429724i
\(506\) 0 0
\(507\) 16.3137 + 23.0711i 0.724517 + 1.02462i
\(508\) 0 0
\(509\) 3.65685 0.162087 0.0810436 0.996711i \(-0.474175\pi\)
0.0810436 + 0.996711i \(0.474175\pi\)
\(510\) 0 0
\(511\) −22.4853 −0.994690
\(512\) 0 0
\(513\) −4.14214 + 1.17157i −0.182880 + 0.0517262i
\(514\) 0 0
\(515\) −16.7574 + 5.58579i −0.738417 + 0.246139i
\(516\) 0 0
\(517\) −2.97056 + 2.97056i −0.130645 + 0.130645i
\(518\) 0 0
\(519\) −0.272078 + 1.58579i −0.0119429 + 0.0696083i
\(520\) 0 0
\(521\) 24.0000i 1.05146i −0.850652 0.525730i \(-0.823792\pi\)
0.850652 0.525730i \(-0.176208\pi\)
\(522\) 0 0
\(523\) −26.8995 26.8995i −1.17623 1.17623i −0.980696 0.195536i \(-0.937355\pi\)
−0.195536 0.980696i \(-0.562645\pi\)
\(524\) 0 0
\(525\) −26.3137 + 13.4853i −1.14842 + 0.588546i
\(526\) 0 0
\(527\) 10.3431 + 10.3431i 0.450555 + 0.450555i
\(528\) 0 0
\(529\) 15.9706i 0.694372i
\(530\) 0 0
\(531\) 4.00000 11.3137i 0.173585 0.490973i
\(532\) 0 0
\(533\) −21.6569 + 21.6569i −0.938062 + 0.938062i
\(534\) 0 0
\(535\) −28.7574 + 9.58579i −1.24329 + 0.414430i
\(536\) 0 0
\(537\) 16.9706 12.0000i 0.732334 0.517838i
\(538\) 0 0
\(539\) −3.85786 −0.166170
\(540\) 0 0
\(541\) −29.3137 −1.26029 −0.630147 0.776476i \(-0.717006\pi\)
−0.630147 + 0.776476i \(0.717006\pi\)
\(542\) 0 0
\(543\) −7.51472 + 5.31371i −0.322487 + 0.228033i
\(544\) 0 0
\(545\) 8.00000 + 4.00000i 0.342682 + 0.171341i
\(546\) 0 0
\(547\) −15.7279 + 15.7279i −0.672477 + 0.672477i −0.958287 0.285809i \(-0.907738\pi\)
0.285809 + 0.958287i \(0.407738\pi\)
\(548\) 0 0
\(549\) 0.343146 0.970563i 0.0146451 0.0414226i
\(550\) 0 0
\(551\) 3.02944i 0.129058i
\(552\) 0 0
\(553\) 2.00000 + 2.00000i 0.0850487 + 0.0850487i
\(554\) 0 0
\(555\) −25.7279 18.8995i −1.09209 0.802239i
\(556\) 0 0
\(557\) 15.6274 + 15.6274i 0.662155 + 0.662155i 0.955888 0.293733i \(-0.0948976\pi\)
−0.293733 + 0.955888i \(0.594898\pi\)
\(558\) 0 0
\(559\) 3.17157i 0.134143i
\(560\) 0 0
\(561\) −0.627417 + 3.65685i −0.0264896 + 0.154393i
\(562\) 0 0
\(563\) 1.44365 1.44365i 0.0608426 0.0608426i −0.676031 0.736873i \(-0.736301\pi\)
0.736873 + 0.676031i \(0.236301\pi\)
\(564\) 0 0
\(565\) 9.48528 + 28.4558i 0.399049 + 1.19715i
\(566\) 0 0
\(567\) −30.5563 + 3.24264i −1.28325 + 0.136178i
\(568\) 0 0
\(569\) 45.3137 1.89965 0.949825 0.312783i \(-0.101261\pi\)
0.949825 + 0.312783i \(0.101261\pi\)
\(570\) 0 0
\(571\) 4.97056 0.208012 0.104006 0.994577i \(-0.466834\pi\)
0.104006 + 0.994577i \(0.466834\pi\)
\(572\) 0 0
\(573\) 4.14214 + 5.85786i 0.173040 + 0.244716i
\(574\) 0 0
\(575\) 30.8995 + 4.41421i 1.28860 + 0.184085i
\(576\) 0 0
\(577\) 14.6569 14.6569i 0.610173 0.610173i −0.332818 0.942991i \(-0.608000\pi\)
0.942991 + 0.332818i \(0.108000\pi\)
\(578\) 0 0
\(579\) 35.3848 + 6.07107i 1.47054 + 0.252305i
\(580\) 0 0
\(581\) 15.6569i 0.649556i
\(582\) 0 0
\(583\) 2.48528 + 2.48528i 0.102930 + 0.102930i
\(584\) 0 0
\(585\) −0.656854 + 36.3137i −0.0271576 + 1.50139i
\(586\) 0 0
\(587\) 10.5563 + 10.5563i 0.435707 + 0.435707i 0.890564 0.454857i \(-0.150310\pi\)
−0.454857 + 0.890564i \(0.650310\pi\)
\(588\) 0 0
\(589\) 4.68629i 0.193095i
\(590\) 0 0
\(591\) 35.3848 + 6.07107i 1.45554 + 0.249730i
\(592\) 0 0
\(593\) −15.4853 + 15.4853i −0.635904 + 0.635904i −0.949543 0.313638i \(-0.898452\pi\)
0.313638 + 0.949543i \(0.398452\pi\)
\(594\) 0 0
\(595\) 8.82843 17.6569i 0.361930 0.723860i
\(596\) 0 0
\(597\) −18.4853 26.1421i −0.756552 1.06993i
\(598\) 0 0
\(599\) 41.9411 1.71367 0.856834 0.515592i \(-0.172428\pi\)
0.856834 + 0.515592i \(0.172428\pi\)
\(600\) 0 0
\(601\) −14.9706 −0.610662 −0.305331 0.952246i \(-0.598767\pi\)
−0.305331 + 0.952246i \(0.598767\pi\)
\(602\) 0 0
\(603\) 18.4142 + 38.5563i 0.749885 + 1.57014i
\(604\) 0 0
\(605\) −10.3137 + 20.6274i −0.419312 + 0.838624i
\(606\) 0 0
\(607\) 33.0416 33.0416i 1.34112 1.34112i 0.446170 0.894948i \(-0.352788\pi\)
0.894948 0.446170i \(-0.147212\pi\)
\(608\) 0 0
\(609\) 3.65685 21.3137i 0.148183 0.863675i
\(610\) 0 0
\(611\) 27.4558i 1.11074i
\(612\) 0 0
\(613\) 9.48528 + 9.48528i 0.383107 + 0.383107i 0.872220 0.489113i \(-0.162680\pi\)
−0.489113 + 0.872220i \(0.662680\pi\)
\(614\) 0 0
\(615\) −21.6569 + 3.31371i −0.873289 + 0.133622i
\(616\) 0 0
\(617\) −12.1716 12.1716i −0.490009 0.490009i 0.418300 0.908309i \(-0.362626\pi\)
−0.908309 + 0.418300i \(0.862626\pi\)
\(618\) 0 0
\(619\) 20.1421i 0.809581i −0.914410 0.404790i \(-0.867344\pi\)
0.914410 0.404790i \(-0.132656\pi\)
\(620\) 0 0
\(621\) 28.3137 + 15.8284i 1.13619 + 0.635173i
\(622\) 0 0
\(623\) 37.7990 37.7990i 1.51438 1.51438i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) −0.970563 + 0.686292i −0.0387605 + 0.0274078i
\(628\) 0 0
\(629\) 21.3137 0.849833
\(630\) 0 0
\(631\) −31.5980 −1.25790 −0.628948 0.777447i \(-0.716514\pi\)
−0.628948 + 0.777447i \(0.716514\pi\)
\(632\) 0 0
\(633\) −29.6569 + 20.9706i −1.17875 + 0.833505i
\(634\) 0 0
\(635\) −5.58579 16.7574i −0.221665 0.664996i
\(636\) 0 0
\(637\) 17.8284 17.8284i 0.706388 0.706388i
\(638\) 0 0
\(639\) −29.6569 10.4853i −1.17321 0.414791i
\(640\) 0 0
\(641\) 20.2843i 0.801181i −0.916257 0.400590i \(-0.868805\pi\)
0.916257 0.400590i \(-0.131195\pi\)
\(642\) 0 0
\(643\) 28.6985 + 28.6985i 1.13176 + 1.13176i 0.989885 + 0.141873i \(0.0453124\pi\)
0.141873 + 0.989885i \(0.454688\pi\)
\(644\) 0 0
\(645\) −1.34315 + 1.82843i −0.0528863 + 0.0719942i
\(646\) 0 0
\(647\) −26.2132 26.2132i −1.03055 1.03055i −0.999518 0.0310289i \(-0.990122\pi\)
−0.0310289 0.999518i \(-0.509878\pi\)
\(648\) 0 0
\(649\) 3.31371i 0.130074i
\(650\) 0 0
\(651\) −5.65685 + 32.9706i −0.221710 + 1.29222i
\(652\) 0 0
\(653\) 26.6569 26.6569i 1.04316 1.04316i 0.0441379 0.999025i \(-0.485946\pi\)
0.999025 0.0441379i \(-0.0140541\pi\)
\(654\) 0 0
\(655\) 17.6569 + 8.82843i 0.689910 + 0.344955i
\(656\) 0 0
\(657\) −17.8284 + 8.51472i −0.695553 + 0.332191i
\(658\) 0 0
\(659\) −10.6274 −0.413985 −0.206993 0.978342i \(-0.566368\pi\)
−0.206993 + 0.978342i \(0.566368\pi\)
\(660\) 0 0
\(661\) −7.65685 −0.297817 −0.148909 0.988851i \(-0.547576\pi\)
−0.148909 + 0.988851i \(0.547576\pi\)
\(662\) 0 0
\(663\) −14.0000 19.7990i −0.543715 0.768929i
\(664\) 0 0
\(665\) 6.00000 2.00000i 0.232670 0.0775567i
\(666\) 0 0
\(667\) −16.1421 + 16.1421i −0.625026 + 0.625026i
\(668\) 0 0
\(669\) 13.4853 + 2.31371i 0.521371 + 0.0894531i
\(670\) 0 0
\(671\) 0.284271i 0.0109742i
\(672\) 0 0
\(673\) −3.68629 3.68629i −0.142096 0.142096i 0.632480 0.774576i \(-0.282037\pi\)
−0.774576 + 0.632480i \(0.782037\pi\)
\(674\) 0 0
\(675\) −15.7574 + 20.6569i −0.606501 + 0.795083i
\(676\) 0 0
\(677\) −35.2843 35.2843i −1.35608 1.35608i −0.878688 0.477397i \(-0.841580\pi\)
−0.477397 0.878688i \(-0.658420\pi\)
\(678\) 0 0
\(679\) 4.82843i 0.185298i
\(680\) 0 0
\(681\) 11.8284 + 2.02944i 0.453266 + 0.0777682i
\(682\) 0 0
\(683\) −22.5563 + 22.5563i −0.863095 + 0.863095i −0.991696 0.128602i \(-0.958951\pi\)
0.128602 + 0.991696i \(0.458951\pi\)
\(684\) 0 0
\(685\) 29.4853 9.82843i 1.12657 0.375525i
\(686\) 0 0
\(687\) −14.3431 20.2843i −0.547225 0.773893i
\(688\) 0 0
\(689\) −22.9706 −0.875109
\(690\) 0 0
\(691\) 33.6569 1.28037 0.640184 0.768222i \(-0.278858\pi\)
0.640184 + 0.768222i \(0.278858\pi\)
\(692\) 0 0
\(693\) −7.65685 + 3.65685i −0.290860 + 0.138912i
\(694\) 0 0
\(695\) 17.6569 + 8.82843i 0.669763 + 0.334881i
\(696\) 0 0
\(697\) 10.3431 10.3431i 0.391775 0.391775i
\(698\) 0 0
\(699\) 4.89949 28.5563i 0.185316 1.08010i
\(700\) 0 0
\(701\) 4.00000i 0.151078i −0.997143 0.0755390i \(-0.975932\pi\)
0.997143 0.0755390i \(-0.0240677\pi\)
\(702\) 0 0
\(703\) 4.82843 + 4.82843i 0.182108 + 0.182108i
\(704\) 0 0
\(705\) 11.6274 15.8284i 0.437914 0.596133i
\(706\) 0 0
\(707\) −23.3137 23.3137i −0.876802 0.876802i
\(708\) 0 0
\(709\) 32.2843i 1.21246i −0.795289 0.606231i \(-0.792681\pi\)
0.795289 0.606231i \(-0.207319\pi\)
\(710\) 0 0
\(711\) 2.34315 + 0.828427i 0.0878748 + 0.0310684i
\(712\) 0 0
\(713\) 24.9706 24.9706i 0.935155 0.935155i
\(714\) 0 0
\(715\) 3.17157 + 9.51472i 0.118610 + 0.355830i
\(716\) 0 0
\(717\) −22.6274 + 16.0000i −0.845036 + 0.597531i
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 26.9706 1.00444
\(722\) 0 0
\(723\) −0.485281 + 0.343146i −0.0180478 + 0.0127617i
\(724\) 0 0
\(725\) −10.9706 14.6274i −0.407436 0.543249i
\(726\) 0 0
\(727\) 12.7574 12.7574i 0.473144 0.473144i −0.429786 0.902931i \(-0.641411\pi\)
0.902931 + 0.429786i \(0.141411\pi\)
\(728\) 0 0
\(729\) −23.0000 + 14.1421i −0.851852 + 0.523783i
\(730\) 0 0
\(731\) 1.51472i 0.0560239i
\(732\) 0 0
\(733\) −5.14214 5.14214i −0.189929 0.189929i 0.605736 0.795665i \(-0.292879\pi\)
−0.795665 + 0.605736i \(0.792879\pi\)
\(734\) 0 0
\(735\) 17.8284 2.72792i 0.657611 0.100621i
\(736\) 0 0
\(737\) 8.34315 + 8.34315i 0.307324 + 0.307324i
\(738\) 0 0
\(739\) 47.1716i 1.73523i 0.497233 + 0.867617i \(0.334350\pi\)
−0.497233 + 0.867617i \(0.665650\pi\)
\(740\) 0 0
\(741\) 1.31371 7.65685i 0.0482603 0.281282i
\(742\) 0 0
\(743\) −37.3848 + 37.3848i −1.37151 + 1.37151i −0.513313 + 0.858202i \(0.671582\pi\)
−0.858202 + 0.513313i \(0.828418\pi\)
\(744\) 0 0
\(745\) 13.3137 26.6274i 0.487777 0.975553i
\(746\) 0 0
\(747\) 5.92893 + 12.4142i 0.216928 + 0.454212i
\(748\) 0 0
\(749\) 46.2843 1.69119
\(750\) 0 0
\(751\) 12.2843 0.448259 0.224130 0.974559i \(-0.428046\pi\)
0.224130 + 0.974559i \(0.428046\pi\)
\(752\) 0 0
\(753\) 26.4853 + 37.4558i 0.965177 + 1.36497i
\(754\) 0 0
\(755\) −13.6569 + 27.3137i −0.497024 + 0.994048i
\(756\) 0 0
\(757\) 14.4558 14.4558i 0.525407 0.525407i −0.393793 0.919199i \(-0.628837\pi\)
0.919199 + 0.393793i \(0.128837\pi\)
\(758\) 0 0
\(759\) 8.82843 + 1.51472i 0.320452 + 0.0549808i
\(760\) 0 0
\(761\) 12.6863i 0.459878i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738526\pi\)
\(762\) 0 0
\(763\) −9.65685 9.65685i −0.349602 0.349602i
\(764\) 0 0
\(765\) 0.313708 17.3431i 0.0113422 0.627043i
\(766\) 0 0
\(767\) 15.3137 + 15.3137i 0.552946 + 0.552946i
\(768\) 0 0
\(769\) 49.9411i 1.80092i 0.434936 + 0.900462i \(0.356771\pi\)
−0.434936 + 0.900462i \(0.643229\pi\)
\(770\) 0 0
\(771\) 22.8995 + 3.92893i 0.824705 + 0.141497i
\(772\) 0 0
\(773\) 10.6569 10.6569i 0.383300 0.383300i −0.488989 0.872290i \(-0.662634\pi\)
0.872290 + 0.488989i \(0.162634\pi\)
\(774\) 0 0
\(775\) 16.9706 + 22.6274i 0.609601 + 0.812801i
\(776\) 0 0
\(777\) 28.1421 + 39.7990i 1.00959 + 1.42778i
\(778\) 0 0
\(779\) 4.68629 0.167904
\(780\) 0 0
\(781\) −8.68629 −0.310820
\(782\) 0 0
\(783\) −5.17157 18.2843i −0.184817 0.653427i
\(784\) 0 0
\(785\) 5.48528 + 16.4558i 0.195778 + 0.587334i
\(786\) 0 0
\(787\) 27.5858 27.5858i 0.983327 0.983327i −0.0165362 0.999863i \(-0.505264\pi\)
0.999863 + 0.0165362i \(0.00526387\pi\)
\(788\) 0 0
\(789\) 2.85786 16.6569i 0.101743 0.593000i
\(790\) 0 0
\(791\) 45.7990i 1.62842i
\(792\) 0 0
\(793\) 1.31371 + 1.31371i 0.0466512 + 0.0466512i
\(794\) 0 0
\(795\) −13.2426 9.72792i −0.469668 0.345014i
\(796\) 0 0
\(797\) −4.65685 4.65685i −0.164954 0.164954i 0.619803 0.784757i \(-0.287212\pi\)
−0.784757 + 0.619803i \(0.787212\pi\)
\(798\) 0 0
\(799\) 13.1127i 0.463894i
\(800\) 0 0
\(801\) 15.6569 44.2843i 0.553208 1.56471i
\(802\) 0 0
\(803\) −3.85786 + 3.85786i −0.136141 + 0.136141i
\(804\) 0 0
\(805\) −42.6274 21.3137i −1.50242 0.751210i
\(806\) 0 0
\(807\) −23.5147 + 16.6274i −0.827757 + 0.585313i
\(808\) 0 0
\(809\) −32.3431 −1.13712 −0.568562 0.822640i \(-0.692500\pi\)
−0.568562 + 0.822640i \(0.692500\pi\)
\(810\) 0 0
\(811\) −33.6569 −1.18185 −0.590926 0.806726i \(-0.701237\pi\)
−0.590926 + 0.806726i \(0.701237\pi\)
\(812\) 0 0
\(813\) −14.6274 + 10.3431i −0.513006 + 0.362750i
\(814\) 0 0
\(815\) −1.24264 + 0.414214i −0.0435278 + 0.0145093i
\(816\) 0 0
\(817\) 0.343146 0.343146i 0.0120052 0.0120052i
\(818\) 0 0
\(819\) 18.4853 52.2843i 0.645928 1.82696i
\(820\) 0 0
\(821\) 37.9411i 1.32415i 0.749436 + 0.662077i \(0.230325\pi\)
−0.749436 + 0.662077i \(0.769675\pi\)
\(822\) 0 0
\(823\) −5.72792 5.72792i −0.199663 0.199663i 0.600193 0.799855i \(-0.295091\pi\)
−0.799855 + 0.600193i \(0.795091\pi\)
\(824\) 0 0
\(825\) −2.20101 + 6.82843i −0.0766293 + 0.237735i
\(826\) 0 0
\(827\) 27.5269 + 27.5269i 0.957205 + 0.957205i 0.999121 0.0419166i \(-0.0133464\pi\)
−0.0419166 + 0.999121i \(0.513346\pi\)
\(828\) 0 0
\(829\) 15.3137i 0.531867i −0.963991 0.265934i \(-0.914320\pi\)
0.963991 0.265934i \(-0.0856802\pi\)
\(830\) 0 0
\(831\) −3.38478 + 19.7279i −0.117417 + 0.684354i
\(832\) 0 0
\(833\) −8.51472 + 8.51472i −0.295018 + 0.295018i
\(834\) 0 0
\(835\) 27.7279 9.24264i 0.959564 0.319855i
\(836\) 0 0
\(837\) 8.00000 + 28.2843i 0.276520 + 0.977647i
\(838\) 0 0
\(839\) 12.6863 0.437979 0.218990 0.975727i \(-0.429724\pi\)
0.218990 + 0.975727i \(0.429724\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) −5.65685 8.00000i −0.194832 0.275535i
\(844\) 0 0
\(845\) −32.6274 16.3137i −1.12242 0.561209i
\(846\) 0 0
\(847\) 24.8995 24.8995i 0.855557 0.855557i
\(848\) 0 0
\(849\) −12.6569 2.17157i −0.434382 0.0745282i
\(850\) 0 0
\(851\) 51.4558i 1.76388i
\(852\) 0 0
\(853\) −24.4558 24.4558i −0.837352 0.837352i 0.151158 0.988510i \(-0.451700\pi\)
−0.988510 + 0.151158i \(0.951700\pi\)
\(854\) 0 0
\(855\) 4.00000 3.85786i 0.136797 0.131936i
\(856\) 0 0
\(857\) 33.4853 + 33.4853i 1.14384 + 1.14384i 0.987742 + 0.156093i \(0.0498900\pi\)
0.156093 + 0.987742i \(0.450110\pi\)
\(858\) 0 0
\(859\) 52.4264i 1.78877i 0.447302 + 0.894383i \(0.352385\pi\)
−0.447302 + 0.894383i \(0.647615\pi\)
\(860\) 0 0
\(861\) 32.9706 + 5.65685i 1.12363 + 0.192785i
\(862\) 0 0
\(863\) 3.58579 3.58579i 0.122062 0.122062i −0.643437 0.765499i \(-0.722492\pi\)
0.765499 + 0.643437i \(0.222492\pi\)
\(864\) 0 0
\(865\) −0.656854 1.97056i −0.0223337 0.0670011i
\(866\) 0 0
\(867\) −10.3137 14.5858i −0.350272 0.495359i
\(868\) 0 0
\(869\) 0.686292 0.0232808
\(870\) 0 0
\(871\) −77.1127 −2.61286
\(872\) 0 0
\(873\) −1.82843 3.82843i −0.0618829 0.129573i
\(874\) 0 0
\(875\) 21.7279 31.3848i 0.734538 1.06100i
\(876\) 0 0
\(877\) −12.4558 + 12.4558i −0.420604 + 0.420604i −0.885412 0.464808i \(-0.846123\pi\)
0.464808 + 0.885412i \(0.346123\pi\)
\(878\) 0 0
\(879\) −6.89949 + 40.2132i −0.232714 + 1.35636i
\(880\) 0 0
\(881\) 8.97056i 0.302226i 0.988516 + 0.151113i \(0.0482857\pi\)
−0.988516 + 0.151113i \(0.951714\pi\)
\(882\) 0 0
\(883\) −31.5858 31.5858i −1.06295 1.06295i −0.997881 0.0650653i \(-0.979274\pi\)
−0.0650653 0.997881i \(-0.520726\pi\)
\(884\) 0 0
\(885\) 2.34315 + 15.3137i 0.0787640 + 0.514765i
\(886\) 0 0
\(887\) 7.72792 + 7.72792i 0.259478 + 0.259478i 0.824842 0.565364i \(-0.191264\pi\)
−0.565364 + 0.824842i \(0.691264\pi\)
\(888\) 0 0
\(889\) 26.9706i 0.904564i
\(890\) 0 0
\(891\) −4.68629 + 5.79899i −0.156997 + 0.194273i
\(892\) 0 0
\(893\) −2.97056 + 2.97056i −0.0994061 + 0.0994061i
\(894\) 0 0
\(895\) −12.0000 + 24.0000i −0.401116 + 0.802232i
\(896\) 0 0
\(897\) −47.7990 + 33.7990i −1.59596 + 1.12852i
\(898\) 0 0
\(899\) −20.6863 −0.689926
\(900\) 0 0
\(901\) 10.9706 0.365482
\(902\) 0 0
\(903\) 2.82843 2.00000i 0.0941242 0.0665558i
\(904\) 0 0
\(905\) 5.31371 10.6274i 0.176634 0.353267i
\(906\) 0 0
\(907\) −28.4142 + 28.4142i −0.943478 + 0.943478i −0.998486 0.0550075i \(-0.982482\pi\)
0.0550075 + 0.998486i \(0.482482\pi\)
\(908\) 0 0
\(909\) −27.3137 9.65685i −0.905939 0.320298i
\(910\) 0 0
\(911\) 40.8284i 1.35271i 0.736578 + 0.676353i \(0.236441\pi\)
−0.736578 + 0.676353i \(0.763559\pi\)
\(912\) 0 0
\(913\) 2.68629 + 2.68629i 0.0889033 + 0.0889033i
\(914\) 0 0
\(915\) 0.201010 + 1.31371i 0.00664519 + 0.0434299i
\(916\) 0 0
\(917\) −21.3137 21.3137i −0.703841 0.703841i
\(918\) 0 0
\(919\) 8.82843i 0.291223i 0.989342 + 0.145611i \(0.0465149\pi\)
−0.989342 + 0.145611i \(0.953485\pi\)
\(920\) 0 0
\(921\) 2.85786 16.6569i 0.0941698 0.548862i
\(922\) 0 0
\(923\) 40.1421 40.1421i 1.32129 1.32129i
\(924\) 0 0
\(925\) 40.7990 + 5.82843i 1.34146 + 0.191638i
\(926\) 0 0
\(927\) 21.3848 10.2132i 0.702368 0.335446i
\(928\) 0 0
\(929\) 11.9411 0.391776 0.195888 0.980626i \(-0.437241\pi\)
0.195888 + 0.980626i \(0.437241\pi\)
\(930\) 0 0
\(931\) −3.85786 −0.126436
\(932\) 0 0
\(933\) 5.51472 + 7.79899i 0.180544 + 0.255327i
\(934\) 0 0
\(935\) −1.51472 4.54416i −0.0495366 0.148610i
\(936\) 0 0
\(937\) −31.0000 + 31.0000i −1.01273 + 1.01273i −0.0128079 + 0.999918i \(0.504077\pi\)
−0.999918 + 0.0128079i \(0.995923\pi\)
\(938\) 0 0
\(939\) −5.58579 0.958369i −0.182285 0.0312752i
\(940\) 0 0
\(941\) 43.5980i 1.42125i 0.703569 + 0.710627i \(0.251589\pi\)
−0.703569 + 0.710627i \(0.748411\pi\)
\(942\) 0 0
\(943\) −24.9706 24.9706i −0.813153 0.813153i
\(944\) 0 0
\(945\) 32.7990 22.3137i 1.06695 0.725865i
\(946\) 0 0
\(947\) −1.72792 1.72792i −0.0561499 0.0561499i 0.678474 0.734624i \(-0.262641\pi\)
−0.734624 + 0.678474i \(0.762641\pi\)
\(948\) 0 0
\(949\) 35.6569i 1.15747i
\(950\) 0 0
\(951\) −11.2426 1.92893i −0.364568 0.0625499i
\(952\) 0 0
\(953\) −15.4853 + 15.4853i −0.501617 + 0.501617i −0.911940 0.410323i \(-0.865416\pi\)
0.410323 + 0.911940i \(0.365416\pi\)
\(954\) 0 0
\(955\) −8.28427 4.14214i −0.268073 0.134036i
\(956\) 0 0
\(957\) −3.02944 4.28427i −0.0979278 0.138491i
\(958\) 0 0
\(959\) −47.4558 −1.53243
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 36.6985 17.5269i 1.18259 0.564797i
\(964\) 0 0
\(965\) −43.9706 + 14.6569i −1.41546 + 0.471821i
\(966\) 0 0
\(967\) 18.4142 18.4142i 0.592161 0.592161i −0.346054 0.938215i \(-0.612478\pi\)
0.938215 + 0.346054i \(0.112478\pi\)
\(968\) 0 0
\(969\) −0.627417 + 3.65685i −0.0201555 + 0.117475i
\(970\) 0 0
\(971\) 42.4853i 1.36342i 0.731624 + 0.681709i \(0.238763\pi\)
−0.731624 + 0.681709i \(0.761237\pi\)
\(972\) 0 0
\(973\) −21.3137 21.3137i −0.683286 0.683286i
\(974\) 0 0
\(975\) −21.3848 41.7279i −0.684861 1.33636i
\(976\) 0 0
\(977\) 23.1421 + 23.1421i 0.740383 + 0.740383i 0.972652 0.232269i \(-0.0746150\pi\)
−0.232269 + 0.972652i \(0.574615\pi\)
\(978\) 0 0
\(979\) 12.9706i 0.414541i
\(980\) 0 0
\(981\) −11.3137 4.00000i −0.361219 0.127710i
\(982\) 0 0
\(983\) −32.6985 + 32.6985i −1.04292 + 1.04292i −0.0438830 + 0.999037i \(0.513973\pi\)
−0.999037 + 0.0438830i \(0.986027\pi\)
\(984\) 0 0
\(985\) −43.9706 + 14.6569i −1.40102 + 0.467006i
\(986\) 0 0
\(987\) −24.4853 + 17.3137i −0.779375 + 0.551101i
\(988\) 0 0
\(989\) −3.65685 −0.116281
\(990\) 0 0
\(991\) 48.9706 1.55560 0.777801 0.628511i \(-0.216335\pi\)
0.777801 + 0.628511i \(0.216335\pi\)
\(992\) 0 0
\(993\) 13.6569 9.65685i 0.433387 0.306451i
\(994\) 0 0
\(995\) 36.9706 + 18.4853i 1.17205 + 0.586023i
\(996\) 0 0
\(997\) −30.7990 + 30.7990i −0.975414 + 0.975414i −0.999705 0.0242911i \(-0.992267\pi\)
0.0242911 + 0.999705i \(0.492267\pi\)
\(998\) 0 0
\(999\) 37.3848 + 20.8995i 1.18280 + 0.661231i
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 240.2.v.c.113.2 4
3.2 odd 2 240.2.v.a.113.2 4
4.3 odd 2 120.2.r.b.113.1 yes 4
5.2 odd 4 240.2.v.a.17.1 4
5.3 odd 4 1200.2.v.j.257.2 4
5.4 even 2 1200.2.v.d.593.1 4
8.3 odd 2 960.2.v.f.833.2 4
8.5 even 2 960.2.v.g.833.1 4
12.11 even 2 120.2.r.c.113.1 yes 4
15.2 even 4 inner 240.2.v.c.17.2 4
15.8 even 4 1200.2.v.d.257.1 4
15.14 odd 2 1200.2.v.j.593.1 4
20.3 even 4 600.2.r.b.257.1 4
20.7 even 4 120.2.r.c.17.2 yes 4
20.19 odd 2 600.2.r.c.593.2 4
24.5 odd 2 960.2.v.i.833.1 4
24.11 even 2 960.2.v.a.833.2 4
40.27 even 4 960.2.v.a.257.1 4
40.37 odd 4 960.2.v.i.257.2 4
60.23 odd 4 600.2.r.c.257.2 4
60.47 odd 4 120.2.r.b.17.1 4
60.59 even 2 600.2.r.b.593.2 4
120.77 even 4 960.2.v.g.257.1 4
120.107 odd 4 960.2.v.f.257.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.r.b.17.1 4 60.47 odd 4
120.2.r.b.113.1 yes 4 4.3 odd 2
120.2.r.c.17.2 yes 4 20.7 even 4
120.2.r.c.113.1 yes 4 12.11 even 2
240.2.v.a.17.1 4 5.2 odd 4
240.2.v.a.113.2 4 3.2 odd 2
240.2.v.c.17.2 4 15.2 even 4 inner
240.2.v.c.113.2 4 1.1 even 1 trivial
600.2.r.b.257.1 4 20.3 even 4
600.2.r.b.593.2 4 60.59 even 2
600.2.r.c.257.2 4 60.23 odd 4
600.2.r.c.593.2 4 20.19 odd 2
960.2.v.a.257.1 4 40.27 even 4
960.2.v.a.833.2 4 24.11 even 2
960.2.v.f.257.2 4 120.107 odd 4
960.2.v.f.833.2 4 8.3 odd 2
960.2.v.g.257.1 4 120.77 even 4
960.2.v.g.833.1 4 8.5 even 2
960.2.v.i.257.2 4 40.37 odd 4
960.2.v.i.833.1 4 24.5 odd 2
1200.2.v.d.257.1 4 15.8 even 4
1200.2.v.d.593.1 4 5.4 even 2
1200.2.v.j.257.2 4 5.3 odd 4
1200.2.v.j.593.1 4 15.14 odd 2