Properties

Label 1280.2.o.p.383.1
Level $1280$
Weight $2$
Character 1280.383
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
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] = [1280,2,Mod(127,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("1280.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.o (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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 383.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.383
Dual form 1280.2.o.p.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 + 2.00000i) q^{3} +(1.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} +5.00000i q^{9} +O(q^{10})\) \(q+(2.00000 + 2.00000i) q^{3} +(1.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} +5.00000i q^{9} +(-1.00000 + 1.00000i) q^{13} +(6.00000 - 2.00000i) q^{15} +(-5.00000 + 5.00000i) q^{17} +4.00000i q^{19} +8.00000i q^{21} +(2.00000 - 2.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(-4.00000 + 4.00000i) q^{27} +4.00000 q^{29} -4.00000i q^{31} +(6.00000 - 2.00000i) q^{35} +(1.00000 + 1.00000i) q^{37} -4.00000 q^{39} +(6.00000 + 6.00000i) q^{43} +(10.0000 + 5.00000i) q^{45} +(2.00000 + 2.00000i) q^{47} +1.00000i q^{49} -20.0000 q^{51} +(7.00000 - 7.00000i) q^{53} +(-8.00000 + 8.00000i) q^{57} -4.00000i q^{59} +4.00000i q^{61} +(-10.0000 + 10.0000i) q^{63} +(1.00000 + 3.00000i) q^{65} +(10.0000 - 10.0000i) q^{67} +8.00000 q^{69} -12.0000i q^{71} +(3.00000 + 3.00000i) q^{73} +(2.00000 - 14.0000i) q^{75} -16.0000 q^{79} -1.00000 q^{81} +(-2.00000 - 2.00000i) q^{83} +(5.00000 + 15.0000i) q^{85} +(8.00000 + 8.00000i) q^{87} -4.00000 q^{91} +(8.00000 - 8.00000i) q^{93} +(8.00000 + 4.00000i) q^{95} +(-3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} - 2 q^{13} + 12 q^{15} - 10 q^{17} + 4 q^{23} - 6 q^{25} - 8 q^{27} + 8 q^{29} + 12 q^{35} + 2 q^{37} - 8 q^{39} + 12 q^{43} + 20 q^{45} + 4 q^{47} - 40 q^{51} + 14 q^{53} - 16 q^{57} - 20 q^{63} + 2 q^{65} + 20 q^{67} + 16 q^{69} + 6 q^{73} + 4 q^{75} - 32 q^{79} - 2 q^{81} - 4 q^{83} + 10 q^{85} + 16 q^{87} - 8 q^{91} + 16 q^{93} + 16 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(-1\)

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
\(3\) 2.00000 + 2.00000i 1.15470 + 1.15470i 0.985599 + 0.169102i \(0.0540867\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) 0 0
\(5\) 1.00000 2.00000i 0.447214 0.894427i
\(6\) 0 0
\(7\) 2.00000 + 2.00000i 0.755929 + 0.755929i 0.975579 0.219650i \(-0.0704915\pi\)
−0.219650 + 0.975579i \(0.570491\pi\)
\(8\) 0 0
\(9\) 5.00000i 1.66667i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.00000i −0.277350 + 0.277350i −0.832050 0.554700i \(-0.812833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 6.00000 2.00000i 1.54919 0.516398i
\(16\) 0 0
\(17\) −5.00000 + 5.00000i −1.21268 + 1.21268i −0.242536 + 0.970143i \(0.577979\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 8.00000i 1.74574i
\(22\) 0 0
\(23\) 2.00000 2.00000i 0.417029 0.417029i −0.467150 0.884178i \(-0.654719\pi\)
0.884178 + 0.467150i \(0.154719\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 2.00000i 1.01419 0.338062i
\(36\) 0 0
\(37\) 1.00000 + 1.00000i 0.164399 + 0.164399i 0.784512 0.620113i \(-0.212913\pi\)
−0.620113 + 0.784512i \(0.712913\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 6.00000 + 6.00000i 0.914991 + 0.914991i 0.996660 0.0816682i \(-0.0260248\pi\)
−0.0816682 + 0.996660i \(0.526025\pi\)
\(44\) 0 0
\(45\) 10.0000 + 5.00000i 1.49071 + 0.745356i
\(46\) 0 0
\(47\) 2.00000 + 2.00000i 0.291730 + 0.291730i 0.837763 0.546033i \(-0.183863\pi\)
−0.546033 + 0.837763i \(0.683863\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) −20.0000 −2.80056
\(52\) 0 0
\(53\) 7.00000 7.00000i 0.961524 0.961524i −0.0377628 0.999287i \(-0.512023\pi\)
0.999287 + 0.0377628i \(0.0120231\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.00000 + 8.00000i −1.05963 + 1.05963i
\(58\) 0 0
\(59\) 4.00000i 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) 4.00000i 0.512148i 0.966657 + 0.256074i \(0.0824290\pi\)
−0.966657 + 0.256074i \(0.917571\pi\)
\(62\) 0 0
\(63\) −10.0000 + 10.0000i −1.25988 + 1.25988i
\(64\) 0 0
\(65\) 1.00000 + 3.00000i 0.124035 + 0.372104i
\(66\) 0 0
\(67\) 10.0000 10.0000i 1.22169 1.22169i 0.254665 0.967029i \(-0.418035\pi\)
0.967029 0.254665i \(-0.0819652\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 3.00000 + 3.00000i 0.351123 + 0.351123i 0.860527 0.509404i \(-0.170134\pi\)
−0.509404 + 0.860527i \(0.670134\pi\)
\(74\) 0 0
\(75\) 2.00000 14.0000i 0.230940 1.61658i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −2.00000 2.00000i −0.219529 0.219529i 0.588771 0.808300i \(-0.299612\pi\)
−0.808300 + 0.588771i \(0.799612\pi\)
\(84\) 0 0
\(85\) 5.00000 + 15.0000i 0.542326 + 1.62698i
\(86\) 0 0
\(87\) 8.00000 + 8.00000i 0.857690 + 0.857690i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 8.00000 8.00000i 0.829561 0.829561i
\(94\) 0 0
\(95\) 8.00000 + 4.00000i 0.820783 + 0.410391i
\(96\) 0 0
\(97\) −3.00000 + 3.00000i −0.304604 + 0.304604i −0.842812 0.538208i \(-0.819101\pi\)
0.538208 + 0.842812i \(0.319101\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000i 0.597022i −0.954406 0.298511i \(-0.903510\pi\)
0.954406 0.298511i \(-0.0964900\pi\)
\(102\) 0 0
\(103\) −6.00000 + 6.00000i −0.591198 + 0.591198i −0.937955 0.346757i \(-0.887283\pi\)
0.346757 + 0.937955i \(0.387283\pi\)
\(104\) 0 0
\(105\) 16.0000 + 8.00000i 1.56144 + 0.780720i
\(106\) 0 0
\(107\) 6.00000 6.00000i 0.580042 0.580042i −0.354873 0.934915i \(-0.615476\pi\)
0.934915 + 0.354873i \(0.115476\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 4.00000i 0.379663i
\(112\) 0 0
\(113\) −9.00000 9.00000i −0.846649 0.846649i 0.143065 0.989713i \(-0.454304\pi\)
−0.989713 + 0.143065i \(0.954304\pi\)
\(114\) 0 0
\(115\) −2.00000 6.00000i −0.186501 0.559503i
\(116\) 0 0
\(117\) −5.00000 5.00000i −0.462250 0.462250i
\(118\) 0 0
\(119\) −20.0000 −1.83340
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.0000 + 2.00000i −0.983870 + 0.178885i
\(126\) 0 0
\(127\) −10.0000 10.0000i −0.887357 0.887357i 0.106912 0.994268i \(-0.465904\pi\)
−0.994268 + 0.106912i \(0.965904\pi\)
\(128\) 0 0
\(129\) 24.0000i 2.11308i
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −8.00000 + 8.00000i −0.693688 + 0.693688i
\(134\) 0 0
\(135\) 4.00000 + 12.0000i 0.344265 + 1.03280i
\(136\) 0 0
\(137\) −1.00000 + 1.00000i −0.0854358 + 0.0854358i −0.748533 0.663097i \(-0.769242\pi\)
0.663097 + 0.748533i \(0.269242\pi\)
\(138\) 0 0
\(139\) 12.0000i 1.01783i 0.860818 + 0.508913i \(0.169953\pi\)
−0.860818 + 0.508913i \(0.830047\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 8.00000i 0.332182 0.664364i
\(146\) 0 0
\(147\) −2.00000 + 2.00000i −0.164957 + 0.164957i
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 12.0000i 0.976546i −0.872691 0.488273i \(-0.837627\pi\)
0.872691 0.488273i \(-0.162373\pi\)
\(152\) 0 0
\(153\) −25.0000 25.0000i −2.02113 2.02113i
\(154\) 0 0
\(155\) −8.00000 4.00000i −0.642575 0.321288i
\(156\) 0 0
\(157\) 9.00000 + 9.00000i 0.718278 + 0.718278i 0.968252 0.249974i \(-0.0804222\pi\)
−0.249974 + 0.968252i \(0.580422\pi\)
\(158\) 0 0
\(159\) 28.0000 2.22054
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 2.00000 + 2.00000i 0.156652 + 0.156652i 0.781081 0.624429i \(-0.214668\pi\)
−0.624429 + 0.781081i \(0.714668\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 2.00000i −0.154765 0.154765i 0.625478 0.780242i \(-0.284904\pi\)
−0.780242 + 0.625478i \(0.784904\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) −20.0000 −1.52944
\(172\) 0 0
\(173\) 13.0000 13.0000i 0.988372 0.988372i −0.0115615 0.999933i \(-0.503680\pi\)
0.999933 + 0.0115615i \(0.00368021\pi\)
\(174\) 0 0
\(175\) 2.00000 14.0000i 0.151186 1.05830i
\(176\) 0 0
\(177\) 8.00000 8.00000i 0.601317 0.601317i
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) 10.0000i 0.743294i −0.928374 0.371647i \(-0.878793\pi\)
0.928374 0.371647i \(-0.121207\pi\)
\(182\) 0 0
\(183\) −8.00000 + 8.00000i −0.591377 + 0.591377i
\(184\) 0 0
\(185\) 3.00000 1.00000i 0.220564 0.0735215i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) 20.0000i 1.44715i −0.690246 0.723575i \(-0.742498\pi\)
0.690246 0.723575i \(-0.257502\pi\)
\(192\) 0 0
\(193\) −5.00000 5.00000i −0.359908 0.359908i 0.503871 0.863779i \(-0.331909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) −4.00000 + 8.00000i −0.286446 + 0.572892i
\(196\) 0 0
\(197\) 5.00000 + 5.00000i 0.356235 + 0.356235i 0.862423 0.506188i \(-0.168946\pi\)
−0.506188 + 0.862423i \(0.668946\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 40.0000 2.82138
\(202\) 0 0
\(203\) 8.00000 + 8.00000i 0.561490 + 0.561490i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.0000 + 10.0000i 0.695048 + 0.695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 24.0000 24.0000i 1.64445 1.64445i
\(214\) 0 0
\(215\) 18.0000 6.00000i 1.22759 0.409197i
\(216\) 0 0
\(217\) 8.00000 8.00000i 0.543075 0.543075i
\(218\) 0 0
\(219\) 12.0000i 0.810885i
\(220\) 0 0
\(221\) 10.0000i 0.672673i
\(222\) 0 0
\(223\) 10.0000 10.0000i 0.669650 0.669650i −0.287985 0.957635i \(-0.592985\pi\)
0.957635 + 0.287985i \(0.0929854\pi\)
\(224\) 0 0
\(225\) 20.0000 15.0000i 1.33333 1.00000i
\(226\) 0 0
\(227\) −10.0000 + 10.0000i −0.663723 + 0.663723i −0.956256 0.292532i \(-0.905502\pi\)
0.292532 + 0.956256i \(0.405502\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.00000 + 5.00000i 0.327561 + 0.327561i 0.851658 0.524097i \(-0.175597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 6.00000 2.00000i 0.391397 0.130466i
\(236\) 0 0
\(237\) −32.0000 32.0000i −2.07862 2.07862i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 0 0
\(243\) 10.0000 + 10.0000i 0.641500 + 0.641500i
\(244\) 0 0
\(245\) 2.00000 + 1.00000i 0.127775 + 0.0638877i
\(246\) 0 0
\(247\) −4.00000 4.00000i −0.254514 0.254514i
\(248\) 0 0
\(249\) 8.00000i 0.506979i
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −20.0000 + 40.0000i −1.25245 + 2.50490i
\(256\) 0 0
\(257\) 7.00000 7.00000i 0.436648 0.436648i −0.454234 0.890882i \(-0.650087\pi\)
0.890882 + 0.454234i \(0.150087\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 20.0000i 1.23797i
\(262\) 0 0
\(263\) 6.00000 6.00000i 0.369976 0.369976i −0.497492 0.867468i \(-0.665746\pi\)
0.867468 + 0.497492i \(0.165746\pi\)
\(264\) 0 0
\(265\) −7.00000 21.0000i −0.430007 1.29002i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 20.0000i 1.21491i 0.794353 + 0.607457i \(0.207810\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) 0 0
\(273\) −8.00000 8.00000i −0.484182 0.484182i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9.00000 9.00000i −0.540758 0.540758i 0.382993 0.923751i \(-0.374893\pi\)
−0.923751 + 0.382993i \(0.874893\pi\)
\(278\) 0 0
\(279\) 20.0000 1.19737
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −6.00000 6.00000i −0.356663 0.356663i 0.505918 0.862581i \(-0.331154\pi\)
−0.862581 + 0.505918i \(0.831154\pi\)
\(284\) 0 0
\(285\) 8.00000 + 24.0000i 0.473879 + 1.42164i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000i 1.94118i
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) 5.00000 5.00000i 0.292103 0.292103i −0.545807 0.837911i \(-0.683777\pi\)
0.837911 + 0.545807i \(0.183777\pi\)
\(294\) 0 0
\(295\) −8.00000 4.00000i −0.465778 0.232889i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000i 0.231326i
\(300\) 0 0
\(301\) 24.0000i 1.38334i
\(302\) 0 0
\(303\) 12.0000 12.0000i 0.689382 0.689382i
\(304\) 0 0
\(305\) 8.00000 + 4.00000i 0.458079 + 0.229039i
\(306\) 0 0
\(307\) 10.0000 10.0000i 0.570730 0.570730i −0.361602 0.932332i \(-0.617770\pi\)
0.932332 + 0.361602i \(0.117770\pi\)
\(308\) 0 0
\(309\) −24.0000 −1.36531
\(310\) 0 0
\(311\) 28.0000i 1.58773i 0.608091 + 0.793867i \(0.291935\pi\)
−0.608091 + 0.793867i \(0.708065\pi\)
\(312\) 0 0
\(313\) −15.0000 15.0000i −0.847850 0.847850i 0.142014 0.989865i \(-0.454642\pi\)
−0.989865 + 0.142014i \(0.954642\pi\)
\(314\) 0 0
\(315\) 10.0000 + 30.0000i 0.563436 + 1.69031i
\(316\) 0 0
\(317\) −11.0000 11.0000i −0.617822 0.617822i 0.327151 0.944972i \(-0.393912\pi\)
−0.944972 + 0.327151i \(0.893912\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −20.0000 20.0000i −1.11283 1.11283i
\(324\) 0 0
\(325\) 7.00000 + 1.00000i 0.388290 + 0.0554700i
\(326\) 0 0
\(327\) −20.0000 20.0000i −1.10600 1.10600i
\(328\) 0 0
\(329\) 8.00000i 0.441054i
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) −5.00000 + 5.00000i −0.273998 + 0.273998i
\(334\) 0 0
\(335\) −10.0000 30.0000i −0.546358 1.63908i
\(336\) 0 0
\(337\) −23.0000 + 23.0000i −1.25289 + 1.25289i −0.298471 + 0.954419i \(0.596477\pi\)
−0.954419 + 0.298471i \(0.903523\pi\)
\(338\) 0 0
\(339\) 36.0000i 1.95525i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 12.0000 12.0000i 0.647939 0.647939i
\(344\) 0 0
\(345\) 8.00000 16.0000i 0.430706 0.861411i
\(346\) 0 0
\(347\) 18.0000 18.0000i 0.966291 0.966291i −0.0331594 0.999450i \(-0.510557\pi\)
0.999450 + 0.0331594i \(0.0105569\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 8.00000i 0.427008i
\(352\) 0 0
\(353\) 9.00000 + 9.00000i 0.479022 + 0.479022i 0.904819 0.425797i \(-0.140006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) −24.0000 12.0000i −1.27379 0.636894i
\(356\) 0 0
\(357\) −40.0000 40.0000i −2.11702 2.11702i
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) −22.0000 22.0000i −1.15470 1.15470i
\(364\) 0 0
\(365\) 9.00000 3.00000i 0.471082 0.157027i
\(366\) 0 0
\(367\) 22.0000 + 22.0000i 1.14839 + 1.14839i 0.986869 + 0.161521i \(0.0516401\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 28.0000 1.45369
\(372\) 0 0
\(373\) −21.0000 + 21.0000i −1.08734 + 1.08734i −0.0915371 + 0.995802i \(0.529178\pi\)
−0.995802 + 0.0915371i \(0.970822\pi\)
\(374\) 0 0
\(375\) −26.0000 18.0000i −1.34263 0.929516i
\(376\) 0 0
\(377\) −4.00000 + 4.00000i −0.206010 + 0.206010i
\(378\) 0 0
\(379\) 28.0000i 1.43826i −0.694874 0.719132i \(-0.744540\pi\)
0.694874 0.719132i \(-0.255460\pi\)
\(380\) 0 0
\(381\) 40.0000i 2.04926i
\(382\) 0 0
\(383\) −22.0000 + 22.0000i −1.12415 + 1.12415i −0.133036 + 0.991111i \(0.542473\pi\)
−0.991111 + 0.133036i \(0.957527\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −30.0000 + 30.0000i −1.52499 + 1.52499i
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 20.0000i 1.01144i
\(392\) 0 0
\(393\) 16.0000 + 16.0000i 0.807093 + 0.807093i
\(394\) 0 0
\(395\) −16.0000 + 32.0000i −0.805047 + 1.61009i
\(396\) 0 0
\(397\) −13.0000 13.0000i −0.652451 0.652451i 0.301131 0.953583i \(-0.402636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) −32.0000 −1.60200
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 4.00000 + 4.00000i 0.199254 + 0.199254i
\(404\) 0 0
\(405\) −1.00000 + 2.00000i −0.0496904 + 0.0993808i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.00000i 0.0988936i 0.998777 + 0.0494468i \(0.0157458\pi\)
−0.998777 + 0.0494468i \(0.984254\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) 8.00000 8.00000i 0.393654 0.393654i
\(414\) 0 0
\(415\) −6.00000 + 2.00000i −0.294528 + 0.0981761i
\(416\) 0 0
\(417\) −24.0000 + 24.0000i −1.17529 + 1.17529i
\(418\) 0 0
\(419\) 12.0000i 0.586238i −0.956076 0.293119i \(-0.905307\pi\)
0.956076 0.293119i \(-0.0946933\pi\)
\(420\) 0 0
\(421\) 20.0000i 0.974740i 0.873195 + 0.487370i \(0.162044\pi\)
−0.873195 + 0.487370i \(0.837956\pi\)
\(422\) 0 0
\(423\) −10.0000 + 10.0000i −0.486217 + 0.486217i
\(424\) 0 0
\(425\) 35.0000 + 5.00000i 1.69775 + 0.242536i
\(426\) 0 0
\(427\) −8.00000 + 8.00000i −0.387147 + 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.00000i 0.192673i 0.995349 + 0.0963366i \(0.0307125\pi\)
−0.995349 + 0.0963366i \(0.969287\pi\)
\(432\) 0 0
\(433\) −19.0000 19.0000i −0.913082 0.913082i 0.0834318 0.996513i \(-0.473412\pi\)
−0.996513 + 0.0834318i \(0.973412\pi\)
\(434\) 0 0
\(435\) 24.0000 8.00000i 1.15071 0.383571i
\(436\) 0 0
\(437\) 8.00000 + 8.00000i 0.382692 + 0.382692i
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) −22.0000 22.0000i −1.04525 1.04525i −0.998926 0.0463251i \(-0.985249\pi\)
−0.0463251 0.998926i \(-0.514751\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −36.0000 36.0000i −1.70274 1.70274i
\(448\) 0 0
\(449\) 26.0000i 1.22702i 0.789689 + 0.613508i \(0.210242\pi\)
−0.789689 + 0.613508i \(0.789758\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 24.0000 24.0000i 1.12762 1.12762i
\(454\) 0 0
\(455\) −4.00000 + 8.00000i −0.187523 + 0.375046i
\(456\) 0 0
\(457\) −15.0000 + 15.0000i −0.701670 + 0.701670i −0.964769 0.263099i \(-0.915256\pi\)
0.263099 + 0.964769i \(0.415256\pi\)
\(458\) 0 0
\(459\) 40.0000i 1.86704i
\(460\) 0 0
\(461\) 14.0000i 0.652045i 0.945362 + 0.326023i \(0.105709\pi\)
−0.945362 + 0.326023i \(0.894291\pi\)
\(462\) 0 0
\(463\) 22.0000 22.0000i 1.02243 1.02243i 0.0226840 0.999743i \(-0.492779\pi\)
0.999743 0.0226840i \(-0.00722117\pi\)
\(464\) 0 0
\(465\) −8.00000 24.0000i −0.370991 1.11297i
\(466\) 0 0
\(467\) −2.00000 + 2.00000i −0.0925490 + 0.0925490i −0.751865 0.659317i \(-0.770846\pi\)
0.659317 + 0.751865i \(0.270846\pi\)
\(468\) 0 0
\(469\) 40.0000 1.84703
\(470\) 0 0
\(471\) 36.0000i 1.65879i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.0000 12.0000i 0.734130 0.550598i
\(476\) 0 0
\(477\) 35.0000 + 35.0000i 1.60254 + 1.60254i
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 16.0000 + 16.0000i 0.728025 + 0.728025i
\(484\) 0 0
\(485\) 3.00000 + 9.00000i 0.136223 + 0.408669i
\(486\) 0 0
\(487\) −6.00000 6.00000i −0.271886 0.271886i 0.557973 0.829859i \(-0.311579\pi\)
−0.829859 + 0.557973i \(0.811579\pi\)
\(488\) 0 0
\(489\) 8.00000i 0.361773i
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 0 0
\(493\) −20.0000 + 20.0000i −0.900755 + 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 24.0000i 1.07655 1.07655i
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 8.00000i 0.357414i
\(502\) 0 0
\(503\) 10.0000 10.0000i 0.445878 0.445878i −0.448104 0.893982i \(-0.647900\pi\)
0.893982 + 0.448104i \(0.147900\pi\)
\(504\) 0 0
\(505\) −12.0000 6.00000i −0.533993 0.266996i
\(506\) 0 0
\(507\) −22.0000 + 22.0000i −0.977054 + 0.977054i
\(508\) 0 0
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 12.0000i 0.530849i
\(512\) 0 0
\(513\) −16.0000 16.0000i −0.706417 0.706417i
\(514\) 0 0
\(515\) 6.00000 + 18.0000i 0.264392 + 0.793175i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 52.0000 2.28255
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −14.0000 14.0000i −0.612177 0.612177i 0.331336 0.943513i \(-0.392501\pi\)
−0.943513 + 0.331336i \(0.892501\pi\)
\(524\) 0 0
\(525\) 32.0000 24.0000i 1.39659 1.04745i
\(526\) 0 0
\(527\) 20.0000 + 20.0000i 0.871214 + 0.871214i
\(528\) 0 0
\(529\) 15.0000i 0.652174i
\(530\) 0 0
\(531\) 20.0000 0.867926
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 18.0000i −0.259403 0.778208i
\(536\) 0 0
\(537\) −24.0000 + 24.0000i −1.03568 + 1.03568i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000i 1.28980i 0.764267 + 0.644900i \(0.223101\pi\)
−0.764267 + 0.644900i \(0.776899\pi\)
\(542\) 0 0
\(543\) 20.0000 20.0000i 0.858282 0.858282i
\(544\) 0 0
\(545\) −10.0000 + 20.0000i −0.428353 + 0.856706i
\(546\) 0 0
\(547\) 6.00000 6.00000i 0.256541 0.256541i −0.567104 0.823646i \(-0.691936\pi\)
0.823646 + 0.567104i \(0.191936\pi\)
\(548\) 0 0
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 16.0000i 0.681623i
\(552\) 0 0
\(553\) −32.0000 32.0000i −1.36078 1.36078i
\(554\) 0 0
\(555\) 8.00000 + 4.00000i 0.339581 + 0.169791i
\(556\) 0 0
\(557\) 15.0000 + 15.0000i 0.635570 + 0.635570i 0.949460 0.313889i \(-0.101632\pi\)
−0.313889 + 0.949460i \(0.601632\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.00000 6.00000i −0.252870 0.252870i 0.569276 0.822146i \(-0.307223\pi\)
−0.822146 + 0.569276i \(0.807223\pi\)
\(564\) 0 0
\(565\) −27.0000 + 9.00000i −1.13590 + 0.378633i
\(566\) 0 0
\(567\) −2.00000 2.00000i −0.0839921 0.0839921i
\(568\) 0 0
\(569\) 2.00000i 0.0838444i −0.999121 0.0419222i \(-0.986652\pi\)
0.999121 0.0419222i \(-0.0133482\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 40.0000 40.0000i 1.67102 1.67102i
\(574\) 0 0
\(575\) −14.0000 2.00000i −0.583840 0.0834058i
\(576\) 0 0
\(577\) 15.0000 15.0000i 0.624458 0.624458i −0.322210 0.946668i \(-0.604426\pi\)
0.946668 + 0.322210i \(0.104426\pi\)
\(578\) 0 0
\(579\) 20.0000i 0.831172i
\(580\) 0 0
\(581\) 8.00000i 0.331896i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −15.0000 + 5.00000i −0.620174 + 0.206725i
\(586\) 0 0
\(587\) −14.0000 + 14.0000i −0.577842 + 0.577842i −0.934308 0.356466i \(-0.883981\pi\)
0.356466 + 0.934308i \(0.383981\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 20.0000i 0.822690i
\(592\) 0 0
\(593\) 1.00000 + 1.00000i 0.0410651 + 0.0410651i 0.727341 0.686276i \(-0.240756\pi\)
−0.686276 + 0.727341i \(0.740756\pi\)
\(594\) 0 0
\(595\) −20.0000 + 40.0000i −0.819920 + 1.63984i
\(596\) 0 0
\(597\) 48.0000 + 48.0000i 1.96451 + 1.96451i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) 50.0000 + 50.0000i 2.03616 + 2.03616i
\(604\) 0 0
\(605\) −11.0000 + 22.0000i −0.447214 + 0.894427i
\(606\) 0 0
\(607\) 18.0000 + 18.0000i 0.730597 + 0.730597i 0.970738 0.240141i \(-0.0771936\pi\)
−0.240141 + 0.970738i \(0.577194\pi\)
\(608\) 0 0
\(609\) 32.0000i 1.29671i
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) 0 0
\(613\) 9.00000 9.00000i 0.363507 0.363507i −0.501596 0.865102i \(-0.667253\pi\)
0.865102 + 0.501596i \(0.167253\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29.0000 29.0000i 1.16750 1.16750i 0.184701 0.982795i \(-0.440868\pi\)
0.982795 0.184701i \(-0.0591318\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) 16.0000i 0.642058i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 4.00000i 0.159237i −0.996825 0.0796187i \(-0.974630\pi\)
0.996825 0.0796187i \(-0.0253703\pi\)
\(632\) 0 0
\(633\) 32.0000 + 32.0000i 1.27189 + 1.27189i
\(634\) 0 0
\(635\) −30.0000 + 10.0000i −1.19051 + 0.396838i
\(636\) 0 0
\(637\) −1.00000 1.00000i −0.0396214 0.0396214i
\(638\) 0 0
\(639\) 60.0000 2.37356
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) −10.0000 10.0000i −0.394362 0.394362i 0.481877 0.876239i \(-0.339955\pi\)
−0.876239 + 0.481877i \(0.839955\pi\)
\(644\) 0 0
\(645\) 48.0000 + 24.0000i 1.89000 + 0.944999i
\(646\) 0 0
\(647\) 10.0000 + 10.0000i 0.393141 + 0.393141i 0.875805 0.482665i \(-0.160331\pi\)
−0.482665 + 0.875805i \(0.660331\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 32.0000 1.25418
\(652\) 0 0
\(653\) 1.00000 1.00000i 0.0391330 0.0391330i −0.687270 0.726403i \(-0.741191\pi\)
0.726403 + 0.687270i \(0.241191\pi\)
\(654\) 0 0
\(655\) 8.00000 16.0000i 0.312586 0.625172i
\(656\) 0 0
\(657\) −15.0000 + 15.0000i −0.585206 + 0.585206i
\(658\) 0 0
\(659\) 20.0000i 0.779089i 0.921008 + 0.389545i \(0.127368\pi\)
−0.921008 + 0.389545i \(0.872632\pi\)
\(660\) 0 0
\(661\) 12.0000i 0.466746i 0.972387 + 0.233373i \(0.0749763\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 0 0
\(663\) 20.0000 20.0000i 0.776736 0.776736i
\(664\) 0 0
\(665\) 8.00000 + 24.0000i 0.310227 + 0.930680i
\(666\) 0 0
\(667\) 8.00000 8.00000i 0.309761 0.309761i
\(668\) 0 0
\(669\) 40.0000 1.54649
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.00000 + 5.00000i 0.192736 + 0.192736i 0.796877 0.604141i \(-0.206484\pi\)
−0.604141 + 0.796877i \(0.706484\pi\)
\(674\) 0 0
\(675\) 28.0000 + 4.00000i 1.07772 + 0.153960i
\(676\) 0 0
\(677\) −3.00000 3.00000i −0.115299 0.115299i 0.647103 0.762402i \(-0.275980\pi\)
−0.762402 + 0.647103i \(0.775980\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) −40.0000 −1.53280
\(682\) 0 0
\(683\) 22.0000 + 22.0000i 0.841807 + 0.841807i 0.989094 0.147287i \(-0.0470541\pi\)
−0.147287 + 0.989094i \(0.547054\pi\)
\(684\) 0 0
\(685\) 1.00000 + 3.00000i 0.0382080 + 0.114624i
\(686\) 0 0
\(687\) −40.0000 40.0000i −1.52610 1.52610i
\(688\) 0 0
\(689\) 14.0000i 0.533358i
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 + 12.0000i 0.910372 + 0.455186i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 20.0000i 0.756469i
\(700\) 0 0
\(701\) 20.0000i 0.755390i −0.925930 0.377695i \(-0.876717\pi\)
0.925930 0.377695i \(-0.123283\pi\)
\(702\) 0 0
\(703\) −4.00000 + 4.00000i −0.150863 + 0.150863i
\(704\) 0 0
\(705\) 16.0000 + 8.00000i 0.602595 + 0.301297i
\(706\) 0 0
\(707\) 12.0000 12.0000i 0.451306 0.451306i
\(708\) 0 0
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 0 0
\(711\) 80.0000i 3.00023i
\(712\) 0 0
\(713\) −8.00000 8.00000i −0.299602 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.0000 16.0000i −0.597531 0.597531i
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) −32.0000 32.0000i −1.19009 1.19009i
\(724\) 0 0
\(725\) −12.0000 16.0000i −0.445669 0.594225i
\(726\) 0 0
\(727\) 18.0000 + 18.0000i 0.667583 + 0.667583i 0.957156 0.289573i \(-0.0935133\pi\)
−0.289573 + 0.957156i \(0.593513\pi\)
\(728\) 0 0
\(729\) 43.0000i 1.59259i
\(730\) 0 0
\(731\) −60.0000 −2.21918
\(732\) 0 0
\(733\) 21.0000 21.0000i 0.775653 0.775653i −0.203436 0.979088i \(-0.565211\pi\)
0.979088 + 0.203436i \(0.0652108\pi\)
\(734\) 0 0
\(735\) 2.00000 + 6.00000i 0.0737711 + 0.221313i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 44.0000i 1.61857i 0.587419 + 0.809283i \(0.300144\pi\)
−0.587419 + 0.809283i \(0.699856\pi\)
\(740\) 0 0
\(741\) 16.0000i 0.587775i
\(742\) 0 0
\(743\) 30.0000 30.0000i 1.10059 1.10059i 0.106254 0.994339i \(-0.466114\pi\)
0.994339 0.106254i \(-0.0338857\pi\)
\(744\) 0 0
\(745\) −18.0000 + 36.0000i −0.659469 + 1.31894i
\(746\) 0 0
\(747\) 10.0000 10.0000i 0.365881 0.365881i
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 44.0000i 1.60558i 0.596260 + 0.802791i \(0.296653\pi\)
−0.596260 + 0.802791i \(0.703347\pi\)
\(752\) 0 0
\(753\) 48.0000 + 48.0000i 1.74922 + 1.74922i
\(754\) 0 0
\(755\) −24.0000 12.0000i −0.873449 0.436725i
\(756\) 0 0
\(757\) −1.00000 1.00000i −0.0363456 0.0363456i 0.688700 0.725046i \(-0.258182\pi\)
−0.725046 + 0.688700i \(0.758182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) −20.0000 20.0000i −0.724049 0.724049i
\(764\) 0 0
\(765\) −75.0000 + 25.0000i −2.71163 + 0.903877i
\(766\) 0 0
\(767\) 4.00000 + 4.00000i 0.144432 + 0.144432i
\(768\) 0 0
\(769\) 40.0000i 1.44244i −0.692708 0.721218i \(-0.743582\pi\)
0.692708 0.721218i \(-0.256418\pi\)
\(770\) 0 0
\(771\) 28.0000 1.00840
\(772\) 0 0
\(773\) 1.00000 1.00000i 0.0359675 0.0359675i −0.688894 0.724862i \(-0.741904\pi\)
0.724862 + 0.688894i \(0.241904\pi\)
\(774\) 0 0
\(775\) −16.0000 + 12.0000i −0.574737 + 0.431053i
\(776\) 0 0
\(777\) −8.00000 + 8.00000i −0.286998 + 0.286998i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −16.0000 + 16.0000i −0.571793 + 0.571793i
\(784\) 0 0
\(785\) 27.0000 9.00000i 0.963671 0.321224i
\(786\) 0 0
\(787\) −30.0000 + 30.0000i −1.06938 + 1.06938i −0.0719783 + 0.997406i \(0.522931\pi\)
−0.997406 + 0.0719783i \(0.977069\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 36.0000i 1.28001i
\(792\) 0 0
\(793\) −4.00000 4.00000i −0.142044 0.142044i
\(794\) 0 0
\(795\) 28.0000 56.0000i 0.993058 1.98612i
\(796\) 0 0
\(797\) 29.0000 + 29.0000i 1.02723 + 1.02723i 0.999619 + 0.0276140i \(0.00879094\pi\)
0.0276140 + 0.999619i \(0.491209\pi\)
\(798\) 0 0
\(799\) −20.0000 −0.707549
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 8.00000 16.0000i 0.281963 0.563926i
\(806\) 0 0
\(807\) 20.0000 + 20.0000i 0.704033 + 0.704033i
\(808\) 0 0
\(809\) 24.0000i 0.843795i 0.906644 + 0.421898i \(0.138636\pi\)
−0.906644 + 0.421898i \(0.861364\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) −40.0000 + 40.0000i −1.40286 + 1.40286i
\(814\) 0 0
\(815\) 6.00000 2.00000i 0.210171 0.0700569i
\(816\) 0 0
\(817\) −24.0000 + 24.0000i −0.839654 + 0.839654i
\(818\) 0 0
\(819\) 20.0000i 0.698857i
\(820\) 0 0
\(821\) 28.0000i 0.977207i −0.872506 0.488603i \(-0.837507\pi\)
0.872506 0.488603i \(-0.162493\pi\)
\(822\) 0 0
\(823\) −30.0000 + 30.0000i −1.04573 + 1.04573i −0.0468315 + 0.998903i \(0.514912\pi\)
−0.998903 + 0.0468315i \(0.985088\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.00000 2.00000i 0.0695468 0.0695468i −0.671478 0.741025i \(-0.734340\pi\)
0.741025 + 0.671478i \(0.234340\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 36.0000i 1.24883i
\(832\) 0 0
\(833\) −5.00000 5.00000i −0.173240 0.173240i
\(834\) 0 0
\(835\) −6.00000 + 2.00000i −0.207639 + 0.0692129i
\(836\) 0 0
\(837\) 16.0000 + 16.0000i 0.553041 + 0.553041i
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −16.0000 16.0000i −0.551069 0.551069i
\(844\) 0 0
\(845\) 22.0000 + 11.0000i 0.756823 + 0.378412i
\(846\) 0 0
\(847\) −22.0000 22.0000i −0.755929 0.755929i
\(848\) 0 0
\(849\) 24.0000i 0.823678i
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) −15.0000 + 15.0000i −0.513590 + 0.513590i −0.915625 0.402034i \(-0.868303\pi\)
0.402034 + 0.915625i \(0.368303\pi\)
\(854\) 0 0
\(855\) −20.0000 + 40.0000i −0.683986 + 1.36797i
\(856\) 0 0
\(857\) 7.00000 7.00000i 0.239115 0.239115i −0.577368 0.816484i \(-0.695920\pi\)
0.816484 + 0.577368i \(0.195920\pi\)
\(858\) 0 0
\(859\) 28.0000i 0.955348i 0.878537 + 0.477674i \(0.158520\pi\)
−0.878537 + 0.477674i \(0.841480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −34.0000 + 34.0000i −1.15737 + 1.15737i −0.172335 + 0.985038i \(0.555131\pi\)
−0.985038 + 0.172335i \(0.944869\pi\)
\(864\) 0 0
\(865\) −13.0000 39.0000i −0.442013 1.32604i
\(866\) 0 0
\(867\) 66.0000 66.0000i 2.24148 2.24148i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 20.0000i 0.677674i
\(872\) 0 0
\(873\) −15.0000 15.0000i −0.507673 0.507673i
\(874\) 0 0
\(875\) −26.0000 18.0000i −0.878960 0.608511i
\(876\) 0 0
\(877\) −15.0000 15.0000i −0.506514 0.506514i 0.406941 0.913455i \(-0.366596\pi\)
−0.913455 + 0.406941i \(0.866596\pi\)
\(878\) 0 0
\(879\) 20.0000 0.674583
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) 10.0000 + 10.0000i 0.336527 + 0.336527i 0.855058 0.518532i \(-0.173521\pi\)
−0.518532 + 0.855058i \(0.673521\pi\)
\(884\) 0 0
\(885\) −8.00000 24.0000i −0.268917 0.806751i
\(886\) 0 0
\(887\) 10.0000 + 10.0000i 0.335767 + 0.335767i 0.854772 0.519004i \(-0.173697\pi\)
−0.519004 + 0.854772i \(0.673697\pi\)
\(888\) 0 0
\(889\) 40.0000i 1.34156i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.00000 + 8.00000i −0.267710 + 0.267710i
\(894\) 0 0
\(895\) 24.0000 + 12.0000i 0.802232 + 0.401116i
\(896\) 0 0
\(897\) −8.00000 + 8.00000i −0.267112 + 0.267112i
\(898\) 0 0
\(899\) 16.0000i 0.533630i
\(900\) 0 0
\(901\) 70.0000i 2.33204i
\(902\) 0 0
\(903\) −48.0000 + 48.0000i −1.59734 + 1.59734i
\(904\) 0 0
\(905\) −20.0000 10.0000i −0.664822 0.332411i
\(906\) 0 0
\(907\) 6.00000 6.00000i 0.199227 0.199227i −0.600442 0.799668i \(-0.705009\pi\)
0.799668 + 0.600442i \(0.205009\pi\)
\(908\) 0 0
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) 4.00000i 0.132526i −0.997802 0.0662630i \(-0.978892\pi\)
0.997802 0.0662630i \(-0.0211076\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 8.00000 + 24.0000i 0.264472 + 0.793416i
\(916\) 0 0
\(917\) 16.0000 + 16.0000i 0.528367 + 0.528367i
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 40.0000 1.31804
\(922\) 0 0
\(923\) 12.0000 + 12.0000i 0.394985 + 0.394985i
\(924\) 0 0
\(925\) 1.00000 7.00000i 0.0328798 0.230159i
\(926\) 0 0
\(927\) −30.0000 30.0000i −0.985329 0.985329i
\(928\) 0 0
\(929\) 6.00000i 0.196854i 0.995144 + 0.0984268i \(0.0313810\pi\)
−0.995144 + 0.0984268i \(0.968619\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) −56.0000 + 56.0000i −1.83336 + 1.83336i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27.0000 + 27.0000i −0.882052 + 0.882052i −0.993743 0.111691i \(-0.964373\pi\)
0.111691 + 0.993743i \(0.464373\pi\)
\(938\) 0 0
\(939\) 60.0000i 1.95803i
\(940\) 0 0
\(941\) 46.0000i 1.49956i −0.661689 0.749779i \(-0.730160\pi\)
0.661689 0.749779i \(-0.269840\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −16.0000 + 32.0000i −0.520480 + 1.04096i
\(946\) 0 0
\(947\) 10.0000 10.0000i 0.324956 0.324956i −0.525708 0.850665i \(-0.676200\pi\)
0.850665 + 0.525708i \(0.176200\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 44.0000i 1.42680i
\(952\) 0 0
\(953\) 9.00000 + 9.00000i 0.291539 + 0.291539i 0.837688 0.546149i \(-0.183907\pi\)
−0.546149 + 0.837688i \(0.683907\pi\)
\(954\) 0 0
\(955\) −40.0000 20.0000i −1.29437 0.647185i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 30.0000 + 30.0000i 0.966736 + 0.966736i
\(964\) 0 0
\(965\) −15.0000 + 5.00000i −0.482867 + 0.160956i
\(966\) 0 0
\(967\) −10.0000 10.0000i −0.321578 0.321578i 0.527794 0.849372i \(-0.323019\pi\)
−0.849372 + 0.527794i \(0.823019\pi\)
\(968\) 0 0
\(969\) 80.0000i 2.56997i
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 0 0
\(973\) −24.0000 + 24.0000i −0.769405 + 0.769405i
\(974\) 0 0
\(975\) 12.0000 + 16.0000i 0.384308 + 0.512410i
\(976\) 0 0
\(977\) 21.0000 21.0000i 0.671850 0.671850i −0.286293 0.958142i \(-0.592423\pi\)
0.958142 + 0.286293i \(0.0924230\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 50.0000i 1.59638i
\(982\) 0 0
\(983\) −34.0000 + 34.0000i −1.08443 + 1.08443i −0.0883413 + 0.996090i \(0.528157\pi\)
−0.996090 + 0.0883413i \(0.971843\pi\)
\(984\) 0 0
\(985\) 15.0000 5.00000i 0.477940 0.159313i
\(986\) 0 0
\(987\) −16.0000 + 16.0000i −0.509286 + 0.509286i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 4.00000i 0.127064i 0.997980 + 0.0635321i \(0.0202365\pi\)
−0.997980 + 0.0635321i \(0.979763\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 24.0000 48.0000i 0.760851 1.52170i
\(996\) 0 0
\(997\) −21.0000 21.0000i −0.665077 0.665077i 0.291496 0.956572i \(-0.405847\pi\)
−0.956572 + 0.291496i \(0.905847\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 1280.2.o.p.383.1 2
4.3 odd 2 1280.2.o.b.383.1 2
5.2 odd 4 1280.2.o.o.127.1 2
8.3 odd 2 1280.2.o.o.383.1 2
8.5 even 2 1280.2.o.a.383.1 2
16.3 odd 4 160.2.n.f.63.1 yes 2
16.5 even 4 320.2.n.h.63.1 2
16.11 odd 4 320.2.n.a.63.1 2
16.13 even 4 160.2.n.a.63.1 2
20.7 even 4 1280.2.o.a.127.1 2
40.27 even 4 inner 1280.2.o.p.127.1 2
40.37 odd 4 1280.2.o.b.127.1 2
48.29 odd 4 1440.2.x.i.703.1 2
48.35 even 4 1440.2.x.j.703.1 2
80.3 even 4 800.2.n.j.607.1 2
80.13 odd 4 800.2.n.a.607.1 2
80.19 odd 4 800.2.n.a.543.1 2
80.27 even 4 320.2.n.h.127.1 2
80.29 even 4 800.2.n.j.543.1 2
80.37 odd 4 320.2.n.a.127.1 2
80.43 even 4 1600.2.n.a.1407.1 2
80.53 odd 4 1600.2.n.n.1407.1 2
80.59 odd 4 1600.2.n.n.1343.1 2
80.67 even 4 160.2.n.a.127.1 yes 2
80.69 even 4 1600.2.n.a.1343.1 2
80.77 odd 4 160.2.n.f.127.1 yes 2
240.77 even 4 1440.2.x.j.127.1 2
240.227 odd 4 1440.2.x.i.127.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.n.a.63.1 2 16.13 even 4
160.2.n.a.127.1 yes 2 80.67 even 4
160.2.n.f.63.1 yes 2 16.3 odd 4
160.2.n.f.127.1 yes 2 80.77 odd 4
320.2.n.a.63.1 2 16.11 odd 4
320.2.n.a.127.1 2 80.37 odd 4
320.2.n.h.63.1 2 16.5 even 4
320.2.n.h.127.1 2 80.27 even 4
800.2.n.a.543.1 2 80.19 odd 4
800.2.n.a.607.1 2 80.13 odd 4
800.2.n.j.543.1 2 80.29 even 4
800.2.n.j.607.1 2 80.3 even 4
1280.2.o.a.127.1 2 20.7 even 4
1280.2.o.a.383.1 2 8.5 even 2
1280.2.o.b.127.1 2 40.37 odd 4
1280.2.o.b.383.1 2 4.3 odd 2
1280.2.o.o.127.1 2 5.2 odd 4
1280.2.o.o.383.1 2 8.3 odd 2
1280.2.o.p.127.1 2 40.27 even 4 inner
1280.2.o.p.383.1 2 1.1 even 1 trivial
1440.2.x.i.127.1 2 240.227 odd 4
1440.2.x.i.703.1 2 48.29 odd 4
1440.2.x.j.127.1 2 240.77 even 4
1440.2.x.j.703.1 2 48.35 even 4
1600.2.n.a.1343.1 2 80.69 even 4
1600.2.n.a.1407.1 2 80.43 even 4
1600.2.n.n.1343.1 2 80.59 odd 4
1600.2.n.n.1407.1 2 80.53 odd 4