Properties

Label 324.3.d.b.163.1
Level $324$
Weight $3$
Character 324.163
Analytic conductor $8.828$
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] = [324,3,Mod(163,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), 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: \(8.82836056527\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 163.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 324.163
Dual form 324.3.d.b.163.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +4.00000 q^{5} +3.46410i q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +4.00000 q^{5} +3.46410i q^{7} +8.00000 q^{8} +(-4.00000 - 6.92820i) q^{10} +12.1244i q^{11} -22.0000 q^{13} +(6.00000 - 3.46410i) q^{14} +(-8.00000 - 13.8564i) q^{16} -11.0000 q^{17} +15.5885i q^{19} +(-8.00000 + 13.8564i) q^{20} +(21.0000 - 12.1244i) q^{22} +24.2487i q^{23} -9.00000 q^{25} +(22.0000 + 38.1051i) q^{26} +(-12.0000 - 6.92820i) q^{28} +34.0000 q^{29} +6.92820i q^{31} +(-16.0000 + 27.7128i) q^{32} +(11.0000 + 19.0526i) q^{34} +13.8564i q^{35} -16.0000 q^{37} +(27.0000 - 15.5885i) q^{38} +32.0000 q^{40} +13.0000 q^{41} +50.2295i q^{43} +(-42.0000 - 24.2487i) q^{44} +(42.0000 - 24.2487i) q^{46} -3.46410i q^{47} +37.0000 q^{49} +(9.00000 + 15.5885i) q^{50} +(44.0000 - 76.2102i) q^{52} +52.0000 q^{53} +48.4974i q^{55} +27.7128i q^{56} +(-34.0000 - 58.8897i) q^{58} -53.6936i q^{59} -16.0000 q^{61} +(12.0000 - 6.92820i) q^{62} +64.0000 q^{64} -88.0000 q^{65} +116.047i q^{67} +(22.0000 - 38.1051i) q^{68} +(24.0000 - 13.8564i) q^{70} -25.0000 q^{73} +(16.0000 + 27.7128i) q^{74} +(-54.0000 - 31.1769i) q^{76} -42.0000 q^{77} -27.7128i q^{79} +(-32.0000 - 55.4256i) q^{80} +(-13.0000 - 22.5167i) q^{82} -34.6410i q^{83} -44.0000 q^{85} +(87.0000 - 50.2295i) q^{86} +96.9948i q^{88} -2.00000 q^{89} -76.2102i q^{91} +(-84.0000 - 48.4974i) q^{92} +(-6.00000 + 3.46410i) q^{94} +62.3538i q^{95} -43.0000 q^{97} +(-37.0000 - 64.0859i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} - 8 q^{10} - 44 q^{13} + 12 q^{14} - 16 q^{16} - 22 q^{17} - 16 q^{20} + 42 q^{22} - 18 q^{25} + 44 q^{26} - 24 q^{28} + 68 q^{29} - 32 q^{32} + 22 q^{34} - 32 q^{37} + 54 q^{38} + 64 q^{40} + 26 q^{41} - 84 q^{44} + 84 q^{46} + 74 q^{49} + 18 q^{50} + 88 q^{52} + 104 q^{53} - 68 q^{58} - 32 q^{61} + 24 q^{62} + 128 q^{64} - 176 q^{65} + 44 q^{68} + 48 q^{70} - 50 q^{73} + 32 q^{74} - 108 q^{76} - 84 q^{77} - 64 q^{80} - 26 q^{82} - 88 q^{85} + 174 q^{86} - 4 q^{89} - 168 q^{92} - 12 q^{94} - 86 q^{97} - 74 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 1.73205i −0.500000 0.866025i
\(3\) 0 0
\(4\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(5\) 4.00000 0.800000 0.400000 0.916515i \(-0.369010\pi\)
0.400000 + 0.916515i \(0.369010\pi\)
\(6\) 0 0
\(7\) 3.46410i 0.494872i 0.968904 + 0.247436i \(0.0795879\pi\)
−0.968904 + 0.247436i \(0.920412\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) −4.00000 6.92820i −0.400000 0.692820i
\(11\) 12.1244i 1.10221i 0.834435 + 0.551107i \(0.185794\pi\)
−0.834435 + 0.551107i \(0.814206\pi\)
\(12\) 0 0
\(13\) −22.0000 −1.69231 −0.846154 0.532939i \(-0.821088\pi\)
−0.846154 + 0.532939i \(0.821088\pi\)
\(14\) 6.00000 3.46410i 0.428571 0.247436i
\(15\) 0 0
\(16\) −8.00000 13.8564i −0.500000 0.866025i
\(17\) −11.0000 −0.647059 −0.323529 0.946218i \(-0.604869\pi\)
−0.323529 + 0.946218i \(0.604869\pi\)
\(18\) 0 0
\(19\) 15.5885i 0.820445i 0.911985 + 0.410223i \(0.134549\pi\)
−0.911985 + 0.410223i \(0.865451\pi\)
\(20\) −8.00000 + 13.8564i −0.400000 + 0.692820i
\(21\) 0 0
\(22\) 21.0000 12.1244i 0.954545 0.551107i
\(23\) 24.2487i 1.05429i 0.849775 + 0.527146i \(0.176738\pi\)
−0.849775 + 0.527146i \(0.823262\pi\)
\(24\) 0 0
\(25\) −9.00000 −0.360000
\(26\) 22.0000 + 38.1051i 0.846154 + 1.46558i
\(27\) 0 0
\(28\) −12.0000 6.92820i −0.428571 0.247436i
\(29\) 34.0000 1.17241 0.586207 0.810161i \(-0.300621\pi\)
0.586207 + 0.810161i \(0.300621\pi\)
\(30\) 0 0
\(31\) 6.92820i 0.223490i 0.993737 + 0.111745i \(0.0356441\pi\)
−0.993737 + 0.111745i \(0.964356\pi\)
\(32\) −16.0000 + 27.7128i −0.500000 + 0.866025i
\(33\) 0 0
\(34\) 11.0000 + 19.0526i 0.323529 + 0.560369i
\(35\) 13.8564i 0.395897i
\(36\) 0 0
\(37\) −16.0000 −0.432432 −0.216216 0.976346i \(-0.569372\pi\)
−0.216216 + 0.976346i \(0.569372\pi\)
\(38\) 27.0000 15.5885i 0.710526 0.410223i
\(39\) 0 0
\(40\) 32.0000 0.800000
\(41\) 13.0000 0.317073 0.158537 0.987353i \(-0.449322\pi\)
0.158537 + 0.987353i \(0.449322\pi\)
\(42\) 0 0
\(43\) 50.2295i 1.16813i 0.811708 + 0.584064i \(0.198538\pi\)
−0.811708 + 0.584064i \(0.801462\pi\)
\(44\) −42.0000 24.2487i −0.954545 0.551107i
\(45\) 0 0
\(46\) 42.0000 24.2487i 0.913043 0.527146i
\(47\) 3.46410i 0.0737043i −0.999321 0.0368521i \(-0.988267\pi\)
0.999321 0.0368521i \(-0.0117331\pi\)
\(48\) 0 0
\(49\) 37.0000 0.755102
\(50\) 9.00000 + 15.5885i 0.180000 + 0.311769i
\(51\) 0 0
\(52\) 44.0000 76.2102i 0.846154 1.46558i
\(53\) 52.0000 0.981132 0.490566 0.871404i \(-0.336790\pi\)
0.490566 + 0.871404i \(0.336790\pi\)
\(54\) 0 0
\(55\) 48.4974i 0.881771i
\(56\) 27.7128i 0.494872i
\(57\) 0 0
\(58\) −34.0000 58.8897i −0.586207 1.01534i
\(59\) 53.6936i 0.910061i −0.890476 0.455030i \(-0.849628\pi\)
0.890476 0.455030i \(-0.150372\pi\)
\(60\) 0 0
\(61\) −16.0000 −0.262295 −0.131148 0.991363i \(-0.541866\pi\)
−0.131148 + 0.991363i \(0.541866\pi\)
\(62\) 12.0000 6.92820i 0.193548 0.111745i
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) −88.0000 −1.35385
\(66\) 0 0
\(67\) 116.047i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 22.0000 38.1051i 0.323529 0.560369i
\(69\) 0 0
\(70\) 24.0000 13.8564i 0.342857 0.197949i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −25.0000 −0.342466 −0.171233 0.985231i \(-0.554775\pi\)
−0.171233 + 0.985231i \(0.554775\pi\)
\(74\) 16.0000 + 27.7128i 0.216216 + 0.374497i
\(75\) 0 0
\(76\) −54.0000 31.1769i −0.710526 0.410223i
\(77\) −42.0000 −0.545455
\(78\) 0 0
\(79\) 27.7128i 0.350795i −0.984498 0.175398i \(-0.943879\pi\)
0.984498 0.175398i \(-0.0561211\pi\)
\(80\) −32.0000 55.4256i −0.400000 0.692820i
\(81\) 0 0
\(82\) −13.0000 22.5167i −0.158537 0.274593i
\(83\) 34.6410i 0.417362i −0.977984 0.208681i \(-0.933083\pi\)
0.977984 0.208681i \(-0.0669170\pi\)
\(84\) 0 0
\(85\) −44.0000 −0.517647
\(86\) 87.0000 50.2295i 1.01163 0.584064i
\(87\) 0 0
\(88\) 96.9948i 1.10221i
\(89\) −2.00000 −0.0224719 −0.0112360 0.999937i \(-0.503577\pi\)
−0.0112360 + 0.999937i \(0.503577\pi\)
\(90\) 0 0
\(91\) 76.2102i 0.837475i
\(92\) −84.0000 48.4974i −0.913043 0.527146i
\(93\) 0 0
\(94\) −6.00000 + 3.46410i −0.0638298 + 0.0368521i
\(95\) 62.3538i 0.656356i
\(96\) 0 0
\(97\) −43.0000 −0.443299 −0.221649 0.975126i \(-0.571144\pi\)
−0.221649 + 0.975126i \(0.571144\pi\)
\(98\) −37.0000 64.0859i −0.377551 0.653938i
\(99\) 0 0
\(100\) 18.0000 31.1769i 0.180000 0.311769i
\(101\) −20.0000 −0.198020 −0.0990099 0.995086i \(-0.531568\pi\)
−0.0990099 + 0.995086i \(0.531568\pi\)
\(102\) 0 0
\(103\) 24.2487i 0.235424i −0.993048 0.117712i \(-0.962444\pi\)
0.993048 0.117712i \(-0.0375560\pi\)
\(104\) −176.000 −1.69231
\(105\) 0 0
\(106\) −52.0000 90.0666i −0.490566 0.849685i
\(107\) 15.5885i 0.145687i 0.997343 + 0.0728433i \(0.0232073\pi\)
−0.997343 + 0.0728433i \(0.976793\pi\)
\(108\) 0 0
\(109\) −88.0000 −0.807339 −0.403670 0.914905i \(-0.632266\pi\)
−0.403670 + 0.914905i \(0.632266\pi\)
\(110\) 84.0000 48.4974i 0.763636 0.440886i
\(111\) 0 0
\(112\) 48.0000 27.7128i 0.428571 0.247436i
\(113\) −50.0000 −0.442478 −0.221239 0.975220i \(-0.571010\pi\)
−0.221239 + 0.975220i \(0.571010\pi\)
\(114\) 0 0
\(115\) 96.9948i 0.843433i
\(116\) −68.0000 + 117.779i −0.586207 + 1.01534i
\(117\) 0 0
\(118\) −93.0000 + 53.6936i −0.788136 + 0.455030i
\(119\) 38.1051i 0.320211i
\(120\) 0 0
\(121\) −26.0000 −0.214876
\(122\) 16.0000 + 27.7128i 0.131148 + 0.227154i
\(123\) 0 0
\(124\) −24.0000 13.8564i −0.193548 0.111745i
\(125\) −136.000 −1.08800
\(126\) 0 0
\(127\) 218.238i 1.71841i 0.511629 + 0.859206i \(0.329042\pi\)
−0.511629 + 0.859206i \(0.670958\pi\)
\(128\) −64.0000 110.851i −0.500000 0.866025i
\(129\) 0 0
\(130\) 88.0000 + 152.420i 0.676923 + 1.17247i
\(131\) 193.990i 1.48084i −0.672146 0.740419i \(-0.734627\pi\)
0.672146 0.740419i \(-0.265373\pi\)
\(132\) 0 0
\(133\) −54.0000 −0.406015
\(134\) 201.000 116.047i 1.50000 0.866025i
\(135\) 0 0
\(136\) −88.0000 −0.647059
\(137\) 169.000 1.23358 0.616788 0.787129i \(-0.288433\pi\)
0.616788 + 0.787129i \(0.288433\pi\)
\(138\) 0 0
\(139\) 195.722i 1.40807i −0.710165 0.704035i \(-0.751380\pi\)
0.710165 0.704035i \(-0.248620\pi\)
\(140\) −48.0000 27.7128i −0.342857 0.197949i
\(141\) 0 0
\(142\) 0 0
\(143\) 266.736i 1.86529i
\(144\) 0 0
\(145\) 136.000 0.937931
\(146\) 25.0000 + 43.3013i 0.171233 + 0.296584i
\(147\) 0 0
\(148\) 32.0000 55.4256i 0.216216 0.374497i
\(149\) 130.000 0.872483 0.436242 0.899830i \(-0.356309\pi\)
0.436242 + 0.899830i \(0.356309\pi\)
\(150\) 0 0
\(151\) 121.244i 0.802937i −0.915873 0.401469i \(-0.868500\pi\)
0.915873 0.401469i \(-0.131500\pi\)
\(152\) 124.708i 0.820445i
\(153\) 0 0
\(154\) 42.0000 + 72.7461i 0.272727 + 0.472377i
\(155\) 27.7128i 0.178792i
\(156\) 0 0
\(157\) −4.00000 −0.0254777 −0.0127389 0.999919i \(-0.504055\pi\)
−0.0127389 + 0.999919i \(0.504055\pi\)
\(158\) −48.0000 + 27.7128i −0.303797 + 0.175398i
\(159\) 0 0
\(160\) −64.0000 + 110.851i −0.400000 + 0.692820i
\(161\) −84.0000 −0.521739
\(162\) 0 0
\(163\) 311.769i 1.91269i −0.292233 0.956347i \(-0.594398\pi\)
0.292233 0.956347i \(-0.405602\pi\)
\(164\) −26.0000 + 45.0333i −0.158537 + 0.274593i
\(165\) 0 0
\(166\) −60.0000 + 34.6410i −0.361446 + 0.208681i
\(167\) 180.133i 1.07864i 0.842100 + 0.539321i \(0.181319\pi\)
−0.842100 + 0.539321i \(0.818681\pi\)
\(168\) 0 0
\(169\) 315.000 1.86391
\(170\) 44.0000 + 76.2102i 0.258824 + 0.448296i
\(171\) 0 0
\(172\) −174.000 100.459i −1.01163 0.584064i
\(173\) −2.00000 −0.0115607 −0.00578035 0.999983i \(-0.501840\pi\)
−0.00578035 + 0.999983i \(0.501840\pi\)
\(174\) 0 0
\(175\) 31.1769i 0.178154i
\(176\) 168.000 96.9948i 0.954545 0.551107i
\(177\) 0 0
\(178\) 2.00000 + 3.46410i 0.0112360 + 0.0194612i
\(179\) 187.061i 1.04504i 0.852628 + 0.522518i \(0.175007\pi\)
−0.852628 + 0.522518i \(0.824993\pi\)
\(180\) 0 0
\(181\) 254.000 1.40331 0.701657 0.712514i \(-0.252444\pi\)
0.701657 + 0.712514i \(0.252444\pi\)
\(182\) −132.000 + 76.2102i −0.725275 + 0.418738i
\(183\) 0 0
\(184\) 193.990i 1.05429i
\(185\) −64.0000 −0.345946
\(186\) 0 0
\(187\) 133.368i 0.713197i
\(188\) 12.0000 + 6.92820i 0.0638298 + 0.0368521i
\(189\) 0 0
\(190\) 108.000 62.3538i 0.568421 0.328178i
\(191\) 3.46410i 0.0181367i −0.999959 0.00906833i \(-0.997113\pi\)
0.999959 0.00906833i \(-0.00288658\pi\)
\(192\) 0 0
\(193\) −67.0000 −0.347150 −0.173575 0.984821i \(-0.555532\pi\)
−0.173575 + 0.984821i \(0.555532\pi\)
\(194\) 43.0000 + 74.4782i 0.221649 + 0.383908i
\(195\) 0 0
\(196\) −74.0000 + 128.172i −0.377551 + 0.653938i
\(197\) 268.000 1.36041 0.680203 0.733024i \(-0.261892\pi\)
0.680203 + 0.733024i \(0.261892\pi\)
\(198\) 0 0
\(199\) 31.1769i 0.156668i −0.996927 0.0783340i \(-0.975040\pi\)
0.996927 0.0783340i \(-0.0249600\pi\)
\(200\) −72.0000 −0.360000
\(201\) 0 0
\(202\) 20.0000 + 34.6410i 0.0990099 + 0.171490i
\(203\) 117.779i 0.580194i
\(204\) 0 0
\(205\) 52.0000 0.253659
\(206\) −42.0000 + 24.2487i −0.203883 + 0.117712i
\(207\) 0 0
\(208\) 176.000 + 304.841i 0.846154 + 1.46558i
\(209\) −189.000 −0.904306
\(210\) 0 0
\(211\) 131.636i 0.623867i 0.950104 + 0.311933i \(0.100977\pi\)
−0.950104 + 0.311933i \(0.899023\pi\)
\(212\) −104.000 + 180.133i −0.490566 + 0.849685i
\(213\) 0 0
\(214\) 27.0000 15.5885i 0.126168 0.0728433i
\(215\) 200.918i 0.934502i
\(216\) 0 0
\(217\) −24.0000 −0.110599
\(218\) 88.0000 + 152.420i 0.403670 + 0.699176i
\(219\) 0 0
\(220\) −168.000 96.9948i −0.763636 0.440886i
\(221\) 242.000 1.09502
\(222\) 0 0
\(223\) 58.8897i 0.264079i −0.991244 0.132040i \(-0.957847\pi\)
0.991244 0.132040i \(-0.0421527\pi\)
\(224\) −96.0000 55.4256i −0.428571 0.247436i
\(225\) 0 0
\(226\) 50.0000 + 86.6025i 0.221239 + 0.383197i
\(227\) 448.601i 1.97622i 0.153760 + 0.988108i \(0.450862\pi\)
−0.153760 + 0.988108i \(0.549138\pi\)
\(228\) 0 0
\(229\) 410.000 1.79039 0.895197 0.445672i \(-0.147035\pi\)
0.895197 + 0.445672i \(0.147035\pi\)
\(230\) 168.000 96.9948i 0.730435 0.421717i
\(231\) 0 0
\(232\) 272.000 1.17241
\(233\) −65.0000 −0.278970 −0.139485 0.990224i \(-0.544545\pi\)
−0.139485 + 0.990224i \(0.544545\pi\)
\(234\) 0 0
\(235\) 13.8564i 0.0589634i
\(236\) 186.000 + 107.387i 0.788136 + 0.455030i
\(237\) 0 0
\(238\) −66.0000 + 38.1051i −0.277311 + 0.160106i
\(239\) 38.1051i 0.159436i −0.996817 0.0797178i \(-0.974598\pi\)
0.996817 0.0797178i \(-0.0254019\pi\)
\(240\) 0 0
\(241\) −223.000 −0.925311 −0.462656 0.886538i \(-0.653103\pi\)
−0.462656 + 0.886538i \(0.653103\pi\)
\(242\) 26.0000 + 45.0333i 0.107438 + 0.186088i
\(243\) 0 0
\(244\) 32.0000 55.4256i 0.131148 0.227154i
\(245\) 148.000 0.604082
\(246\) 0 0
\(247\) 342.946i 1.38845i
\(248\) 55.4256i 0.223490i
\(249\) 0 0
\(250\) 136.000 + 235.559i 0.544000 + 0.942236i
\(251\) 109.119i 0.434738i 0.976090 + 0.217369i \(0.0697475\pi\)
−0.976090 + 0.217369i \(0.930253\pi\)
\(252\) 0 0
\(253\) −294.000 −1.16206
\(254\) 378.000 218.238i 1.48819 0.859206i
\(255\) 0 0
\(256\) −128.000 + 221.703i −0.500000 + 0.866025i
\(257\) −437.000 −1.70039 −0.850195 0.526469i \(-0.823516\pi\)
−0.850195 + 0.526469i \(0.823516\pi\)
\(258\) 0 0
\(259\) 55.4256i 0.213999i
\(260\) 176.000 304.841i 0.676923 1.17247i
\(261\) 0 0
\(262\) −336.000 + 193.990i −1.28244 + 0.740419i
\(263\) 315.233i 1.19861i −0.800522 0.599303i \(-0.795445\pi\)
0.800522 0.599303i \(-0.204555\pi\)
\(264\) 0 0
\(265\) 208.000 0.784906
\(266\) 54.0000 + 93.5307i 0.203008 + 0.351619i
\(267\) 0 0
\(268\) −402.000 232.095i −1.50000 0.866025i
\(269\) 304.000 1.13011 0.565056 0.825053i \(-0.308855\pi\)
0.565056 + 0.825053i \(0.308855\pi\)
\(270\) 0 0
\(271\) 311.769i 1.15044i −0.817999 0.575220i \(-0.804917\pi\)
0.817999 0.575220i \(-0.195083\pi\)
\(272\) 88.0000 + 152.420i 0.323529 + 0.560369i
\(273\) 0 0
\(274\) −169.000 292.717i −0.616788 1.06831i
\(275\) 109.119i 0.396797i
\(276\) 0 0
\(277\) −34.0000 −0.122744 −0.0613718 0.998115i \(-0.519548\pi\)
−0.0613718 + 0.998115i \(0.519548\pi\)
\(278\) −339.000 + 195.722i −1.21942 + 0.704035i
\(279\) 0 0
\(280\) 110.851i 0.395897i
\(281\) −218.000 −0.775801 −0.387900 0.921701i \(-0.626800\pi\)
−0.387900 + 0.921701i \(0.626800\pi\)
\(282\) 0 0
\(283\) 6.92820i 0.0244813i 0.999925 + 0.0122406i \(0.00389641\pi\)
−0.999925 + 0.0122406i \(0.996104\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −462.000 + 266.736i −1.61538 + 0.932643i
\(287\) 45.0333i 0.156911i
\(288\) 0 0
\(289\) −168.000 −0.581315
\(290\) −136.000 235.559i −0.468966 0.812272i
\(291\) 0 0
\(292\) 50.0000 86.6025i 0.171233 0.296584i
\(293\) 202.000 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(294\) 0 0
\(295\) 214.774i 0.728048i
\(296\) −128.000 −0.432432
\(297\) 0 0
\(298\) −130.000 225.167i −0.436242 0.755593i
\(299\) 533.472i 1.78419i
\(300\) 0 0
\(301\) −174.000 −0.578073
\(302\) −210.000 + 121.244i −0.695364 + 0.401469i
\(303\) 0 0
\(304\) 216.000 124.708i 0.710526 0.410223i
\(305\) −64.0000 −0.209836
\(306\) 0 0
\(307\) 109.119i 0.355437i 0.984081 + 0.177719i \(0.0568717\pi\)
−0.984081 + 0.177719i \(0.943128\pi\)
\(308\) 84.0000 145.492i 0.272727 0.472377i
\(309\) 0 0
\(310\) 48.0000 27.7128i 0.154839 0.0893962i
\(311\) 273.664i 0.879949i 0.898010 + 0.439974i \(0.145012\pi\)
−0.898010 + 0.439974i \(0.854988\pi\)
\(312\) 0 0
\(313\) −79.0000 −0.252396 −0.126198 0.992005i \(-0.540277\pi\)
−0.126198 + 0.992005i \(0.540277\pi\)
\(314\) 4.00000 + 6.92820i 0.0127389 + 0.0220643i
\(315\) 0 0
\(316\) 96.0000 + 55.4256i 0.303797 + 0.175398i
\(317\) 502.000 1.58360 0.791798 0.610783i \(-0.209145\pi\)
0.791798 + 0.610783i \(0.209145\pi\)
\(318\) 0 0
\(319\) 412.228i 1.29225i
\(320\) 256.000 0.800000
\(321\) 0 0
\(322\) 84.0000 + 145.492i 0.260870 + 0.451839i
\(323\) 171.473i 0.530876i
\(324\) 0 0
\(325\) 198.000 0.609231
\(326\) −540.000 + 311.769i −1.65644 + 0.956347i
\(327\) 0 0
\(328\) 104.000 0.317073
\(329\) 12.0000 0.0364742
\(330\) 0 0
\(331\) 408.764i 1.23494i 0.786596 + 0.617468i \(0.211842\pi\)
−0.786596 + 0.617468i \(0.788158\pi\)
\(332\) 120.000 + 69.2820i 0.361446 + 0.208681i
\(333\) 0 0
\(334\) 312.000 180.133i 0.934132 0.539321i
\(335\) 464.190i 1.38564i
\(336\) 0 0
\(337\) −337.000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) −315.000 545.596i −0.931953 1.61419i
\(339\) 0 0
\(340\) 88.0000 152.420i 0.258824 0.448296i
\(341\) −84.0000 −0.246334
\(342\) 0 0
\(343\) 297.913i 0.868550i
\(344\) 401.836i 1.16813i
\(345\) 0 0
\(346\) 2.00000 + 3.46410i 0.00578035 + 0.0100119i
\(347\) 271.932i 0.783666i −0.920036 0.391833i \(-0.871841\pi\)
0.920036 0.391833i \(-0.128159\pi\)
\(348\) 0 0
\(349\) 272.000 0.779370 0.389685 0.920948i \(-0.372584\pi\)
0.389685 + 0.920948i \(0.372584\pi\)
\(350\) −54.0000 + 31.1769i −0.154286 + 0.0890769i
\(351\) 0 0
\(352\) −336.000 193.990i −0.954545 0.551107i
\(353\) −461.000 −1.30595 −0.652975 0.757380i \(-0.726479\pi\)
−0.652975 + 0.757380i \(0.726479\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.00000 6.92820i 0.0112360 0.0194612i
\(357\) 0 0
\(358\) 324.000 187.061i 0.905028 0.522518i
\(359\) 530.008i 1.47634i −0.674612 0.738172i \(-0.735689\pi\)
0.674612 0.738172i \(-0.264311\pi\)
\(360\) 0 0
\(361\) 118.000 0.326870
\(362\) −254.000 439.941i −0.701657 1.21531i
\(363\) 0 0
\(364\) 264.000 + 152.420i 0.725275 + 0.418738i
\(365\) −100.000 −0.273973
\(366\) 0 0
\(367\) 96.9948i 0.264291i 0.991230 + 0.132146i \(0.0421866\pi\)
−0.991230 + 0.132146i \(0.957813\pi\)
\(368\) 336.000 193.990i 0.913043 0.527146i
\(369\) 0 0
\(370\) 64.0000 + 110.851i 0.172973 + 0.299598i
\(371\) 180.133i 0.485534i
\(372\) 0 0
\(373\) −346.000 −0.927614 −0.463807 0.885936i \(-0.653517\pi\)
−0.463807 + 0.885936i \(0.653517\pi\)
\(374\) −231.000 + 133.368i −0.617647 + 0.356599i
\(375\) 0 0
\(376\) 27.7128i 0.0737043i
\(377\) −748.000 −1.98408
\(378\) 0 0
\(379\) 327.358i 0.863740i 0.901936 + 0.431870i \(0.142146\pi\)
−0.901936 + 0.431870i \(0.857854\pi\)
\(380\) −216.000 124.708i −0.568421 0.328178i
\(381\) 0 0
\(382\) −6.00000 + 3.46410i −0.0157068 + 0.00906833i
\(383\) 630.466i 1.64613i −0.567950 0.823063i \(-0.692263\pi\)
0.567950 0.823063i \(-0.307737\pi\)
\(384\) 0 0
\(385\) −168.000 −0.436364
\(386\) 67.0000 + 116.047i 0.173575 + 0.300641i
\(387\) 0 0
\(388\) 86.0000 148.956i 0.221649 0.383908i
\(389\) −146.000 −0.375321 −0.187661 0.982234i \(-0.560091\pi\)
−0.187661 + 0.982234i \(0.560091\pi\)
\(390\) 0 0
\(391\) 266.736i 0.682189i
\(392\) 296.000 0.755102
\(393\) 0 0
\(394\) −268.000 464.190i −0.680203 1.17815i
\(395\) 110.851i 0.280636i
\(396\) 0 0
\(397\) 488.000 1.22922 0.614610 0.788831i \(-0.289314\pi\)
0.614610 + 0.788831i \(0.289314\pi\)
\(398\) −54.0000 + 31.1769i −0.135678 + 0.0783340i
\(399\) 0 0
\(400\) 72.0000 + 124.708i 0.180000 + 0.311769i
\(401\) 445.000 1.10973 0.554863 0.831942i \(-0.312771\pi\)
0.554863 + 0.831942i \(0.312771\pi\)
\(402\) 0 0
\(403\) 152.420i 0.378215i
\(404\) 40.0000 69.2820i 0.0990099 0.171490i
\(405\) 0 0
\(406\) 204.000 117.779i 0.502463 0.290097i
\(407\) 193.990i 0.476633i
\(408\) 0 0
\(409\) −67.0000 −0.163814 −0.0819071 0.996640i \(-0.526101\pi\)
−0.0819071 + 0.996640i \(0.526101\pi\)
\(410\) −52.0000 90.0666i −0.126829 0.219675i
\(411\) 0 0
\(412\) 84.0000 + 48.4974i 0.203883 + 0.117712i
\(413\) 186.000 0.450363
\(414\) 0 0
\(415\) 138.564i 0.333889i
\(416\) 352.000 609.682i 0.846154 1.46558i
\(417\) 0 0
\(418\) 189.000 + 327.358i 0.452153 + 0.783152i
\(419\) 616.610i 1.47162i 0.677186 + 0.735812i \(0.263199\pi\)
−0.677186 + 0.735812i \(0.736801\pi\)
\(420\) 0 0
\(421\) 272.000 0.646081 0.323040 0.946385i \(-0.395295\pi\)
0.323040 + 0.946385i \(0.395295\pi\)
\(422\) 228.000 131.636i 0.540284 0.311933i
\(423\) 0 0
\(424\) 416.000 0.981132
\(425\) 99.0000 0.232941
\(426\) 0 0
\(427\) 55.4256i 0.129802i
\(428\) −54.0000 31.1769i −0.126168 0.0728433i
\(429\) 0 0
\(430\) 348.000 200.918i 0.809302 0.467251i
\(431\) 405.300i 0.940371i 0.882568 + 0.470185i \(0.155813\pi\)
−0.882568 + 0.470185i \(0.844187\pi\)
\(432\) 0 0
\(433\) −439.000 −1.01386 −0.506928 0.861988i \(-0.669219\pi\)
−0.506928 + 0.861988i \(0.669219\pi\)
\(434\) 24.0000 + 41.5692i 0.0552995 + 0.0957816i
\(435\) 0 0
\(436\) 176.000 304.841i 0.403670 0.699176i
\(437\) −378.000 −0.864989
\(438\) 0 0
\(439\) 845.241i 1.92538i 0.270611 + 0.962689i \(0.412774\pi\)
−0.270611 + 0.962689i \(0.587226\pi\)
\(440\) 387.979i 0.881771i
\(441\) 0 0
\(442\) −242.000 419.156i −0.547511 0.948317i
\(443\) 330.822i 0.746776i −0.927675 0.373388i \(-0.878196\pi\)
0.927675 0.373388i \(-0.121804\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.0179775
\(446\) −102.000 + 58.8897i −0.228700 + 0.132040i
\(447\) 0 0
\(448\) 221.703i 0.494872i
\(449\) −47.0000 −0.104677 −0.0523385 0.998629i \(-0.516667\pi\)
−0.0523385 + 0.998629i \(0.516667\pi\)
\(450\) 0 0
\(451\) 157.617i 0.349483i
\(452\) 100.000 173.205i 0.221239 0.383197i
\(453\) 0 0
\(454\) 777.000 448.601i 1.71145 0.988108i
\(455\) 304.841i 0.669980i
\(456\) 0 0
\(457\) −331.000 −0.724289 −0.362144 0.932122i \(-0.617955\pi\)
−0.362144 + 0.932122i \(0.617955\pi\)
\(458\) −410.000 710.141i −0.895197 1.55053i
\(459\) 0 0
\(460\) −336.000 193.990i −0.730435 0.421717i
\(461\) 538.000 1.16703 0.583514 0.812103i \(-0.301677\pi\)
0.583514 + 0.812103i \(0.301677\pi\)
\(462\) 0 0
\(463\) 568.113i 1.22703i 0.789685 + 0.613513i \(0.210244\pi\)
−0.789685 + 0.613513i \(0.789756\pi\)
\(464\) −272.000 471.118i −0.586207 1.01534i
\(465\) 0 0
\(466\) 65.0000 + 112.583i 0.139485 + 0.241595i
\(467\) 639.127i 1.36858i −0.729210 0.684290i \(-0.760112\pi\)
0.729210 0.684290i \(-0.239888\pi\)
\(468\) 0 0
\(469\) −402.000 −0.857143
\(470\) −24.0000 + 13.8564i −0.0510638 + 0.0294817i
\(471\) 0 0
\(472\) 429.549i 0.910061i
\(473\) −609.000 −1.28753
\(474\) 0 0
\(475\) 140.296i 0.295360i
\(476\) 132.000 + 76.2102i 0.277311 + 0.160106i
\(477\) 0 0
\(478\) −66.0000 + 38.1051i −0.138075 + 0.0797178i
\(479\) 121.244i 0.253118i 0.991959 + 0.126559i \(0.0403933\pi\)
−0.991959 + 0.126559i \(0.959607\pi\)
\(480\) 0 0
\(481\) 352.000 0.731809
\(482\) 223.000 + 386.247i 0.462656 + 0.801343i
\(483\) 0 0
\(484\) 52.0000 90.0666i 0.107438 0.186088i
\(485\) −172.000 −0.354639
\(486\) 0 0
\(487\) 405.300i 0.832238i 0.909310 + 0.416119i \(0.136610\pi\)
−0.909310 + 0.416119i \(0.863390\pi\)
\(488\) −128.000 −0.262295
\(489\) 0 0
\(490\) −148.000 256.344i −0.302041 0.523150i
\(491\) 725.729i 1.47806i 0.673670 + 0.739032i \(0.264717\pi\)
−0.673670 + 0.739032i \(0.735283\pi\)
\(492\) 0 0
\(493\) −374.000 −0.758621
\(494\) −594.000 + 342.946i −1.20243 + 0.694223i
\(495\) 0 0
\(496\) 96.0000 55.4256i 0.193548 0.111745i
\(497\) 0 0
\(498\) 0 0
\(499\) 521.347i 1.04478i 0.852705 + 0.522392i \(0.174960\pi\)
−0.852705 + 0.522392i \(0.825040\pi\)
\(500\) 272.000 471.118i 0.544000 0.942236i
\(501\) 0 0
\(502\) 189.000 109.119i 0.376494 0.217369i
\(503\) 872.954i 1.73549i 0.497006 + 0.867747i \(0.334433\pi\)
−0.497006 + 0.867747i \(0.665567\pi\)
\(504\) 0 0
\(505\) −80.0000 −0.158416
\(506\) 294.000 + 509.223i 0.581028 + 1.00637i
\(507\) 0 0
\(508\) −756.000 436.477i −1.48819 0.859206i
\(509\) 760.000 1.49312 0.746562 0.665316i \(-0.231703\pi\)
0.746562 + 0.665316i \(0.231703\pi\)
\(510\) 0 0
\(511\) 86.6025i 0.169477i
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) 437.000 + 756.906i 0.850195 + 1.47258i
\(515\) 96.9948i 0.188340i
\(516\) 0 0
\(517\) 42.0000 0.0812379
\(518\) −96.0000 + 55.4256i −0.185328 + 0.106999i
\(519\) 0 0
\(520\) −704.000 −1.35385
\(521\) 745.000 1.42994 0.714971 0.699154i \(-0.246440\pi\)
0.714971 + 0.699154i \(0.246440\pi\)
\(522\) 0 0
\(523\) 561.184i 1.07301i 0.843897 + 0.536505i \(0.180256\pi\)
−0.843897 + 0.536505i \(0.819744\pi\)
\(524\) 672.000 + 387.979i 1.28244 + 0.740419i
\(525\) 0 0
\(526\) −546.000 + 315.233i −1.03802 + 0.599303i
\(527\) 76.2102i 0.144611i
\(528\) 0 0
\(529\) −59.0000 −0.111531
\(530\) −208.000 360.267i −0.392453 0.679748i
\(531\) 0 0
\(532\) 108.000 187.061i 0.203008 0.351619i
\(533\) −286.000 −0.536585
\(534\) 0 0
\(535\) 62.3538i 0.116549i
\(536\) 928.379i 1.73205i
\(537\) 0 0
\(538\) −304.000 526.543i −0.565056 0.978705i
\(539\) 448.601i 0.832284i
\(540\) 0 0
\(541\) −520.000 −0.961183 −0.480591 0.876945i \(-0.659578\pi\)
−0.480591 + 0.876945i \(0.659578\pi\)
\(542\) −540.000 + 311.769i −0.996310 + 0.575220i
\(543\) 0 0
\(544\) 176.000 304.841i 0.323529 0.560369i
\(545\) −352.000 −0.645872
\(546\) 0 0
\(547\) 386.247i 0.706119i −0.935601 0.353060i \(-0.885141\pi\)
0.935601 0.353060i \(-0.114859\pi\)
\(548\) −338.000 + 585.433i −0.616788 + 1.06831i
\(549\) 0 0
\(550\) −189.000 + 109.119i −0.343636 + 0.198399i
\(551\) 530.008i 0.961901i
\(552\) 0 0
\(553\) 96.0000 0.173599
\(554\) 34.0000 + 58.8897i 0.0613718 + 0.106299i
\(555\) 0 0
\(556\) 678.000 + 391.443i 1.21942 + 0.704035i
\(557\) 934.000 1.67684 0.838420 0.545025i \(-0.183480\pi\)
0.838420 + 0.545025i \(0.183480\pi\)
\(558\) 0 0
\(559\) 1105.05i 1.97683i
\(560\) 192.000 110.851i 0.342857 0.197949i
\(561\) 0 0
\(562\) 218.000 + 377.587i 0.387900 + 0.671863i
\(563\) 708.409i 1.25827i −0.777294 0.629137i \(-0.783408\pi\)
0.777294 0.629137i \(-0.216592\pi\)
\(564\) 0 0
\(565\) −200.000 −0.353982
\(566\) 12.0000 6.92820i 0.0212014 0.0122406i
\(567\) 0 0
\(568\) 0 0
\(569\) −695.000 −1.22144 −0.610721 0.791846i \(-0.709120\pi\)
−0.610721 + 0.791846i \(0.709120\pi\)
\(570\) 0 0
\(571\) 538.668i 0.943376i −0.881765 0.471688i \(-0.843645\pi\)
0.881765 0.471688i \(-0.156355\pi\)
\(572\) 924.000 + 533.472i 1.61538 + 0.932643i
\(573\) 0 0
\(574\) 78.0000 45.0333i 0.135889 0.0784553i
\(575\) 218.238i 0.379545i
\(576\) 0 0
\(577\) 227.000 0.393414 0.196707 0.980462i \(-0.436975\pi\)
0.196707 + 0.980462i \(0.436975\pi\)
\(578\) 168.000 + 290.985i 0.290657 + 0.503433i
\(579\) 0 0
\(580\) −272.000 + 471.118i −0.468966 + 0.812272i
\(581\) 120.000 0.206540
\(582\) 0 0
\(583\) 630.466i 1.08142i
\(584\) −200.000 −0.342466
\(585\) 0 0
\(586\) −202.000 349.874i −0.344710 0.597055i
\(587\) 143.760i 0.244907i −0.992474 0.122453i \(-0.960924\pi\)
0.992474 0.122453i \(-0.0390762\pi\)
\(588\) 0 0
\(589\) −108.000 −0.183362
\(590\) −372.000 + 214.774i −0.630508 + 0.364024i
\(591\) 0 0
\(592\) 128.000 + 221.703i 0.216216 + 0.374497i
\(593\) −506.000 −0.853288 −0.426644 0.904420i \(-0.640304\pi\)
−0.426644 + 0.904420i \(0.640304\pi\)
\(594\) 0 0
\(595\) 152.420i 0.256169i
\(596\) −260.000 + 450.333i −0.436242 + 0.755593i
\(597\) 0 0
\(598\) −924.000 + 533.472i −1.54515 + 0.892093i
\(599\) 55.4256i 0.0925303i 0.998929 + 0.0462651i \(0.0147319\pi\)
−0.998929 + 0.0462651i \(0.985268\pi\)
\(600\) 0 0
\(601\) 335.000 0.557404 0.278702 0.960378i \(-0.410096\pi\)
0.278702 + 0.960378i \(0.410096\pi\)
\(602\) 174.000 + 301.377i 0.289037 + 0.500626i
\(603\) 0 0
\(604\) 420.000 + 242.487i 0.695364 + 0.401469i
\(605\) −104.000 −0.171901
\(606\) 0 0
\(607\) 630.466i 1.03866i 0.854574 + 0.519330i \(0.173818\pi\)
−0.854574 + 0.519330i \(0.826182\pi\)
\(608\) −432.000 249.415i −0.710526 0.410223i
\(609\) 0 0
\(610\) 64.0000 + 110.851i 0.104918 + 0.181723i
\(611\) 76.2102i 0.124730i
\(612\) 0 0
\(613\) −340.000 −0.554649 −0.277325 0.960776i \(-0.589448\pi\)
−0.277325 + 0.960776i \(0.589448\pi\)
\(614\) 189.000 109.119i 0.307818 0.177719i
\(615\) 0 0
\(616\) −336.000 −0.545455
\(617\) 391.000 0.633712 0.316856 0.948474i \(-0.397373\pi\)
0.316856 + 0.948474i \(0.397373\pi\)
\(618\) 0 0
\(619\) 12.1244i 0.0195870i −0.999952 0.00979350i \(-0.996883\pi\)
0.999952 0.00979350i \(-0.00311742\pi\)
\(620\) −96.0000 55.4256i −0.154839 0.0893962i
\(621\) 0 0
\(622\) 474.000 273.664i 0.762058 0.439974i
\(623\) 6.92820i 0.0111207i
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 79.0000 + 136.832i 0.126198 + 0.218581i
\(627\) 0 0
\(628\) 8.00000 13.8564i 0.0127389 0.0220643i
\(629\) 176.000 0.279809
\(630\) 0 0
\(631\) 436.477i 0.691722i −0.938286 0.345861i \(-0.887587\pi\)
0.938286 0.345861i \(-0.112413\pi\)
\(632\) 221.703i 0.350795i
\(633\) 0 0
\(634\) −502.000 869.490i −0.791798 1.37143i
\(635\) 872.954i 1.37473i
\(636\) 0 0
\(637\) −814.000 −1.27786
\(638\) 714.000 412.228i 1.11912 0.646126i
\(639\) 0 0
\(640\) −256.000 443.405i −0.400000 0.692820i
\(641\) 421.000 0.656786 0.328393 0.944541i \(-0.393493\pi\)
0.328393 + 0.944541i \(0.393493\pi\)
\(642\) 0 0
\(643\) 413.960i 0.643795i −0.946775 0.321897i \(-0.895679\pi\)
0.946775 0.321897i \(-0.104321\pi\)
\(644\) 168.000 290.985i 0.260870 0.451839i
\(645\) 0 0
\(646\) −297.000 + 171.473i −0.459752 + 0.265438i
\(647\) 405.300i 0.626430i 0.949682 + 0.313215i \(0.101406\pi\)
−0.949682 + 0.313215i \(0.898594\pi\)
\(648\) 0 0
\(649\) 651.000 1.00308
\(650\) −198.000 342.946i −0.304615 0.527609i
\(651\) 0 0
\(652\) 1080.00 + 623.538i 1.65644 + 0.956347i
\(653\) 886.000 1.35681 0.678407 0.734686i \(-0.262670\pi\)
0.678407 + 0.734686i \(0.262670\pi\)
\(654\) 0 0
\(655\) 775.959i 1.18467i
\(656\) −104.000 180.133i −0.158537 0.274593i
\(657\) 0 0
\(658\) −12.0000 20.7846i −0.0182371 0.0315876i
\(659\) 838.313i 1.27210i 0.771649 + 0.636049i \(0.219432\pi\)
−0.771649 + 0.636049i \(0.780568\pi\)
\(660\) 0 0
\(661\) 248.000 0.375189 0.187595 0.982247i \(-0.439931\pi\)
0.187595 + 0.982247i \(0.439931\pi\)
\(662\) 708.000 408.764i 1.06949 0.617468i
\(663\) 0 0
\(664\) 277.128i 0.417362i
\(665\) −216.000 −0.324812
\(666\) 0 0
\(667\) 824.456i 1.23607i
\(668\) −624.000 360.267i −0.934132 0.539321i
\(669\) 0 0
\(670\) 804.000 464.190i 1.20000 0.692820i
\(671\) 193.990i 0.289105i
\(672\) 0 0
\(673\) 1154.00 1.71471 0.857355 0.514725i \(-0.172106\pi\)
0.857355 + 0.514725i \(0.172106\pi\)
\(674\) 337.000 + 583.701i 0.500000 + 0.866025i
\(675\) 0 0
\(676\) −630.000 + 1091.19i −0.931953 + 1.61419i
\(677\) 1132.00 1.67208 0.836041 0.548666i \(-0.184864\pi\)
0.836041 + 0.548666i \(0.184864\pi\)
\(678\) 0 0
\(679\) 148.956i 0.219376i
\(680\) −352.000 −0.517647
\(681\) 0 0
\(682\) 84.0000 + 145.492i 0.123167 + 0.213332i
\(683\) 795.011i 1.16400i −0.813189 0.582000i \(-0.802271\pi\)
0.813189 0.582000i \(-0.197729\pi\)
\(684\) 0 0
\(685\) 676.000 0.986861
\(686\) 516.000 297.913i 0.752187 0.434275i
\(687\) 0 0
\(688\) 696.000 401.836i 1.01163 0.584064i
\(689\) −1144.00 −1.66038
\(690\) 0 0
\(691\) 900.666i 1.30342i −0.758466 0.651712i \(-0.774051\pi\)
0.758466 0.651712i \(-0.225949\pi\)
\(692\) 4.00000 6.92820i 0.00578035 0.0100119i
\(693\) 0 0
\(694\) −471.000 + 271.932i −0.678674 + 0.391833i
\(695\) 782.887i 1.12646i
\(696\) 0 0
\(697\) −143.000 −0.205165
\(698\) −272.000 471.118i −0.389685 0.674954i
\(699\) 0 0
\(700\) 108.000 + 62.3538i 0.154286 + 0.0890769i
\(701\) 142.000 0.202568 0.101284 0.994858i \(-0.467705\pi\)
0.101284 + 0.994858i \(0.467705\pi\)
\(702\) 0 0
\(703\) 249.415i 0.354787i
\(704\) 775.959i 1.10221i
\(705\) 0 0
\(706\) 461.000 + 798.475i 0.652975 + 1.13099i
\(707\) 69.2820i 0.0979944i
\(708\) 0 0
\(709\) 740.000 1.04372 0.521862 0.853030i \(-0.325238\pi\)
0.521862 + 0.853030i \(0.325238\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −16.0000 −0.0224719
\(713\) −168.000 −0.235624
\(714\) 0 0
\(715\) 1066.94i 1.49223i
\(716\) −648.000 374.123i −0.905028 0.522518i
\(717\) 0 0
\(718\) −918.000 + 530.008i −1.27855 + 0.738172i
\(719\) 124.708i 0.173446i 0.996232 + 0.0867230i \(0.0276395\pi\)
−0.996232 + 0.0867230i \(0.972360\pi\)
\(720\) 0 0
\(721\) 84.0000 0.116505
\(722\) −118.000 204.382i −0.163435 0.283078i
\(723\) 0 0
\(724\) −508.000 + 879.882i −0.701657 + 1.21531i
\(725\) −306.000 −0.422069
\(726\) 0 0
\(727\) 814.064i 1.11976i 0.828574 + 0.559879i \(0.189152\pi\)
−0.828574 + 0.559879i \(0.810848\pi\)
\(728\) 609.682i 0.837475i
\(729\) 0 0
\(730\) 100.000 + 173.205i 0.136986 + 0.237267i
\(731\) 552.524i 0.755847i
\(732\) 0 0
\(733\) 914.000 1.24693 0.623465 0.781851i \(-0.285724\pi\)
0.623465 + 0.781851i \(0.285724\pi\)
\(734\) 168.000 96.9948i 0.228883 0.132146i
\(735\) 0 0
\(736\) −672.000 387.979i −0.913043 0.527146i
\(737\) −1407.00 −1.90909
\(738\) 0 0
\(739\) 358.535i 0.485162i −0.970131 0.242581i \(-0.922006\pi\)
0.970131 0.242581i \(-0.0779940\pi\)
\(740\) 128.000 221.703i 0.172973 0.299598i
\(741\) 0 0
\(742\) 312.000 180.133i 0.420485 0.242767i
\(743\) 398.372i 0.536166i 0.963396 + 0.268083i \(0.0863902\pi\)
−0.963396 + 0.268083i \(0.913610\pi\)
\(744\) 0 0
\(745\) 520.000 0.697987
\(746\) 346.000 + 599.290i 0.463807 + 0.803337i
\(747\) 0 0
\(748\) 462.000 + 266.736i 0.617647 + 0.356599i
\(749\) −54.0000 −0.0720961
\(750\) 0 0
\(751\) 1115.44i 1.48527i −0.669694 0.742637i \(-0.733575\pi\)
0.669694 0.742637i \(-0.266425\pi\)
\(752\) −48.0000 + 27.7128i −0.0638298 + 0.0368521i
\(753\) 0 0
\(754\) 748.000 + 1295.57i 0.992042 + 1.71827i
\(755\) 484.974i 0.642350i
\(756\) 0 0
\(757\) 758.000 1.00132 0.500661 0.865644i \(-0.333091\pi\)
0.500661 + 0.865644i \(0.333091\pi\)
\(758\) 567.000 327.358i 0.748021 0.431870i
\(759\) 0 0
\(760\) 498.831i 0.656356i
\(761\) −374.000 −0.491459 −0.245729 0.969338i \(-0.579027\pi\)
−0.245729 + 0.969338i \(0.579027\pi\)
\(762\) 0 0
\(763\) 304.841i 0.399529i
\(764\) 12.0000 + 6.92820i 0.0157068 + 0.00906833i
\(765\) 0 0
\(766\) −1092.00 + 630.466i −1.42559 + 0.823063i
\(767\) 1181.26i 1.54010i
\(768\) 0 0
\(769\) −22.0000 −0.0286086 −0.0143043 0.999898i \(-0.504553\pi\)
−0.0143043 + 0.999898i \(0.504553\pi\)
\(770\) 168.000 + 290.985i 0.218182 + 0.377902i
\(771\) 0 0
\(772\) 134.000 232.095i 0.173575 0.300641i
\(773\) −1334.00 −1.72574 −0.862872 0.505423i \(-0.831337\pi\)
−0.862872 + 0.505423i \(0.831337\pi\)
\(774\) 0 0
\(775\) 62.3538i 0.0804566i
\(776\) −344.000 −0.443299
\(777\) 0 0
\(778\) 146.000 + 252.879i 0.187661 + 0.325038i
\(779\) 202.650i 0.260141i
\(780\) 0 0
\(781\) 0 0
\(782\) −462.000 + 266.736i −0.590793 + 0.341094i
\(783\) 0 0
\(784\) −296.000 512.687i −0.377551 0.653938i
\(785\) −16.0000 −0.0203822
\(786\) 0 0
\(787\) 879.882i 1.11802i 0.829161 + 0.559010i \(0.188819\pi\)
−0.829161 + 0.559010i \(0.811181\pi\)
\(788\) −536.000 + 928.379i −0.680203 + 1.17815i
\(789\) 0 0
\(790\) −192.000 + 110.851i −0.243038 + 0.140318i
\(791\) 173.205i 0.218970i
\(792\) 0 0
\(793\) 352.000 0.443884
\(794\) −488.000 845.241i −0.614610 1.06454i
\(795\) 0 0
\(796\) 108.000 + 62.3538i 0.135678 + 0.0783340i
\(797\) 832.000 1.04391 0.521957 0.852972i \(-0.325202\pi\)
0.521957 + 0.852972i \(0.325202\pi\)
\(798\) 0 0
\(799\) 38.1051i 0.0476910i
\(800\) 144.000 249.415i 0.180000 0.311769i
\(801\) 0 0
\(802\) −445.000 770.763i −0.554863 0.961051i
\(803\) 303.109i 0.377471i
\(804\) 0 0
\(805\) −336.000 −0.417391
\(806\) −264.000 + 152.420i −0.327543 + 0.189107i
\(807\) 0 0
\(808\) −160.000 −0.198020
\(809\) 493.000 0.609394 0.304697 0.952449i \(-0.401445\pi\)
0.304697 + 0.952449i \(0.401445\pi\)
\(810\) 0 0
\(811\) 327.358i 0.403647i −0.979422 0.201823i \(-0.935313\pi\)
0.979422 0.201823i \(-0.0646867\pi\)
\(812\) −408.000 235.559i −0.502463 0.290097i
\(813\) 0 0
\(814\) −336.000 + 193.990i −0.412776 + 0.238317i
\(815\) 1247.08i 1.53016i
\(816\) 0 0
\(817\) −783.000 −0.958384
\(818\) 67.0000 + 116.047i 0.0819071 + 0.141867i
\(819\) 0 0
\(820\) −104.000 + 180.133i −0.126829 + 0.219675i
\(821\) −758.000 −0.923264 −0.461632 0.887071i \(-0.652736\pi\)
−0.461632 + 0.887071i \(0.652736\pi\)
\(822\) 0 0
\(823\) 866.025i 1.05228i −0.850398 0.526139i \(-0.823639\pi\)
0.850398 0.526139i \(-0.176361\pi\)
\(824\) 193.990i 0.235424i
\(825\) 0 0
\(826\) −186.000 322.161i −0.225182 0.390026i
\(827\) 436.477i 0.527783i 0.964552 + 0.263892i \(0.0850061\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(828\) 0 0
\(829\) −718.000 −0.866104 −0.433052 0.901369i \(-0.642563\pi\)
−0.433052 + 0.901369i \(0.642563\pi\)
\(830\) −240.000 + 138.564i −0.289157 + 0.166945i
\(831\) 0 0
\(832\) −1408.00 −1.69231
\(833\) −407.000 −0.488595
\(834\) 0 0
\(835\) 720.533i 0.862914i
\(836\) 378.000 654.715i 0.452153 0.783152i
\(837\) 0 0
\(838\) 1068.00 616.610i 1.27446 0.735812i
\(839\) 907.595i 1.08176i −0.841101 0.540879i \(-0.818092\pi\)
0.841101 0.540879i \(-0.181908\pi\)
\(840\) 0 0
\(841\) 315.000 0.374554
\(842\) −272.000 471.118i −0.323040 0.559522i
\(843\) 0 0
\(844\) −456.000 263.272i −0.540284 0.311933i
\(845\) 1260.00 1.49112
\(846\) 0 0
\(847\) 90.0666i 0.106336i
\(848\) −416.000 720.533i −0.490566 0.849685i
\(849\) 0 0
\(850\) −99.0000 171.473i −0.116471 0.201733i
\(851\) 387.979i 0.455910i
\(852\) 0 0
\(853\) 146.000 0.171161 0.0855803 0.996331i \(-0.472726\pi\)
0.0855803 + 0.996331i \(0.472726\pi\)
\(854\) −96.0000 + 55.4256i −0.112412 + 0.0649012i
\(855\) 0 0
\(856\) 124.708i 0.145687i
\(857\) −146.000 −0.170362 −0.0851809 0.996366i \(-0.527147\pi\)
−0.0851809 + 0.996366i \(0.527147\pi\)
\(858\) 0 0
\(859\) 84.8705i 0.0988015i 0.998779 + 0.0494008i \(0.0157312\pi\)
−0.998779 + 0.0494008i \(0.984269\pi\)
\(860\) −696.000 401.836i −0.809302 0.467251i
\(861\) 0 0
\(862\) 702.000 405.300i 0.814385 0.470185i
\(863\) 1184.72i 1.37280i −0.727226 0.686398i \(-0.759191\pi\)
0.727226 0.686398i \(-0.240809\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.00924855
\(866\) 439.000 + 760.370i 0.506928 + 0.878026i
\(867\) 0 0
\(868\) 48.0000 83.1384i 0.0552995 0.0957816i
\(869\) 336.000 0.386651
\(870\) 0 0
\(871\) 2553.04i 2.93116i
\(872\) −704.000 −0.807339
\(873\) 0 0
\(874\) 378.000 + 654.715i 0.432494 + 0.749102i
\(875\) 471.118i 0.538420i
\(876\) 0 0
\(877\) −1480.00 −1.68757 −0.843786 0.536680i \(-0.819678\pi\)
−0.843786 + 0.536680i \(0.819678\pi\)
\(878\) 1464.00 845.241i 1.66743 0.962689i
\(879\) 0 0
\(880\) 672.000 387.979i 0.763636 0.440886i
\(881\) 142.000 0.161180 0.0805902 0.996747i \(-0.474319\pi\)
0.0805902 + 0.996747i \(0.474319\pi\)
\(882\) 0 0
\(883\) 1200.31i 1.35936i −0.733511 0.679678i \(-0.762120\pi\)
0.733511 0.679678i \(-0.237880\pi\)
\(884\) −484.000 + 838.313i −0.547511 + 0.948317i
\(885\) 0 0
\(886\) −573.000 + 330.822i −0.646727 + 0.373388i
\(887\) 630.466i 0.710785i −0.934717 0.355393i \(-0.884347\pi\)
0.934717 0.355393i \(-0.115653\pi\)
\(888\) 0 0
\(889\) −756.000 −0.850394
\(890\) 8.00000 + 13.8564i 0.00898876 + 0.0155690i
\(891\) 0 0
\(892\) 204.000 + 117.779i 0.228700 + 0.132040i
\(893\) 54.0000 0.0604703
\(894\) 0 0
\(895\) 748.246i 0.836029i
\(896\) 384.000 221.703i 0.428571 0.247436i
\(897\) 0 0
\(898\) 47.0000 + 81.4064i 0.0523385 + 0.0906530i
\(899\) 235.559i 0.262023i
\(900\) 0 0
\(901\) −572.000 −0.634850
\(902\) 273.000 157.617i 0.302661 0.174741i
\(903\) 0 0
\(904\) −400.000 −0.442478
\(905\) 1016.00 1.12265
\(906\) 0 0
\(907\) 642.591i 0.708479i 0.935155 + 0.354240i \(0.115260\pi\)
−0.935155 + 0.354240i \(0.884740\pi\)
\(908\) −1554.00 897.202i −1.71145 0.988108i
\(909\) 0 0
\(910\) −528.000 + 304.841i −0.580220 + 0.334990i
\(911\) 401.836i 0.441093i 0.975376 + 0.220547i \(0.0707841\pi\)
−0.975376 + 0.220547i \(0.929216\pi\)
\(912\) 0 0
\(913\) 420.000 0.460022
\(914\) 331.000 + 573.309i 0.362144 + 0.627253i
\(915\) 0 0
\(916\) −820.000 + 1420.28i −0.895197 + 1.55053i
\(917\) 672.000 0.732824
\(918\) 0 0
\(919\) 779.423i 0.848121i 0.905634 + 0.424060i \(0.139396\pi\)
−0.905634 + 0.424060i \(0.860604\pi\)
\(920\) 775.959i 0.843433i
\(921\) 0 0
\(922\) −538.000 931.843i −0.583514 1.01068i
\(923\) 0 0
\(924\) 0 0
\(925\) 144.000 0.155676
\(926\) 984.000 568.113i 1.06263 0.613513i
\(927\) 0 0
\(928\) −544.000 + 942.236i −0.586207 + 1.01534i
\(929\) −758.000 −0.815931 −0.407966 0.912997i \(-0.633762\pi\)
−0.407966 + 0.912997i \(0.633762\pi\)
\(930\) 0 0
\(931\) 576.773i 0.619520i
\(932\) 130.000 225.167i 0.139485 0.241595i
\(933\) 0 0
\(934\) −1107.00 + 639.127i −1.18522 + 0.684290i
\(935\) 533.472i 0.570558i
\(936\) 0 0
\(937\) −754.000 −0.804696 −0.402348 0.915487i \(-0.631806\pi\)
−0.402348 + 0.915487i \(0.631806\pi\)
\(938\) 402.000 + 696.284i 0.428571 + 0.742307i
\(939\) 0 0
\(940\) 48.0000 + 27.7128i 0.0510638 + 0.0294817i
\(941\) −1796.00 −1.90861 −0.954304 0.298838i \(-0.903401\pi\)
−0.954304 + 0.298838i \(0.903401\pi\)
\(942\) 0 0
\(943\) 315.233i 0.334288i
\(944\) −744.000 + 429.549i −0.788136 + 0.455030i
\(945\) 0 0
\(946\) 609.000 + 1054.82i 0.643763 + 1.11503i
\(947\) 105.655i 0.111568i 0.998443 + 0.0557841i \(0.0177659\pi\)
−0.998443 + 0.0557841i \(0.982234\pi\)
\(948\) 0 0
\(949\) 550.000 0.579557
\(950\) −243.000 + 140.296i −0.255789 + 0.147680i
\(951\) 0 0
\(952\) 304.841i 0.320211i
\(953\) 1213.00 1.27282 0.636411 0.771350i \(-0.280418\pi\)
0.636411 + 0.771350i \(0.280418\pi\)
\(954\) 0 0
\(955\) 13.8564i 0.0145093i
\(956\) 132.000 + 76.2102i 0.138075 + 0.0797178i
\(957\) 0 0
\(958\) 210.000 121.244i 0.219207 0.126559i
\(959\) 585.433i 0.610462i
\(960\) 0 0
\(961\) 913.000 0.950052
\(962\) −352.000 609.682i −0.365904 0.633765i
\(963\) 0 0
\(964\) 446.000 772.495i 0.462656 0.801343i
\(965\) −268.000 −0.277720
\(966\) 0 0
\(967\) 349.874i 0.361814i 0.983500 + 0.180907i \(0.0579033\pi\)
−0.983500 + 0.180907i \(0.942097\pi\)
\(968\) −208.000 −0.214876
\(969\) 0 0
\(970\) 172.000 + 297.913i 0.177320 + 0.307127i
\(971\) 1434.14i 1.47697i 0.674270 + 0.738485i \(0.264458\pi\)
−0.674270 + 0.738485i \(0.735542\pi\)
\(972\) 0 0
\(973\) 678.000 0.696814
\(974\) 702.000 405.300i 0.720739 0.416119i
\(975\) 0 0
\(976\) 128.000 + 221.703i 0.131148 + 0.227154i
\(977\) 157.000 0.160696 0.0803480 0.996767i \(-0.474397\pi\)
0.0803480 + 0.996767i \(0.474397\pi\)
\(978\) 0 0
\(979\) 24.2487i 0.0247689i
\(980\) −296.000 + 512.687i −0.302041 + 0.523150i
\(981\) 0 0
\(982\) 1257.00 725.729i 1.28004 0.739032i
\(983\) 1406.43i 1.43075i −0.698742 0.715374i \(-0.746256\pi\)
0.698742 0.715374i \(-0.253744\pi\)
\(984\) 0 0
\(985\) 1072.00 1.08832
\(986\) 374.000 + 647.787i 0.379310 + 0.656985i
\(987\) 0 0
\(988\) 1188.00 + 685.892i 1.20243 + 0.694223i
\(989\) −1218.00 −1.23155
\(990\) 0 0
\(991\) 249.415i 0.251680i −0.992051 0.125840i \(-0.959837\pi\)
0.992051 0.125840i \(-0.0401627\pi\)
\(992\) −192.000 110.851i −0.193548 0.111745i
\(993\) 0 0
\(994\) 0 0
\(995\) 124.708i 0.125334i
\(996\) 0 0
\(997\) −412.000 −0.413240 −0.206620 0.978421i \(-0.566246\pi\)
−0.206620 + 0.978421i \(0.566246\pi\)
\(998\) 903.000 521.347i 0.904810 0.522392i
\(999\) 0 0
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 324.3.d.b.163.1 2
3.2 odd 2 324.3.d.c.163.2 2
4.3 odd 2 inner 324.3.d.b.163.2 2
9.2 odd 6 108.3.f.b.91.1 2
9.4 even 3 36.3.f.b.7.1 yes 2
9.5 odd 6 108.3.f.a.19.1 2
9.7 even 3 36.3.f.a.31.1 yes 2
12.11 even 2 324.3.d.c.163.1 2
36.7 odd 6 36.3.f.b.31.1 yes 2
36.11 even 6 108.3.f.a.91.1 2
36.23 even 6 108.3.f.b.19.1 2
36.31 odd 6 36.3.f.a.7.1 2
72.5 odd 6 1728.3.o.a.127.1 2
72.11 even 6 1728.3.o.a.1279.1 2
72.13 even 6 576.3.o.b.511.1 2
72.29 odd 6 1728.3.o.b.1279.1 2
72.43 odd 6 576.3.o.b.319.1 2
72.59 even 6 1728.3.o.b.127.1 2
72.61 even 6 576.3.o.a.319.1 2
72.67 odd 6 576.3.o.a.511.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.a.7.1 2 36.31 odd 6
36.3.f.a.31.1 yes 2 9.7 even 3
36.3.f.b.7.1 yes 2 9.4 even 3
36.3.f.b.31.1 yes 2 36.7 odd 6
108.3.f.a.19.1 2 9.5 odd 6
108.3.f.a.91.1 2 36.11 even 6
108.3.f.b.19.1 2 36.23 even 6
108.3.f.b.91.1 2 9.2 odd 6
324.3.d.b.163.1 2 1.1 even 1 trivial
324.3.d.b.163.2 2 4.3 odd 2 inner
324.3.d.c.163.1 2 12.11 even 2
324.3.d.c.163.2 2 3.2 odd 2
576.3.o.a.319.1 2 72.61 even 6
576.3.o.a.511.1 2 72.67 odd 6
576.3.o.b.319.1 2 72.43 odd 6
576.3.o.b.511.1 2 72.13 even 6
1728.3.o.a.127.1 2 72.5 odd 6
1728.3.o.a.1279.1 2 72.11 even 6
1728.3.o.b.127.1 2 72.59 even 6
1728.3.o.b.1279.1 2 72.29 odd 6