Properties

Label 320.2.l.a.81.5
Level $320$
Weight $2$
Character 320.81
Analytic conductor $2.555$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,2,Mod(81,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.l (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: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 81.5
Root \(-0.296075 + 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 320.81
Dual form 320.2.l.a.241.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.120009 + 0.120009i) q^{3} +(-0.707107 + 0.707107i) q^{5} -2.66881i q^{7} -2.97120i q^{9} +O(q^{10})\) \(q+(0.120009 + 0.120009i) q^{3} +(-0.707107 + 0.707107i) q^{5} -2.66881i q^{7} -2.97120i q^{9} +(3.49714 - 3.49714i) q^{11} +(2.94072 + 2.94072i) q^{13} -0.169718 q^{15} +1.85116 q^{17} +(3.44856 + 3.44856i) q^{19} +(0.320281 - 0.320281i) q^{21} +0.707288i q^{23} -1.00000i q^{25} +(0.716597 - 0.716597i) q^{27} +(-3.49909 - 3.49909i) q^{29} -6.84272 q^{31} +0.839377 q^{33} +(1.88714 + 1.88714i) q^{35} +(-0.0975060 + 0.0975060i) q^{37} +0.705826i q^{39} -10.2052i q^{41} +(-4.43844 + 4.43844i) q^{43} +(2.10095 + 2.10095i) q^{45} +1.89428 q^{47} -0.122561 q^{49} +(0.222155 + 0.222155i) q^{51} +(-7.43897 + 7.43897i) q^{53} +4.94571i q^{55} +0.827717i q^{57} +(-0.959574 + 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} -7.92956 q^{63} -4.15881 q^{65} +(-3.49691 - 3.49691i) q^{67} +(-0.0848809 + 0.0848809i) q^{69} -7.86777i q^{71} +15.6564i q^{73} +(0.120009 - 0.120009i) q^{75} +(-9.33322 - 9.33322i) q^{77} +6.70212 q^{79} -8.74159 q^{81} +(3.87327 + 3.87327i) q^{83} +(-1.30896 + 1.30896i) q^{85} -0.839845i q^{87} +10.5055i q^{89} +(7.84824 - 7.84824i) q^{91} +(-0.821187 - 0.821187i) q^{93} -4.87701 q^{95} +4.79937 q^{97} +(-10.3907 - 10.3907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{11} + 8 q^{15} + 8 q^{19} - 24 q^{27} - 16 q^{29} - 16 q^{37} - 8 q^{43} + 40 q^{47} - 16 q^{49} + 32 q^{51} + 16 q^{53} + 8 q^{59} + 16 q^{61} - 40 q^{63} - 40 q^{67} + 16 q^{69} + 16 q^{77} - 16 q^{79} - 16 q^{81} - 40 q^{83} - 16 q^{85} - 32 q^{91} - 48 q^{93} - 32 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 0.120009 + 0.120009i 0.0692872 + 0.0692872i 0.740901 0.671614i \(-0.234399\pi\)
−0.671614 + 0.740901i \(0.734399\pi\)
\(4\) 0 0
\(5\) −0.707107 + 0.707107i −0.316228 + 0.316228i
\(6\) 0 0
\(7\) 2.66881i 1.00872i −0.863495 0.504358i \(-0.831729\pi\)
0.863495 0.504358i \(-0.168271\pi\)
\(8\) 0 0
\(9\) 2.97120i 0.990399i
\(10\) 0 0
\(11\) 3.49714 3.49714i 1.05443 1.05443i 0.0559977 0.998431i \(-0.482166\pi\)
0.998431 0.0559977i \(-0.0178339\pi\)
\(12\) 0 0
\(13\) 2.94072 + 2.94072i 0.815610 + 0.815610i 0.985468 0.169858i \(-0.0543310\pi\)
−0.169858 + 0.985468i \(0.554331\pi\)
\(14\) 0 0
\(15\) −0.169718 −0.0438211
\(16\) 0 0
\(17\) 1.85116 0.448971 0.224486 0.974477i \(-0.427930\pi\)
0.224486 + 0.974477i \(0.427930\pi\)
\(18\) 0 0
\(19\) 3.44856 + 3.44856i 0.791155 + 0.791155i 0.981682 0.190527i \(-0.0610197\pi\)
−0.190527 + 0.981682i \(0.561020\pi\)
\(20\) 0 0
\(21\) 0.320281 0.320281i 0.0698911 0.0698911i
\(22\) 0 0
\(23\) 0.707288i 0.147480i 0.997278 + 0.0737399i \(0.0234935\pi\)
−0.997278 + 0.0737399i \(0.976507\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 0.716597 0.716597i 0.137909 0.137909i
\(28\) 0 0
\(29\) −3.49909 3.49909i −0.649766 0.649766i 0.303171 0.952936i \(-0.401955\pi\)
−0.952936 + 0.303171i \(0.901955\pi\)
\(30\) 0 0
\(31\) −6.84272 −1.22899 −0.614494 0.788921i \(-0.710640\pi\)
−0.614494 + 0.788921i \(0.710640\pi\)
\(32\) 0 0
\(33\) 0.839377 0.146117
\(34\) 0 0
\(35\) 1.88714 + 1.88714i 0.318984 + 0.318984i
\(36\) 0 0
\(37\) −0.0975060 + 0.0975060i −0.0160299 + 0.0160299i −0.715076 0.699046i \(-0.753608\pi\)
0.699046 + 0.715076i \(0.253608\pi\)
\(38\) 0 0
\(39\) 0.705826i 0.113023i
\(40\) 0 0
\(41\) 10.2052i 1.59379i −0.604117 0.796896i \(-0.706474\pi\)
0.604117 0.796896i \(-0.293526\pi\)
\(42\) 0 0
\(43\) −4.43844 + 4.43844i −0.676855 + 0.676855i −0.959287 0.282432i \(-0.908859\pi\)
0.282432 + 0.959287i \(0.408859\pi\)
\(44\) 0 0
\(45\) 2.10095 + 2.10095i 0.313192 + 0.313192i
\(46\) 0 0
\(47\) 1.89428 0.276310 0.138155 0.990411i \(-0.455883\pi\)
0.138155 + 0.990411i \(0.455883\pi\)
\(48\) 0 0
\(49\) −0.122561 −0.0175087
\(50\) 0 0
\(51\) 0.222155 + 0.222155i 0.0311079 + 0.0311079i
\(52\) 0 0
\(53\) −7.43897 + 7.43897i −1.02182 + 1.02182i −0.0220650 + 0.999757i \(0.507024\pi\)
−0.999757 + 0.0220650i \(0.992976\pi\)
\(54\) 0 0
\(55\) 4.94571i 0.666879i
\(56\) 0 0
\(57\) 0.827717i 0.109634i
\(58\) 0 0
\(59\) −0.959574 + 0.959574i −0.124926 + 0.124926i −0.766805 0.641880i \(-0.778155\pi\)
0.641880 + 0.766805i \(0.278155\pi\)
\(60\) 0 0
\(61\) 6.49825 + 6.49825i 0.832015 + 0.832015i 0.987792 0.155777i \(-0.0497881\pi\)
−0.155777 + 0.987792i \(0.549788\pi\)
\(62\) 0 0
\(63\) −7.92956 −0.999031
\(64\) 0 0
\(65\) −4.15881 −0.515837
\(66\) 0 0
\(67\) −3.49691 3.49691i −0.427216 0.427216i 0.460463 0.887679i \(-0.347683\pi\)
−0.887679 + 0.460463i \(0.847683\pi\)
\(68\) 0 0
\(69\) −0.0848809 + 0.0848809i −0.0102185 + 0.0102185i
\(70\) 0 0
\(71\) 7.86777i 0.933733i −0.884328 0.466866i \(-0.845383\pi\)
0.884328 0.466866i \(-0.154617\pi\)
\(72\) 0 0
\(73\) 15.6564i 1.83244i 0.400675 + 0.916220i \(0.368776\pi\)
−0.400675 + 0.916220i \(0.631224\pi\)
\(74\) 0 0
\(75\) 0.120009 0.120009i 0.0138574 0.0138574i
\(76\) 0 0
\(77\) −9.33322 9.33322i −1.06362 1.06362i
\(78\) 0 0
\(79\) 6.70212 0.754047 0.377024 0.926204i \(-0.376948\pi\)
0.377024 + 0.926204i \(0.376948\pi\)
\(80\) 0 0
\(81\) −8.74159 −0.971288
\(82\) 0 0
\(83\) 3.87327 + 3.87327i 0.425147 + 0.425147i 0.886971 0.461825i \(-0.152805\pi\)
−0.461825 + 0.886971i \(0.652805\pi\)
\(84\) 0 0
\(85\) −1.30896 + 1.30896i −0.141977 + 0.141977i
\(86\) 0 0
\(87\) 0.839845i 0.0900408i
\(88\) 0 0
\(89\) 10.5055i 1.11358i 0.830653 + 0.556790i \(0.187967\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(90\) 0 0
\(91\) 7.84824 7.84824i 0.822719 0.822719i
\(92\) 0 0
\(93\) −0.821187 0.821187i −0.0851531 0.0851531i
\(94\) 0 0
\(95\) −4.87701 −0.500370
\(96\) 0 0
\(97\) 4.79937 0.487303 0.243651 0.969863i \(-0.421655\pi\)
0.243651 + 0.969863i \(0.421655\pi\)
\(98\) 0 0
\(99\) −10.3907 10.3907i −1.04430 1.04430i
\(100\) 0 0
\(101\) 0.372979 0.372979i 0.0371128 0.0371128i −0.688307 0.725420i \(-0.741646\pi\)
0.725420 + 0.688307i \(0.241646\pi\)
\(102\) 0 0
\(103\) 10.3013i 1.01502i 0.861647 + 0.507508i \(0.169433\pi\)
−0.861647 + 0.507508i \(0.830567\pi\)
\(104\) 0 0
\(105\) 0.452946i 0.0442030i
\(106\) 0 0
\(107\) −14.5069 + 14.5069i −1.40244 + 1.40244i −0.610165 + 0.792274i \(0.708897\pi\)
−0.792274 + 0.610165i \(0.791103\pi\)
\(108\) 0 0
\(109\) 0.796284 + 0.796284i 0.0762701 + 0.0762701i 0.744213 0.667943i \(-0.232825\pi\)
−0.667943 + 0.744213i \(0.732825\pi\)
\(110\) 0 0
\(111\) −0.0234032 −0.00222133
\(112\) 0 0
\(113\) 0.842524 0.0792580 0.0396290 0.999214i \(-0.487382\pi\)
0.0396290 + 0.999214i \(0.487382\pi\)
\(114\) 0 0
\(115\) −0.500128 0.500128i −0.0466372 0.0466372i
\(116\) 0 0
\(117\) 8.73747 8.73747i 0.807779 0.807779i
\(118\) 0 0
\(119\) 4.94039i 0.452885i
\(120\) 0 0
\(121\) 13.4600i 1.22364i
\(122\) 0 0
\(123\) 1.22472 1.22472i 0.110429 0.110429i
\(124\) 0 0
\(125\) 0.707107 + 0.707107i 0.0632456 + 0.0632456i
\(126\) 0 0
\(127\) 21.1693 1.87847 0.939234 0.343277i \(-0.111537\pi\)
0.939234 + 0.343277i \(0.111537\pi\)
\(128\) 0 0
\(129\) −1.06530 −0.0937947
\(130\) 0 0
\(131\) −4.67248 4.67248i −0.408237 0.408237i 0.472887 0.881123i \(-0.343212\pi\)
−0.881123 + 0.472887i \(0.843212\pi\)
\(132\) 0 0
\(133\) 9.20357 9.20357i 0.798051 0.798051i
\(134\) 0 0
\(135\) 1.01342i 0.0872214i
\(136\) 0 0
\(137\) 10.2840i 0.878623i 0.898335 + 0.439312i \(0.144778\pi\)
−0.898335 + 0.439312i \(0.855222\pi\)
\(138\) 0 0
\(139\) −4.98588 + 4.98588i −0.422897 + 0.422897i −0.886200 0.463303i \(-0.846664\pi\)
0.463303 + 0.886200i \(0.346664\pi\)
\(140\) 0 0
\(141\) 0.227331 + 0.227331i 0.0191447 + 0.0191447i
\(142\) 0 0
\(143\) 20.5683 1.72001
\(144\) 0 0
\(145\) 4.94847 0.410948
\(146\) 0 0
\(147\) −0.0147084 0.0147084i −0.00121313 0.00121313i
\(148\) 0 0
\(149\) 8.79493 8.79493i 0.720509 0.720509i −0.248200 0.968709i \(-0.579839\pi\)
0.968709 + 0.248200i \(0.0798390\pi\)
\(150\) 0 0
\(151\) 22.1838i 1.80529i 0.430385 + 0.902645i \(0.358378\pi\)
−0.430385 + 0.902645i \(0.641622\pi\)
\(152\) 0 0
\(153\) 5.50015i 0.444661i
\(154\) 0 0
\(155\) 4.83853 4.83853i 0.388640 0.388640i
\(156\) 0 0
\(157\) −3.72187 3.72187i −0.297038 0.297038i 0.542815 0.839852i \(-0.317359\pi\)
−0.839852 + 0.542815i \(0.817359\pi\)
\(158\) 0 0
\(159\) −1.78549 −0.141598
\(160\) 0 0
\(161\) 1.88762 0.148765
\(162\) 0 0
\(163\) −2.11630 2.11630i −0.165761 0.165761i 0.619352 0.785113i \(-0.287395\pi\)
−0.785113 + 0.619352i \(0.787395\pi\)
\(164\) 0 0
\(165\) −0.593529 + 0.593529i −0.0462062 + 0.0462062i
\(166\) 0 0
\(167\) 18.1604i 1.40530i −0.711538 0.702648i \(-0.752001\pi\)
0.711538 0.702648i \(-0.247999\pi\)
\(168\) 0 0
\(169\) 4.29572i 0.330440i
\(170\) 0 0
\(171\) 10.2464 10.2464i 0.783559 0.783559i
\(172\) 0 0
\(173\) −8.53542 8.53542i −0.648936 0.648936i 0.303800 0.952736i \(-0.401745\pi\)
−0.952736 + 0.303800i \(0.901745\pi\)
\(174\) 0 0
\(175\) −2.66881 −0.201743
\(176\) 0 0
\(177\) −0.230315 −0.0173115
\(178\) 0 0
\(179\) 2.42499 + 2.42499i 0.181252 + 0.181252i 0.791901 0.610649i \(-0.209091\pi\)
−0.610649 + 0.791901i \(0.709091\pi\)
\(180\) 0 0
\(181\) 4.46593 4.46593i 0.331950 0.331950i −0.521377 0.853327i \(-0.674581\pi\)
0.853327 + 0.521377i \(0.174581\pi\)
\(182\) 0 0
\(183\) 1.55970i 0.115296i
\(184\) 0 0
\(185\) 0.137894i 0.0101382i
\(186\) 0 0
\(187\) 6.47376 6.47376i 0.473408 0.473408i
\(188\) 0 0
\(189\) −1.91246 1.91246i −0.139111 0.139111i
\(190\) 0 0
\(191\) −7.75030 −0.560792 −0.280396 0.959884i \(-0.590466\pi\)
−0.280396 + 0.959884i \(0.590466\pi\)
\(192\) 0 0
\(193\) −11.3388 −0.816181 −0.408091 0.912941i \(-0.633805\pi\)
−0.408091 + 0.912941i \(0.633805\pi\)
\(194\) 0 0
\(195\) −0.499094 0.499094i −0.0357409 0.0357409i
\(196\) 0 0
\(197\) 1.10001 1.10001i 0.0783725 0.0783725i −0.666834 0.745206i \(-0.732351\pi\)
0.745206 + 0.666834i \(0.232351\pi\)
\(198\) 0 0
\(199\) 14.2722i 1.01173i −0.862614 0.505864i \(-0.831174\pi\)
0.862614 0.505864i \(-0.168826\pi\)
\(200\) 0 0
\(201\) 0.839321i 0.0592012i
\(202\) 0 0
\(203\) −9.33843 + 9.33843i −0.655429 + 0.655429i
\(204\) 0 0
\(205\) 7.21620 + 7.21620i 0.504001 + 0.504001i
\(206\) 0 0
\(207\) 2.10149 0.146064
\(208\) 0 0
\(209\) 24.1203 1.66843
\(210\) 0 0
\(211\) 12.4716 + 12.4716i 0.858577 + 0.858577i 0.991171 0.132593i \(-0.0423305\pi\)
−0.132593 + 0.991171i \(0.542330\pi\)
\(212\) 0 0
\(213\) 0.944203 0.944203i 0.0646957 0.0646957i
\(214\) 0 0
\(215\) 6.27690i 0.428081i
\(216\) 0 0
\(217\) 18.2619i 1.23970i
\(218\) 0 0
\(219\) −1.87890 + 1.87890i −0.126965 + 0.126965i
\(220\) 0 0
\(221\) 5.44374 + 5.44374i 0.366186 + 0.366186i
\(222\) 0 0
\(223\) −3.08673 −0.206703 −0.103351 0.994645i \(-0.532957\pi\)
−0.103351 + 0.994645i \(0.532957\pi\)
\(224\) 0 0
\(225\) −2.97120 −0.198080
\(226\) 0 0
\(227\) −8.31678 8.31678i −0.552004 0.552004i 0.375015 0.927019i \(-0.377638\pi\)
−0.927019 + 0.375015i \(0.877638\pi\)
\(228\) 0 0
\(229\) −9.98910 + 9.98910i −0.660098 + 0.660098i −0.955403 0.295305i \(-0.904579\pi\)
0.295305 + 0.955403i \(0.404579\pi\)
\(230\) 0 0
\(231\) 2.24014i 0.147390i
\(232\) 0 0
\(233\) 13.9015i 0.910718i −0.890308 0.455359i \(-0.849511\pi\)
0.890308 0.455359i \(-0.150489\pi\)
\(234\) 0 0
\(235\) −1.33946 + 1.33946i −0.0873768 + 0.0873768i
\(236\) 0 0
\(237\) 0.804314 + 0.804314i 0.0522458 + 0.0522458i
\(238\) 0 0
\(239\) 10.7687 0.696569 0.348284 0.937389i \(-0.386764\pi\)
0.348284 + 0.937389i \(0.386764\pi\)
\(240\) 0 0
\(241\) −12.4707 −0.803305 −0.401653 0.915792i \(-0.631564\pi\)
−0.401653 + 0.915792i \(0.631564\pi\)
\(242\) 0 0
\(243\) −3.19886 3.19886i −0.205207 0.205207i
\(244\) 0 0
\(245\) 0.0866638 0.0866638i 0.00553675 0.00553675i
\(246\) 0 0
\(247\) 20.2826i 1.29055i
\(248\) 0 0
\(249\) 0.929654i 0.0589144i
\(250\) 0 0
\(251\) −3.69093 + 3.69093i −0.232969 + 0.232969i −0.813931 0.580962i \(-0.802677\pi\)
0.580962 + 0.813931i \(0.302677\pi\)
\(252\) 0 0
\(253\) 2.47349 + 2.47349i 0.155507 + 0.155507i
\(254\) 0 0
\(255\) −0.314175 −0.0196744
\(256\) 0 0
\(257\) 3.11011 0.194003 0.0970016 0.995284i \(-0.469075\pi\)
0.0970016 + 0.995284i \(0.469075\pi\)
\(258\) 0 0
\(259\) 0.260225 + 0.260225i 0.0161696 + 0.0161696i
\(260\) 0 0
\(261\) −10.3965 + 10.3965i −0.643527 + 0.643527i
\(262\) 0 0
\(263\) 17.9512i 1.10692i −0.832877 0.553458i \(-0.813308\pi\)
0.832877 0.553458i \(-0.186692\pi\)
\(264\) 0 0
\(265\) 10.5203i 0.646257i
\(266\) 0 0
\(267\) −1.26075 + 1.26075i −0.0771568 + 0.0771568i
\(268\) 0 0
\(269\) 1.62436 + 1.62436i 0.0990392 + 0.0990392i 0.754890 0.655851i \(-0.227690\pi\)
−0.655851 + 0.754890i \(0.727690\pi\)
\(270\) 0 0
\(271\) −18.1808 −1.10440 −0.552201 0.833711i \(-0.686212\pi\)
−0.552201 + 0.833711i \(0.686212\pi\)
\(272\) 0 0
\(273\) 1.88372 0.114008
\(274\) 0 0
\(275\) −3.49714 3.49714i −0.210886 0.210886i
\(276\) 0 0
\(277\) −13.8675 + 13.8675i −0.833218 + 0.833218i −0.987956 0.154737i \(-0.950547\pi\)
0.154737 + 0.987956i \(0.450547\pi\)
\(278\) 0 0
\(279\) 20.3310i 1.21719i
\(280\) 0 0
\(281\) 10.7377i 0.640556i 0.947324 + 0.320278i \(0.103776\pi\)
−0.947324 + 0.320278i \(0.896224\pi\)
\(282\) 0 0
\(283\) 16.3679 16.3679i 0.972971 0.972971i −0.0266735 0.999644i \(-0.508491\pi\)
0.999644 + 0.0266735i \(0.00849143\pi\)
\(284\) 0 0
\(285\) −0.585284 0.585284i −0.0346692 0.0346692i
\(286\) 0 0
\(287\) −27.2359 −1.60768
\(288\) 0 0
\(289\) −13.5732 −0.798425
\(290\) 0 0
\(291\) 0.575968 + 0.575968i 0.0337638 + 0.0337638i
\(292\) 0 0
\(293\) −4.22052 + 4.22052i −0.246566 + 0.246566i −0.819560 0.572994i \(-0.805782\pi\)
0.572994 + 0.819560i \(0.305782\pi\)
\(294\) 0 0
\(295\) 1.35704i 0.0790101i
\(296\) 0 0
\(297\) 5.01208i 0.290831i
\(298\) 0 0
\(299\) −2.07994 + 2.07994i −0.120286 + 0.120286i
\(300\) 0 0
\(301\) 11.8454 + 11.8454i 0.682755 + 0.682755i
\(302\) 0 0
\(303\) 0.0895217 0.00514289
\(304\) 0 0
\(305\) −9.18991 −0.526213
\(306\) 0 0
\(307\) 12.6363 + 12.6363i 0.721190 + 0.721190i 0.968848 0.247658i \(-0.0796608\pi\)
−0.247658 + 0.968848i \(0.579661\pi\)
\(308\) 0 0
\(309\) −1.23625 + 1.23625i −0.0703276 + 0.0703276i
\(310\) 0 0
\(311\) 8.56815i 0.485855i 0.970044 + 0.242928i \(0.0781078\pi\)
−0.970044 + 0.242928i \(0.921892\pi\)
\(312\) 0 0
\(313\) 19.1825i 1.08426i −0.840295 0.542129i \(-0.817618\pi\)
0.840295 0.542129i \(-0.182382\pi\)
\(314\) 0 0
\(315\) 5.60705 5.60705i 0.315921 0.315921i
\(316\) 0 0
\(317\) −9.41764 9.41764i −0.528947 0.528947i 0.391311 0.920258i \(-0.372022\pi\)
−0.920258 + 0.391311i \(0.872022\pi\)
\(318\) 0 0
\(319\) −24.4737 −1.37026
\(320\) 0 0
\(321\) −3.48193 −0.194342
\(322\) 0 0
\(323\) 6.38383 + 6.38383i 0.355206 + 0.355206i
\(324\) 0 0
\(325\) 2.94072 2.94072i 0.163122 0.163122i
\(326\) 0 0
\(327\) 0.191122i 0.0105691i
\(328\) 0 0
\(329\) 5.05549i 0.278718i
\(330\) 0 0
\(331\) −12.8579 + 12.8579i −0.706733 + 0.706733i −0.965847 0.259114i \(-0.916569\pi\)
0.259114 + 0.965847i \(0.416569\pi\)
\(332\) 0 0
\(333\) 0.289709 + 0.289709i 0.0158760 + 0.0158760i
\(334\) 0 0
\(335\) 4.94538 0.270195
\(336\) 0 0
\(337\) 3.31961 0.180831 0.0904153 0.995904i \(-0.471181\pi\)
0.0904153 + 0.995904i \(0.471181\pi\)
\(338\) 0 0
\(339\) 0.101110 + 0.101110i 0.00549156 + 0.00549156i
\(340\) 0 0
\(341\) −23.9300 + 23.9300i −1.29588 + 1.29588i
\(342\) 0 0
\(343\) 18.3546i 0.991055i
\(344\) 0 0
\(345\) 0.120040i 0.00646272i
\(346\) 0 0
\(347\) −17.8860 + 17.8860i −0.960171 + 0.960171i −0.999237 0.0390656i \(-0.987562\pi\)
0.0390656 + 0.999237i \(0.487562\pi\)
\(348\) 0 0
\(349\) −3.68796 3.68796i −0.197412 0.197412i 0.601478 0.798890i \(-0.294579\pi\)
−0.798890 + 0.601478i \(0.794579\pi\)
\(350\) 0 0
\(351\) 4.21463 0.224960
\(352\) 0 0
\(353\) 33.0951 1.76148 0.880738 0.473604i \(-0.157047\pi\)
0.880738 + 0.473604i \(0.157047\pi\)
\(354\) 0 0
\(355\) 5.56335 + 5.56335i 0.295272 + 0.295272i
\(356\) 0 0
\(357\) 0.592891 0.592891i 0.0313791 0.0313791i
\(358\) 0 0
\(359\) 6.52522i 0.344388i 0.985063 + 0.172194i \(0.0550856\pi\)
−0.985063 + 0.172194i \(0.944914\pi\)
\(360\) 0 0
\(361\) 4.78519i 0.251852i
\(362\) 0 0
\(363\) 1.61532 1.61532i 0.0847825 0.0847825i
\(364\) 0 0
\(365\) −11.0707 11.0707i −0.579469 0.579469i
\(366\) 0 0
\(367\) 11.0338 0.575959 0.287980 0.957636i \(-0.407016\pi\)
0.287980 + 0.957636i \(0.407016\pi\)
\(368\) 0 0
\(369\) −30.3218 −1.57849
\(370\) 0 0
\(371\) 19.8532 + 19.8532i 1.03073 + 1.03073i
\(372\) 0 0
\(373\) 6.84468 6.84468i 0.354404 0.354404i −0.507341 0.861745i \(-0.669372\pi\)
0.861745 + 0.507341i \(0.169372\pi\)
\(374\) 0 0
\(375\) 0.169718i 0.00876421i
\(376\) 0 0
\(377\) 20.5797i 1.05991i
\(378\) 0 0
\(379\) 10.1072 10.1072i 0.519171 0.519171i −0.398150 0.917321i \(-0.630347\pi\)
0.917321 + 0.398150i \(0.130347\pi\)
\(380\) 0 0
\(381\) 2.54050 + 2.54050i 0.130154 + 0.130154i
\(382\) 0 0
\(383\) −29.5283 −1.50883 −0.754413 0.656400i \(-0.772078\pi\)
−0.754413 + 0.656400i \(0.772078\pi\)
\(384\) 0 0
\(385\) 13.1992 0.672692
\(386\) 0 0
\(387\) 13.1875 + 13.1875i 0.670356 + 0.670356i
\(388\) 0 0
\(389\) 0.990949 0.990949i 0.0502431 0.0502431i −0.681539 0.731782i \(-0.738689\pi\)
0.731782 + 0.681539i \(0.238689\pi\)
\(390\) 0 0
\(391\) 1.30930i 0.0662142i
\(392\) 0 0
\(393\) 1.12148i 0.0565711i
\(394\) 0 0
\(395\) −4.73911 + 4.73911i −0.238451 + 0.238451i
\(396\) 0 0
\(397\) 17.0024 + 17.0024i 0.853326 + 0.853326i 0.990541 0.137216i \(-0.0438153\pi\)
−0.137216 + 0.990541i \(0.543815\pi\)
\(398\) 0 0
\(399\) 2.20902 0.110589
\(400\) 0 0
\(401\) 26.7791 1.33728 0.668642 0.743585i \(-0.266876\pi\)
0.668642 + 0.743585i \(0.266876\pi\)
\(402\) 0 0
\(403\) −20.1225 20.1225i −1.00238 1.00238i
\(404\) 0 0
\(405\) 6.18124 6.18124i 0.307148 0.307148i
\(406\) 0 0
\(407\) 0.681985i 0.0338048i
\(408\) 0 0
\(409\) 13.1970i 0.652550i −0.945275 0.326275i \(-0.894206\pi\)
0.945275 0.326275i \(-0.105794\pi\)
\(410\) 0 0
\(411\) −1.23417 + 1.23417i −0.0608773 + 0.0608773i
\(412\) 0 0
\(413\) 2.56092 + 2.56092i 0.126015 + 0.126015i
\(414\) 0 0
\(415\) −5.47763 −0.268886
\(416\) 0 0
\(417\) −1.19670 −0.0586026
\(418\) 0 0
\(419\) 9.92468 + 9.92468i 0.484852 + 0.484852i 0.906677 0.421825i \(-0.138610\pi\)
−0.421825 + 0.906677i \(0.638610\pi\)
\(420\) 0 0
\(421\) 15.7930 15.7930i 0.769702 0.769702i −0.208352 0.978054i \(-0.566810\pi\)
0.978054 + 0.208352i \(0.0668100\pi\)
\(422\) 0 0
\(423\) 5.62829i 0.273657i
\(424\) 0 0
\(425\) 1.85116i 0.0897943i
\(426\) 0 0
\(427\) 17.3426 17.3426i 0.839268 0.839268i
\(428\) 0 0
\(429\) 2.46838 + 2.46838i 0.119174 + 0.119174i
\(430\) 0 0
\(431\) 0.285215 0.0137383 0.00686917 0.999976i \(-0.497813\pi\)
0.00686917 + 0.999976i \(0.497813\pi\)
\(432\) 0 0
\(433\) −18.1101 −0.870318 −0.435159 0.900354i \(-0.643308\pi\)
−0.435159 + 0.900354i \(0.643308\pi\)
\(434\) 0 0
\(435\) 0.593860 + 0.593860i 0.0284734 + 0.0284734i
\(436\) 0 0
\(437\) −2.43913 + 2.43913i −0.116679 + 0.116679i
\(438\) 0 0
\(439\) 11.5931i 0.553308i −0.960970 0.276654i \(-0.910774\pi\)
0.960970 0.276654i \(-0.0892256\pi\)
\(440\) 0 0
\(441\) 0.364153i 0.0173406i
\(442\) 0 0
\(443\) 22.6855 22.6855i 1.07782 1.07782i 0.0811145 0.996705i \(-0.474152\pi\)
0.996705 0.0811145i \(-0.0258479\pi\)
\(444\) 0 0
\(445\) −7.42850 7.42850i −0.352145 0.352145i
\(446\) 0 0
\(447\) 2.11094 0.0998440
\(448\) 0 0
\(449\) 12.1999 0.575747 0.287873 0.957669i \(-0.407052\pi\)
0.287873 + 0.957669i \(0.407052\pi\)
\(450\) 0 0
\(451\) −35.6892 35.6892i −1.68054 1.68054i
\(452\) 0 0
\(453\) −2.66225 + 2.66225i −0.125083 + 0.125083i
\(454\) 0 0
\(455\) 11.0991i 0.520333i
\(456\) 0 0
\(457\) 1.70660i 0.0798314i −0.999203 0.0399157i \(-0.987291\pi\)
0.999203 0.0399157i \(-0.0127089\pi\)
\(458\) 0 0
\(459\) 1.32653 1.32653i 0.0619172 0.0619172i
\(460\) 0 0
\(461\) −4.74710 4.74710i −0.221094 0.221094i 0.587865 0.808959i \(-0.299969\pi\)
−0.808959 + 0.587865i \(0.799969\pi\)
\(462\) 0 0
\(463\) 11.1761 0.519398 0.259699 0.965690i \(-0.416377\pi\)
0.259699 + 0.965690i \(0.416377\pi\)
\(464\) 0 0
\(465\) 1.16133 0.0538556
\(466\) 0 0
\(467\) −2.06471 2.06471i −0.0955435 0.0955435i 0.657719 0.753263i \(-0.271521\pi\)
−0.753263 + 0.657719i \(0.771521\pi\)
\(468\) 0 0
\(469\) −9.33260 + 9.33260i −0.430940 + 0.430940i
\(470\) 0 0
\(471\) 0.893315i 0.0411618i
\(472\) 0 0
\(473\) 31.0437i 1.42739i
\(474\) 0 0
\(475\) 3.44856 3.44856i 0.158231 0.158231i
\(476\) 0 0
\(477\) 22.1026 + 22.1026i 1.01201 + 1.01201i
\(478\) 0 0
\(479\) 41.6214 1.90173 0.950864 0.309608i \(-0.100198\pi\)
0.950864 + 0.309608i \(0.100198\pi\)
\(480\) 0 0
\(481\) −0.573477 −0.0261483
\(482\) 0 0
\(483\) 0.226531 + 0.226531i 0.0103075 + 0.0103075i
\(484\) 0 0
\(485\) −3.39367 + 3.39367i −0.154099 + 0.154099i
\(486\) 0 0
\(487\) 8.25627i 0.374127i −0.982348 0.187064i \(-0.940103\pi\)
0.982348 0.187064i \(-0.0598970\pi\)
\(488\) 0 0
\(489\) 0.507950i 0.0229703i
\(490\) 0 0
\(491\) −4.28512 + 4.28512i −0.193385 + 0.193385i −0.797157 0.603772i \(-0.793664\pi\)
0.603772 + 0.797157i \(0.293664\pi\)
\(492\) 0 0
\(493\) −6.47737 6.47737i −0.291726 0.291726i
\(494\) 0 0
\(495\) 14.6947 0.660476
\(496\) 0 0
\(497\) −20.9976 −0.941871
\(498\) 0 0
\(499\) 15.1287 + 15.1287i 0.677253 + 0.677253i 0.959378 0.282125i \(-0.0910392\pi\)
−0.282125 + 0.959378i \(0.591039\pi\)
\(500\) 0 0
\(501\) 2.17941 2.17941i 0.0973689 0.0973689i
\(502\) 0 0
\(503\) 18.6439i 0.831291i −0.909527 0.415646i \(-0.863556\pi\)
0.909527 0.415646i \(-0.136444\pi\)
\(504\) 0 0
\(505\) 0.527472i 0.0234722i
\(506\) 0 0
\(507\) −0.515524 + 0.515524i −0.0228952 + 0.0228952i
\(508\) 0 0
\(509\) 11.6243 + 11.6243i 0.515239 + 0.515239i 0.916127 0.400888i \(-0.131298\pi\)
−0.400888 + 0.916127i \(0.631298\pi\)
\(510\) 0 0
\(511\) 41.7839 1.84841
\(512\) 0 0
\(513\) 4.94246 0.218215
\(514\) 0 0
\(515\) −7.28411 7.28411i −0.320976 0.320976i
\(516\) 0 0
\(517\) 6.62459 6.62459i 0.291349 0.291349i
\(518\) 0 0
\(519\) 2.04865i 0.0899259i
\(520\) 0 0
\(521\) 36.9052i 1.61684i 0.588603 + 0.808422i \(0.299678\pi\)
−0.588603 + 0.808422i \(0.700322\pi\)
\(522\) 0 0
\(523\) −6.04158 + 6.04158i −0.264180 + 0.264180i −0.826750 0.562570i \(-0.809813\pi\)
0.562570 + 0.826750i \(0.309813\pi\)
\(524\) 0 0
\(525\) −0.320281 0.320281i −0.0139782 0.0139782i
\(526\) 0 0
\(527\) −12.6669 −0.551780
\(528\) 0 0
\(529\) 22.4997 0.978250
\(530\) 0 0
\(531\) 2.85108 + 2.85108i 0.123726 + 0.123726i
\(532\) 0 0
\(533\) 30.0108 30.0108i 1.29991 1.29991i
\(534\) 0 0
\(535\) 20.5159i 0.886981i
\(536\) 0 0
\(537\) 0.582041i 0.0251169i
\(538\) 0 0
\(539\) −0.428614 + 0.428614i −0.0184617 + 0.0184617i
\(540\) 0 0
\(541\) −28.4222 28.4222i −1.22197 1.22197i −0.966932 0.255035i \(-0.917913\pi\)
−0.255035 0.966932i \(-0.582087\pi\)
\(542\) 0 0
\(543\) 1.07190 0.0459997
\(544\) 0 0
\(545\) −1.12612 −0.0482375
\(546\) 0 0
\(547\) −23.3562 23.3562i −0.998640 0.998640i 0.00135902 0.999999i \(-0.499567\pi\)
−0.999999 + 0.00135902i \(0.999567\pi\)
\(548\) 0 0
\(549\) 19.3076 19.3076i 0.824027 0.824027i
\(550\) 0 0
\(551\) 24.1337i 1.02813i
\(552\) 0 0
\(553\) 17.8867i 0.760620i
\(554\) 0 0
\(555\) 0.0165485 0.0165485i 0.000702447 0.000702447i
\(556\) 0 0
\(557\) −4.89520 4.89520i −0.207416 0.207416i 0.595752 0.803168i \(-0.296854\pi\)
−0.803168 + 0.595752i \(0.796854\pi\)
\(558\) 0 0
\(559\) −26.1044 −1.10410
\(560\) 0 0
\(561\) 1.55382 0.0656022
\(562\) 0 0
\(563\) −1.28613 1.28613i −0.0542040 0.0542040i 0.679485 0.733689i \(-0.262203\pi\)
−0.733689 + 0.679485i \(0.762203\pi\)
\(564\) 0 0
\(565\) −0.595755 + 0.595755i −0.0250636 + 0.0250636i
\(566\) 0 0
\(567\) 23.3297i 0.979754i
\(568\) 0 0
\(569\) 11.4799i 0.481261i −0.970617 0.240631i \(-0.922646\pi\)
0.970617 0.240631i \(-0.0773543\pi\)
\(570\) 0 0
\(571\) −28.7069 + 28.7069i −1.20134 + 1.20134i −0.227587 + 0.973758i \(0.573083\pi\)
−0.973758 + 0.227587i \(0.926917\pi\)
\(572\) 0 0
\(573\) −0.930105 0.930105i −0.0388557 0.0388557i
\(574\) 0 0
\(575\) 0.707288 0.0294960
\(576\) 0 0
\(577\) −20.3419 −0.846842 −0.423421 0.905933i \(-0.639171\pi\)
−0.423421 + 0.905933i \(0.639171\pi\)
\(578\) 0 0
\(579\) −1.36075 1.36075i −0.0565509 0.0565509i
\(580\) 0 0
\(581\) 10.3370 10.3370i 0.428852 0.428852i
\(582\) 0 0
\(583\) 52.0303i 2.15488i
\(584\) 0 0
\(585\) 12.3566i 0.510884i
\(586\) 0 0
\(587\) 25.8136 25.8136i 1.06544 1.06544i 0.0677360 0.997703i \(-0.478422\pi\)
0.997703 0.0677360i \(-0.0215775\pi\)
\(588\) 0 0
\(589\) −23.5975 23.5975i −0.972320 0.972320i
\(590\) 0 0
\(591\) 0.264022 0.0108604
\(592\) 0 0
\(593\) 4.02945 0.165470 0.0827349 0.996572i \(-0.473635\pi\)
0.0827349 + 0.996572i \(0.473635\pi\)
\(594\) 0 0
\(595\) 3.49338 + 3.49338i 0.143215 + 0.143215i
\(596\) 0 0
\(597\) 1.71279 1.71279i 0.0700997 0.0700997i
\(598\) 0 0
\(599\) 31.6701i 1.29400i 0.762489 + 0.647002i \(0.223977\pi\)
−0.762489 + 0.647002i \(0.776023\pi\)
\(600\) 0 0
\(601\) 19.4667i 0.794065i 0.917805 + 0.397032i \(0.129960\pi\)
−0.917805 + 0.397032i \(0.870040\pi\)
\(602\) 0 0
\(603\) −10.3900 + 10.3900i −0.423114 + 0.423114i
\(604\) 0 0
\(605\) 9.51768 + 9.51768i 0.386949 + 0.386949i
\(606\) 0 0
\(607\) −13.6128 −0.552528 −0.276264 0.961082i \(-0.589096\pi\)
−0.276264 + 0.961082i \(0.589096\pi\)
\(608\) 0 0
\(609\) −2.24139 −0.0908257
\(610\) 0 0
\(611\) 5.57057 + 5.57057i 0.225361 + 0.225361i
\(612\) 0 0
\(613\) −11.1480 + 11.1480i −0.450265 + 0.450265i −0.895442 0.445177i \(-0.853141\pi\)
0.445177 + 0.895442i \(0.353141\pi\)
\(614\) 0 0
\(615\) 1.73202i 0.0698416i
\(616\) 0 0
\(617\) 1.96695i 0.0791863i 0.999216 + 0.0395932i \(0.0126062\pi\)
−0.999216 + 0.0395932i \(0.987394\pi\)
\(618\) 0 0
\(619\) −7.84144 + 7.84144i −0.315174 + 0.315174i −0.846910 0.531736i \(-0.821540\pi\)
0.531736 + 0.846910i \(0.321540\pi\)
\(620\) 0 0
\(621\) 0.506840 + 0.506840i 0.0203388 + 0.0203388i
\(622\) 0 0
\(623\) 28.0372 1.12329
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 2.89464 + 2.89464i 0.115601 + 0.115601i
\(628\) 0 0
\(629\) −0.180499 + 0.180499i −0.00719696 + 0.00719696i
\(630\) 0 0
\(631\) 0.220729i 0.00878708i −0.999990 0.00439354i \(-0.998601\pi\)
0.999990 0.00439354i \(-0.00139851\pi\)
\(632\) 0 0
\(633\) 2.99339i 0.118977i
\(634\) 0 0
\(635\) −14.9689 + 14.9689i −0.594024 + 0.594024i
\(636\) 0 0
\(637\) −0.360418 0.360418i −0.0142803 0.0142803i
\(638\) 0 0
\(639\) −23.3767 −0.924767
\(640\) 0 0
\(641\) −19.2037 −0.758502 −0.379251 0.925294i \(-0.623818\pi\)
−0.379251 + 0.925294i \(0.623818\pi\)
\(642\) 0 0
\(643\) 7.17110 + 7.17110i 0.282801 + 0.282801i 0.834225 0.551424i \(-0.185915\pi\)
−0.551424 + 0.834225i \(0.685915\pi\)
\(644\) 0 0
\(645\) 0.753283 0.753283i 0.0296605 0.0296605i
\(646\) 0 0
\(647\) 26.4735i 1.04078i 0.853928 + 0.520391i \(0.174214\pi\)
−0.853928 + 0.520391i \(0.825786\pi\)
\(648\) 0 0
\(649\) 6.71153i 0.263451i
\(650\) 0 0
\(651\) −2.19159 + 2.19159i −0.0858953 + 0.0858953i
\(652\) 0 0
\(653\) 10.5746 + 10.5746i 0.413815 + 0.413815i 0.883065 0.469250i \(-0.155476\pi\)
−0.469250 + 0.883065i \(0.655476\pi\)
\(654\) 0 0
\(655\) 6.60789 0.258192
\(656\) 0 0
\(657\) 46.5182 1.81485
\(658\) 0 0
\(659\) −24.1291 24.1291i −0.939937 0.939937i 0.0583584 0.998296i \(-0.481413\pi\)
−0.998296 + 0.0583584i \(0.981413\pi\)
\(660\) 0 0
\(661\) 23.4294 23.4294i 0.911299 0.911299i −0.0850756 0.996374i \(-0.527113\pi\)
0.996374 + 0.0850756i \(0.0271132\pi\)
\(662\) 0 0
\(663\) 1.30659i 0.0507439i
\(664\) 0 0
\(665\) 13.0158i 0.504732i
\(666\) 0 0
\(667\) 2.47487 2.47487i 0.0958273 0.0958273i
\(668\) 0 0
\(669\) −0.370435 0.370435i −0.0143218 0.0143218i
\(670\) 0 0
\(671\) 45.4506 1.75460
\(672\) 0 0
\(673\) −41.8069 −1.61154 −0.805769 0.592230i \(-0.798248\pi\)
−0.805769 + 0.592230i \(0.798248\pi\)
\(674\) 0 0
\(675\) −0.716597 0.716597i −0.0275818 0.0275818i
\(676\) 0 0
\(677\) −22.7350 + 22.7350i −0.873776 + 0.873776i −0.992882 0.119106i \(-0.961997\pi\)
0.119106 + 0.992882i \(0.461997\pi\)
\(678\) 0 0
\(679\) 12.8086i 0.491550i
\(680\) 0 0
\(681\) 1.99617i 0.0764936i
\(682\) 0 0
\(683\) −34.4402 + 34.4402i −1.31782 + 1.31782i −0.402315 + 0.915501i \(0.631794\pi\)
−0.915501 + 0.402315i \(0.868206\pi\)
\(684\) 0 0
\(685\) −7.27190 7.27190i −0.277845 0.277845i
\(686\) 0 0
\(687\) −2.39756 −0.0914727
\(688\) 0 0
\(689\) −43.7519 −1.66682
\(690\) 0 0
\(691\) −16.0991 16.0991i −0.612438 0.612438i 0.331143 0.943581i \(-0.392566\pi\)
−0.943581 + 0.331143i \(0.892566\pi\)
\(692\) 0 0
\(693\) −27.7308 + 27.7308i −1.05341 + 1.05341i
\(694\) 0 0
\(695\) 7.05110i 0.267463i
\(696\) 0 0
\(697\) 18.8915i 0.715567i
\(698\) 0 0
\(699\) 1.66830 1.66830i 0.0631010 0.0631010i
\(700\) 0 0
\(701\) −30.0507 30.0507i −1.13500 1.13500i −0.989334 0.145666i \(-0.953467\pi\)
−0.145666 0.989334i \(-0.546533\pi\)
\(702\) 0 0
\(703\) −0.672512 −0.0253643
\(704\) 0 0
\(705\) −0.321495 −0.0121082
\(706\) 0 0
\(707\) −0.995412 0.995412i −0.0374363 0.0374363i
\(708\) 0 0
\(709\) −12.9188 + 12.9188i −0.485176 + 0.485176i −0.906780 0.421604i \(-0.861467\pi\)
0.421604 + 0.906780i \(0.361467\pi\)
\(710\) 0 0
\(711\) 19.9133i 0.746807i
\(712\) 0 0
\(713\) 4.83977i 0.181251i
\(714\) 0 0
\(715\) −14.5440 + 14.5440i −0.543913 + 0.543913i
\(716\) 0 0
\(717\) 1.29234 + 1.29234i 0.0482633 + 0.0482633i
\(718\) 0 0
\(719\) −17.0356 −0.635319 −0.317659 0.948205i \(-0.602897\pi\)
−0.317659 + 0.948205i \(0.602897\pi\)
\(720\) 0 0
\(721\) 27.4922 1.02386
\(722\) 0 0
\(723\) −1.49659 1.49659i −0.0556588 0.0556588i
\(724\) 0 0
\(725\) −3.49909 + 3.49909i −0.129953 + 0.129953i
\(726\) 0 0
\(727\) 31.7051i 1.17588i −0.808905 0.587939i \(-0.799939\pi\)
0.808905 0.587939i \(-0.200061\pi\)
\(728\) 0 0
\(729\) 25.4570i 0.942852i
\(730\) 0 0
\(731\) −8.21624 + 8.21624i −0.303888 + 0.303888i
\(732\) 0 0
\(733\) 3.87657 + 3.87657i 0.143184 + 0.143184i 0.775065 0.631881i \(-0.217717\pi\)
−0.631881 + 0.775065i \(0.717717\pi\)
\(734\) 0 0
\(735\) 0.0208008 0.000767251
\(736\) 0 0
\(737\) −24.4584 −0.900937
\(738\) 0 0
\(739\) −11.3024 11.3024i −0.415766 0.415766i 0.467975 0.883742i \(-0.344984\pi\)
−0.883742 + 0.467975i \(0.844984\pi\)
\(740\) 0 0
\(741\) −2.43409 + 2.43409i −0.0894184 + 0.0894184i
\(742\) 0 0
\(743\) 30.7210i 1.12704i 0.826102 + 0.563521i \(0.190554\pi\)
−0.826102 + 0.563521i \(0.809446\pi\)
\(744\) 0 0
\(745\) 12.4379i 0.455690i
\(746\) 0 0
\(747\) 11.5082 11.5082i 0.421065 0.421065i
\(748\) 0 0
\(749\) 38.7163 + 38.7163i 1.41466 + 1.41466i
\(750\) 0 0
\(751\) −16.4695 −0.600981 −0.300491 0.953785i \(-0.597150\pi\)
−0.300491 + 0.953785i \(0.597150\pi\)
\(752\) 0 0
\(753\) −0.885888 −0.0322836
\(754\) 0 0
\(755\) −15.6863 15.6863i −0.570883 0.570883i
\(756\) 0 0
\(757\) 21.9737 21.9737i 0.798649 0.798649i −0.184233 0.982883i \(-0.558980\pi\)
0.982883 + 0.184233i \(0.0589802\pi\)
\(758\) 0 0
\(759\) 0.593681i 0.0215493i
\(760\) 0 0
\(761\) 5.91749i 0.214509i −0.994232 0.107254i \(-0.965794\pi\)
0.994232 0.107254i \(-0.0342060\pi\)
\(762\) 0 0
\(763\) 2.12513 2.12513i 0.0769349 0.0769349i
\(764\) 0 0
\(765\) 3.88919 + 3.88919i 0.140614 + 0.140614i
\(766\) 0 0
\(767\) −5.64368 −0.203782
\(768\) 0 0
\(769\) 10.7206 0.386596 0.193298 0.981140i \(-0.438082\pi\)
0.193298 + 0.981140i \(0.438082\pi\)
\(770\) 0 0
\(771\) 0.373240 + 0.373240i 0.0134419 + 0.0134419i
\(772\) 0 0
\(773\) −16.8329 + 16.8329i −0.605438 + 0.605438i −0.941750 0.336312i \(-0.890820\pi\)
0.336312 + 0.941750i \(0.390820\pi\)
\(774\) 0 0
\(775\) 6.84272i 0.245798i
\(776\) 0 0
\(777\) 0.0624587i 0.00224069i
\(778\) 0 0
\(779\) 35.1934 35.1934i 1.26094 1.26094i
\(780\) 0 0
\(781\) −27.5147 27.5147i −0.984554 0.984554i
\(782\) 0 0
\(783\) −5.01488 −0.179217
\(784\) 0 0
\(785\) 5.26352 0.187863
\(786\) 0 0
\(787\) 25.4619 + 25.4619i 0.907619 + 0.907619i 0.996080 0.0884603i \(-0.0281946\pi\)
−0.0884603 + 0.996080i \(0.528195\pi\)
\(788\) 0 0
\(789\) 2.15430 2.15430i 0.0766951 0.0766951i
\(790\) 0 0
\(791\) 2.24854i 0.0799489i
\(792\) 0 0
\(793\) 38.2191i 1.35720i
\(794\) 0 0
\(795\) 1.26253 1.26253i 0.0447773 0.0447773i
\(796\) 0 0
\(797\) 7.14518 + 7.14518i 0.253095 + 0.253095i 0.822238 0.569143i \(-0.192725\pi\)
−0.569143 + 0.822238i \(0.692725\pi\)
\(798\) 0 0
\(799\) 3.50662 0.124055
\(800\) 0 0
\(801\) 31.2139 1.10289
\(802\) 0 0
\(803\) 54.7526 + 54.7526i 1.93218 + 1.93218i
\(804\) 0 0
\(805\) −1.33475 + 1.33475i −0.0470437 + 0.0470437i
\(806\) 0 0
\(807\) 0.389876i 0.0137243i
\(808\) 0 0
\(809\) 12.4413i 0.437412i 0.975791 + 0.218706i \(0.0701835\pi\)
−0.975791 + 0.218706i \(0.929817\pi\)
\(810\) 0 0
\(811\) 30.6494 30.6494i 1.07624 1.07624i 0.0794022 0.996843i \(-0.474699\pi\)
0.996843 0.0794022i \(-0.0253011\pi\)
\(812\) 0 0
\(813\) −2.18185 2.18185i −0.0765210 0.0765210i
\(814\) 0 0
\(815\) 2.99290 0.104837
\(816\) 0 0
\(817\) −30.6125 −1.07099
\(818\) 0 0
\(819\) −23.3187 23.3187i −0.814820 0.814820i
\(820\) 0 0
\(821\) 8.84907 8.84907i 0.308835 0.308835i −0.535623 0.844457i \(-0.679923\pi\)
0.844457 + 0.535623i \(0.179923\pi\)
\(822\) 0 0
\(823\) 11.7501i 0.409583i −0.978806 0.204792i \(-0.934348\pi\)
0.978806 0.204792i \(-0.0656516\pi\)
\(824\) 0 0
\(825\) 0.839377i 0.0292233i
\(826\) 0 0
\(827\) 3.40407 3.40407i 0.118371 0.118371i −0.645440 0.763811i \(-0.723326\pi\)
0.763811 + 0.645440i \(0.223326\pi\)
\(828\) 0 0
\(829\) 24.8718 + 24.8718i 0.863834 + 0.863834i 0.991781 0.127947i \(-0.0408389\pi\)
−0.127947 + 0.991781i \(0.540839\pi\)
\(830\) 0 0
\(831\) −3.32845 −0.115463
\(832\) 0 0
\(833\) −0.226880 −0.00786092
\(834\) 0 0
\(835\) 12.8414 + 12.8414i 0.444393 + 0.444393i
\(836\) 0 0
\(837\) −4.90347 + 4.90347i −0.169489 + 0.169489i
\(838\) 0 0
\(839\) 3.26196i 0.112615i −0.998413 0.0563076i \(-0.982067\pi\)
0.998413 0.0563076i \(-0.0179328\pi\)
\(840\) 0 0
\(841\) 4.51268i 0.155610i
\(842\) 0 0
\(843\) −1.28862 + 1.28862i −0.0443823 + 0.0443823i
\(844\) 0 0
\(845\) −3.03753 3.03753i −0.104494 0.104494i
\(846\) 0 0
\(847\) −35.9223 −1.23430
\(848\) 0 0
\(849\) 3.92859 0.134829
\(850\) 0 0
\(851\) −0.0689649 0.0689649i −0.00236409 0.00236409i
\(852\) 0 0
\(853\) 4.02276 4.02276i 0.137737 0.137737i −0.634877 0.772613i \(-0.718949\pi\)
0.772613 + 0.634877i \(0.218949\pi\)
\(854\) 0 0
\(855\) 14.4905i 0.495566i
\(856\) 0 0
\(857\) 44.6563i 1.52543i −0.646736 0.762714i \(-0.723866\pi\)
0.646736 0.762714i \(-0.276134\pi\)
\(858\) 0 0
\(859\) −5.22864 + 5.22864i −0.178399 + 0.178399i −0.790658 0.612259i \(-0.790261\pi\)
0.612259 + 0.790658i \(0.290261\pi\)
\(860\) 0 0
\(861\) −3.26855 3.26855i −0.111392 0.111392i
\(862\) 0 0
\(863\) −36.9653 −1.25831 −0.629157 0.777278i \(-0.716600\pi\)
−0.629157 + 0.777278i \(0.716600\pi\)
\(864\) 0 0
\(865\) 12.0709 0.410423
\(866\) 0 0
\(867\) −1.62891 1.62891i −0.0553206 0.0553206i
\(868\) 0 0
\(869\) 23.4383 23.4383i 0.795089 0.795089i
\(870\) 0 0
\(871\) 20.5669i 0.696883i
\(872\) 0 0
\(873\) 14.2599i 0.482624i
\(874\) 0 0
\(875\) 1.88714 1.88714i 0.0637968 0.0637968i
\(876\) 0 0
\(877\) 9.40192 + 9.40192i 0.317480 + 0.317480i 0.847799 0.530318i \(-0.177928\pi\)
−0.530318 + 0.847799i \(0.677928\pi\)
\(878\) 0 0
\(879\) −1.01300 −0.0341677
\(880\) 0 0
\(881\) −10.3069 −0.347248 −0.173624 0.984812i \(-0.555548\pi\)
−0.173624 + 0.984812i \(0.555548\pi\)
\(882\) 0 0
\(883\) −34.3375 34.3375i −1.15555 1.15555i −0.985422 0.170128i \(-0.945582\pi\)
−0.170128 0.985422i \(-0.554418\pi\)
\(884\) 0 0
\(885\) 0.162857 0.162857i 0.00547438 0.00547438i
\(886\) 0 0
\(887\) 6.79523i 0.228161i −0.993471 0.114081i \(-0.963608\pi\)
0.993471 0.114081i \(-0.0363922\pi\)
\(888\) 0 0
\(889\) 56.4968i 1.89484i
\(890\) 0 0
\(891\) −30.5706 + 30.5706i −1.02415 + 1.02415i
\(892\) 0 0
\(893\) 6.53256 + 6.53256i 0.218604 + 0.218604i
\(894\) 0 0
\(895\) −3.42945 −0.114634
\(896\) 0 0
\(897\) −0.499223 −0.0166686
\(898\) 0 0
\(899\) 23.9433 + 23.9433i 0.798554 + 0.798554i
\(900\) 0 0
\(901\) −13.7707 + 13.7707i −0.458769 + 0.458769i
\(902\) 0 0
\(903\) 2.84310i 0.0946123i
\(904\) 0 0
\(905\) 6.31578i 0.209944i
\(906\) 0 0
\(907\) 3.82391 3.82391i 0.126971 0.126971i −0.640766 0.767737i \(-0.721383\pi\)
0.767737 + 0.640766i \(0.221383\pi\)
\(908\) 0 0
\(909\) −1.10819 1.10819i −0.0367565 0.0367565i
\(910\) 0 0
\(911\) 18.9169 0.626743 0.313372 0.949631i \(-0.398541\pi\)
0.313372 + 0.949631i \(0.398541\pi\)
\(912\) 0 0
\(913\) 27.0908 0.896574
\(914\) 0 0
\(915\) −1.10287 1.10287i −0.0364598 0.0364598i
\(916\) 0 0
\(917\) −12.4700 + 12.4700i −0.411795 + 0.411795i
\(918\) 0 0
\(919\) 48.9075i 1.61331i −0.591022 0.806655i \(-0.701276\pi\)
0.591022 0.806655i \(-0.298724\pi\)
\(920\) 0 0
\(921\) 3.03293i 0.0999384i
\(922\) 0 0
\(923\) 23.1369 23.1369i 0.761562 0.761562i
\(924\) 0 0
\(925\) 0.0975060 + 0.0975060i 0.00320598 + 0.00320598i
\(926\) 0 0
\(927\) 30.6071 1.00527
\(928\) 0 0
\(929\) 35.4660 1.16360 0.581801 0.813331i \(-0.302348\pi\)
0.581801 + 0.813331i \(0.302348\pi\)
\(930\) 0 0
\(931\) −0.422660 0.422660i −0.0138521 0.0138521i
\(932\) 0 0
\(933\) −1.02825 + 1.02825i −0.0336635 + 0.0336635i
\(934\) 0 0
\(935\) 9.15528i 0.299410i
\(936\) 0 0
\(937\) 56.4991i 1.84575i −0.385105 0.922873i \(-0.625835\pi\)
0.385105 0.922873i \(-0.374165\pi\)
\(938\) 0 0
\(939\) 2.30207 2.30207i 0.0751252 0.0751252i
\(940\) 0 0
\(941\) −2.86034 2.86034i −0.0932445 0.0932445i 0.658946 0.752190i \(-0.271003\pi\)
−0.752190 + 0.658946i \(0.771003\pi\)
\(942\) 0 0
\(943\) 7.21805 0.235052
\(944\) 0 0
\(945\) 2.70463 0.0879816
\(946\) 0 0
\(947\) −5.86681 5.86681i −0.190646 0.190646i 0.605329 0.795975i \(-0.293041\pi\)
−0.795975 + 0.605329i \(0.793041\pi\)
\(948\) 0 0
\(949\) −46.0411 + 46.0411i −1.49456 + 1.49456i
\(950\) 0 0
\(951\) 2.26040i 0.0732985i
\(952\) 0 0
\(953\) 25.0238i 0.810599i −0.914184 0.405299i \(-0.867167\pi\)
0.914184 0.405299i \(-0.132833\pi\)
\(954\) 0 0
\(955\) 5.48029 5.48029i 0.177338 0.177338i
\(956\) 0 0
\(957\) −2.93706 2.93706i −0.0949416 0.0949416i
\(958\) 0 0
\(959\) 27.4461 0.886281
\(960\) 0 0
\(961\) 15.8228 0.510412
\(962\) 0 0
\(963\) 43.1030 + 43.1030i 1.38897 + 1.38897i
\(964\) 0 0
\(965\) 8.01771 8.01771i 0.258099 0.258099i
\(966\) 0 0
\(967\) 16.6523i 0.535502i 0.963488 + 0.267751i \(0.0862804\pi\)
−0.963488 + 0.267751i \(0.913720\pi\)
\(968\) 0 0
\(969\) 1.53223i 0.0492224i
\(970\) 0 0
\(971\) 30.6552 30.6552i 0.983771 0.983771i −0.0160991 0.999870i \(-0.505125\pi\)
0.999870 + 0.0160991i \(0.00512472\pi\)
\(972\) 0 0
\(973\) 13.3064 + 13.3064i 0.426583 + 0.426583i
\(974\) 0 0
\(975\) 0.705826 0.0226045
\(976\) 0 0
\(977\) −13.4307 −0.429687 −0.214844 0.976648i \(-0.568924\pi\)
−0.214844 + 0.976648i \(0.568924\pi\)
\(978\) 0 0
\(979\) 36.7392 + 36.7392i 1.17419 + 1.17419i
\(980\) 0 0
\(981\) 2.36591 2.36591i 0.0755378 0.0755378i
\(982\) 0 0
\(983\) 7.94549i 0.253422i 0.991940 + 0.126711i \(0.0404420\pi\)
−0.991940 + 0.126711i \(0.959558\pi\)
\(984\) 0 0
\(985\) 1.55565i 0.0495672i
\(986\) 0 0
\(987\) 0.606704 0.606704i 0.0193116 0.0193116i
\(988\) 0 0
\(989\) −3.13925 3.13925i −0.0998225 0.0998225i
\(990\) 0 0
\(991\) 25.0787 0.796652 0.398326 0.917244i \(-0.369591\pi\)
0.398326 + 0.917244i \(0.369591\pi\)
\(992\) 0 0
\(993\) −3.08612 −0.0979350
\(994\) 0 0
\(995\) 10.0919 + 10.0919i 0.319936 + 0.319936i
\(996\) 0 0
\(997\) 36.1819 36.1819i 1.14589 1.14589i 0.158539 0.987353i \(-0.449322\pi\)
0.987353 0.158539i \(-0.0506783\pi\)
\(998\) 0 0
\(999\) 0.139745i 0.00442134i
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 320.2.l.a.81.5 16
3.2 odd 2 2880.2.t.c.721.6 16
4.3 odd 2 80.2.l.a.61.7 yes 16
5.2 odd 4 1600.2.q.g.849.5 16
5.3 odd 4 1600.2.q.h.849.4 16
5.4 even 2 1600.2.l.i.401.4 16
8.3 odd 2 640.2.l.b.161.5 16
8.5 even 2 640.2.l.a.161.4 16
12.11 even 2 720.2.t.c.541.2 16
16.3 odd 4 640.2.l.b.481.5 16
16.5 even 4 inner 320.2.l.a.241.5 16
16.11 odd 4 80.2.l.a.21.7 16
16.13 even 4 640.2.l.a.481.4 16
20.3 even 4 400.2.q.g.349.3 16
20.7 even 4 400.2.q.h.349.6 16
20.19 odd 2 400.2.l.h.301.2 16
32.5 even 8 5120.2.a.u.1.4 8
32.11 odd 8 5120.2.a.v.1.4 8
32.21 even 8 5120.2.a.t.1.5 8
32.27 odd 8 5120.2.a.s.1.5 8
48.5 odd 4 2880.2.t.c.2161.7 16
48.11 even 4 720.2.t.c.181.2 16
80.27 even 4 400.2.q.g.149.3 16
80.37 odd 4 1600.2.q.h.49.4 16
80.43 even 4 400.2.q.h.149.6 16
80.53 odd 4 1600.2.q.g.49.5 16
80.59 odd 4 400.2.l.h.101.2 16
80.69 even 4 1600.2.l.i.1201.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 16 16.11 odd 4
80.2.l.a.61.7 yes 16 4.3 odd 2
320.2.l.a.81.5 16 1.1 even 1 trivial
320.2.l.a.241.5 16 16.5 even 4 inner
400.2.l.h.101.2 16 80.59 odd 4
400.2.l.h.301.2 16 20.19 odd 2
400.2.q.g.149.3 16 80.27 even 4
400.2.q.g.349.3 16 20.3 even 4
400.2.q.h.149.6 16 80.43 even 4
400.2.q.h.349.6 16 20.7 even 4
640.2.l.a.161.4 16 8.5 even 2
640.2.l.a.481.4 16 16.13 even 4
640.2.l.b.161.5 16 8.3 odd 2
640.2.l.b.481.5 16 16.3 odd 4
720.2.t.c.181.2 16 48.11 even 4
720.2.t.c.541.2 16 12.11 even 2
1600.2.l.i.401.4 16 5.4 even 2
1600.2.l.i.1201.4 16 80.69 even 4
1600.2.q.g.49.5 16 80.53 odd 4
1600.2.q.g.849.5 16 5.2 odd 4
1600.2.q.h.49.4 16 80.37 odd 4
1600.2.q.h.849.4 16 5.3 odd 4
2880.2.t.c.721.6 16 3.2 odd 2
2880.2.t.c.2161.7 16 48.5 odd 4
5120.2.a.s.1.5 8 32.27 odd 8
5120.2.a.t.1.5 8 32.21 even 8
5120.2.a.u.1.4 8 32.5 even 8
5120.2.a.v.1.4 8 32.11 odd 8