Properties

Label 315.2.i.b.106.1
Level $315$
Weight $2$
Character 315.106
Analytic conductor $2.515$
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] = [315,2,Mod(106,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.i (of order \(3\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 106.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 315.106
Dual form 315.2.i.b.211.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +(3.00000 + 1.73205i) q^{12} +(0.500000 + 0.866025i) q^{13} +(1.50000 + 0.866025i) q^{15} +(-2.00000 + 3.46410i) q^{16} -3.00000 q^{17} +2.00000 q^{19} +(-1.00000 + 1.73205i) q^{20} +1.73205i q^{21} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} -5.19615i q^{27} -2.00000 q^{28} +(4.50000 - 7.79423i) q^{29} +(2.00000 + 3.46410i) q^{31} -1.00000 q^{35} +6.00000 q^{36} -4.00000 q^{37} +(1.50000 + 0.866025i) q^{39} +(2.00000 - 3.46410i) q^{43} +3.00000 q^{45} +6.92820i q^{48} +(-0.500000 - 0.866025i) q^{49} +(-4.50000 + 2.59808i) q^{51} +(-1.00000 + 1.73205i) q^{52} -6.00000 q^{53} +(3.00000 - 1.73205i) q^{57} +(-3.00000 - 5.19615i) q^{59} +3.46410i q^{60} +(-7.00000 + 12.1244i) q^{61} +(1.50000 + 2.59808i) q^{63} -8.00000 q^{64} +(-0.500000 + 0.866025i) q^{65} +(-7.00000 - 12.1244i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(-9.00000 - 5.19615i) q^{69} +9.00000 q^{71} +11.0000 q^{73} +1.73205i q^{75} +(2.00000 + 3.46410i) q^{76} +(-4.00000 + 6.92820i) q^{79} -4.00000 q^{80} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 + 7.79423i) q^{83} +(-3.00000 + 1.73205i) q^{84} +(-1.50000 - 2.59808i) q^{85} -15.5885i q^{87} -12.0000 q^{89} -1.00000 q^{91} +(6.00000 - 10.3923i) q^{92} +(6.00000 + 3.46410i) q^{93} +(1.00000 + 1.73205i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 2 q^{4} + q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 2 q^{4} + q^{5} - q^{7} + 3 q^{9} + 6 q^{12} + q^{13} + 3 q^{15} - 4 q^{16} - 6 q^{17} + 4 q^{19} - 2 q^{20} - 6 q^{23} - q^{25} - 4 q^{28} + 9 q^{29} + 4 q^{31} - 2 q^{35} + 12 q^{36} - 8 q^{37} + 3 q^{39} + 4 q^{43} + 6 q^{45} - q^{49} - 9 q^{51} - 2 q^{52} - 12 q^{53} + 6 q^{57} - 6 q^{59} - 14 q^{61} + 3 q^{63} - 16 q^{64} - q^{65} - 14 q^{67} - 6 q^{68} - 18 q^{69} + 18 q^{71} + 22 q^{73} + 4 q^{76} - 8 q^{79} - 8 q^{80} - 9 q^{81} - 9 q^{83} - 6 q^{84} - 3 q^{85} - 24 q^{89} - 2 q^{91} + 12 q^{92} + 12 q^{93} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 1.50000 0.866025i 0.866025 0.500000i
\(4\) 1.00000 + 1.73205i 0.500000 + 0.866025i
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 3.00000 + 1.73205i 0.866025 + 0.500000i
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) 1.50000 + 0.866025i 0.387298 + 0.223607i
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 + 1.73205i −0.223607 + 0.387298i
\(21\) 1.73205i 0.377964i
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) −2.00000 −0.377964
\(29\) 4.50000 7.79423i 0.835629 1.44735i −0.0578882 0.998323i \(-0.518437\pi\)
0.893517 0.449029i \(-0.148230\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 6.00000 1.00000
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 1.50000 + 0.866025i 0.240192 + 0.138675i
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 2.00000 3.46410i 0.304997 0.528271i −0.672264 0.740312i \(-0.734678\pi\)
0.977261 + 0.212041i \(0.0680112\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 6.92820i 1.00000i
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) −4.50000 + 2.59808i −0.630126 + 0.363803i
\(52\) −1.00000 + 1.73205i −0.138675 + 0.240192i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 1.73205i 0.397360 0.229416i
\(58\) 0 0
\(59\) −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 3.46410i 0.447214i
\(61\) −7.00000 + 12.1244i −0.896258 + 1.55236i −0.0640184 + 0.997949i \(0.520392\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 1.50000 + 2.59808i 0.188982 + 0.327327i
\(64\) −8.00000 −1.00000
\(65\) −0.500000 + 0.866025i −0.0620174 + 0.107417i
\(66\) 0 0
\(67\) −7.00000 12.1244i −0.855186 1.48123i −0.876472 0.481452i \(-0.840109\pi\)
0.0212861 0.999773i \(-0.493224\pi\)
\(68\) −3.00000 5.19615i −0.363803 0.630126i
\(69\) −9.00000 5.19615i −1.08347 0.625543i
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 1.73205i 0.200000i
\(76\) 2.00000 + 3.46410i 0.229416 + 0.397360i
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) −4.00000 −0.447214
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) −4.50000 + 7.79423i −0.493939 + 0.855528i −0.999976 0.00698436i \(-0.997777\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(84\) −3.00000 + 1.73205i −0.327327 + 0.188982i
\(85\) −1.50000 2.59808i −0.162698 0.281801i
\(86\) 0 0
\(87\) 15.5885i 1.67126i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 6.00000 10.3923i 0.625543 1.08347i
\(93\) 6.00000 + 3.46410i 0.622171 + 0.359211i
\(94\) 0 0
\(95\) 1.00000 + 1.73205i 0.102598 + 0.177705i
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −6.00000 + 10.3923i −0.597022 + 1.03407i 0.396236 + 0.918149i \(0.370316\pi\)
−0.993258 + 0.115924i \(0.963017\pi\)
\(102\) 0 0
\(103\) −5.50000 9.52628i −0.541931 0.938652i −0.998793 0.0491146i \(-0.984360\pi\)
0.456862 0.889538i \(-0.348973\pi\)
\(104\) 0 0
\(105\) −1.50000 + 0.866025i −0.146385 + 0.0845154i
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 9.00000 5.19615i 0.866025 0.500000i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −6.00000 + 3.46410i −0.569495 + 0.328798i
\(112\) −2.00000 3.46410i −0.188982 0.327327i
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 3.00000 5.19615i 0.279751 0.484544i
\(116\) 18.0000 1.67126
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 1.50000 2.59808i 0.137505 0.238165i
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −4.00000 + 6.92820i −0.359211 + 0.622171i
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 6.92820i 0.609994i
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) −1.00000 + 1.73205i −0.0867110 + 0.150188i
\(134\) 0 0
\(135\) 4.50000 2.59808i 0.387298 0.223607i
\(136\) 0 0
\(137\) 3.00000 5.19615i 0.256307 0.443937i −0.708942 0.705266i \(-0.750827\pi\)
0.965250 + 0.261329i \(0.0841608\pi\)
\(138\) 0 0
\(139\) −4.00000 6.92820i −0.339276 0.587643i 0.645021 0.764165i \(-0.276849\pi\)
−0.984297 + 0.176522i \(0.943515\pi\)
\(140\) −1.00000 1.73205i −0.0845154 0.146385i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 + 10.3923i 0.500000 + 0.866025i
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) −1.50000 0.866025i −0.123718 0.0714286i
\(148\) −4.00000 6.92820i −0.328798 0.569495i
\(149\) 10.5000 + 18.1865i 0.860194 + 1.48990i 0.871742 + 0.489966i \(0.162991\pi\)
−0.0115483 + 0.999933i \(0.503676\pi\)
\(150\) 0 0
\(151\) −2.50000 + 4.33013i −0.203447 + 0.352381i −0.949637 0.313353i \(-0.898548\pi\)
0.746190 + 0.665733i \(0.231881\pi\)
\(152\) 0 0
\(153\) −4.50000 + 7.79423i −0.363803 + 0.630126i
\(154\) 0 0
\(155\) −2.00000 + 3.46410i −0.160644 + 0.278243i
\(156\) 3.46410i 0.277350i
\(157\) 6.50000 + 11.2583i 0.518756 + 0.898513i 0.999762 + 0.0217953i \(0.00693820\pi\)
−0.481006 + 0.876717i \(0.659728\pi\)
\(158\) 0 0
\(159\) −9.00000 + 5.19615i −0.713746 + 0.412082i
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.50000 + 12.9904i 0.580367 + 1.00523i 0.995436 + 0.0954356i \(0.0304244\pi\)
−0.415068 + 0.909790i \(0.636242\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) 3.00000 5.19615i 0.229416 0.397360i
\(172\) 8.00000 0.609994
\(173\) −10.5000 + 18.1865i −0.798300 + 1.38270i 0.122422 + 0.992478i \(0.460934\pi\)
−0.920722 + 0.390218i \(0.872399\pi\)
\(174\) 0 0
\(175\) −0.500000 0.866025i −0.0377964 0.0654654i
\(176\) 0 0
\(177\) −9.00000 5.19615i −0.676481 0.390567i
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 3.00000 + 5.19615i 0.223607 + 0.387298i
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 24.2487i 1.79252i
\(184\) 0 0
\(185\) −2.00000 3.46410i −0.147043 0.254686i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.50000 + 2.59808i 0.327327 + 0.188982i
\(190\) 0 0
\(191\) 1.50000 2.59808i 0.108536 0.187990i −0.806641 0.591041i \(-0.798717\pi\)
0.915177 + 0.403051i \(0.132050\pi\)
\(192\) −12.0000 + 6.92820i −0.866025 + 0.500000i
\(193\) −13.0000 22.5167i −0.935760 1.62078i −0.773272 0.634074i \(-0.781381\pi\)
−0.162488 0.986710i \(-0.551952\pi\)
\(194\) 0 0
\(195\) 1.73205i 0.124035i
\(196\) 1.00000 1.73205i 0.0714286 0.123718i
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 26.0000 1.84309 0.921546 0.388270i \(-0.126927\pi\)
0.921546 + 0.388270i \(0.126927\pi\)
\(200\) 0 0
\(201\) −21.0000 12.1244i −1.48123 0.855186i
\(202\) 0 0
\(203\) 4.50000 + 7.79423i 0.315838 + 0.547048i
\(204\) −9.00000 5.19615i −0.630126 0.363803i
\(205\) 0 0
\(206\) 0 0
\(207\) −18.0000 −1.25109
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −10.0000 17.3205i −0.688428 1.19239i −0.972346 0.233544i \(-0.924968\pi\)
0.283918 0.958849i \(-0.408366\pi\)
\(212\) −6.00000 10.3923i −0.412082 0.713746i
\(213\) 13.5000 7.79423i 0.925005 0.534052i
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 16.5000 9.52628i 1.11497 0.643726i
\(220\) 0 0
\(221\) −1.50000 2.59808i −0.100901 0.174766i
\(222\) 0 0
\(223\) 9.50000 16.4545i 0.636167 1.10187i −0.350100 0.936713i \(-0.613852\pi\)
0.986267 0.165161i \(-0.0528144\pi\)
\(224\) 0 0
\(225\) 1.50000 + 2.59808i 0.100000 + 0.173205i
\(226\) 0 0
\(227\) 1.50000 2.59808i 0.0995585 0.172440i −0.811943 0.583736i \(-0.801590\pi\)
0.911502 + 0.411296i \(0.134924\pi\)
\(228\) 6.00000 + 3.46410i 0.397360 + 0.229416i
\(229\) −7.00000 12.1244i −0.462573 0.801200i 0.536515 0.843891i \(-0.319740\pi\)
−0.999088 + 0.0426906i \(0.986407\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 10.3923i 0.390567 0.676481i
\(237\) 13.8564i 0.900070i
\(238\) 0 0
\(239\) 4.50000 + 7.79423i 0.291081 + 0.504167i 0.974066 0.226266i \(-0.0726518\pi\)
−0.682985 + 0.730433i \(0.739318\pi\)
\(240\) −6.00000 + 3.46410i −0.387298 + 0.223607i
\(241\) −10.0000 + 17.3205i −0.644157 + 1.11571i 0.340339 + 0.940303i \(0.389458\pi\)
−0.984496 + 0.175409i \(0.943875\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) −28.0000 −1.79252
\(245\) 0.500000 0.866025i 0.0319438 0.0553283i
\(246\) 0 0
\(247\) 1.00000 + 1.73205i 0.0636285 + 0.110208i
\(248\) 0 0
\(249\) 15.5885i 0.987878i
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) −3.00000 + 5.19615i −0.188982 + 0.327327i
\(253\) 0 0
\(254\) 0 0
\(255\) −4.50000 2.59808i −0.281801 0.162698i
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 13.5000 + 23.3827i 0.842107 + 1.45857i 0.888110 + 0.459631i \(0.152018\pi\)
−0.0460033 + 0.998941i \(0.514648\pi\)
\(258\) 0 0
\(259\) 2.00000 3.46410i 0.124274 0.215249i
\(260\) −2.00000 −0.124035
\(261\) −13.5000 23.3827i −0.835629 1.44735i
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) −3.00000 5.19615i −0.184289 0.319197i
\(266\) 0 0
\(267\) −18.0000 + 10.3923i −1.10158 + 0.635999i
\(268\) 14.0000 24.2487i 0.855186 1.48123i
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 6.00000 10.3923i 0.363803 0.630126i
\(273\) −1.50000 + 0.866025i −0.0907841 + 0.0524142i
\(274\) 0 0
\(275\) 0 0
\(276\) 20.7846i 1.25109i
\(277\) −13.0000 + 22.5167i −0.781094 + 1.35290i 0.150210 + 0.988654i \(0.452005\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 0 0
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) −3.00000 + 5.19615i −0.178965 + 0.309976i −0.941526 0.336939i \(-0.890608\pi\)
0.762561 + 0.646916i \(0.223942\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 9.00000 + 15.5885i 0.534052 + 0.925005i
\(285\) 3.00000 + 1.73205i 0.177705 + 0.102598i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 3.46410i 0.203069i
\(292\) 11.0000 + 19.0526i 0.643726 + 1.11497i
\(293\) −10.5000 18.1865i −0.613417 1.06247i −0.990660 0.136355i \(-0.956461\pi\)
0.377244 0.926114i \(-0.376872\pi\)
\(294\) 0 0
\(295\) 3.00000 5.19615i 0.174667 0.302532i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 5.19615i 0.173494 0.300501i
\(300\) −3.00000 + 1.73205i −0.173205 + 0.100000i
\(301\) 2.00000 + 3.46410i 0.115278 + 0.199667i
\(302\) 0 0
\(303\) 20.7846i 1.19404i
\(304\) −4.00000 + 6.92820i −0.229416 + 0.397360i
\(305\) −14.0000 −0.801638
\(306\) 0 0
\(307\) −13.0000 −0.741949 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(308\) 0 0
\(309\) −16.5000 9.52628i −0.938652 0.541931i
\(310\) 0 0
\(311\) 3.00000 + 5.19615i 0.170114 + 0.294647i 0.938460 0.345389i \(-0.112253\pi\)
−0.768345 + 0.640036i \(0.778920\pi\)
\(312\) 0 0
\(313\) 0.500000 0.866025i 0.0282617 0.0489506i −0.851549 0.524276i \(-0.824336\pi\)
0.879810 + 0.475325i \(0.157669\pi\)
\(314\) 0 0
\(315\) −1.50000 + 2.59808i −0.0845154 + 0.146385i
\(316\) −16.0000 −0.900070
\(317\) −12.0000 + 20.7846i −0.673987 + 1.16738i 0.302777 + 0.953062i \(0.402086\pi\)
−0.976764 + 0.214318i \(0.931247\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −4.00000 6.92820i −0.223607 0.387298i
\(321\) −9.00000 + 5.19615i −0.502331 + 0.290021i
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 9.00000 15.5885i 0.500000 0.866025i
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 21.0000 12.1244i 1.16130 0.670478i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.50000 16.4545i 0.522167 0.904420i −0.477500 0.878632i \(-0.658457\pi\)
0.999667 0.0257885i \(-0.00820965\pi\)
\(332\) −18.0000 −0.987878
\(333\) −6.00000 + 10.3923i −0.328798 + 0.569495i
\(334\) 0 0
\(335\) 7.00000 12.1244i 0.382451 0.662424i
\(336\) −6.00000 3.46410i −0.327327 0.188982i
\(337\) 11.0000 + 19.0526i 0.599208 + 1.03786i 0.992938 + 0.118633i \(0.0378512\pi\)
−0.393730 + 0.919226i \(0.628816\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 3.00000 5.19615i 0.162698 0.281801i
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 10.3923i 0.559503i
\(346\) 0 0
\(347\) −6.00000 10.3923i −0.322097 0.557888i 0.658824 0.752297i \(-0.271054\pi\)
−0.980921 + 0.194409i \(0.937721\pi\)
\(348\) 27.0000 15.5885i 1.44735 0.835629i
\(349\) 2.00000 3.46410i 0.107058 0.185429i −0.807519 0.589841i \(-0.799190\pi\)
0.914577 + 0.404412i \(0.132524\pi\)
\(350\) 0 0
\(351\) 4.50000 2.59808i 0.240192 0.138675i
\(352\) 0 0
\(353\) −16.5000 + 28.5788i −0.878206 + 1.52110i −0.0248989 + 0.999690i \(0.507926\pi\)
−0.853307 + 0.521408i \(0.825407\pi\)
\(354\) 0 0
\(355\) 4.50000 + 7.79423i 0.238835 + 0.413675i
\(356\) −12.0000 20.7846i −0.635999 1.10158i
\(357\) 5.19615i 0.275010i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 16.5000 + 9.52628i 0.866025 + 0.500000i
\(364\) −1.00000 1.73205i −0.0524142 0.0907841i
\(365\) 5.50000 + 9.52628i 0.287883 + 0.498628i
\(366\) 0 0
\(367\) −8.50000 + 14.7224i −0.443696 + 0.768505i −0.997960 0.0638362i \(-0.979666\pi\)
0.554264 + 0.832341i \(0.313000\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 3.00000 5.19615i 0.155752 0.269771i
\(372\) 13.8564i 0.718421i
\(373\) 11.0000 + 19.0526i 0.569558 + 0.986504i 0.996610 + 0.0822766i \(0.0262191\pi\)
−0.427051 + 0.904227i \(0.640448\pi\)
\(374\) 0 0
\(375\) −1.50000 + 0.866025i −0.0774597 + 0.0447214i
\(376\) 0 0
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) −2.00000 + 3.46410i −0.102598 + 0.177705i
\(381\) 12.0000 6.92820i 0.614779 0.354943i
\(382\) 0 0
\(383\) −7.50000 12.9904i −0.383232 0.663777i 0.608290 0.793715i \(-0.291856\pi\)
−0.991522 + 0.129937i \(0.958522\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.00000 10.3923i −0.304997 0.528271i
\(388\) −4.00000 −0.203069
\(389\) 9.00000 15.5885i 0.456318 0.790366i −0.542445 0.840091i \(-0.682501\pi\)
0.998763 + 0.0497253i \(0.0158346\pi\)
\(390\) 0 0
\(391\) 9.00000 + 15.5885i 0.455150 + 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 29.0000 1.45547 0.727734 0.685859i \(-0.240573\pi\)
0.727734 + 0.685859i \(0.240573\pi\)
\(398\) 0 0
\(399\) 3.46410i 0.173422i
\(400\) −2.00000 3.46410i −0.100000 0.173205i
\(401\) −16.5000 28.5788i −0.823971 1.42716i −0.902703 0.430263i \(-0.858421\pi\)
0.0787327 0.996896i \(-0.474913\pi\)
\(402\) 0 0
\(403\) −2.00000 + 3.46410i −0.0996271 + 0.172559i
\(404\) −24.0000 −1.19404
\(405\) 4.50000 7.79423i 0.223607 0.387298i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 17.0000 + 29.4449i 0.840596 + 1.45595i 0.889392 + 0.457146i \(0.151128\pi\)
−0.0487958 + 0.998809i \(0.515538\pi\)
\(410\) 0 0
\(411\) 10.3923i 0.512615i
\(412\) 11.0000 19.0526i 0.541931 0.938652i
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) −12.0000 6.92820i −0.587643 0.339276i
\(418\) 0 0
\(419\) 6.00000 + 10.3923i 0.293119 + 0.507697i 0.974546 0.224189i \(-0.0719734\pi\)
−0.681426 + 0.731887i \(0.738640\pi\)
\(420\) −3.00000 1.73205i −0.146385 0.0845154i
\(421\) 5.00000 8.66025i 0.243685 0.422075i −0.718076 0.695965i \(-0.754977\pi\)
0.961761 + 0.273890i \(0.0883103\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.50000 2.59808i 0.0727607 0.126025i
\(426\) 0 0
\(427\) −7.00000 12.1244i −0.338754 0.586739i
\(428\) −6.00000 10.3923i −0.290021 0.502331i
\(429\) 0 0
\(430\) 0 0
\(431\) −9.00000 −0.433515 −0.216757 0.976226i \(-0.569548\pi\)
−0.216757 + 0.976226i \(0.569548\pi\)
\(432\) 18.0000 + 10.3923i 0.866025 + 0.500000i
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 13.5000 7.79423i 0.647275 0.373705i
\(436\) 14.0000 + 24.2487i 0.670478 + 1.16130i
\(437\) −6.00000 10.3923i −0.287019 0.497131i
\(438\) 0 0
\(439\) 5.00000 8.66025i 0.238637 0.413331i −0.721686 0.692220i \(-0.756633\pi\)
0.960323 + 0.278889i \(0.0899661\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −3.00000 + 5.19615i −0.142534 + 0.246877i −0.928450 0.371457i \(-0.878858\pi\)
0.785916 + 0.618333i \(0.212192\pi\)
\(444\) −12.0000 6.92820i −0.569495 0.328798i
\(445\) −6.00000 10.3923i −0.284427 0.492642i
\(446\) 0 0
\(447\) 31.5000 + 18.1865i 1.48990 + 0.860194i
\(448\) 4.00000 6.92820i 0.188982 0.327327i
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.66025i 0.406894i
\(454\) 0 0
\(455\) −0.500000 0.866025i −0.0234404 0.0405999i
\(456\) 0 0
\(457\) −1.00000 + 1.73205i −0.0467780 + 0.0810219i −0.888466 0.458942i \(-0.848229\pi\)
0.841688 + 0.539964i \(0.181562\pi\)
\(458\) 0 0
\(459\) 15.5885i 0.727607i
\(460\) 12.0000 0.559503
\(461\) 12.0000 20.7846i 0.558896 0.968036i −0.438693 0.898637i \(-0.644559\pi\)
0.997589 0.0693989i \(-0.0221081\pi\)
\(462\) 0 0
\(463\) 2.00000 + 3.46410i 0.0929479 + 0.160990i 0.908750 0.417340i \(-0.137038\pi\)
−0.815802 + 0.578331i \(0.803704\pi\)
\(464\) 18.0000 + 31.1769i 0.835629 + 1.44735i
\(465\) 6.92820i 0.321288i
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 3.00000 + 5.19615i 0.138675 + 0.240192i
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 19.5000 + 11.2583i 0.898513 + 0.518756i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.00000 + 1.73205i −0.0458831 + 0.0794719i
\(476\) 6.00000 0.275010
\(477\) −9.00000 + 15.5885i −0.412082 + 0.713746i
\(478\) 0 0
\(479\) 9.00000 15.5885i 0.411220 0.712255i −0.583803 0.811895i \(-0.698436\pi\)
0.995023 + 0.0996406i \(0.0317693\pi\)
\(480\) 0 0
\(481\) −2.00000 3.46410i −0.0911922 0.157949i
\(482\) 0 0
\(483\) 9.00000 5.19615i 0.409514 0.236433i
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) 21.0000 12.1244i 0.949653 0.548282i
\(490\) 0 0
\(491\) 13.5000 + 23.3827i 0.609246 + 1.05525i 0.991365 + 0.131132i \(0.0418613\pi\)
−0.382118 + 0.924113i \(0.624805\pi\)
\(492\) 0 0
\(493\) −13.5000 + 23.3827i −0.608009 + 1.05310i
\(494\) 0 0
\(495\) 0 0
\(496\) −16.0000 −0.718421
\(497\) −4.50000 + 7.79423i −0.201853 + 0.349619i
\(498\) 0 0
\(499\) 12.5000 + 21.6506i 0.559577 + 0.969216i 0.997532 + 0.0702185i \(0.0223697\pi\)
−0.437955 + 0.898997i \(0.644297\pi\)
\(500\) −1.00000 1.73205i −0.0447214 0.0774597i
\(501\) 22.5000 + 12.9904i 1.00523 + 0.580367i
\(502\) 0 0
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 20.7846i 0.923077i
\(508\) 8.00000 + 13.8564i 0.354943 + 0.614779i
\(509\) −9.00000 15.5885i −0.398918 0.690946i 0.594675 0.803966i \(-0.297281\pi\)
−0.993593 + 0.113020i \(0.963948\pi\)
\(510\) 0 0
\(511\) −5.50000 + 9.52628i −0.243306 + 0.421418i
\(512\) 0 0
\(513\) 10.3923i 0.458831i
\(514\) 0 0
\(515\) 5.50000 9.52628i 0.242359 0.419778i
\(516\) 12.0000 6.92820i 0.528271 0.304997i
\(517\) 0 0
\(518\) 0 0
\(519\) 36.3731i 1.59660i
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) 0 0
\(525\) −1.50000 0.866025i −0.0654654 0.0377964i
\(526\) 0 0
\(527\) −6.00000 10.3923i −0.261364 0.452696i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 5.19615i −0.129701 0.224649i
\(536\) 0 0
\(537\) 22.5000 12.9904i 0.970947 0.560576i
\(538\) 0 0
\(539\) 0 0
\(540\) 9.00000 + 5.19615i 0.387298 + 0.223607i
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) 3.00000 1.73205i 0.128742 0.0743294i
\(544\) 0 0
\(545\) 7.00000 + 12.1244i 0.299847 + 0.519350i
\(546\) 0 0
\(547\) −1.00000 + 1.73205i −0.0427569 + 0.0740571i −0.886612 0.462514i \(-0.846947\pi\)
0.843855 + 0.536571i \(0.180281\pi\)
\(548\) 12.0000 0.512615
\(549\) 21.0000 + 36.3731i 0.896258 + 1.55236i
\(550\) 0 0
\(551\) 9.00000 15.5885i 0.383413 0.664091i
\(552\) 0 0
\(553\) −4.00000 6.92820i −0.170097 0.294617i
\(554\) 0 0
\(555\) −6.00000 3.46410i −0.254686 0.147043i
\(556\) 8.00000 13.8564i 0.339276 0.587643i
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 2.00000 3.46410i 0.0845154 0.146385i
\(561\) 0 0
\(562\) 0 0
\(563\) −10.5000 18.1865i −0.442522 0.766471i 0.555354 0.831614i \(-0.312583\pi\)
−0.997876 + 0.0651433i \(0.979250\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) 7.50000 12.9904i 0.314416 0.544585i −0.664897 0.746935i \(-0.731525\pi\)
0.979313 + 0.202350i \(0.0648579\pi\)
\(570\) 0 0
\(571\) −2.50000 4.33013i −0.104622 0.181210i 0.808962 0.587861i \(-0.200030\pi\)
−0.913584 + 0.406651i \(0.866697\pi\)
\(572\) 0 0
\(573\) 5.19615i 0.217072i
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) −12.0000 + 20.7846i −0.500000 + 0.866025i
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) −39.0000 22.5167i −1.62078 0.935760i
\(580\) 9.00000 + 15.5885i 0.373705 + 0.647275i
\(581\) −4.50000 7.79423i −0.186691 0.323359i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.50000 + 2.59808i 0.0620174 + 0.107417i
\(586\) 0 0
\(587\) −6.00000 + 10.3923i −0.247647 + 0.428936i −0.962872 0.269957i \(-0.912990\pi\)
0.715226 + 0.698893i \(0.246324\pi\)
\(588\) 3.46410i 0.142857i
\(589\) 4.00000 + 6.92820i 0.164817 + 0.285472i
\(590\) 0 0
\(591\) 36.0000 20.7846i 1.48084 0.854965i
\(592\) 8.00000 13.8564i 0.328798 0.569495i
\(593\) 3.00000 0.123195 0.0615976 0.998101i \(-0.480380\pi\)
0.0615976 + 0.998101i \(0.480380\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) −21.0000 + 36.3731i −0.860194 + 1.48990i
\(597\) 39.0000 22.5167i 1.59616 0.921546i
\(598\) 0 0
\(599\) −13.5000 23.3827i −0.551595 0.955391i −0.998160 0.0606393i \(-0.980686\pi\)
0.446565 0.894751i \(-0.352647\pi\)
\(600\) 0 0
\(601\) −1.00000 + 1.73205i −0.0407909 + 0.0706518i −0.885700 0.464258i \(-0.846321\pi\)
0.844909 + 0.534910i \(0.179654\pi\)
\(602\) 0 0
\(603\) −42.0000 −1.71037
\(604\) −10.0000 −0.406894
\(605\) −5.50000 + 9.52628i −0.223607 + 0.387298i
\(606\) 0 0
\(607\) −20.5000 35.5070i −0.832069 1.44119i −0.896394 0.443257i \(-0.853823\pi\)
0.0643251 0.997929i \(-0.479511\pi\)
\(608\) 0 0
\(609\) 13.5000 + 7.79423i 0.547048 + 0.315838i
\(610\) 0 0
\(611\) 0 0
\(612\) −18.0000 −0.727607
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9.00000 15.5885i −0.362326 0.627568i 0.626017 0.779809i \(-0.284684\pi\)
−0.988343 + 0.152242i \(0.951351\pi\)
\(618\) 0 0
\(619\) 8.00000 13.8564i 0.321547 0.556936i −0.659260 0.751915i \(-0.729130\pi\)
0.980807 + 0.194979i \(0.0624638\pi\)
\(620\) −8.00000 −0.321288
\(621\) −27.0000 + 15.5885i −1.08347 + 0.625543i
\(622\) 0 0
\(623\) 6.00000 10.3923i 0.240385 0.416359i
\(624\) −6.00000 + 3.46410i −0.240192 + 0.138675i
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) −13.0000 + 22.5167i −0.518756 + 0.898513i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) −30.0000 17.3205i −1.19239 0.688428i
\(634\) 0 0
\(635\) 4.00000 + 6.92820i 0.158735 + 0.274937i
\(636\) −18.0000 10.3923i −0.713746 0.412082i
\(637\) 0.500000 0.866025i 0.0198107 0.0343132i
\(638\) 0 0
\(639\) 13.5000 23.3827i 0.534052 0.925005i
\(640\) 0 0
\(641\) −15.0000 + 25.9808i −0.592464 + 1.02618i 0.401435 + 0.915888i \(0.368512\pi\)
−0.993899 + 0.110291i \(0.964822\pi\)
\(642\) 0 0
\(643\) 2.00000 + 3.46410i 0.0788723 + 0.136611i 0.902764 0.430137i \(-0.141535\pi\)
−0.823891 + 0.566748i \(0.808201\pi\)
\(644\) 6.00000 + 10.3923i 0.236433 + 0.409514i
\(645\) 6.00000 3.46410i 0.236250 0.136399i
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 + 3.46410i −0.235159 + 0.135769i
\(652\) 14.0000 + 24.2487i 0.548282 + 0.949653i
\(653\) −15.0000 25.9808i −0.586995 1.01671i −0.994623 0.103558i \(-0.966977\pi\)
0.407628 0.913148i \(-0.366356\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.5000 28.5788i 0.643726 1.11497i
\(658\) 0 0
\(659\) 22.5000 38.9711i 0.876476 1.51810i 0.0212930 0.999773i \(-0.493222\pi\)
0.855183 0.518327i \(-0.173445\pi\)
\(660\) 0 0
\(661\) −7.00000 12.1244i −0.272268 0.471583i 0.697174 0.716902i \(-0.254441\pi\)
−0.969442 + 0.245319i \(0.921107\pi\)
\(662\) 0 0
\(663\) −4.50000 2.59808i −0.174766 0.100901i
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) −15.0000 + 25.9808i −0.580367 + 1.00523i
\(669\) 32.9090i 1.27233i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.00000 8.66025i 0.192736 0.333828i −0.753420 0.657539i \(-0.771597\pi\)
0.946156 + 0.323711i \(0.104931\pi\)
\(674\) 0 0
\(675\) 4.50000 + 2.59808i 0.173205 + 0.100000i
\(676\) 24.0000 0.923077
\(677\) −19.5000 + 33.7750i −0.749446 + 1.29808i 0.198643 + 0.980072i \(0.436347\pi\)
−0.948089 + 0.318006i \(0.896987\pi\)
\(678\) 0 0
\(679\) −1.00000 1.73205i −0.0383765 0.0664700i
\(680\) 0 0
\(681\) 5.19615i 0.199117i
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 12.0000 0.458831
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) −21.0000 12.1244i −0.801200 0.462573i
\(688\) 8.00000 + 13.8564i 0.304997 + 0.528271i
\(689\) −3.00000 5.19615i −0.114291 0.197958i
\(690\) 0 0
\(691\) −1.00000 + 1.73205i −0.0380418 + 0.0658903i −0.884419 0.466693i \(-0.845445\pi\)
0.846378 + 0.532583i \(0.178779\pi\)
\(692\) −42.0000 −1.59660
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 6.92820i 0.151729 0.262802i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −36.0000 + 20.7846i −1.36165 + 0.786146i
\(700\) 1.00000 1.73205i 0.0377964 0.0654654i
\(701\) −33.0000 −1.24639 −0.623196 0.782065i \(-0.714166\pi\)
−0.623196 + 0.782065i \(0.714166\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 10.3923i −0.225653 0.390843i
\(708\) 20.7846i 0.781133i
\(709\) 12.5000 21.6506i 0.469447 0.813107i −0.529943 0.848034i \(-0.677787\pi\)
0.999390 + 0.0349269i \(0.0111198\pi\)
\(710\) 0 0
\(711\) 12.0000 + 20.7846i 0.450035 + 0.779484i
\(712\) 0 0
\(713\) 12.0000 20.7846i 0.449404 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 15.0000 + 25.9808i 0.560576 + 0.970947i
\(717\) 13.5000 + 7.79423i 0.504167 + 0.291081i
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −6.00000 + 10.3923i −0.223607 + 0.387298i
\(721\) 11.0000 0.409661
\(722\) 0 0
\(723\) 34.6410i 1.28831i
\(724\) 2.00000 + 3.46410i 0.0743294 + 0.128742i
\(725\) 4.50000 + 7.79423i 0.167126 + 0.289470i
\(726\) 0 0
\(727\) 6.50000 11.2583i 0.241072 0.417548i −0.719948 0.694028i \(-0.755834\pi\)
0.961020 + 0.276479i \(0.0891678\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −6.00000 + 10.3923i −0.221918 + 0.384373i
\(732\) −42.0000 + 24.2487i −1.55236 + 0.896258i
\(733\) −5.50000 9.52628i −0.203147 0.351861i 0.746394 0.665505i \(-0.231784\pi\)
−0.949541 + 0.313644i \(0.898450\pi\)
\(734\) 0 0
\(735\) 1.73205i 0.0638877i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 17.0000 0.625355 0.312678 0.949859i \(-0.398774\pi\)
0.312678 + 0.949859i \(0.398774\pi\)
\(740\) 4.00000 6.92820i 0.147043 0.254686i
\(741\) 3.00000 + 1.73205i 0.110208 + 0.0636285i
\(742\) 0 0
\(743\) −3.00000 5.19615i −0.110059 0.190628i 0.805735 0.592277i \(-0.201771\pi\)
−0.915794 + 0.401648i \(0.868437\pi\)
\(744\) 0 0
\(745\) −10.5000 + 18.1865i −0.384690 + 0.666303i
\(746\) 0 0
\(747\) 13.5000 + 23.3827i 0.493939 + 0.855528i
\(748\) 0 0
\(749\) 3.00000 5.19615i 0.109618 0.189863i
\(750\) 0 0
\(751\) 6.50000 + 11.2583i 0.237188 + 0.410822i 0.959906 0.280321i \(-0.0904408\pi\)
−0.722718 + 0.691143i \(0.757107\pi\)
\(752\) 0 0
\(753\) 36.0000 20.7846i 1.31191 0.757433i
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 10.3923i 0.377964i
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 5.19615i −0.108750 0.188360i 0.806514 0.591215i \(-0.201351\pi\)
−0.915264 + 0.402854i \(0.868018\pi\)
\(762\) 0 0
\(763\) −7.00000 + 12.1244i −0.253417 + 0.438931i
\(764\) 6.00000 0.217072
\(765\) −9.00000 −0.325396
\(766\) 0 0
\(767\) 3.00000 5.19615i 0.108324 0.187622i
\(768\) −24.0000 13.8564i −0.866025 0.500000i
\(769\) −7.00000 12.1244i −0.252426 0.437215i 0.711767 0.702416i \(-0.247895\pi\)
−0.964193 + 0.265200i \(0.914562\pi\)
\(770\) 0 0
\(771\) 40.5000 + 23.3827i 1.45857 + 0.842107i
\(772\) 26.0000 45.0333i 0.935760 1.62078i
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 6.92820i 0.248548i
\(778\) 0 0
\(779\) 0 0
\(780\) −3.00000 + 1.73205i −0.107417 + 0.0620174i
\(781\) 0 0
\(782\) 0 0
\(783\) −40.5000 23.3827i −1.44735 0.835629i
\(784\) 4.00000 0.142857
\(785\) −6.50000 + 11.2583i −0.231995 + 0.401827i
\(786\) 0 0
\(787\) −14.5000 25.1147i −0.516869 0.895244i −0.999808 0.0195896i \(-0.993764\pi\)
0.482939 0.875654i \(-0.339569\pi\)
\(788\) 24.0000 + 41.5692i 0.854965 + 1.48084i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) 0 0
\(795\) −9.00000 5.19615i −0.319197 0.184289i
\(796\) 26.0000 + 45.0333i 0.921546 + 1.59616i
\(797\) 19.5000 + 33.7750i 0.690725 + 1.19637i 0.971601 + 0.236627i \(0.0760420\pi\)
−0.280875 + 0.959744i \(0.590625\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 + 31.1769i −0.635999 + 1.10158i
\(802\) 0 0
\(803\) 0 0
\(804\) 48.4974i 1.71037i
\(805\) 3.00000 + 5.19615i 0.105736 + 0.183140i
\(806\) 0 0
\(807\) −36.0000 + 20.7846i −1.26726 + 0.731653i
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) −9.00000 + 15.5885i −0.315838 + 0.547048i
\(813\) 21.0000 12.1244i 0.736502 0.425220i
\(814\) 0 0
\(815\) 7.00000 + 12.1244i 0.245199 + 0.424698i
\(816\) 20.7846i 0.727607i
\(817\) 4.00000 6.92820i 0.139942 0.242387i
\(818\) 0 0
\(819\) −1.50000 + 2.59808i −0.0524142 + 0.0907841i
\(820\) 0 0
\(821\) 25.5000 44.1673i 0.889956 1.54145i 0.0500305 0.998748i \(-0.484068\pi\)
0.839926 0.542702i \(-0.182599\pi\)
\(822\) 0 0
\(823\) 17.0000 + 29.4449i 0.592583 + 1.02638i 0.993883 + 0.110437i \(0.0352250\pi\)
−0.401300 + 0.915947i \(0.631442\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −18.0000 31.1769i −0.625543 1.08347i
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) 45.0333i 1.56219i
\(832\) −4.00000 6.92820i −0.138675 0.240192i
\(833\) 1.50000 + 2.59808i 0.0519719 + 0.0900180i
\(834\) 0 0
\(835\) −7.50000 + 12.9904i −0.259548 + 0.449551i
\(836\) 0 0
\(837\) 18.0000 10.3923i 0.622171 0.359211i
\(838\) 0 0
\(839\) −24.0000 + 41.5692i −0.828572 + 1.43513i 0.0705865 + 0.997506i \(0.477513\pi\)
−0.899158 + 0.437623i \(0.855820\pi\)
\(840\) 0 0
\(841\) −26.0000 45.0333i −0.896552 1.55287i
\(842\) 0 0
\(843\) 10.3923i 0.357930i
\(844\) 20.0000 34.6410i 0.688428 1.19239i
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 12.0000 20.7846i 0.412082 0.713746i
\(849\) 6.00000 + 3.46410i 0.205919 + 0.118888i
\(850\) 0 0
\(851\) 12.0000 + 20.7846i 0.411355 + 0.712487i
\(852\) 27.0000 + 15.5885i 0.925005 + 0.534052i
\(853\) 5.00000 8.66025i 0.171197 0.296521i −0.767642 0.640879i \(-0.778570\pi\)
0.938839 + 0.344358i \(0.111903\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 16.5000 28.5788i 0.563629 0.976235i −0.433546 0.901131i \(-0.642738\pi\)
0.997176 0.0751033i \(-0.0239287\pi\)
\(858\) 0 0
\(859\) −28.0000 48.4974i −0.955348 1.65471i −0.733571 0.679613i \(-0.762148\pi\)
−0.221777 0.975097i \(-0.571186\pi\)
\(860\) 4.00000 + 6.92820i 0.136399 + 0.236250i
\(861\) 0 0
\(862\) 0 0
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 0 0
\(867\) −12.0000 + 6.92820i −0.407541 + 0.235294i
\(868\) −4.00000 6.92820i −0.135769 0.235159i
\(869\) 0 0
\(870\) 0 0
\(871\) 7.00000 12.1244i 0.237186 0.410818i
\(872\) 0 0
\(873\) 3.00000 + 5.19615i 0.101535 + 0.175863i
\(874\) 0 0
\(875\) 0.500000 0.866025i 0.0169031 0.0292770i
\(876\) 33.0000 + 19.0526i 1.11497 + 0.643726i
\(877\) −7.00000 12.1244i −0.236373 0.409410i 0.723298 0.690536i \(-0.242625\pi\)
−0.959671 + 0.281126i \(0.909292\pi\)
\(878\) 0 0
\(879\) −31.5000 18.1865i −1.06247 0.613417i
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 3.00000 5.19615i 0.100901 0.174766i
\(885\) 10.3923i 0.349334i
\(886\) 0 0
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) −4.00000 + 6.92820i −0.134156 + 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 38.0000 1.27233
\(893\) 0 0
\(894\) 0 0
\(895\) 7.50000 + 12.9904i 0.250697 + 0.434221i
\(896\) 0 0
\(897\) 10.3923i 0.346989i
\(898\) 0 0
\(899\) 36.0000 1.20067
\(900\) −3.00000 + 5.19615i −0.100000 + 0.173205i
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 6.00000 + 3.46410i 0.199667 + 0.115278i
\(904\) 0 0
\(905\) 1.00000 + 1.73205i 0.0332411 + 0.0575753i
\(906\) 0 0
\(907\) 14.0000 24.2487i 0.464862 0.805165i −0.534333 0.845274i \(-0.679437\pi\)
0.999195 + 0.0401089i \(0.0127705\pi\)
\(908\) 6.00000 0.199117
\(909\) 18.0000 + 31.1769i 0.597022 + 1.03407i
\(910\) 0 0
\(911\) −25.5000 + 44.1673i −0.844853 + 1.46333i 0.0408964 + 0.999163i \(0.486979\pi\)
−0.885749 + 0.464164i \(0.846355\pi\)
\(912\) 13.8564i 0.458831i
\(913\) 0 0
\(914\) 0 0
\(915\) −21.0000 + 12.1244i −0.694239 + 0.400819i
\(916\) 14.0000 24.2487i 0.462573 0.801200i
\(917\) 0 0
\(918\) 0 0
\(919\) −49.0000 −1.61636 −0.808180 0.588935i \(-0.799547\pi\)
−0.808180 + 0.588935i \(0.799547\pi\)
\(920\) 0 0
\(921\) −19.5000 + 11.2583i −0.642547 + 0.370975i
\(922\) 0 0
\(923\) 4.50000 + 7.79423i 0.148119 + 0.256550i
\(924\) 0 0
\(925\) 2.00000 3.46410i 0.0657596 0.113899i
\(926\) 0 0
\(927\) −33.0000 −1.08386
\(928\) 0 0
\(929\) 3.00000 5.19615i 0.0984268 0.170480i −0.812607 0.582812i \(-0.801952\pi\)
0.911034 + 0.412332i \(0.135286\pi\)
\(930\) 0 0
\(931\) −1.00000 1.73205i −0.0327737 0.0567657i
\(932\) −24.0000 41.5692i −0.786146 1.36165i
\(933\) 9.00000 + 5.19615i 0.294647 + 0.170114i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) 1.73205i 0.0565233i
\(940\) 0 0
\(941\) 27.0000 + 46.7654i 0.880175 + 1.52451i 0.851146 + 0.524929i \(0.175908\pi\)
0.0290288 + 0.999579i \(0.490759\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 24.0000 0.781133
\(945\) 5.19615i 0.169031i
\(946\) 0 0
\(947\) −12.0000 + 20.7846i −0.389948 + 0.675409i −0.992442 0.122714i \(-0.960840\pi\)
0.602494 + 0.798123i \(0.294174\pi\)
\(948\) −24.0000 + 13.8564i −0.779484 + 0.450035i
\(949\) 5.50000 + 9.52628i 0.178538 + 0.309236i
\(950\) 0 0
\(951\) 41.5692i 1.34797i
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) −9.00000 + 15.5885i −0.291081 + 0.504167i
\(957\) 0 0
\(958\) 0 0
\(959\) 3.00000 + 5.19615i 0.0968751 + 0.167793i
\(960\) −12.0000 6.92820i −0.387298 0.223607i
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) −9.00000 + 15.5885i −0.290021 + 0.502331i
\(964\) −40.0000 −1.28831
\(965\) 13.0000 22.5167i 0.418485 0.724837i
\(966\) 0 0
\(967\) 17.0000 + 29.4449i 0.546683 + 0.946883i 0.998499 + 0.0547717i \(0.0174431\pi\)
−0.451816 + 0.892111i \(0.649224\pi\)
\(968\) 0 0
\(969\) −9.00000 + 5.19615i −0.289122 + 0.166924i
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 31.1769i 1.00000i
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) −1.50000 + 0.866025i −0.0480384 + 0.0277350i
\(976\) −28.0000 48.4974i −0.896258 1.55236i
\(977\) −6.00000 10.3923i −0.191957 0.332479i 0.753942 0.656941i \(-0.228150\pi\)
−0.945899 + 0.324462i \(0.894817\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 21.0000 36.3731i 0.670478 1.16130i
\(982\) 0 0
\(983\) 24.0000 41.5692i 0.765481 1.32585i −0.174511 0.984655i \(-0.555834\pi\)
0.939992 0.341197i \(-0.110832\pi\)
\(984\) 0 0
\(985\) 12.0000 + 20.7846i 0.382352 + 0.662253i
\(986\) 0 0
\(987\) 0 0
\(988\) −2.00000 + 3.46410i −0.0636285 + 0.110208i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 0 0
\(993\) 32.9090i 1.04433i
\(994\) 0 0
\(995\) 13.0000 + 22.5167i 0.412128 + 0.713826i
\(996\) −27.0000 + 15.5885i −0.855528 + 0.493939i
\(997\) −8.50000 + 14.7224i −0.269198 + 0.466264i −0.968655 0.248410i \(-0.920092\pi\)
0.699457 + 0.714675i \(0.253425\pi\)
\(998\) 0 0
\(999\) 20.7846i 0.657596i
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 315.2.i.b.106.1 2
3.2 odd 2 945.2.i.a.316.1 2
9.2 odd 6 2835.2.a.f.1.1 1
9.4 even 3 inner 315.2.i.b.211.1 yes 2
9.5 odd 6 945.2.i.a.631.1 2
9.7 even 3 2835.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.i.b.106.1 2 1.1 even 1 trivial
315.2.i.b.211.1 yes 2 9.4 even 3 inner
945.2.i.a.316.1 2 3.2 odd 2
945.2.i.a.631.1 2 9.5 odd 6
2835.2.a.c.1.1 1 9.7 even 3
2835.2.a.f.1.1 1 9.2 odd 6