Properties

Label 768.2.j.f.577.2
Level $768$
Weight $2$
Character 768.577
Analytic conductor $6.133$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(193,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.j (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 577.2
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 768.577
Dual form 768.2.j.f.193.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{3} +(2.73205 - 2.73205i) q^{5} +2.44949i q^{7} +1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 - 0.707107i) q^{3} +(2.73205 - 2.73205i) q^{5} +2.44949i q^{7} +1.00000i q^{9} +(1.03528 - 1.03528i) q^{11} +(3.00000 + 3.00000i) q^{13} -3.86370 q^{15} +3.46410 q^{17} +(5.27792 + 5.27792i) q^{19} +(1.73205 - 1.73205i) q^{21} -2.82843i q^{23} -9.92820i q^{25} +(0.707107 - 0.707107i) q^{27} +(-6.19615 - 6.19615i) q^{29} -7.34847 q^{31} -1.46410 q^{33} +(6.69213 + 6.69213i) q^{35} +(0.464102 - 0.464102i) q^{37} -4.24264i q^{39} +4.53590i q^{41} +(2.44949 - 2.44949i) q^{43} +(2.73205 + 2.73205i) q^{45} -2.82843 q^{47} +1.00000 q^{49} +(-2.44949 - 2.44949i) q^{51} +(3.26795 - 3.26795i) q^{53} -5.65685i q^{55} -7.46410i q^{57} +(9.79796 - 9.79796i) q^{59} +(0.464102 + 0.464102i) q^{61} -2.44949 q^{63} +16.3923 q^{65} +(-10.5558 - 10.5558i) q^{67} +(-2.00000 + 2.00000i) q^{69} +6.41473i q^{71} +4.00000i q^{73} +(-7.02030 + 7.02030i) q^{75} +(2.53590 + 2.53590i) q^{77} -2.44949 q^{79} -1.00000 q^{81} +(3.86370 + 3.86370i) q^{83} +(9.46410 - 9.46410i) q^{85} +8.76268i q^{87} +4.92820i q^{89} +(-7.34847 + 7.34847i) q^{91} +(5.19615 + 5.19615i) q^{93} +28.8391 q^{95} +1.07180 q^{97} +(1.03528 + 1.03528i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 24 q^{13} - 8 q^{29} + 16 q^{33} - 24 q^{37} + 8 q^{45} + 8 q^{49} + 40 q^{53} - 24 q^{61} + 48 q^{65} - 16 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −0.707107 0.707107i −0.408248 0.408248i
\(4\) 0 0
\(5\) 2.73205 2.73205i 1.22181 1.22181i 0.254822 0.966988i \(-0.417983\pi\)
0.966988 0.254822i \(-0.0820170\pi\)
\(6\) 0 0
\(7\) 2.44949i 0.925820i 0.886405 + 0.462910i \(0.153195\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.03528 1.03528i 0.312148 0.312148i −0.533594 0.845741i \(-0.679159\pi\)
0.845741 + 0.533594i \(0.179159\pi\)
\(12\) 0 0
\(13\) 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 0.155747i \(-0.0497784\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 0 0
\(15\) −3.86370 −0.997604
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 5.27792 + 5.27792i 1.21084 + 1.21084i 0.970752 + 0.240085i \(0.0771754\pi\)
0.240085 + 0.970752i \(0.422825\pi\)
\(20\) 0 0
\(21\) 1.73205 1.73205i 0.377964 0.377964i
\(22\) 0 0
\(23\) 2.82843i 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) 9.92820i 1.98564i
\(26\) 0 0
\(27\) 0.707107 0.707107i 0.136083 0.136083i
\(28\) 0 0
\(29\) −6.19615 6.19615i −1.15060 1.15060i −0.986433 0.164163i \(-0.947508\pi\)
−0.164163 0.986433i \(-0.552492\pi\)
\(30\) 0 0
\(31\) −7.34847 −1.31982 −0.659912 0.751343i \(-0.729406\pi\)
−0.659912 + 0.751343i \(0.729406\pi\)
\(32\) 0 0
\(33\) −1.46410 −0.254867
\(34\) 0 0
\(35\) 6.69213 + 6.69213i 1.13118 + 1.13118i
\(36\) 0 0
\(37\) 0.464102 0.464102i 0.0762978 0.0762978i −0.667928 0.744226i \(-0.732819\pi\)
0.744226 + 0.667928i \(0.232819\pi\)
\(38\) 0 0
\(39\) 4.24264i 0.679366i
\(40\) 0 0
\(41\) 4.53590i 0.708388i 0.935172 + 0.354194i \(0.115245\pi\)
−0.935172 + 0.354194i \(0.884755\pi\)
\(42\) 0 0
\(43\) 2.44949 2.44949i 0.373544 0.373544i −0.495222 0.868766i \(-0.664913\pi\)
0.868766 + 0.495222i \(0.164913\pi\)
\(44\) 0 0
\(45\) 2.73205 + 2.73205i 0.407270 + 0.407270i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.44949 2.44949i −0.342997 0.342997i
\(52\) 0 0
\(53\) 3.26795 3.26795i 0.448887 0.448887i −0.446097 0.894985i \(-0.647186\pi\)
0.894985 + 0.446097i \(0.147186\pi\)
\(54\) 0 0
\(55\) 5.65685i 0.762770i
\(56\) 0 0
\(57\) 7.46410i 0.988644i
\(58\) 0 0
\(59\) 9.79796 9.79796i 1.27559 1.27559i 0.332473 0.943113i \(-0.392117\pi\)
0.943113 0.332473i \(-0.107883\pi\)
\(60\) 0 0
\(61\) 0.464102 + 0.464102i 0.0594221 + 0.0594221i 0.736193 0.676771i \(-0.236621\pi\)
−0.676771 + 0.736193i \(0.736621\pi\)
\(62\) 0 0
\(63\) −2.44949 −0.308607
\(64\) 0 0
\(65\) 16.3923 2.03322
\(66\) 0 0
\(67\) −10.5558 10.5558i −1.28960 1.28960i −0.935029 0.354571i \(-0.884627\pi\)
−0.354571 0.935029i \(-0.615373\pi\)
\(68\) 0 0
\(69\) −2.00000 + 2.00000i −0.240772 + 0.240772i
\(70\) 0 0
\(71\) 6.41473i 0.761288i 0.924722 + 0.380644i \(0.124298\pi\)
−0.924722 + 0.380644i \(0.875702\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) −7.02030 + 7.02030i −0.810634 + 0.810634i
\(76\) 0 0
\(77\) 2.53590 + 2.53590i 0.288992 + 0.288992i
\(78\) 0 0
\(79\) −2.44949 −0.275589 −0.137795 0.990461i \(-0.544001\pi\)
−0.137795 + 0.990461i \(0.544001\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 3.86370 + 3.86370i 0.424097 + 0.424097i 0.886611 0.462515i \(-0.153053\pi\)
−0.462515 + 0.886611i \(0.653053\pi\)
\(84\) 0 0
\(85\) 9.46410 9.46410i 1.02653 1.02653i
\(86\) 0 0
\(87\) 8.76268i 0.939458i
\(88\) 0 0
\(89\) 4.92820i 0.522388i 0.965286 + 0.261194i \(0.0841163\pi\)
−0.965286 + 0.261194i \(0.915884\pi\)
\(90\) 0 0
\(91\) −7.34847 + 7.34847i −0.770329 + 0.770329i
\(92\) 0 0
\(93\) 5.19615 + 5.19615i 0.538816 + 0.538816i
\(94\) 0 0
\(95\) 28.8391 2.95883
\(96\) 0 0
\(97\) 1.07180 0.108824 0.0544122 0.998519i \(-0.482671\pi\)
0.0544122 + 0.998519i \(0.482671\pi\)
\(98\) 0 0
\(99\) 1.03528 + 1.03528i 0.104049 + 0.104049i
\(100\) 0 0
\(101\) −10.1962 + 10.1962i −1.01456 + 1.01456i −0.0146626 + 0.999892i \(0.504667\pi\)
−0.999892 + 0.0146626i \(0.995333\pi\)
\(102\) 0 0
\(103\) 3.20736i 0.316031i −0.987437 0.158016i \(-0.949490\pi\)
0.987437 0.158016i \(-0.0505096\pi\)
\(104\) 0 0
\(105\) 9.46410i 0.923602i
\(106\) 0 0
\(107\) 2.82843 2.82843i 0.273434 0.273434i −0.557047 0.830481i \(-0.688066\pi\)
0.830481 + 0.557047i \(0.188066\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.00000i 0.0957826 + 0.0957826i 0.753374 0.657592i \(-0.228425\pi\)
−0.657592 + 0.753374i \(0.728425\pi\)
\(110\) 0 0
\(111\) −0.656339 −0.0622969
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −7.72741 7.72741i −0.720584 0.720584i
\(116\) 0 0
\(117\) −3.00000 + 3.00000i −0.277350 + 0.277350i
\(118\) 0 0
\(119\) 8.48528i 0.777844i
\(120\) 0 0
\(121\) 8.85641i 0.805128i
\(122\) 0 0
\(123\) 3.20736 3.20736i 0.289198 0.289198i
\(124\) 0 0
\(125\) −13.4641 13.4641i −1.20427 1.20427i
\(126\) 0 0
\(127\) −14.5211 −1.28854 −0.644268 0.764799i \(-0.722838\pi\)
−0.644268 + 0.764799i \(0.722838\pi\)
\(128\) 0 0
\(129\) −3.46410 −0.304997
\(130\) 0 0
\(131\) 10.5558 + 10.5558i 0.922267 + 0.922267i 0.997189 0.0749219i \(-0.0238708\pi\)
−0.0749219 + 0.997189i \(0.523871\pi\)
\(132\) 0 0
\(133\) −12.9282 + 12.9282i −1.12102 + 1.12102i
\(134\) 0 0
\(135\) 3.86370i 0.332535i
\(136\) 0 0
\(137\) 14.3923i 1.22962i 0.788676 + 0.614809i \(0.210767\pi\)
−0.788676 + 0.614809i \(0.789233\pi\)
\(138\) 0 0
\(139\) −4.89898 + 4.89898i −0.415526 + 0.415526i −0.883658 0.468132i \(-0.844927\pi\)
0.468132 + 0.883658i \(0.344927\pi\)
\(140\) 0 0
\(141\) 2.00000 + 2.00000i 0.168430 + 0.168430i
\(142\) 0 0
\(143\) 6.21166 0.519445
\(144\) 0 0
\(145\) −33.8564 −2.81162
\(146\) 0 0
\(147\) −0.707107 0.707107i −0.0583212 0.0583212i
\(148\) 0 0
\(149\) 3.80385 3.80385i 0.311623 0.311623i −0.533915 0.845538i \(-0.679280\pi\)
0.845538 + 0.533915i \(0.179280\pi\)
\(150\) 0 0
\(151\) 6.59059i 0.536335i −0.963372 0.268167i \(-0.913582\pi\)
0.963372 0.268167i \(-0.0864181\pi\)
\(152\) 0 0
\(153\) 3.46410i 0.280056i
\(154\) 0 0
\(155\) −20.0764 + 20.0764i −1.61257 + 1.61257i
\(156\) 0 0
\(157\) −1.53590 1.53590i −0.122578 0.122578i 0.643157 0.765735i \(-0.277624\pi\)
−0.765735 + 0.643157i \(0.777624\pi\)
\(158\) 0 0
\(159\) −4.62158 −0.366515
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) −3.20736 3.20736i −0.251220 0.251220i 0.570251 0.821471i \(-0.306846\pi\)
−0.821471 + 0.570251i \(0.806846\pi\)
\(164\) 0 0
\(165\) −4.00000 + 4.00000i −0.311400 + 0.311400i
\(166\) 0 0
\(167\) 7.72741i 0.597965i −0.954259 0.298982i \(-0.903353\pi\)
0.954259 0.298982i \(-0.0966472\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) −5.27792 + 5.27792i −0.403612 + 0.403612i
\(172\) 0 0
\(173\) −12.1962 12.1962i −0.927256 0.927256i 0.0702716 0.997528i \(-0.477613\pi\)
−0.997528 + 0.0702716i \(0.977613\pi\)
\(174\) 0 0
\(175\) 24.3190 1.83835
\(176\) 0 0
\(177\) −13.8564 −1.04151
\(178\) 0 0
\(179\) −13.3843 13.3843i −1.00039 1.00039i −1.00000 0.000386700i \(-0.999877\pi\)
−0.000386700 1.00000i \(-0.500123\pi\)
\(180\) 0 0
\(181\) −5.00000 + 5.00000i −0.371647 + 0.371647i −0.868077 0.496430i \(-0.834644\pi\)
0.496430 + 0.868077i \(0.334644\pi\)
\(182\) 0 0
\(183\) 0.656339i 0.0485180i
\(184\) 0 0
\(185\) 2.53590i 0.186443i
\(186\) 0 0
\(187\) 3.58630 3.58630i 0.262256 0.262256i
\(188\) 0 0
\(189\) 1.73205 + 1.73205i 0.125988 + 0.125988i
\(190\) 0 0
\(191\) 7.72741 0.559136 0.279568 0.960126i \(-0.409809\pi\)
0.279568 + 0.960126i \(0.409809\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) −11.5911 11.5911i −0.830057 0.830057i
\(196\) 0 0
\(197\) −6.19615 + 6.19615i −0.441458 + 0.441458i −0.892502 0.451044i \(-0.851052\pi\)
0.451044 + 0.892502i \(0.351052\pi\)
\(198\) 0 0
\(199\) 18.6622i 1.32293i 0.749977 + 0.661463i \(0.230064\pi\)
−0.749977 + 0.661463i \(0.769936\pi\)
\(200\) 0 0
\(201\) 14.9282i 1.05295i
\(202\) 0 0
\(203\) 15.1774 15.1774i 1.06525 1.06525i
\(204\) 0 0
\(205\) 12.3923 + 12.3923i 0.865516 + 0.865516i
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 10.9282 0.755920
\(210\) 0 0
\(211\) 4.89898 + 4.89898i 0.337260 + 0.337260i 0.855335 0.518075i \(-0.173351\pi\)
−0.518075 + 0.855335i \(0.673351\pi\)
\(212\) 0 0
\(213\) 4.53590 4.53590i 0.310795 0.310795i
\(214\) 0 0
\(215\) 13.3843i 0.912799i
\(216\) 0 0
\(217\) 18.0000i 1.22192i
\(218\) 0 0
\(219\) 2.82843 2.82843i 0.191127 0.191127i
\(220\) 0 0
\(221\) 10.3923 + 10.3923i 0.699062 + 0.699062i
\(222\) 0 0
\(223\) 14.5211 0.972403 0.486201 0.873847i \(-0.338382\pi\)
0.486201 + 0.873847i \(0.338382\pi\)
\(224\) 0 0
\(225\) 9.92820 0.661880
\(226\) 0 0
\(227\) −8.00481 8.00481i −0.531298 0.531298i 0.389661 0.920958i \(-0.372592\pi\)
−0.920958 + 0.389661i \(0.872592\pi\)
\(228\) 0 0
\(229\) −11.9282 + 11.9282i −0.788238 + 0.788238i −0.981205 0.192967i \(-0.938189\pi\)
0.192967 + 0.981205i \(0.438189\pi\)
\(230\) 0 0
\(231\) 3.58630i 0.235961i
\(232\) 0 0
\(233\) 12.9282i 0.846955i 0.905907 + 0.423477i \(0.139191\pi\)
−0.905907 + 0.423477i \(0.860809\pi\)
\(234\) 0 0
\(235\) −7.72741 + 7.72741i −0.504080 + 0.504080i
\(236\) 0 0
\(237\) 1.73205 + 1.73205i 0.112509 + 0.112509i
\(238\) 0 0
\(239\) 7.72741 0.499844 0.249922 0.968266i \(-0.419595\pi\)
0.249922 + 0.968266i \(0.419595\pi\)
\(240\) 0 0
\(241\) 26.7846 1.72535 0.862674 0.505760i \(-0.168788\pi\)
0.862674 + 0.505760i \(0.168788\pi\)
\(242\) 0 0
\(243\) 0.707107 + 0.707107i 0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 2.73205 2.73205i 0.174544 0.174544i
\(246\) 0 0
\(247\) 31.6675i 2.01495i
\(248\) 0 0
\(249\) 5.46410i 0.346273i
\(250\) 0 0
\(251\) −4.62158 + 4.62158i −0.291711 + 0.291711i −0.837756 0.546045i \(-0.816133\pi\)
0.546045 + 0.837756i \(0.316133\pi\)
\(252\) 0 0
\(253\) −2.92820 2.92820i −0.184095 0.184095i
\(254\) 0 0
\(255\) −13.3843 −0.838155
\(256\) 0 0
\(257\) −22.7846 −1.42126 −0.710632 0.703563i \(-0.751591\pi\)
−0.710632 + 0.703563i \(0.751591\pi\)
\(258\) 0 0
\(259\) 1.13681 + 1.13681i 0.0706381 + 0.0706381i
\(260\) 0 0
\(261\) 6.19615 6.19615i 0.383532 0.383532i
\(262\) 0 0
\(263\) 22.6274i 1.39527i 0.716455 + 0.697633i \(0.245763\pi\)
−0.716455 + 0.697633i \(0.754237\pi\)
\(264\) 0 0
\(265\) 17.8564i 1.09691i
\(266\) 0 0
\(267\) 3.48477 3.48477i 0.213264 0.213264i
\(268\) 0 0
\(269\) 7.26795 + 7.26795i 0.443135 + 0.443135i 0.893064 0.449929i \(-0.148551\pi\)
−0.449929 + 0.893064i \(0.648551\pi\)
\(270\) 0 0
\(271\) −3.20736 −0.194834 −0.0974168 0.995244i \(-0.531058\pi\)
−0.0974168 + 0.995244i \(0.531058\pi\)
\(272\) 0 0
\(273\) 10.3923 0.628971
\(274\) 0 0
\(275\) −10.2784 10.2784i −0.619813 0.619813i
\(276\) 0 0
\(277\) −8.85641 + 8.85641i −0.532130 + 0.532130i −0.921206 0.389076i \(-0.872794\pi\)
0.389076 + 0.921206i \(0.372794\pi\)
\(278\) 0 0
\(279\) 7.34847i 0.439941i
\(280\) 0 0
\(281\) 0.928203i 0.0553720i −0.999617 0.0276860i \(-0.991186\pi\)
0.999617 0.0276860i \(-0.00881385\pi\)
\(282\) 0 0
\(283\) 6.96953 6.96953i 0.414296 0.414296i −0.468936 0.883232i \(-0.655363\pi\)
0.883232 + 0.468936i \(0.155363\pi\)
\(284\) 0 0
\(285\) −20.3923 20.3923i −1.20794 1.20794i
\(286\) 0 0
\(287\) −11.1106 −0.655840
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −0.757875 0.757875i −0.0444274 0.0444274i
\(292\) 0 0
\(293\) 7.66025 7.66025i 0.447517 0.447517i −0.447011 0.894528i \(-0.647512\pi\)
0.894528 + 0.447011i \(0.147512\pi\)
\(294\) 0 0
\(295\) 53.5370i 3.11705i
\(296\) 0 0
\(297\) 1.46410i 0.0849558i
\(298\) 0 0
\(299\) 8.48528 8.48528i 0.490716 0.490716i
\(300\) 0 0
\(301\) 6.00000 + 6.00000i 0.345834 + 0.345834i
\(302\) 0 0
\(303\) 14.4195 0.828381
\(304\) 0 0
\(305\) 2.53590 0.145205
\(306\) 0 0
\(307\) 16.2127 + 16.2127i 0.925307 + 0.925307i 0.997398 0.0720911i \(-0.0229672\pi\)
−0.0720911 + 0.997398i \(0.522967\pi\)
\(308\) 0 0
\(309\) −2.26795 + 2.26795i −0.129019 + 0.129019i
\(310\) 0 0
\(311\) 32.4254i 1.83867i −0.393471 0.919337i \(-0.628726\pi\)
0.393471 0.919337i \(-0.371274\pi\)
\(312\) 0 0
\(313\) 3.85641i 0.217977i −0.994043 0.108988i \(-0.965239\pi\)
0.994043 0.108988i \(-0.0347612\pi\)
\(314\) 0 0
\(315\) −6.69213 + 6.69213i −0.377059 + 0.377059i
\(316\) 0 0
\(317\) −4.73205 4.73205i −0.265778 0.265778i 0.561618 0.827397i \(-0.310179\pi\)
−0.827397 + 0.561618i \(0.810179\pi\)
\(318\) 0 0
\(319\) −12.8295 −0.718312
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 18.2832 + 18.2832i 1.01731 + 1.01731i
\(324\) 0 0
\(325\) 29.7846 29.7846i 1.65215 1.65215i
\(326\) 0 0
\(327\) 1.41421i 0.0782062i
\(328\) 0 0
\(329\) 6.92820i 0.381964i
\(330\) 0 0
\(331\) −0.757875 + 0.757875i −0.0416566 + 0.0416566i −0.727628 0.685972i \(-0.759377\pi\)
0.685972 + 0.727628i \(0.259377\pi\)
\(332\) 0 0
\(333\) 0.464102 + 0.464102i 0.0254326 + 0.0254326i
\(334\) 0 0
\(335\) −57.6781 −3.15129
\(336\) 0 0
\(337\) −9.85641 −0.536913 −0.268456 0.963292i \(-0.586514\pi\)
−0.268456 + 0.963292i \(0.586514\pi\)
\(338\) 0 0
\(339\) 12.7279 + 12.7279i 0.691286 + 0.691286i
\(340\) 0 0
\(341\) −7.60770 + 7.60770i −0.411980 + 0.411980i
\(342\) 0 0
\(343\) 19.5959i 1.05808i
\(344\) 0 0
\(345\) 10.9282i 0.588355i
\(346\) 0 0
\(347\) 1.79315 1.79315i 0.0962614 0.0962614i −0.657336 0.753598i \(-0.728317\pi\)
0.753598 + 0.657336i \(0.228317\pi\)
\(348\) 0 0
\(349\) −19.3923 19.3923i −1.03805 1.03805i −0.999247 0.0387993i \(-0.987647\pi\)
−0.0387993 0.999247i \(-0.512353\pi\)
\(350\) 0 0
\(351\) 4.24264 0.226455
\(352\) 0 0
\(353\) −0.928203 −0.0494033 −0.0247016 0.999695i \(-0.507864\pi\)
−0.0247016 + 0.999695i \(0.507864\pi\)
\(354\) 0 0
\(355\) 17.5254 + 17.5254i 0.930150 + 0.930150i
\(356\) 0 0
\(357\) 6.00000 6.00000i 0.317554 0.317554i
\(358\) 0 0
\(359\) 18.8380i 0.994234i 0.867684 + 0.497117i \(0.165608\pi\)
−0.867684 + 0.497117i \(0.834392\pi\)
\(360\) 0 0
\(361\) 36.7128i 1.93225i
\(362\) 0 0
\(363\) 6.26243 6.26243i 0.328692 0.328692i
\(364\) 0 0
\(365\) 10.9282 + 10.9282i 0.572008 + 0.572008i
\(366\) 0 0
\(367\) −16.3886 −0.855476 −0.427738 0.903903i \(-0.640689\pi\)
−0.427738 + 0.903903i \(0.640689\pi\)
\(368\) 0 0
\(369\) −4.53590 −0.236129
\(370\) 0 0
\(371\) 8.00481 + 8.00481i 0.415589 + 0.415589i
\(372\) 0 0
\(373\) 9.39230 9.39230i 0.486315 0.486315i −0.420826 0.907141i \(-0.638260\pi\)
0.907141 + 0.420826i \(0.138260\pi\)
\(374\) 0 0
\(375\) 19.0411i 0.983279i
\(376\) 0 0
\(377\) 37.1769i 1.91471i
\(378\) 0 0
\(379\) 9.41902 9.41902i 0.483823 0.483823i −0.422527 0.906350i \(-0.638857\pi\)
0.906350 + 0.422527i \(0.138857\pi\)
\(380\) 0 0
\(381\) 10.2679 + 10.2679i 0.526043 + 0.526043i
\(382\) 0 0
\(383\) −28.8391 −1.47361 −0.736804 0.676106i \(-0.763666\pi\)
−0.736804 + 0.676106i \(0.763666\pi\)
\(384\) 0 0
\(385\) 13.8564 0.706188
\(386\) 0 0
\(387\) 2.44949 + 2.44949i 0.124515 + 0.124515i
\(388\) 0 0
\(389\) 6.73205 6.73205i 0.341329 0.341329i −0.515538 0.856867i \(-0.672408\pi\)
0.856867 + 0.515538i \(0.172408\pi\)
\(390\) 0 0
\(391\) 9.79796i 0.495504i
\(392\) 0 0
\(393\) 14.9282i 0.753028i
\(394\) 0 0
\(395\) −6.69213 + 6.69213i −0.336718 + 0.336718i
\(396\) 0 0
\(397\) −16.3205 16.3205i −0.819103 0.819103i 0.166875 0.985978i \(-0.446632\pi\)
−0.985978 + 0.166875i \(0.946632\pi\)
\(398\) 0 0
\(399\) 18.2832 0.915307
\(400\) 0 0
\(401\) 18.3923 0.918468 0.459234 0.888315i \(-0.348124\pi\)
0.459234 + 0.888315i \(0.348124\pi\)
\(402\) 0 0
\(403\) −22.0454 22.0454i −1.09816 1.09816i
\(404\) 0 0
\(405\) −2.73205 + 2.73205i −0.135757 + 0.135757i
\(406\) 0 0
\(407\) 0.960947i 0.0476324i
\(408\) 0 0
\(409\) 9.07180i 0.448571i 0.974523 + 0.224286i \(0.0720049\pi\)
−0.974523 + 0.224286i \(0.927995\pi\)
\(410\) 0 0
\(411\) 10.1769 10.1769i 0.501989 0.501989i
\(412\) 0 0
\(413\) 24.0000 + 24.0000i 1.18096 + 1.18096i
\(414\) 0 0
\(415\) 21.1117 1.03633
\(416\) 0 0
\(417\) 6.92820 0.339276
\(418\) 0 0
\(419\) −0.277401 0.277401i −0.0135519 0.0135519i 0.700298 0.713850i \(-0.253050\pi\)
−0.713850 + 0.700298i \(0.753050\pi\)
\(420\) 0 0
\(421\) 13.9282 13.9282i 0.678819 0.678819i −0.280914 0.959733i \(-0.590638\pi\)
0.959733 + 0.280914i \(0.0906375\pi\)
\(422\) 0 0
\(423\) 2.82843i 0.137523i
\(424\) 0 0
\(425\) 34.3923i 1.66827i
\(426\) 0 0
\(427\) −1.13681 + 1.13681i −0.0550142 + 0.0550142i
\(428\) 0 0
\(429\) −4.39230 4.39230i −0.212062 0.212062i
\(430\) 0 0
\(431\) 4.34418 0.209252 0.104626 0.994512i \(-0.466636\pi\)
0.104626 + 0.994512i \(0.466636\pi\)
\(432\) 0 0
\(433\) 29.8564 1.43481 0.717404 0.696658i \(-0.245330\pi\)
0.717404 + 0.696658i \(0.245330\pi\)
\(434\) 0 0
\(435\) 23.9401 + 23.9401i 1.14784 + 1.14784i
\(436\) 0 0
\(437\) 14.9282 14.9282i 0.714113 0.714113i
\(438\) 0 0
\(439\) 5.83272i 0.278381i −0.990266 0.139190i \(-0.955550\pi\)
0.990266 0.139190i \(-0.0444500\pi\)
\(440\) 0 0
\(441\) 1.00000i 0.0476190i
\(442\) 0 0
\(443\) −24.9754 + 24.9754i −1.18662 + 1.18662i −0.208619 + 0.977997i \(0.566897\pi\)
−0.977997 + 0.208619i \(0.933103\pi\)
\(444\) 0 0
\(445\) 13.4641 + 13.4641i 0.638260 + 0.638260i
\(446\) 0 0
\(447\) −5.37945 −0.254439
\(448\) 0 0
\(449\) −6.39230 −0.301672 −0.150836 0.988559i \(-0.548196\pi\)
−0.150836 + 0.988559i \(0.548196\pi\)
\(450\) 0 0
\(451\) 4.69591 + 4.69591i 0.221122 + 0.221122i
\(452\) 0 0
\(453\) −4.66025 + 4.66025i −0.218958 + 0.218958i
\(454\) 0 0
\(455\) 40.1528i 1.88239i
\(456\) 0 0
\(457\) 20.7846i 0.972263i −0.873886 0.486132i \(-0.838408\pi\)
0.873886 0.486132i \(-0.161592\pi\)
\(458\) 0 0
\(459\) 2.44949 2.44949i 0.114332 0.114332i
\(460\) 0 0
\(461\) 22.0526 + 22.0526i 1.02709 + 1.02709i 0.999623 + 0.0274674i \(0.00874423\pi\)
0.0274674 + 0.999623i \(0.491256\pi\)
\(462\) 0 0
\(463\) −29.2180 −1.35788 −0.678938 0.734196i \(-0.737560\pi\)
−0.678938 + 0.734196i \(0.737560\pi\)
\(464\) 0 0
\(465\) 28.3923 1.31666
\(466\) 0 0
\(467\) −21.3891 21.3891i −0.989768 0.989768i 0.0101801 0.999948i \(-0.496760\pi\)
−0.999948 + 0.0101801i \(0.996760\pi\)
\(468\) 0 0
\(469\) 25.8564 25.8564i 1.19394 1.19394i
\(470\) 0 0
\(471\) 2.17209i 0.100085i
\(472\) 0 0
\(473\) 5.07180i 0.233201i
\(474\) 0 0
\(475\) 52.4002 52.4002i 2.40429 2.40429i
\(476\) 0 0
\(477\) 3.26795 + 3.26795i 0.149629 + 0.149629i
\(478\) 0 0
\(479\) 29.0421 1.32697 0.663485 0.748190i \(-0.269077\pi\)
0.663485 + 0.748190i \(0.269077\pi\)
\(480\) 0 0
\(481\) 2.78461 0.126967
\(482\) 0 0
\(483\) −4.89898 4.89898i −0.222911 0.222911i
\(484\) 0 0
\(485\) 2.92820 2.92820i 0.132963 0.132963i
\(486\) 0 0
\(487\) 11.4896i 0.520642i 0.965522 + 0.260321i \(0.0838284\pi\)
−0.965522 + 0.260321i \(0.916172\pi\)
\(488\) 0 0
\(489\) 4.53590i 0.205120i
\(490\) 0 0
\(491\) −4.14110 + 4.14110i −0.186885 + 0.186885i −0.794348 0.607463i \(-0.792187\pi\)
0.607463 + 0.794348i \(0.292187\pi\)
\(492\) 0 0
\(493\) −21.4641 21.4641i −0.966695 0.966695i
\(494\) 0 0
\(495\) 5.65685 0.254257
\(496\) 0 0
\(497\) −15.7128 −0.704816
\(498\) 0 0
\(499\) 2.82843 + 2.82843i 0.126618 + 0.126618i 0.767576 0.640958i \(-0.221463\pi\)
−0.640958 + 0.767576i \(0.721463\pi\)
\(500\) 0 0
\(501\) −5.46410 + 5.46410i −0.244118 + 0.244118i
\(502\) 0 0
\(503\) 23.3853i 1.04270i −0.853343 0.521349i \(-0.825429\pi\)
0.853343 0.521349i \(-0.174571\pi\)
\(504\) 0 0
\(505\) 55.7128i 2.47919i
\(506\) 0 0
\(507\) 3.53553 3.53553i 0.157019 0.157019i
\(508\) 0 0
\(509\) 5.12436 + 5.12436i 0.227133 + 0.227133i 0.811494 0.584361i \(-0.198655\pi\)
−0.584361 + 0.811494i \(0.698655\pi\)
\(510\) 0 0
\(511\) −9.79796 −0.433436
\(512\) 0 0
\(513\) 7.46410 0.329548
\(514\) 0 0
\(515\) −8.76268 8.76268i −0.386130 0.386130i
\(516\) 0 0
\(517\) −2.92820 + 2.92820i −0.128782 + 0.128782i
\(518\) 0 0
\(519\) 17.2480i 0.757102i
\(520\) 0 0
\(521\) 21.3205i 0.934068i −0.884239 0.467034i \(-0.845322\pi\)
0.884239 0.467034i \(-0.154678\pi\)
\(522\) 0 0
\(523\) −22.2485 + 22.2485i −0.972858 + 0.972858i −0.999641 0.0267832i \(-0.991474\pi\)
0.0267832 + 0.999641i \(0.491474\pi\)
\(524\) 0 0
\(525\) −17.1962 17.1962i −0.750502 0.750502i
\(526\) 0 0
\(527\) −25.4558 −1.10887
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 9.79796 + 9.79796i 0.425195 + 0.425195i
\(532\) 0 0
\(533\) −13.6077 + 13.6077i −0.589415 + 0.589415i
\(534\) 0 0
\(535\) 15.4548i 0.668170i
\(536\) 0 0
\(537\) 18.9282i 0.816812i
\(538\) 0 0
\(539\) 1.03528 1.03528i 0.0445925 0.0445925i
\(540\) 0 0
\(541\) 19.9282 + 19.9282i 0.856780 + 0.856780i 0.990957 0.134177i \(-0.0428391\pi\)
−0.134177 + 0.990957i \(0.542839\pi\)
\(542\) 0 0
\(543\) 7.07107 0.303449
\(544\) 0 0
\(545\) 5.46410 0.234056
\(546\) 0 0
\(547\) −16.3886 16.3886i −0.700724 0.700724i 0.263842 0.964566i \(-0.415010\pi\)
−0.964566 + 0.263842i \(0.915010\pi\)
\(548\) 0 0
\(549\) −0.464102 + 0.464102i −0.0198074 + 0.0198074i
\(550\) 0 0
\(551\) 65.4056i 2.78637i
\(552\) 0 0
\(553\) 6.00000i 0.255146i
\(554\) 0 0
\(555\) −1.79315 + 1.79315i −0.0761150 + 0.0761150i
\(556\) 0 0
\(557\) −19.8038 19.8038i −0.839116 0.839116i 0.149627 0.988743i \(-0.452193\pi\)
−0.988743 + 0.149627i \(0.952193\pi\)
\(558\) 0 0
\(559\) 14.6969 0.621614
\(560\) 0 0
\(561\) −5.07180 −0.214131
\(562\) 0 0
\(563\) 1.03528 + 1.03528i 0.0436317 + 0.0436317i 0.728586 0.684954i \(-0.240178\pi\)
−0.684954 + 0.728586i \(0.740178\pi\)
\(564\) 0 0
\(565\) −49.1769 + 49.1769i −2.06889 + 2.06889i
\(566\) 0 0
\(567\) 2.44949i 0.102869i
\(568\) 0 0
\(569\) 3.46410i 0.145223i 0.997360 + 0.0726113i \(0.0231333\pi\)
−0.997360 + 0.0726113i \(0.976867\pi\)
\(570\) 0 0
\(571\) −22.4243 + 22.4243i −0.938430 + 0.938430i −0.998211 0.0597818i \(-0.980959\pi\)
0.0597818 + 0.998211i \(0.480959\pi\)
\(572\) 0 0
\(573\) −5.46410 5.46410i −0.228266 0.228266i
\(574\) 0 0
\(575\) −28.0812 −1.17107
\(576\) 0 0
\(577\) −21.8564 −0.909894 −0.454947 0.890518i \(-0.650342\pi\)
−0.454947 + 0.890518i \(0.650342\pi\)
\(578\) 0 0
\(579\) −4.24264 4.24264i −0.176318 0.176318i
\(580\) 0 0
\(581\) −9.46410 + 9.46410i −0.392637 + 0.392637i
\(582\) 0 0
\(583\) 6.76646i 0.280238i
\(584\) 0 0
\(585\) 16.3923i 0.677738i
\(586\) 0 0
\(587\) 30.3548 30.3548i 1.25288 1.25288i 0.298453 0.954424i \(-0.403529\pi\)
0.954424 0.298453i \(-0.0964708\pi\)
\(588\) 0 0
\(589\) −38.7846 38.7846i −1.59809 1.59809i
\(590\) 0 0
\(591\) 8.76268 0.360449
\(592\) 0 0
\(593\) −37.7128 −1.54868 −0.774340 0.632770i \(-0.781918\pi\)
−0.774340 + 0.632770i \(0.781918\pi\)
\(594\) 0 0
\(595\) 23.1822 + 23.1822i 0.950378 + 0.950378i
\(596\) 0 0
\(597\) 13.1962 13.1962i 0.540083 0.540083i
\(598\) 0 0
\(599\) 6.96953i 0.284767i −0.989812 0.142384i \(-0.954523\pi\)
0.989812 0.142384i \(-0.0454767\pi\)
\(600\) 0 0
\(601\) 19.7128i 0.804102i −0.915617 0.402051i \(-0.868297\pi\)
0.915617 0.402051i \(-0.131703\pi\)
\(602\) 0 0
\(603\) 10.5558 10.5558i 0.429867 0.429867i
\(604\) 0 0
\(605\) 24.1962 + 24.1962i 0.983713 + 0.983713i
\(606\) 0 0
\(607\) 31.8434 1.29248 0.646241 0.763133i \(-0.276340\pi\)
0.646241 + 0.763133i \(0.276340\pi\)
\(608\) 0 0
\(609\) −21.4641 −0.869769
\(610\) 0 0
\(611\) −8.48528 8.48528i −0.343278 0.343278i
\(612\) 0 0
\(613\) −4.32051 + 4.32051i −0.174504 + 0.174504i −0.788955 0.614451i \(-0.789377\pi\)
0.614451 + 0.788955i \(0.289377\pi\)
\(614\) 0 0
\(615\) 17.5254i 0.706691i
\(616\) 0 0
\(617\) 39.8564i 1.60456i −0.596949 0.802279i \(-0.703621\pi\)
0.596949 0.802279i \(-0.296379\pi\)
\(618\) 0 0
\(619\) 8.48528 8.48528i 0.341052 0.341052i −0.515711 0.856763i \(-0.672472\pi\)
0.856763 + 0.515711i \(0.172472\pi\)
\(620\) 0 0
\(621\) −2.00000 2.00000i −0.0802572 0.0802572i
\(622\) 0 0
\(623\) −12.0716 −0.483638
\(624\) 0 0
\(625\) −23.9282 −0.957128
\(626\) 0 0
\(627\) −7.72741 7.72741i −0.308603 0.308603i
\(628\) 0 0
\(629\) 1.60770 1.60770i 0.0641030 0.0641030i
\(630\) 0 0
\(631\) 23.5612i 0.937955i 0.883210 + 0.468977i \(0.155377\pi\)
−0.883210 + 0.468977i \(0.844623\pi\)
\(632\) 0 0
\(633\) 6.92820i 0.275371i
\(634\) 0 0
\(635\) −39.6723 + 39.6723i −1.57435 + 1.57435i
\(636\) 0 0
\(637\) 3.00000 + 3.00000i 0.118864 + 0.118864i
\(638\) 0 0
\(639\) −6.41473 −0.253763
\(640\) 0 0
\(641\) −25.6077 −1.01144 −0.505722 0.862697i \(-0.668774\pi\)
−0.505722 + 0.862697i \(0.668774\pi\)
\(642\) 0 0
\(643\) 20.1779 + 20.1779i 0.795740 + 0.795740i 0.982421 0.186681i \(-0.0597730\pi\)
−0.186681 + 0.982421i \(0.559773\pi\)
\(644\) 0 0
\(645\) −9.46410 + 9.46410i −0.372649 + 0.372649i
\(646\) 0 0
\(647\) 40.9107i 1.60836i 0.594383 + 0.804182i \(0.297396\pi\)
−0.594383 + 0.804182i \(0.702604\pi\)
\(648\) 0 0
\(649\) 20.2872i 0.796342i
\(650\) 0 0
\(651\) −12.7279 + 12.7279i −0.498847 + 0.498847i
\(652\) 0 0
\(653\) −17.6603 17.6603i −0.691099 0.691099i 0.271375 0.962474i \(-0.412522\pi\)
−0.962474 + 0.271375i \(0.912522\pi\)
\(654\) 0 0
\(655\) 57.6781 2.25367
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −16.9706 16.9706i −0.661079 0.661079i 0.294555 0.955634i \(-0.404829\pi\)
−0.955634 + 0.294555i \(0.904829\pi\)
\(660\) 0 0
\(661\) −4.60770 + 4.60770i −0.179219 + 0.179219i −0.791015 0.611797i \(-0.790447\pi\)
0.611797 + 0.791015i \(0.290447\pi\)
\(662\) 0 0
\(663\) 14.6969i 0.570782i
\(664\) 0 0
\(665\) 70.6410i 2.73934i
\(666\) 0 0
\(667\) −17.5254 + 17.5254i −0.678585 + 0.678585i
\(668\) 0 0
\(669\) −10.2679 10.2679i −0.396982 0.396982i
\(670\) 0 0
\(671\) 0.960947 0.0370969
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) −7.02030 7.02030i −0.270211 0.270211i
\(676\) 0 0
\(677\) 2.33975 2.33975i 0.0899237 0.0899237i −0.660714 0.750638i \(-0.729746\pi\)
0.750638 + 0.660714i \(0.229746\pi\)
\(678\) 0 0
\(679\) 2.62536i 0.100752i
\(680\) 0 0
\(681\) 11.3205i 0.433803i
\(682\) 0 0
\(683\) −17.8028 + 17.8028i −0.681204 + 0.681204i −0.960271 0.279068i \(-0.909975\pi\)
0.279068 + 0.960271i \(0.409975\pi\)
\(684\) 0 0
\(685\) 39.3205 + 39.3205i 1.50236 + 1.50236i
\(686\) 0 0
\(687\) 16.8690 0.643594
\(688\) 0 0
\(689\) 19.6077 0.746994
\(690\) 0 0
\(691\) 18.6622 + 18.6622i 0.709943 + 0.709943i 0.966523 0.256580i \(-0.0825958\pi\)
−0.256580 + 0.966523i \(0.582596\pi\)
\(692\) 0 0
\(693\) −2.53590 + 2.53590i −0.0963308 + 0.0963308i
\(694\) 0 0
\(695\) 26.7685i 1.01539i
\(696\) 0 0
\(697\) 15.7128i 0.595165i
\(698\) 0 0
\(699\) 9.14162 9.14162i 0.345768 0.345768i
\(700\) 0 0
\(701\) 10.5885 + 10.5885i 0.399921 + 0.399921i 0.878205 0.478284i \(-0.158741\pi\)
−0.478284 + 0.878205i \(0.658741\pi\)
\(702\) 0 0
\(703\) 4.89898 0.184769
\(704\) 0 0
\(705\) 10.9282 0.411580
\(706\) 0 0
\(707\) −24.9754 24.9754i −0.939295 0.939295i
\(708\) 0 0
\(709\) −34.7128 + 34.7128i −1.30367 + 1.30367i −0.377766 + 0.925901i \(0.623308\pi\)
−0.925901 + 0.377766i \(0.876692\pi\)
\(710\) 0 0
\(711\) 2.44949i 0.0918630i
\(712\) 0 0
\(713\) 20.7846i 0.778390i
\(714\) 0 0
\(715\) 16.9706 16.9706i 0.634663 0.634663i
\(716\) 0 0
\(717\) −5.46410 5.46410i −0.204061 0.204061i
\(718\) 0 0
\(719\) −2.27362 −0.0847919 −0.0423959 0.999101i \(-0.513499\pi\)
−0.0423959 + 0.999101i \(0.513499\pi\)
\(720\) 0 0
\(721\) 7.85641 0.292588
\(722\) 0 0
\(723\) −18.9396 18.9396i −0.704371 0.704371i
\(724\) 0 0
\(725\) −61.5167 + 61.5167i −2.28467 + 2.28467i
\(726\) 0 0
\(727\) 8.10634i 0.300648i 0.988637 + 0.150324i \(0.0480316\pi\)
−0.988637 + 0.150324i \(0.951968\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 8.48528 8.48528i 0.313839 0.313839i
\(732\) 0 0
\(733\) −5.00000 5.00000i −0.184679 0.184679i 0.608712 0.793391i \(-0.291686\pi\)
−0.793391 + 0.608712i \(0.791686\pi\)
\(734\) 0 0
\(735\) −3.86370 −0.142515
\(736\) 0 0
\(737\) −21.8564 −0.805091
\(738\) 0 0
\(739\) −25.4558 25.4558i −0.936408 0.936408i 0.0616872 0.998096i \(-0.480352\pi\)
−0.998096 + 0.0616872i \(0.980352\pi\)
\(740\) 0 0
\(741\) 22.3923 22.3923i 0.822602 0.822602i
\(742\) 0 0
\(743\) 27.3233i 1.00240i 0.865333 + 0.501198i \(0.167107\pi\)
−0.865333 + 0.501198i \(0.832893\pi\)
\(744\) 0 0
\(745\) 20.7846i 0.761489i
\(746\) 0 0
\(747\) −3.86370 + 3.86370i −0.141366 + 0.141366i
\(748\) 0 0
\(749\) 6.92820 + 6.92820i 0.253151 + 0.253151i
\(750\) 0 0
\(751\) 46.1886 1.68545 0.842723 0.538348i \(-0.180951\pi\)
0.842723 + 0.538348i \(0.180951\pi\)
\(752\) 0 0
\(753\) 6.53590 0.238181
\(754\) 0 0
\(755\) −18.0058 18.0058i −0.655300 0.655300i
\(756\) 0 0
\(757\) 23.7846 23.7846i 0.864466 0.864466i −0.127387 0.991853i \(-0.540659\pi\)
0.991853 + 0.127387i \(0.0406589\pi\)
\(758\) 0 0
\(759\) 4.14110i 0.150313i
\(760\) 0 0
\(761\) 52.2487i 1.89401i 0.321212 + 0.947007i \(0.395910\pi\)
−0.321212 + 0.947007i \(0.604090\pi\)
\(762\) 0 0
\(763\) −2.44949 + 2.44949i −0.0886775 + 0.0886775i
\(764\) 0 0
\(765\) 9.46410 + 9.46410i 0.342175 + 0.342175i
\(766\) 0 0
\(767\) 58.7878 2.12270
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 16.1112 + 16.1112i 0.580229 + 0.580229i
\(772\) 0 0
\(773\) 34.5885 34.5885i 1.24406 1.24406i 0.285759 0.958302i \(-0.407754\pi\)
0.958302 0.285759i \(-0.0922456\pi\)
\(774\) 0 0
\(775\) 72.9571i 2.62070i
\(776\) 0 0
\(777\) 1.60770i 0.0576757i
\(778\) 0 0
\(779\) −23.9401 + 23.9401i −0.857743 + 0.857743i
\(780\) 0 0
\(781\) 6.64102 + 6.64102i 0.237634 + 0.237634i
\(782\) 0 0
\(783\) −8.76268 −0.313153
\(784\) 0 0
\(785\) −8.39230 −0.299534
\(786\) 0 0
\(787\) −16.5916 16.5916i −0.591428 0.591428i 0.346589 0.938017i \(-0.387340\pi\)
−0.938017 + 0.346589i \(0.887340\pi\)
\(788\) 0 0
\(789\) 16.0000 16.0000i 0.569615 0.569615i
\(790\) 0 0
\(791\) 44.0908i 1.56769i
\(792\) 0 0
\(793\) 2.78461i 0.0988844i
\(794\) 0 0
\(795\) −12.6264 + 12.6264i −0.447812 + 0.447812i
\(796\) 0 0
\(797\) −15.5167 15.5167i −0.549628 0.549628i 0.376705 0.926333i \(-0.377057\pi\)
−0.926333 + 0.376705i \(0.877057\pi\)
\(798\) 0 0
\(799\) −9.79796 −0.346627
\(800\) 0 0
\(801\) −4.92820 −0.174129
\(802\) 0 0
\(803\) 4.14110 + 4.14110i 0.146136 + 0.146136i
\(804\) 0 0
\(805\) 18.9282 18.9282i 0.667132 0.667132i
\(806\) 0 0
\(807\) 10.2784i 0.361818i
\(808\) 0 0
\(809\) 6.67949i 0.234838i −0.993082 0.117419i \(-0.962538\pi\)
0.993082 0.117419i \(-0.0374621\pi\)
\(810\) 0 0
\(811\) −38.4612 + 38.4612i −1.35055 + 1.35055i −0.465513 + 0.885041i \(0.654130\pi\)
−0.885041 + 0.465513i \(0.845870\pi\)
\(812\) 0 0
\(813\) 2.26795 + 2.26795i 0.0795405 + 0.0795405i
\(814\) 0 0
\(815\) −17.5254 −0.613887
\(816\) 0 0
\(817\) 25.8564 0.904601
\(818\) 0 0
\(819\) −7.34847 7.34847i −0.256776 0.256776i
\(820\) 0 0
\(821\) 17.8038 17.8038i 0.621359 0.621359i −0.324520 0.945879i \(-0.605203\pi\)
0.945879 + 0.324520i \(0.105203\pi\)
\(822\) 0 0
\(823\) 9.97382i 0.347666i −0.984775 0.173833i \(-0.944385\pi\)
0.984775 0.173833i \(-0.0556152\pi\)
\(824\) 0 0
\(825\) 14.5359i 0.506075i
\(826\) 0 0
\(827\) 31.6675 31.6675i 1.10119 1.10119i 0.106919 0.994268i \(-0.465901\pi\)
0.994268 0.106919i \(-0.0340986\pi\)
\(828\) 0 0
\(829\) 1.14359 + 1.14359i 0.0397186 + 0.0397186i 0.726687 0.686969i \(-0.241059\pi\)
−0.686969 + 0.726687i \(0.741059\pi\)
\(830\) 0 0
\(831\) 12.5249 0.434482
\(832\) 0 0
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) −21.1117 21.1117i −0.730599 0.730599i
\(836\) 0 0
\(837\) −5.19615 + 5.19615i −0.179605 + 0.179605i
\(838\) 0 0
\(839\) 26.0106i 0.897987i −0.893535 0.448994i \(-0.851783\pi\)
0.893535 0.448994i \(-0.148217\pi\)
\(840\) 0 0
\(841\) 47.7846i 1.64775i
\(842\) 0 0
\(843\) −0.656339 + 0.656339i −0.0226055 + 0.0226055i
\(844\) 0 0
\(845\) 13.6603 + 13.6603i 0.469927 + 0.469927i
\(846\) 0 0
\(847\) −21.6937 −0.745404
\(848\) 0 0
\(849\) −9.85641 −0.338271
\(850\) 0 0
\(851\) −1.31268 1.31268i −0.0449980 0.0449980i
\(852\) 0 0
\(853\) −33.2487 + 33.2487i −1.13841 + 1.13841i −0.149680 + 0.988735i \(0.547824\pi\)
−0.988735 + 0.149680i \(0.952176\pi\)
\(854\) 0 0
\(855\) 28.8391i 0.986276i
\(856\) 0 0
\(857\) 43.1769i 1.47490i −0.675404 0.737448i \(-0.736031\pi\)
0.675404 0.737448i \(-0.263969\pi\)
\(858\) 0 0
\(859\) −22.8033 + 22.8033i −0.778038 + 0.778038i −0.979497 0.201459i \(-0.935432\pi\)
0.201459 + 0.979497i \(0.435432\pi\)
\(860\) 0 0
\(861\) 7.85641 + 7.85641i 0.267746 + 0.267746i
\(862\) 0 0
\(863\) 37.1213 1.26362 0.631812 0.775122i \(-0.282312\pi\)
0.631812 + 0.775122i \(0.282312\pi\)
\(864\) 0 0
\(865\) −66.6410 −2.26586
\(866\) 0 0
\(867\) 3.53553 + 3.53553i 0.120073 + 0.120073i
\(868\) 0 0
\(869\) −2.53590 + 2.53590i −0.0860245 + 0.0860245i
\(870\) 0 0
\(871\) 63.3350i 2.14602i
\(872\) 0 0
\(873\) 1.07180i 0.0362748i
\(874\) 0 0
\(875\) 32.9802 32.9802i 1.11493 1.11493i
\(876\) 0 0
\(877\) −25.2487 25.2487i −0.852588 0.852588i 0.137863 0.990451i \(-0.455977\pi\)
−0.990451 + 0.137863i \(0.955977\pi\)
\(878\) 0 0
\(879\) −10.8332 −0.365396
\(880\) 0 0
\(881\) 36.6410 1.23447 0.617234 0.786780i \(-0.288253\pi\)
0.617234 + 0.786780i \(0.288253\pi\)
\(882\) 0 0
\(883\) −3.96524 3.96524i −0.133441 0.133441i 0.637232 0.770672i \(-0.280080\pi\)
−0.770672 + 0.637232i \(0.780080\pi\)
\(884\) 0 0
\(885\) −37.8564 + 37.8564i −1.27253 + 1.27253i
\(886\) 0 0
\(887\) 36.5665i 1.22778i 0.789391 + 0.613891i \(0.210397\pi\)
−0.789391 + 0.613891i \(0.789603\pi\)
\(888\) 0 0
\(889\) 35.5692i 1.19295i
\(890\) 0 0
\(891\) −1.03528 + 1.03528i −0.0346831 + 0.0346831i
\(892\) 0 0
\(893\) −14.9282 14.9282i −0.499553 0.499553i
\(894\) 0 0
\(895\) −73.1330 −2.44457
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 45.5322 + 45.5322i 1.51859 + 1.51859i
\(900\) 0 0
\(901\) 11.3205 11.3205i 0.377141 0.377141i
\(902\) 0 0
\(903\) 8.48528i 0.282372i
\(904\) 0 0
\(905\) 27.3205i 0.908164i
\(906\) 0 0
\(907\) 29.9759 29.9759i 0.995333 0.995333i −0.00465619 0.999989i \(-0.501482\pi\)
0.999989 + 0.00465619i \(0.00148212\pi\)
\(908\) 0 0
\(909\) −10.1962 10.1962i −0.338185 0.338185i
\(910\) 0 0
\(911\) 22.0726 0.731298 0.365649 0.930753i \(-0.380847\pi\)
0.365649 + 0.930753i \(0.380847\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 0 0
\(915\) −1.79315 1.79315i −0.0592797 0.0592797i
\(916\) 0 0
\(917\) −25.8564 + 25.8564i −0.853854 + 0.853854i
\(918\) 0 0
\(919\) 0.582009i 0.0191987i −0.999954 0.00959936i \(-0.996944\pi\)
0.999954 0.00959936i \(-0.00305562\pi\)
\(920\) 0 0
\(921\) 22.9282i 0.755510i
\(922\) 0 0
\(923\) −19.2442 + 19.2442i −0.633430 + 0.633430i
\(924\) 0 0
\(925\) −4.60770 4.60770i −0.151500 0.151500i
\(926\) 0 0
\(927\) 3.20736 0.105344
\(928\) 0 0
\(929\) −45.3205 −1.48692 −0.743459 0.668782i \(-0.766816\pi\)
−0.743459 + 0.668782i \(0.766816\pi\)
\(930\) 0 0
\(931\) 5.27792 + 5.27792i 0.172977 + 0.172977i
\(932\) 0 0
\(933\) −22.9282 + 22.9282i −0.750636 + 0.750636i
\(934\) 0 0
\(935\) 19.5959i 0.640855i
\(936\) 0 0
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 0 0
\(939\) −2.72689 + 2.72689i −0.0889887 + 0.0889887i
\(940\) 0 0
\(941\) −25.6603 25.6603i −0.836500 0.836500i 0.151896 0.988396i \(-0.451462\pi\)
−0.988396 + 0.151896i \(0.951462\pi\)
\(942\) 0 0
\(943\) 12.8295 0.417785
\(944\) 0 0
\(945\) 9.46410 0.307867
\(946\) 0 0
\(947\) 20.5569 + 20.5569i 0.668008 + 0.668008i 0.957255 0.289246i \(-0.0934047\pi\)
−0.289246 + 0.957255i \(0.593405\pi\)
\(948\) 0 0
\(949\) −12.0000 + 12.0000i −0.389536 + 0.389536i
\(950\) 0 0
\(951\) 6.69213i 0.217007i
\(952\) 0 0
\(953\) 46.3923i 1.50279i −0.659850 0.751397i \(-0.729380\pi\)
0.659850 0.751397i \(-0.270620\pi\)
\(954\) 0 0
\(955\) 21.1117 21.1117i 0.683158 0.683158i
\(956\) 0 0
\(957\) 9.07180 + 9.07180i 0.293250 + 0.293250i
\(958\) 0 0
\(959\) −35.2538 −1.13840
\(960\) 0 0
\(961\) 23.0000 0.741935
\(962\) 0 0
\(963\) 2.82843 + 2.82843i 0.0911448 + 0.0911448i
\(964\) 0 0
\(965\) 16.3923 16.3923i 0.527687 0.527687i
\(966\) 0 0
\(967\) 10.3800i 0.333797i −0.985974 0.166899i \(-0.946625\pi\)
0.985974 0.166899i \(-0.0533753\pi\)
\(968\) 0 0
\(969\) 25.8564i 0.830627i
\(970\) 0 0
\(971\) −15.7322 + 15.7322i −0.504871 + 0.504871i −0.912948 0.408077i \(-0.866200\pi\)
0.408077 + 0.912948i \(0.366200\pi\)
\(972\) 0 0
\(973\) −12.0000 12.0000i −0.384702 0.384702i
\(974\) 0 0
\(975\) −42.1218 −1.34898
\(976\) 0 0
\(977\) 35.1769 1.12541 0.562705 0.826658i \(-0.309761\pi\)
0.562705 + 0.826658i \(0.309761\pi\)
\(978\) 0 0
\(979\) 5.10205 + 5.10205i 0.163062 + 0.163062i
\(980\) 0 0
\(981\) −1.00000 + 1.00000i −0.0319275 + 0.0319275i
\(982\) 0 0
\(983\) 23.7370i 0.757093i 0.925582 + 0.378547i \(0.123576\pi\)
−0.925582 + 0.378547i \(0.876424\pi\)
\(984\) 0 0
\(985\) 33.8564i 1.07875i
\(986\) 0 0
\(987\) −4.89898 + 4.89898i −0.155936 + 0.155936i
\(988\) 0 0
\(989\) −6.92820 6.92820i −0.220304 0.220304i
\(990\) 0 0
\(991\) −12.2474 −0.389053 −0.194527 0.980897i \(-0.562317\pi\)
−0.194527 + 0.980897i \(0.562317\pi\)
\(992\) 0 0
\(993\) 1.07180 0.0340124
\(994\) 0 0
\(995\) 50.9860 + 50.9860i 1.61637 + 1.61637i
\(996\) 0 0
\(997\) 12.4641 12.4641i 0.394742 0.394742i −0.481632 0.876374i \(-0.659956\pi\)
0.876374 + 0.481632i \(0.159956\pi\)
\(998\) 0 0
\(999\) 0.656339i 0.0207656i
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 768.2.j.f.577.2 yes 8
3.2 odd 2 2304.2.k.e.577.2 8
4.3 odd 2 inner 768.2.j.f.577.4 yes 8
8.3 odd 2 768.2.j.e.577.1 yes 8
8.5 even 2 768.2.j.e.577.3 yes 8
12.11 even 2 2304.2.k.e.577.1 8
16.3 odd 4 768.2.j.e.193.1 8
16.5 even 4 inner 768.2.j.f.193.2 yes 8
16.11 odd 4 inner 768.2.j.f.193.4 yes 8
16.13 even 4 768.2.j.e.193.3 yes 8
24.5 odd 2 2304.2.k.l.577.4 8
24.11 even 2 2304.2.k.l.577.3 8
32.3 odd 8 3072.2.d.h.1537.8 8
32.5 even 8 3072.2.a.r.1.4 4
32.11 odd 8 3072.2.a.r.1.1 4
32.13 even 8 3072.2.d.h.1537.5 8
32.19 odd 8 3072.2.d.h.1537.1 8
32.21 even 8 3072.2.a.l.1.1 4
32.27 odd 8 3072.2.a.l.1.4 4
32.29 even 8 3072.2.d.h.1537.4 8
48.5 odd 4 2304.2.k.e.1729.1 8
48.11 even 4 2304.2.k.e.1729.2 8
48.29 odd 4 2304.2.k.l.1729.3 8
48.35 even 4 2304.2.k.l.1729.4 8
96.5 odd 8 9216.2.a.bc.1.1 4
96.11 even 8 9216.2.a.bc.1.4 4
96.53 odd 8 9216.2.a.bi.1.4 4
96.59 even 8 9216.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.e.193.1 8 16.3 odd 4
768.2.j.e.193.3 yes 8 16.13 even 4
768.2.j.e.577.1 yes 8 8.3 odd 2
768.2.j.e.577.3 yes 8 8.5 even 2
768.2.j.f.193.2 yes 8 16.5 even 4 inner
768.2.j.f.193.4 yes 8 16.11 odd 4 inner
768.2.j.f.577.2 yes 8 1.1 even 1 trivial
768.2.j.f.577.4 yes 8 4.3 odd 2 inner
2304.2.k.e.577.1 8 12.11 even 2
2304.2.k.e.577.2 8 3.2 odd 2
2304.2.k.e.1729.1 8 48.5 odd 4
2304.2.k.e.1729.2 8 48.11 even 4
2304.2.k.l.577.3 8 24.11 even 2
2304.2.k.l.577.4 8 24.5 odd 2
2304.2.k.l.1729.3 8 48.29 odd 4
2304.2.k.l.1729.4 8 48.35 even 4
3072.2.a.l.1.1 4 32.21 even 8
3072.2.a.l.1.4 4 32.27 odd 8
3072.2.a.r.1.1 4 32.11 odd 8
3072.2.a.r.1.4 4 32.5 even 8
3072.2.d.h.1537.1 8 32.19 odd 8
3072.2.d.h.1537.4 8 32.29 even 8
3072.2.d.h.1537.5 8 32.13 even 8
3072.2.d.h.1537.8 8 32.3 odd 8
9216.2.a.bc.1.1 4 96.5 odd 8
9216.2.a.bc.1.4 4 96.11 even 8
9216.2.a.bi.1.1 4 96.59 even 8
9216.2.a.bi.1.4 4 96.53 odd 8