Properties

Label 1764.2.bm.b.1685.4
Level $1764$
Weight $2$
Character 1764.1685
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1685,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1685");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.bm (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1685.4
Root \(-0.604587 + 1.62311i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1685
Dual form 1764.2.bm.b.1697.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.604587 + 1.62311i) q^{3} +0.533560 q^{5} +(-2.26895 - 1.96262i) q^{9} +O(q^{10})\) \(q+(-0.604587 + 1.62311i) q^{3} +0.533560 q^{5} +(-2.26895 - 1.96262i) q^{9} +3.92524i q^{11} +(0.116911 - 0.0674987i) q^{13} +(-0.322584 + 0.866025i) q^{15} +(2.16266 + 3.74584i) q^{17} +(1.93067 + 1.11467i) q^{19} -1.96732i q^{23} -4.71531 q^{25} +(4.55732 - 2.49617i) q^{27} +(-5.16548 - 2.98229i) q^{29} +(-0.800341 - 0.462077i) q^{31} +(-6.37108 - 2.37315i) q^{33} +(-3.89936 + 6.75388i) q^{37} +(0.0388745 + 0.230568i) q^{39} +(4.59027 + 7.95059i) q^{41} +(3.24544 - 5.62127i) q^{43} +(-1.21062 - 1.04718i) q^{45} +(3.04329 + 5.27114i) q^{47} +(-7.38742 + 1.24554i) q^{51} +(-9.54072 + 5.50834i) q^{53} +2.09435i q^{55} +(-2.97650 + 2.45977i) q^{57} +(1.89588 - 3.28377i) q^{59} +(-9.35116 + 5.39889i) q^{61} +(0.0623791 - 0.0360146i) q^{65} +(-5.75701 + 9.97144i) q^{67} +(3.19316 + 1.18941i) q^{69} +3.22884i q^{71} +(-0.329991 + 0.190521i) q^{73} +(2.85082 - 7.65345i) q^{75} +(-4.60310 - 7.97280i) q^{79} +(1.29625 + 8.90616i) q^{81} +(1.28020 - 2.21737i) q^{83} +(1.15391 + 1.99863i) q^{85} +(7.96356 - 6.58107i) q^{87} +(8.56670 - 14.8380i) q^{89} +(1.23388 - 1.01967i) q^{93} +(1.03013 + 0.594746i) q^{95} +(-13.6747 - 7.89507i) q^{97} +(7.70375 - 8.90616i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{9} - 12 q^{15} + 16 q^{25} - 12 q^{29} - 2 q^{37} + 18 q^{39} + 4 q^{43} + 6 q^{51} - 36 q^{53} - 42 q^{57} + 24 q^{65} + 14 q^{67} + 20 q^{79} + 54 q^{81} + 6 q^{85} + 30 q^{93} + 60 q^{95} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −0.604587 + 1.62311i −0.349059 + 0.937101i
\(4\) 0 0
\(5\) 0.533560 0.238615 0.119308 0.992857i \(-0.461932\pi\)
0.119308 + 0.992857i \(0.461932\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.26895 1.96262i −0.756316 0.654206i
\(10\) 0 0
\(11\) 3.92524i 1.18350i 0.806120 + 0.591752i \(0.201563\pi\)
−0.806120 + 0.591752i \(0.798437\pi\)
\(12\) 0 0
\(13\) 0.116911 0.0674987i 0.0324253 0.0187208i −0.483700 0.875234i \(-0.660707\pi\)
0.516125 + 0.856513i \(0.327374\pi\)
\(14\) 0 0
\(15\) −0.322584 + 0.866025i −0.0832908 + 0.223607i
\(16\) 0 0
\(17\) 2.16266 + 3.74584i 0.524523 + 0.908500i 0.999592 + 0.0285519i \(0.00908959\pi\)
−0.475069 + 0.879948i \(0.657577\pi\)
\(18\) 0 0
\(19\) 1.93067 + 1.11467i 0.442927 + 0.255724i 0.704838 0.709368i \(-0.251020\pi\)
−0.261912 + 0.965092i \(0.584353\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.96732i 0.410214i −0.978740 0.205107i \(-0.934246\pi\)
0.978740 0.205107i \(-0.0657542\pi\)
\(24\) 0 0
\(25\) −4.71531 −0.943063
\(26\) 0 0
\(27\) 4.55732 2.49617i 0.877056 0.480388i
\(28\) 0 0
\(29\) −5.16548 2.98229i −0.959205 0.553798i −0.0632771 0.997996i \(-0.520155\pi\)
−0.895928 + 0.444198i \(0.853489\pi\)
\(30\) 0 0
\(31\) −0.800341 0.462077i −0.143745 0.0829915i 0.426402 0.904534i \(-0.359781\pi\)
−0.570148 + 0.821542i \(0.693114\pi\)
\(32\) 0 0
\(33\) −6.37108 2.37315i −1.10906 0.413112i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.89936 + 6.75388i −0.641050 + 1.11033i 0.344149 + 0.938915i \(0.388168\pi\)
−0.985199 + 0.171416i \(0.945166\pi\)
\(38\) 0 0
\(39\) 0.0388745 + 0.230568i 0.00622491 + 0.0369204i
\(40\) 0 0
\(41\) 4.59027 + 7.95059i 0.716880 + 1.24167i 0.962230 + 0.272239i \(0.0877640\pi\)
−0.245349 + 0.969435i \(0.578903\pi\)
\(42\) 0 0
\(43\) 3.24544 5.62127i 0.494926 0.857236i −0.505057 0.863086i \(-0.668529\pi\)
0.999983 + 0.00584958i \(0.00186199\pi\)
\(44\) 0 0
\(45\) −1.21062 1.04718i −0.180469 0.156104i
\(46\) 0 0
\(47\) 3.04329 + 5.27114i 0.443910 + 0.768874i 0.997976 0.0635985i \(-0.0202577\pi\)
−0.554066 + 0.832473i \(0.686924\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −7.38742 + 1.24554i −1.03445 + 0.174411i
\(52\) 0 0
\(53\) −9.54072 + 5.50834i −1.31052 + 0.756628i −0.982182 0.187932i \(-0.939821\pi\)
−0.328337 + 0.944561i \(0.606488\pi\)
\(54\) 0 0
\(55\) 2.09435i 0.282402i
\(56\) 0 0
\(57\) −2.97650 + 2.45977i −0.394246 + 0.325804i
\(58\) 0 0
\(59\) 1.89588 3.28377i 0.246823 0.427510i −0.715820 0.698285i \(-0.753947\pi\)
0.962643 + 0.270775i \(0.0872801\pi\)
\(60\) 0 0
\(61\) −9.35116 + 5.39889i −1.19729 + 0.691258i −0.959951 0.280168i \(-0.909610\pi\)
−0.237342 + 0.971426i \(0.576276\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.0623791 0.0360146i 0.00773718 0.00446706i
\(66\) 0 0
\(67\) −5.75701 + 9.97144i −0.703331 + 1.21820i 0.263960 + 0.964534i \(0.414971\pi\)
−0.967291 + 0.253671i \(0.918362\pi\)
\(68\) 0 0
\(69\) 3.19316 + 1.18941i 0.384412 + 0.143189i
\(70\) 0 0
\(71\) 3.22884i 0.383192i 0.981474 + 0.191596i \(0.0613664\pi\)
−0.981474 + 0.191596i \(0.938634\pi\)
\(72\) 0 0
\(73\) −0.329991 + 0.190521i −0.0386226 + 0.0222988i −0.519187 0.854661i \(-0.673765\pi\)
0.480564 + 0.876959i \(0.340432\pi\)
\(74\) 0 0
\(75\) 2.85082 7.65345i 0.329184 0.883745i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.60310 7.97280i −0.517889 0.897011i −0.999784 0.0207814i \(-0.993385\pi\)
0.481895 0.876229i \(-0.339949\pi\)
\(80\) 0 0
\(81\) 1.29625 + 8.90616i 0.144028 + 0.989574i
\(82\) 0 0
\(83\) 1.28020 2.21737i 0.140520 0.243388i −0.787172 0.616733i \(-0.788456\pi\)
0.927693 + 0.373345i \(0.121789\pi\)
\(84\) 0 0
\(85\) 1.15391 + 1.99863i 0.125159 + 0.216782i
\(86\) 0 0
\(87\) 7.96356 6.58107i 0.853783 0.705565i
\(88\) 0 0
\(89\) 8.56670 14.8380i 0.908068 1.57282i 0.0913236 0.995821i \(-0.470890\pi\)
0.816745 0.576999i \(-0.195776\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.23388 1.01967i 0.127947 0.105735i
\(94\) 0 0
\(95\) 1.03013 + 0.594746i 0.105689 + 0.0610197i
\(96\) 0 0
\(97\) −13.6747 7.89507i −1.38845 0.801622i −0.395310 0.918548i \(-0.629363\pi\)
−0.993141 + 0.116925i \(0.962696\pi\)
\(98\) 0 0
\(99\) 7.70375 8.90616i 0.774256 0.895103i
\(100\) 0 0
\(101\) −14.7372 −1.46641 −0.733205 0.680008i \(-0.761976\pi\)
−0.733205 + 0.680008i \(0.761976\pi\)
\(102\) 0 0
\(103\) 12.8682i 1.26794i 0.773357 + 0.633970i \(0.218576\pi\)
−0.773357 + 0.633970i \(0.781424\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.6679 7.89119i −1.32133 0.762870i −0.337389 0.941365i \(-0.609544\pi\)
−0.983941 + 0.178496i \(0.942877\pi\)
\(108\) 0 0
\(109\) 1.54170 + 2.67030i 0.147668 + 0.255768i 0.930365 0.366634i \(-0.119490\pi\)
−0.782697 + 0.622403i \(0.786157\pi\)
\(110\) 0 0
\(111\) −8.60477 10.4124i −0.816728 0.988299i
\(112\) 0 0
\(113\) 7.96173 4.59671i 0.748977 0.432422i −0.0763472 0.997081i \(-0.524326\pi\)
0.825324 + 0.564659i \(0.190992\pi\)
\(114\) 0 0
\(115\) 1.04968i 0.0978833i
\(116\) 0 0
\(117\) −0.397739 0.0763010i −0.0367710 0.00705403i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.40749 −0.400681
\(122\) 0 0
\(123\) −15.6799 + 2.64368i −1.41381 + 0.238373i
\(124\) 0 0
\(125\) −5.18371 −0.463645
\(126\) 0 0
\(127\) −10.1065 −0.896810 −0.448405 0.893831i \(-0.648008\pi\)
−0.448405 + 0.893831i \(0.648008\pi\)
\(128\) 0 0
\(129\) 7.16177 + 8.66625i 0.630559 + 0.763021i
\(130\) 0 0
\(131\) 15.6365 1.36616 0.683082 0.730342i \(-0.260639\pi\)
0.683082 + 0.730342i \(0.260639\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.43160 1.33186i 0.209279 0.114628i
\(136\) 0 0
\(137\) 2.46980i 0.211009i −0.994419 0.105505i \(-0.966354\pi\)
0.994419 0.105505i \(-0.0336458\pi\)
\(138\) 0 0
\(139\) −16.8526 + 9.72984i −1.42942 + 0.825274i −0.997074 0.0764359i \(-0.975646\pi\)
−0.432342 + 0.901710i \(0.642313\pi\)
\(140\) 0 0
\(141\) −10.3956 + 1.75273i −0.875464 + 0.147606i
\(142\) 0 0
\(143\) 0.264948 + 0.458904i 0.0221561 + 0.0383755i
\(144\) 0 0
\(145\) −2.75610 1.59123i −0.228881 0.132145i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.8544i 1.29884i 0.760429 + 0.649422i \(0.224989\pi\)
−0.760429 + 0.649422i \(0.775011\pi\)
\(150\) 0 0
\(151\) 8.33096 0.677964 0.338982 0.940793i \(-0.389917\pi\)
0.338982 + 0.940793i \(0.389917\pi\)
\(152\) 0 0
\(153\) 2.44469 12.7436i 0.197641 1.03026i
\(154\) 0 0
\(155\) −0.427030 0.246546i −0.0342999 0.0198031i
\(156\) 0 0
\(157\) −7.73794 4.46750i −0.617555 0.356545i 0.158362 0.987381i \(-0.449379\pi\)
−0.775916 + 0.630836i \(0.782712\pi\)
\(158\) 0 0
\(159\) −3.17242 18.8159i −0.251589 1.49220i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.10310 12.3029i 0.556358 0.963640i −0.441439 0.897291i \(-0.645532\pi\)
0.997797 0.0663485i \(-0.0211349\pi\)
\(164\) 0 0
\(165\) −3.39936 1.26622i −0.264639 0.0985750i
\(166\) 0 0
\(167\) 6.27308 + 10.8653i 0.485425 + 0.840781i 0.999860 0.0167485i \(-0.00533145\pi\)
−0.514434 + 0.857530i \(0.671998\pi\)
\(168\) 0 0
\(169\) −6.49089 + 11.2425i −0.499299 + 0.864811i
\(170\) 0 0
\(171\) −2.19292 6.31831i −0.167696 0.483174i
\(172\) 0 0
\(173\) 10.6787 + 18.4960i 0.811886 + 1.40623i 0.911543 + 0.411206i \(0.134892\pi\)
−0.0996566 + 0.995022i \(0.531774\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.18367 + 5.06254i 0.314464 + 0.380524i
\(178\) 0 0
\(179\) 5.03259 2.90557i 0.376153 0.217172i −0.299990 0.953942i \(-0.596983\pi\)
0.676143 + 0.736770i \(0.263650\pi\)
\(180\) 0 0
\(181\) 7.38877i 0.549203i −0.961558 0.274602i \(-0.911454\pi\)
0.961558 0.274602i \(-0.0885460\pi\)
\(182\) 0 0
\(183\) −3.10939 18.4420i −0.229853 1.36327i
\(184\) 0 0
\(185\) −2.08054 + 3.60360i −0.152964 + 0.264942i
\(186\) 0 0
\(187\) −14.7033 + 8.48897i −1.07521 + 0.620775i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.86109 + 3.96125i −0.496451 + 0.286626i −0.727247 0.686376i \(-0.759200\pi\)
0.230796 + 0.973002i \(0.425867\pi\)
\(192\) 0 0
\(193\) 3.16548 5.48277i 0.227856 0.394659i −0.729316 0.684177i \(-0.760162\pi\)
0.957173 + 0.289518i \(0.0934951\pi\)
\(194\) 0 0
\(195\) 0.0207419 + 0.123022i 0.00148536 + 0.00880979i
\(196\) 0 0
\(197\) 15.1580i 1.07996i 0.841677 + 0.539981i \(0.181569\pi\)
−0.841677 + 0.539981i \(0.818431\pi\)
\(198\) 0 0
\(199\) 7.40524 4.27542i 0.524944 0.303076i −0.214011 0.976831i \(-0.568653\pi\)
0.738955 + 0.673755i \(0.235320\pi\)
\(200\) 0 0
\(201\) −12.7041 15.3728i −0.896077 1.08432i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.44919 + 4.24212i 0.171059 + 0.296282i
\(206\) 0 0
\(207\) −3.86109 + 4.46374i −0.268364 + 0.310251i
\(208\) 0 0
\(209\) −4.37536 + 7.57835i −0.302650 + 0.524205i
\(210\) 0 0
\(211\) −2.80782 4.86329i −0.193299 0.334803i 0.753043 0.657971i \(-0.228585\pi\)
−0.946341 + 0.323169i \(0.895252\pi\)
\(212\) 0 0
\(213\) −5.24075 1.95211i −0.359090 0.133757i
\(214\) 0 0
\(215\) 1.73164 2.99929i 0.118097 0.204550i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.109727 0.650797i −0.00741464 0.0439768i
\(220\) 0 0
\(221\) 0.505679 + 0.291954i 0.0340156 + 0.0196389i
\(222\) 0 0
\(223\) 6.00510 + 3.46705i 0.402131 + 0.232171i 0.687403 0.726276i \(-0.258751\pi\)
−0.285272 + 0.958447i \(0.592084\pi\)
\(224\) 0 0
\(225\) 10.6988 + 9.25436i 0.713254 + 0.616957i
\(226\) 0 0
\(227\) 14.5767 0.967487 0.483743 0.875210i \(-0.339277\pi\)
0.483743 + 0.875210i \(0.339277\pi\)
\(228\) 0 0
\(229\) 24.5631i 1.62317i 0.584233 + 0.811586i \(0.301396\pi\)
−0.584233 + 0.811586i \(0.698604\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.10556 + 5.25710i 0.596525 + 0.344404i 0.767673 0.640841i \(-0.221414\pi\)
−0.171148 + 0.985245i \(0.554748\pi\)
\(234\) 0 0
\(235\) 1.62378 + 2.81247i 0.105924 + 0.183465i
\(236\) 0 0
\(237\) 15.7237 2.65107i 1.02136 0.172205i
\(238\) 0 0
\(239\) 16.9075 9.76154i 1.09365 0.631422i 0.159108 0.987261i \(-0.449138\pi\)
0.934547 + 0.355839i \(0.115805\pi\)
\(240\) 0 0
\(241\) 13.1382i 0.846305i −0.906059 0.423152i \(-0.860923\pi\)
0.906059 0.423152i \(-0.139077\pi\)
\(242\) 0 0
\(243\) −15.2393 3.28059i −0.977605 0.210450i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.300956 0.0191494
\(248\) 0 0
\(249\) 2.82504 + 3.41850i 0.179030 + 0.216639i
\(250\) 0 0
\(251\) 22.6864 1.43195 0.715977 0.698124i \(-0.245981\pi\)
0.715977 + 0.698124i \(0.245981\pi\)
\(252\) 0 0
\(253\) 7.72218 0.485489
\(254\) 0 0
\(255\) −3.94163 + 0.664573i −0.246835 + 0.0416172i
\(256\) 0 0
\(257\) 13.6096 0.848944 0.424472 0.905441i \(-0.360460\pi\)
0.424472 + 0.905441i \(0.360460\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 5.86711 + 16.9045i 0.363165 + 1.04636i
\(262\) 0 0
\(263\) 22.8550i 1.40930i −0.709554 0.704651i \(-0.751104\pi\)
0.709554 0.704651i \(-0.248896\pi\)
\(264\) 0 0
\(265\) −5.09055 + 2.93903i −0.312710 + 0.180543i
\(266\) 0 0
\(267\) 18.9043 + 22.8755i 1.15692 + 1.39996i
\(268\) 0 0
\(269\) 5.33875 + 9.24698i 0.325509 + 0.563798i 0.981615 0.190870i \(-0.0611310\pi\)
−0.656106 + 0.754669i \(0.727798\pi\)
\(270\) 0 0
\(271\) −3.90987 2.25737i −0.237508 0.137125i 0.376523 0.926407i \(-0.377120\pi\)
−0.614031 + 0.789282i \(0.710453\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 18.5087i 1.11612i
\(276\) 0 0
\(277\) −6.69904 −0.402507 −0.201253 0.979539i \(-0.564501\pi\)
−0.201253 + 0.979539i \(0.564501\pi\)
\(278\) 0 0
\(279\) 0.909051 + 2.61919i 0.0544235 + 0.156807i
\(280\) 0 0
\(281\) −15.1414 8.74187i −0.903258 0.521496i −0.0250023 0.999687i \(-0.507959\pi\)
−0.878256 + 0.478191i \(0.841293\pi\)
\(282\) 0 0
\(283\) −7.42049 4.28422i −0.441102 0.254670i 0.262963 0.964806i \(-0.415300\pi\)
−0.704065 + 0.710135i \(0.748634\pi\)
\(284\) 0 0
\(285\) −1.58814 + 1.31244i −0.0940733 + 0.0777420i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.854223 + 1.47956i −0.0502484 + 0.0870328i
\(290\) 0 0
\(291\) 21.0821 17.4222i 1.23585 1.02131i
\(292\) 0 0
\(293\) 12.1436 + 21.0333i 0.709434 + 1.22878i 0.965067 + 0.262002i \(0.0843826\pi\)
−0.255633 + 0.966774i \(0.582284\pi\)
\(294\) 0 0
\(295\) 1.01157 1.75209i 0.0588958 0.102010i
\(296\) 0 0
\(297\) 9.79806 + 17.8885i 0.568541 + 1.03800i
\(298\) 0 0
\(299\) −0.132791 0.230001i −0.00767951 0.0133013i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 8.90995 23.9201i 0.511863 1.37417i
\(304\) 0 0
\(305\) −4.98941 + 2.88064i −0.285693 + 0.164945i
\(306\) 0 0
\(307\) 12.4777i 0.712139i 0.934460 + 0.356069i \(0.115883\pi\)
−0.934460 + 0.356069i \(0.884117\pi\)
\(308\) 0 0
\(309\) −20.8864 7.77994i −1.18819 0.442586i
\(310\) 0 0
\(311\) −9.07984 + 15.7267i −0.514871 + 0.891782i 0.484980 + 0.874525i \(0.338827\pi\)
−0.999851 + 0.0172571i \(0.994507\pi\)
\(312\) 0 0
\(313\) −2.76700 + 1.59753i −0.156400 + 0.0902977i −0.576157 0.817339i \(-0.695449\pi\)
0.419757 + 0.907636i \(0.362115\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.5893 13.0419i 1.26874 0.732508i 0.293991 0.955808i \(-0.405016\pi\)
0.974750 + 0.223300i \(0.0716831\pi\)
\(318\) 0 0
\(319\) 11.7062 20.2757i 0.655421 1.13522i
\(320\) 0 0
\(321\) 21.0717 17.4136i 1.17611 0.971933i
\(322\) 0 0
\(323\) 9.64266i 0.536532i
\(324\) 0 0
\(325\) −0.551272 + 0.318277i −0.0305791 + 0.0176548i
\(326\) 0 0
\(327\) −5.26627 + 0.887911i −0.291226 + 0.0491016i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.06484 + 13.9687i 0.443283 + 0.767789i 0.997931 0.0642960i \(-0.0204802\pi\)
−0.554647 + 0.832085i \(0.687147\pi\)
\(332\) 0 0
\(333\) 22.1027 7.67126i 1.21122 0.420383i
\(334\) 0 0
\(335\) −3.07171 + 5.32036i −0.167826 + 0.290683i
\(336\) 0 0
\(337\) −4.16548 7.21482i −0.226908 0.393016i 0.729982 0.683466i \(-0.239528\pi\)
−0.956890 + 0.290450i \(0.906195\pi\)
\(338\) 0 0
\(339\) 2.64739 + 15.7019i 0.143786 + 0.852808i
\(340\) 0 0
\(341\) 1.81376 3.14153i 0.0982207 0.170123i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.70375 + 0.634624i 0.0917266 + 0.0341670i
\(346\) 0 0
\(347\) 30.1403 + 17.4015i 1.61801 + 0.934161i 0.987433 + 0.158037i \(0.0505166\pi\)
0.630581 + 0.776124i \(0.282817\pi\)
\(348\) 0 0
\(349\) −19.6825 11.3637i −1.05358 0.608283i −0.129929 0.991523i \(-0.541475\pi\)
−0.923649 + 0.383240i \(0.874808\pi\)
\(350\) 0 0
\(351\) 0.364313 0.599443i 0.0194456 0.0319959i
\(352\) 0 0
\(353\) −8.05659 −0.428809 −0.214404 0.976745i \(-0.568781\pi\)
−0.214404 + 0.976745i \(0.568781\pi\)
\(354\) 0 0
\(355\) 1.72278i 0.0914357i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.4341 + 8.91086i 0.814579 + 0.470297i 0.848543 0.529126i \(-0.177480\pi\)
−0.0339648 + 0.999423i \(0.510813\pi\)
\(360\) 0 0
\(361\) −7.01500 12.1503i −0.369211 0.639492i
\(362\) 0 0
\(363\) 2.66471 7.15383i 0.139861 0.375478i
\(364\) 0 0
\(365\) −0.176070 + 0.101654i −0.00921594 + 0.00532083i
\(366\) 0 0
\(367\) 23.7739i 1.24099i 0.784211 + 0.620494i \(0.213068\pi\)
−0.784211 + 0.620494i \(0.786932\pi\)
\(368\) 0 0
\(369\) 5.18888 27.0484i 0.270122 1.40809i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.5372 0.545593 0.272797 0.962072i \(-0.412051\pi\)
0.272797 + 0.962072i \(0.412051\pi\)
\(374\) 0 0
\(375\) 3.13400 8.41371i 0.161839 0.434482i
\(376\) 0 0
\(377\) −0.805203 −0.0414700
\(378\) 0 0
\(379\) 24.0049 1.23305 0.616525 0.787336i \(-0.288540\pi\)
0.616525 + 0.787336i \(0.288540\pi\)
\(380\) 0 0
\(381\) 6.11028 16.4040i 0.313039 0.840401i
\(382\) 0 0
\(383\) 36.1960 1.84953 0.924764 0.380542i \(-0.124263\pi\)
0.924764 + 0.380542i \(0.124263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −18.3962 + 6.38481i −0.935130 + 0.324558i
\(388\) 0 0
\(389\) 21.4810i 1.08913i −0.838718 0.544565i \(-0.816695\pi\)
0.838718 0.544565i \(-0.183305\pi\)
\(390\) 0 0
\(391\) 7.36925 4.25464i 0.372679 0.215166i
\(392\) 0 0
\(393\) −9.45360 + 25.3796i −0.476871 + 1.28023i
\(394\) 0 0
\(395\) −2.45603 4.25397i −0.123576 0.214041i
\(396\) 0 0
\(397\) 18.4505 + 10.6524i 0.926006 + 0.534630i 0.885546 0.464551i \(-0.153784\pi\)
0.0404601 + 0.999181i \(0.487118\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.1081i 0.904276i 0.891948 + 0.452138i \(0.149339\pi\)
−0.891948 + 0.452138i \(0.850661\pi\)
\(402\) 0 0
\(403\) −0.124758 −0.00621465
\(404\) 0 0
\(405\) 0.691630 + 4.75198i 0.0343674 + 0.236128i
\(406\) 0 0
\(407\) −26.5106 15.3059i −1.31408 0.758685i
\(408\) 0 0
\(409\) 17.6807 + 10.2080i 0.874254 + 0.504751i 0.868760 0.495234i \(-0.164918\pi\)
0.00549461 + 0.999985i \(0.498251\pi\)
\(410\) 0 0
\(411\) 4.00875 + 1.49321i 0.197737 + 0.0736546i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.683065 1.18310i 0.0335303 0.0580762i
\(416\) 0 0
\(417\) −5.60371 33.2360i −0.274415 1.62758i
\(418\) 0 0
\(419\) −12.6789 21.9606i −0.619407 1.07284i −0.989594 0.143887i \(-0.954040\pi\)
0.370187 0.928957i \(-0.379294\pi\)
\(420\) 0 0
\(421\) −3.21875 + 5.57503i −0.156872 + 0.271710i −0.933739 0.357954i \(-0.883474\pi\)
0.776867 + 0.629665i \(0.216808\pi\)
\(422\) 0 0
\(423\) 3.44016 17.9328i 0.167266 0.871921i
\(424\) 0 0
\(425\) −10.1976 17.6628i −0.494658 0.856773i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.905034 + 0.152592i −0.0436955 + 0.00736720i
\(430\) 0 0
\(431\) −13.1109 + 7.56961i −0.631532 + 0.364615i −0.781345 0.624099i \(-0.785466\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(432\) 0 0
\(433\) 8.44792i 0.405981i −0.979181 0.202991i \(-0.934934\pi\)
0.979181 0.202991i \(-0.0650661\pi\)
\(434\) 0 0
\(435\) 4.24904 3.51140i 0.203726 0.168359i
\(436\) 0 0
\(437\) 2.19292 3.79824i 0.104901 0.181695i
\(438\) 0 0
\(439\) −23.6831 + 13.6734i −1.13033 + 0.652598i −0.944018 0.329893i \(-0.892987\pi\)
−0.186314 + 0.982490i \(0.559654\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14.6520 8.45931i 0.696135 0.401914i −0.109771 0.993957i \(-0.535012\pi\)
0.805906 + 0.592043i \(0.201678\pi\)
\(444\) 0 0
\(445\) 4.57085 7.91695i 0.216679 0.375299i
\(446\) 0 0
\(447\) −25.7334 9.58537i −1.21715 0.453372i
\(448\) 0 0
\(449\) 22.5985i 1.06649i 0.845962 + 0.533244i \(0.179027\pi\)
−0.845962 + 0.533244i \(0.820973\pi\)
\(450\) 0 0
\(451\) −31.2079 + 18.0179i −1.46952 + 0.848431i
\(452\) 0 0
\(453\) −5.03679 + 13.5220i −0.236649 + 0.635321i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.4018 + 30.1408i 0.814022 + 1.40993i 0.910028 + 0.414547i \(0.136060\pi\)
−0.0960053 + 0.995381i \(0.530607\pi\)
\(458\) 0 0
\(459\) 19.2062 + 11.6726i 0.896469 + 0.544831i
\(460\) 0 0
\(461\) −13.8264 + 23.9479i −0.643958 + 1.11537i 0.340584 + 0.940214i \(0.389375\pi\)
−0.984541 + 0.175153i \(0.943958\pi\)
\(462\) 0 0
\(463\) 10.6272 + 18.4069i 0.493889 + 0.855440i 0.999975 0.00704260i \(-0.00224175\pi\)
−0.506087 + 0.862483i \(0.668908\pi\)
\(464\) 0 0
\(465\) 0.658347 0.544057i 0.0305301 0.0252300i
\(466\) 0 0
\(467\) −4.40378 + 7.62758i −0.203783 + 0.352962i −0.949744 0.313027i \(-0.898657\pi\)
0.745961 + 0.665989i \(0.231990\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 11.9295 9.85850i 0.549682 0.454256i
\(472\) 0 0
\(473\) 22.0648 + 12.7391i 1.01454 + 0.585746i
\(474\) 0 0
\(475\) −9.10373 5.25604i −0.417708 0.241164i
\(476\) 0 0
\(477\) 32.4582 + 6.22666i 1.48616 + 0.285099i
\(478\) 0 0
\(479\) 13.6628 0.624269 0.312134 0.950038i \(-0.398956\pi\)
0.312134 + 0.950038i \(0.398956\pi\)
\(480\) 0 0
\(481\) 1.05281i 0.0480038i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.29625 4.21249i −0.331306 0.191280i
\(486\) 0 0
\(487\) −8.31028 14.3938i −0.376575 0.652246i 0.613987 0.789316i \(-0.289565\pi\)
−0.990561 + 0.137070i \(0.956232\pi\)
\(488\) 0 0
\(489\) 15.6745 + 18.9673i 0.708826 + 0.857730i
\(490\) 0 0
\(491\) 17.8129 10.2843i 0.803883 0.464122i −0.0409440 0.999161i \(-0.513037\pi\)
0.844827 + 0.535039i \(0.179703\pi\)
\(492\) 0 0
\(493\) 25.7988i 1.16192i
\(494\) 0 0
\(495\) 4.11041 4.75198i 0.184749 0.213585i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.69218 −0.120518 −0.0602592 0.998183i \(-0.519193\pi\)
−0.0602592 + 0.998183i \(0.519193\pi\)
\(500\) 0 0
\(501\) −21.4281 + 3.61286i −0.957339 + 0.161411i
\(502\) 0 0
\(503\) 27.3871 1.22113 0.610566 0.791965i \(-0.290942\pi\)
0.610566 + 0.791965i \(0.290942\pi\)
\(504\) 0 0
\(505\) −7.86321 −0.349908
\(506\) 0 0
\(507\) −14.3235 17.3325i −0.636131 0.769763i
\(508\) 0 0
\(509\) −5.92234 −0.262503 −0.131252 0.991349i \(-0.541900\pi\)
−0.131252 + 0.991349i \(0.541900\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 11.5811 + 0.260636i 0.511318 + 0.0115073i
\(514\) 0 0
\(515\) 6.86596i 0.302550i
\(516\) 0 0
\(517\) −20.6905 + 11.9456i −0.909966 + 0.525369i
\(518\) 0 0
\(519\) −36.4772 + 6.15019i −1.60117 + 0.269963i
\(520\) 0 0
\(521\) −19.5943 33.9383i −0.858442 1.48686i −0.873415 0.486976i \(-0.838100\pi\)
0.0149735 0.999888i \(-0.495234\pi\)
\(522\) 0 0
\(523\) −19.9496 11.5179i −0.872333 0.503642i −0.00421050 0.999991i \(-0.501340\pi\)
−0.868123 + 0.496349i \(0.834674\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.99727i 0.174124i
\(528\) 0 0
\(529\) 19.1297 0.831725
\(530\) 0 0
\(531\) −10.7464 + 3.72980i −0.466356 + 0.161860i
\(532\) 0 0
\(533\) 1.07331 + 0.619675i 0.0464901 + 0.0268411i
\(534\) 0 0
\(535\) −7.29267 4.21043i −0.315290 0.182033i
\(536\) 0 0
\(537\) 1.67340 + 9.92509i 0.0722127 + 0.428299i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.66302 2.88044i 0.0714990 0.123840i −0.828059 0.560640i \(-0.810555\pi\)
0.899558 + 0.436800i \(0.143888\pi\)
\(542\) 0 0
\(543\) 11.9928 + 4.46716i 0.514659 + 0.191704i
\(544\) 0 0
\(545\) 0.822590 + 1.42477i 0.0352359 + 0.0610303i
\(546\) 0 0
\(547\) 13.8937 24.0646i 0.594051 1.02893i −0.399629 0.916677i \(-0.630861\pi\)
0.993680 0.112249i \(-0.0358055\pi\)
\(548\) 0 0
\(549\) 31.8133 + 6.10295i 1.35776 + 0.260467i
\(550\) 0 0
\(551\) −6.64857 11.5157i −0.283238 0.490583i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −4.59116 5.55563i −0.194884 0.235824i
\(556\) 0 0
\(557\) 3.13213 1.80833i 0.132712 0.0766216i −0.432174 0.901790i \(-0.642253\pi\)
0.564886 + 0.825169i \(0.308920\pi\)
\(558\) 0 0
\(559\) 0.876252i 0.0370615i
\(560\) 0 0
\(561\) −4.88906 28.9974i −0.206416 1.22427i
\(562\) 0 0
\(563\) −4.75452 + 8.23506i −0.200379 + 0.347067i −0.948651 0.316326i \(-0.897551\pi\)
0.748272 + 0.663393i \(0.230884\pi\)
\(564\) 0 0
\(565\) 4.24807 2.45262i 0.178718 0.103183i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.09742 + 5.25240i −0.381384 + 0.220192i −0.678420 0.734674i \(-0.737335\pi\)
0.297036 + 0.954866i \(0.404002\pi\)
\(570\) 0 0
\(571\) −2.24201 + 3.88328i −0.0938252 + 0.162510i −0.909118 0.416539i \(-0.863243\pi\)
0.815292 + 0.579049i \(0.196576\pi\)
\(572\) 0 0
\(573\) −2.28141 13.5312i −0.0953071 0.565274i
\(574\) 0 0
\(575\) 9.27651i 0.386857i
\(576\) 0 0
\(577\) 40.8602 23.5906i 1.70103 0.982090i 0.756310 0.654213i \(-0.227000\pi\)
0.944720 0.327877i \(-0.106333\pi\)
\(578\) 0 0
\(579\) 6.98531 + 8.45272i 0.290300 + 0.351283i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −21.6215 37.4496i −0.895473 1.55100i
\(584\) 0 0
\(585\) −0.212218 0.0407112i −0.00877414 0.00168320i
\(586\) 0 0
\(587\) 5.65373 9.79255i 0.233354 0.404182i −0.725439 0.688287i \(-0.758363\pi\)
0.958793 + 0.284105i \(0.0916964\pi\)
\(588\) 0 0
\(589\) −1.03013 1.78424i −0.0424458 0.0735183i
\(590\) 0 0
\(591\) −24.6030 9.16433i −1.01203 0.376970i
\(592\) 0 0
\(593\) −4.72490 + 8.18376i −0.194028 + 0.336067i −0.946582 0.322465i \(-0.895489\pi\)
0.752553 + 0.658531i \(0.228822\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.46234 + 14.6044i 0.100777 + 0.597717i
\(598\) 0 0
\(599\) −31.6406 18.2677i −1.29280 0.746398i −0.313650 0.949539i \(-0.601552\pi\)
−0.979150 + 0.203140i \(0.934885\pi\)
\(600\) 0 0
\(601\) 1.92247 + 1.10994i 0.0784193 + 0.0452754i 0.538697 0.842500i \(-0.318917\pi\)
−0.460278 + 0.887775i \(0.652250\pi\)
\(602\) 0 0
\(603\) 32.6325 11.3259i 1.32890 0.461225i
\(604\) 0 0
\(605\) −2.35166 −0.0956087
\(606\) 0 0
\(607\) 1.98331i 0.0804999i 0.999190 + 0.0402499i \(0.0128154\pi\)
−0.999190 + 0.0402499i \(0.987185\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.711589 + 0.410836i 0.0287878 + 0.0166207i
\(612\) 0 0
\(613\) 11.5683 + 20.0368i 0.467238 + 0.809280i 0.999299 0.0374258i \(-0.0119158\pi\)
−0.532061 + 0.846706i \(0.678582\pi\)
\(614\) 0 0
\(615\) −8.36616 + 1.41056i −0.337356 + 0.0568794i
\(616\) 0 0
\(617\) −1.19807 + 0.691704i −0.0482323 + 0.0278470i −0.523922 0.851766i \(-0.675532\pi\)
0.475690 + 0.879613i \(0.342198\pi\)
\(618\) 0 0
\(619\) 6.03389i 0.242522i −0.992621 0.121261i \(-0.961306\pi\)
0.992621 0.121261i \(-0.0386939\pi\)
\(620\) 0 0
\(621\) −4.91075 8.96568i −0.197062 0.359780i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 20.8107 0.832430
\(626\) 0 0
\(627\) −9.65518 11.6835i −0.385591 0.466592i
\(628\) 0 0
\(629\) −33.7320 −1.34498
\(630\) 0 0
\(631\) −20.4727 −0.815004 −0.407502 0.913204i \(-0.633600\pi\)
−0.407502 + 0.913204i \(0.633600\pi\)
\(632\) 0 0
\(633\) 9.59121 1.61711i 0.381217 0.0642744i
\(634\) 0 0
\(635\) −5.39245 −0.213993
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.33698 7.32607i 0.250687 0.289815i
\(640\) 0 0
\(641\) 46.4474i 1.83456i −0.398240 0.917281i \(-0.630379\pi\)
0.398240 0.917281i \(-0.369621\pi\)
\(642\) 0 0
\(643\) 9.74133 5.62416i 0.384161 0.221795i −0.295466 0.955353i \(-0.595475\pi\)
0.679627 + 0.733558i \(0.262142\pi\)
\(644\) 0 0
\(645\) 3.82124 + 4.62397i 0.150461 + 0.182069i
\(646\) 0 0
\(647\) 2.32507 + 4.02715i 0.0914081 + 0.158323i 0.908104 0.418745i \(-0.137530\pi\)
−0.816696 + 0.577068i \(0.804197\pi\)
\(648\) 0 0
\(649\) 12.8896 + 7.44179i 0.505959 + 0.292116i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.46371i 0.135545i 0.997701 + 0.0677727i \(0.0215893\pi\)
−0.997701 + 0.0677727i \(0.978411\pi\)
\(654\) 0 0
\(655\) 8.34299 0.325988
\(656\) 0 0
\(657\) 1.12265 + 0.215366i 0.0437989 + 0.00840222i
\(658\) 0 0
\(659\) −1.59819 0.922715i −0.0622566 0.0359439i 0.468549 0.883438i \(-0.344777\pi\)
−0.530805 + 0.847494i \(0.678110\pi\)
\(660\) 0 0
\(661\) −17.5196 10.1149i −0.681433 0.393426i 0.118962 0.992899i \(-0.462043\pi\)
−0.800395 + 0.599473i \(0.795377\pi\)
\(662\) 0 0
\(663\) −0.779599 + 0.644259i −0.0302771 + 0.0250209i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.86711 + 10.1621i −0.227175 + 0.393479i
\(668\) 0 0
\(669\) −9.25799 + 7.65078i −0.357935 + 0.295796i
\(670\) 0 0
\(671\) −21.1919 36.7055i −0.818106 1.41700i
\(672\) 0 0
\(673\) −7.31596 + 12.6716i −0.282009 + 0.488455i −0.971880 0.235478i \(-0.924334\pi\)
0.689870 + 0.723933i \(0.257668\pi\)
\(674\) 0 0
\(675\) −21.4892 + 11.7702i −0.827119 + 0.453036i
\(676\) 0 0
\(677\) 7.71449 + 13.3619i 0.296492 + 0.513539i 0.975331 0.220748i \(-0.0708499\pi\)
−0.678839 + 0.734287i \(0.737517\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8.81286 + 23.6595i −0.337710 + 0.906633i
\(682\) 0 0
\(683\) −13.6137 + 7.85987i −0.520913 + 0.300750i −0.737308 0.675556i \(-0.763903\pi\)
0.216395 + 0.976306i \(0.430570\pi\)
\(684\) 0 0
\(685\) 1.31779i 0.0503501i
\(686\) 0 0
\(687\) −39.8685 14.8505i −1.52108 0.566582i
\(688\) 0 0
\(689\) −0.743611 + 1.28797i −0.0283293 + 0.0490678i
\(690\) 0 0
\(691\) 41.5878 24.0107i 1.58207 0.913411i 0.587517 0.809212i \(-0.300106\pi\)
0.994556 0.104199i \(-0.0332278\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.99187 + 5.19146i −0.341081 + 0.196923i
\(696\) 0 0
\(697\) −19.8544 + 34.3889i −0.752040 + 1.30257i
\(698\) 0 0
\(699\) −14.0379 + 11.6009i −0.530963 + 0.438787i
\(700\) 0 0
\(701\) 24.7005i 0.932923i 0.884541 + 0.466462i \(0.154471\pi\)
−0.884541 + 0.466462i \(0.845529\pi\)
\(702\) 0 0
\(703\) −15.0568 + 8.69302i −0.567876 + 0.327864i
\(704\) 0 0
\(705\) −5.54666 + 0.935185i −0.208899 + 0.0352211i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0432 29.5196i −0.640070 1.10863i −0.985417 0.170159i \(-0.945572\pi\)
0.345347 0.938475i \(-0.387761\pi\)
\(710\) 0 0
\(711\) −5.20338 + 27.1240i −0.195142 + 1.01723i
\(712\) 0 0
\(713\) −0.909051 + 1.57452i −0.0340442 + 0.0589663i
\(714\) 0 0
\(715\) 0.141366 + 0.244853i 0.00528679 + 0.00915698i
\(716\) 0 0
\(717\) 5.62197 + 33.3444i 0.209956 + 1.24527i
\(718\) 0 0
\(719\) −9.23791 + 16.0005i −0.344516 + 0.596719i −0.985266 0.171031i \(-0.945290\pi\)
0.640750 + 0.767750i \(0.278624\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 21.3247 + 7.94318i 0.793073 + 0.295410i
\(724\) 0 0
\(725\) 24.3569 + 14.0624i 0.904591 + 0.522266i
\(726\) 0 0
\(727\) 39.2911 + 22.6847i 1.45723 + 0.841330i 0.998874 0.0474398i \(-0.0151062\pi\)
0.458353 + 0.888770i \(0.348440\pi\)
\(728\) 0 0
\(729\) 14.5383 22.7517i 0.538454 0.842655i
\(730\) 0 0
\(731\) 28.0752 1.03840
\(732\) 0 0
\(733\) 50.0773i 1.84965i −0.380394 0.924825i \(-0.624212\pi\)
0.380394 0.924825i \(-0.375788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −39.1403 22.5976i −1.44175 0.832395i
\(738\) 0 0
\(739\) 8.47021 + 14.6708i 0.311582 + 0.539675i 0.978705 0.205272i \(-0.0658079\pi\)
−0.667123 + 0.744947i \(0.732475\pi\)
\(740\) 0 0
\(741\) −0.181954 + 0.488484i −0.00668426 + 0.0179449i
\(742\) 0 0
\(743\) 34.4723 19.9026i 1.26467 0.730156i 0.290693 0.956816i \(-0.406114\pi\)
0.973974 + 0.226661i \(0.0727808\pi\)
\(744\) 0 0
\(745\) 8.45928i 0.309924i
\(746\) 0 0
\(747\) −7.25657 + 2.51856i −0.265504 + 0.0921493i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29.4055 −1.07302 −0.536512 0.843893i \(-0.680258\pi\)
−0.536512 + 0.843893i \(0.680258\pi\)
\(752\) 0 0
\(753\) −13.7159 + 36.8225i −0.499836 + 1.34189i
\(754\) 0 0
\(755\) 4.44507 0.161773
\(756\) 0 0
\(757\) −27.4010 −0.995908 −0.497954 0.867203i \(-0.665915\pi\)
−0.497954 + 0.867203i \(0.665915\pi\)
\(758\) 0 0
\(759\) −4.66873 + 12.5339i −0.169464 + 0.454952i
\(760\) 0 0
\(761\) −12.2213 −0.443023 −0.221511 0.975158i \(-0.571099\pi\)
−0.221511 + 0.975158i \(0.571099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.30439 6.79948i 0.0471603 0.245836i
\(766\) 0 0
\(767\) 0.511878i 0.0184829i
\(768\) 0 0
\(769\) 29.4039 16.9764i 1.06033 0.612184i 0.134809 0.990872i \(-0.456958\pi\)
0.925524 + 0.378688i \(0.123625\pi\)
\(770\) 0 0
\(771\) −8.22820 + 22.0898i −0.296331 + 0.795546i
\(772\) 0 0
\(773\) 15.1047 + 26.1622i 0.543279 + 0.940987i 0.998713 + 0.0507175i \(0.0161508\pi\)
−0.455434 + 0.890270i \(0.650516\pi\)
\(774\) 0 0
\(775\) 3.77386 + 2.17884i 0.135561 + 0.0782662i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.4666i 0.733294i
\(780\) 0 0
\(781\) −12.6740 −0.453510
\(782\) 0 0
\(783\) −30.9850 0.697326i −1.10731 0.0249204i
\(784\) 0 0
\(785\) −4.12866 2.38368i −0.147358 0.0850772i
\(786\) 0 0
\(787\) 33.3310 + 19.2436i 1.18812 + 0.685962i 0.957879 0.287172i \(-0.0927151\pi\)
0.230241 + 0.973134i \(0.426048\pi\)
\(788\) 0 0
\(789\) 37.0961 + 13.8179i 1.32066 + 0.491929i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.728836 + 1.26238i −0.0258817 + 0.0448285i
\(794\) 0 0
\(795\) −1.69268 10.0394i −0.0600331 0.356061i
\(796\) 0 0
\(797\) 6.72949 + 11.6558i 0.238371 + 0.412870i 0.960247 0.279152i \(-0.0900533\pi\)
−0.721876 + 0.692022i \(0.756720\pi\)
\(798\) 0 0
\(799\) −13.1632 + 22.7994i −0.465682 + 0.806584i
\(800\) 0 0
\(801\) −48.5587 + 16.8534i −1.71574 + 0.595486i
\(802\) 0 0
\(803\) −0.747839 1.29529i −0.0263907 0.0457100i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −18.2366 + 3.07475i −0.641958 + 0.108236i
\(808\) 0 0
\(809\) −44.8465 + 25.8921i −1.57672 + 0.910318i −0.581404 + 0.813615i \(0.697497\pi\)
−0.995313 + 0.0967036i \(0.969170\pi\)
\(810\) 0 0
\(811\) 27.2471i 0.956775i 0.878149 + 0.478387i \(0.158779\pi\)
−0.878149 + 0.478387i \(0.841221\pi\)
\(812\) 0 0
\(813\) 6.02781 4.98137i 0.211404 0.174704i
\(814\) 0 0
\(815\) 3.78993 6.56436i 0.132756 0.229939i
\(816\) 0 0
\(817\) 12.5318 7.23523i 0.438432 0.253129i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −22.3465 + 12.9017i −0.779897 + 0.450274i −0.836394 0.548129i \(-0.815340\pi\)
0.0564968 + 0.998403i \(0.482007\pi\)
\(822\) 0 0
\(823\) 0.570514 0.988159i 0.0198869 0.0344451i −0.855911 0.517124i \(-0.827003\pi\)
0.875798 + 0.482679i \(0.160336\pi\)
\(824\) 0 0
\(825\) 30.0416 + 11.1901i 1.04592 + 0.389591i
\(826\) 0 0
\(827\) 23.1713i 0.805746i 0.915256 + 0.402873i \(0.131988\pi\)
−0.915256 + 0.402873i \(0.868012\pi\)
\(828\) 0 0
\(829\) −8.31700 + 4.80182i −0.288861 + 0.166774i −0.637428 0.770510i \(-0.720002\pi\)
0.348567 + 0.937284i \(0.386668\pi\)
\(830\) 0 0
\(831\) 4.05016 10.8733i 0.140498 0.377189i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.34707 + 5.79729i 0.115830 + 0.200623i
\(836\) 0 0
\(837\) −4.80083 0.108044i −0.165941 0.00373454i
\(838\) 0 0
\(839\) 15.7821 27.3354i 0.544859 0.943723i −0.453757 0.891126i \(-0.649917\pi\)
0.998616 0.0525978i \(-0.0167501\pi\)
\(840\) 0 0
\(841\) 3.28812 + 5.69519i 0.113383 + 0.196386i
\(842\) 0 0
\(843\) 23.3433 19.2908i 0.803985 0.664411i
\(844\) 0 0
\(845\) −3.46328 + 5.99858i −0.119140 + 0.206357i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 11.4401 9.45405i 0.392622 0.324462i
\(850\) 0 0
\(851\) 13.2870 + 7.67126i 0.455473 + 0.262967i
\(852\) 0 0
\(853\) −7.50412 4.33250i −0.256936 0.148342i 0.366000 0.930615i \(-0.380727\pi\)
−0.622936 + 0.782273i \(0.714060\pi\)
\(854\) 0 0
\(855\) −1.17005 3.37120i −0.0400150 0.115293i
\(856\) 0 0
\(857\) −22.8878 −0.781834 −0.390917 0.920426i \(-0.627842\pi\)
−0.390917 + 0.920426i \(0.627842\pi\)
\(858\) 0 0
\(859\) 13.2701i 0.452769i 0.974038 + 0.226384i \(0.0726905\pi\)
−0.974038 + 0.226384i \(0.927309\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.646706 + 0.373376i 0.0220141 + 0.0127099i 0.510967 0.859601i \(-0.329288\pi\)
−0.488953 + 0.872310i \(0.662621\pi\)
\(864\) 0 0
\(865\) 5.69773 + 9.86875i 0.193729 + 0.335548i
\(866\) 0 0
\(867\) −1.88503 2.28102i −0.0640189 0.0774674i
\(868\) 0 0
\(869\) 31.2951 18.0683i 1.06162 0.612924i
\(870\) 0 0
\(871\) 1.55436i 0.0526676i
\(872\) 0 0
\(873\) 15.5321 + 44.7516i 0.525681 + 1.51461i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.37919 0.147875 0.0739373 0.997263i \(-0.476444\pi\)
0.0739373 + 0.997263i \(0.476444\pi\)
\(878\) 0 0
\(879\) −41.4810 + 6.99384i −1.39912 + 0.235896i
\(880\) 0 0
\(881\) 14.1505 0.476742 0.238371 0.971174i \(-0.423387\pi\)
0.238371 + 0.971174i \(0.423387\pi\)
\(882\) 0 0
\(883\) 23.0261 0.774890 0.387445 0.921893i \(-0.373358\pi\)
0.387445 + 0.921893i \(0.373358\pi\)
\(884\) 0 0
\(885\) 2.23224 + 2.70117i 0.0750360 + 0.0907989i
\(886\) 0 0
\(887\) −52.3623 −1.75815 −0.879077 0.476679i \(-0.841840\pi\)
−0.879077 + 0.476679i \(0.841840\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −34.9588 + 5.08811i −1.17116 + 0.170458i
\(892\) 0 0
\(893\) 13.5691i 0.454073i
\(894\) 0 0
\(895\) 2.68519 1.55029i 0.0897560 0.0518206i
\(896\) 0 0
\(897\) 0.453600 0.0764785i 0.0151453 0.00255354i
\(898\) 0 0
\(899\) 2.75610 + 4.77370i 0.0919209 + 0.159212i
\(900\) 0 0
\(901\) −41.2667 23.8254i −1.37479 0.793738i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.94236i 0.131048i
\(906\) 0 0
\(907\) −24.3804 −0.809539 −0.404770 0.914419i \(-0.632648\pi\)
−0.404770 + 0.914419i \(0.632648\pi\)
\(908\) 0 0
\(909\) 33.4380 + 28.9236i 1.10907 + 0.959335i
\(910\) 0 0
\(911\) −46.8606 27.0550i −1.55256 0.896372i −0.997932 0.0642741i \(-0.979527\pi\)
−0.554629 0.832098i \(-0.687140\pi\)
\(912\) 0 0
\(913\) 8.70372 + 5.02509i 0.288051 + 0.166306i
\(914\) 0 0
\(915\) −1.65905 9.83994i −0.0548464 0.325298i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3.25947 + 5.64556i −0.107520 + 0.186230i −0.914765 0.403986i \(-0.867624\pi\)
0.807245 + 0.590216i \(0.200958\pi\)
\(920\) 0 0
\(921\) −20.2526 7.54385i −0.667346 0.248578i
\(922\) 0 0
\(923\) 0.217942 + 0.377487i 0.00717365 + 0.0124251i
\(924\) 0 0
\(925\) 18.3867 31.8467i 0.604550 1.04711i
\(926\) 0 0
\(927\) 25.2554 29.1973i 0.829495 0.958964i
\(928\) 0 0
\(929\) −13.9048 24.0838i −0.456202 0.790165i 0.542554 0.840021i \(-0.317457\pi\)
−0.998756 + 0.0498555i \(0.984124\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −20.0366 24.2457i −0.655970 0.793770i
\(934\) 0 0
\(935\) −7.84511 + 4.52938i −0.256563 + 0.148126i
\(936\) 0 0
\(937\) 54.8174i 1.79081i −0.445256 0.895403i \(-0.646887\pi\)
0.445256 0.895403i \(-0.353113\pi\)
\(938\) 0 0
\(939\) −0.920066 5.45698i −0.0300252 0.178082i
\(940\) 0 0
\(941\) −2.56526 + 4.44317i −0.0836252 + 0.144843i −0.904805 0.425827i \(-0.859983\pi\)
0.821179 + 0.570670i \(0.193317\pi\)
\(942\) 0 0
\(943\) 15.6413 9.03052i 0.509351 0.294074i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.606033 0.349893i 0.0196934 0.0113700i −0.490121 0.871654i \(-0.663047\pi\)
0.509814 + 0.860284i \(0.329714\pi\)
\(948\) 0 0
\(949\) −0.0257198 + 0.0445480i −0.000834899 + 0.00144609i
\(950\) 0 0
\(951\) 7.51125 + 44.5498i 0.243569 + 1.44463i
\(952\) 0 0
\(953\) 0.162845i 0.00527506i 0.999997 + 0.00263753i \(0.000839552\pi\)
−0.999997 + 0.00263753i \(0.999160\pi\)
\(954\) 0 0
\(955\) −3.66081 + 2.11357i −0.118461 + 0.0683934i
\(956\) 0 0
\(957\) 25.8323 + 31.2589i 0.835038 + 1.01046i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0730 26.1071i −0.486225 0.842166i
\(962\) 0 0
\(963\) 15.5245 + 44.7296i 0.500268 + 1.44139i
\(964\) 0 0
\(965\) 1.68897 2.92539i 0.0543700 0.0941716i
\(966\) 0 0
\(967\) −15.6968 27.1876i −0.504773 0.874293i −0.999985 0.00552073i \(-0.998243\pi\)
0.495211 0.868773i \(-0.335091\pi\)
\(968\) 0 0
\(969\) −15.6511 5.82983i −0.502785 0.187281i
\(970\) 0 0
\(971\) −1.46120 + 2.53087i −0.0468920 + 0.0812194i −0.888519 0.458840i \(-0.848265\pi\)
0.841627 + 0.540060i \(0.181598\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −0.183306 1.08720i −0.00587048 0.0348183i
\(976\) 0 0
\(977\) 24.7013 + 14.2613i 0.790264 + 0.456259i 0.840056 0.542500i \(-0.182522\pi\)
−0.0497913 + 0.998760i \(0.515856\pi\)
\(978\) 0 0
\(979\) 58.2425 + 33.6263i 1.86144 + 1.07470i
\(980\) 0 0
\(981\) 1.74275 9.08454i 0.0556416 0.290047i
\(982\) 0 0
\(983\) 11.2579 0.359073 0.179536 0.983751i \(-0.442540\pi\)
0.179536 + 0.983751i \(0.442540\pi\)
\(984\) 0 0
\(985\) 8.08771i 0.257696i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.0588 6.38481i −0.351650 0.203025i
\(990\) 0 0
\(991\) −10.5811 18.3270i −0.336120 0.582177i 0.647579 0.761998i \(-0.275781\pi\)
−0.983699 + 0.179821i \(0.942448\pi\)
\(992\) 0 0
\(993\) −27.5486 + 4.64478i −0.874228 + 0.147398i
\(994\) 0 0
\(995\) 3.95114 2.28119i 0.125260 0.0723187i
\(996\) 0 0
\(997\) 16.9251i 0.536024i 0.963416 + 0.268012i \(0.0863667\pi\)
−0.963416 + 0.268012i \(0.913633\pi\)
\(998\) 0 0
\(999\) −0.911756 + 40.5130i −0.0288467 + 1.28178i
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 1764.2.bm.b.1685.4 16
3.2 odd 2 5292.2.bm.b.4625.4 16
7.2 even 3 252.2.x.a.209.3 yes 16
7.3 odd 6 1764.2.w.a.1109.1 16
7.4 even 3 1764.2.w.a.1109.8 16
7.5 odd 6 252.2.x.a.209.6 yes 16
7.6 odd 2 inner 1764.2.bm.b.1685.5 16
9.4 even 3 5292.2.w.a.1097.4 16
9.5 odd 6 1764.2.w.a.509.1 16
21.2 odd 6 756.2.x.a.629.5 16
21.5 even 6 756.2.x.a.629.4 16
21.11 odd 6 5292.2.w.a.521.5 16
21.17 even 6 5292.2.w.a.521.4 16
21.20 even 2 5292.2.bm.b.4625.5 16
28.19 even 6 1008.2.cc.c.209.3 16
28.23 odd 6 1008.2.cc.c.209.6 16
63.2 odd 6 2268.2.f.b.1133.8 16
63.4 even 3 5292.2.bm.b.2285.5 16
63.5 even 6 252.2.x.a.41.3 16
63.13 odd 6 5292.2.w.a.1097.5 16
63.16 even 3 2268.2.f.b.1133.10 16
63.23 odd 6 252.2.x.a.41.6 yes 16
63.31 odd 6 5292.2.bm.b.2285.4 16
63.32 odd 6 inner 1764.2.bm.b.1697.5 16
63.40 odd 6 756.2.x.a.125.5 16
63.41 even 6 1764.2.w.a.509.8 16
63.47 even 6 2268.2.f.b.1133.9 16
63.58 even 3 756.2.x.a.125.4 16
63.59 even 6 inner 1764.2.bm.b.1697.4 16
63.61 odd 6 2268.2.f.b.1133.7 16
84.23 even 6 3024.2.cc.c.2897.5 16
84.47 odd 6 3024.2.cc.c.2897.4 16
252.23 even 6 1008.2.cc.c.545.3 16
252.103 even 6 3024.2.cc.c.881.5 16
252.131 odd 6 1008.2.cc.c.545.6 16
252.247 odd 6 3024.2.cc.c.881.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.3 16 63.5 even 6
252.2.x.a.41.6 yes 16 63.23 odd 6
252.2.x.a.209.3 yes 16 7.2 even 3
252.2.x.a.209.6 yes 16 7.5 odd 6
756.2.x.a.125.4 16 63.58 even 3
756.2.x.a.125.5 16 63.40 odd 6
756.2.x.a.629.4 16 21.5 even 6
756.2.x.a.629.5 16 21.2 odd 6
1008.2.cc.c.209.3 16 28.19 even 6
1008.2.cc.c.209.6 16 28.23 odd 6
1008.2.cc.c.545.3 16 252.23 even 6
1008.2.cc.c.545.6 16 252.131 odd 6
1764.2.w.a.509.1 16 9.5 odd 6
1764.2.w.a.509.8 16 63.41 even 6
1764.2.w.a.1109.1 16 7.3 odd 6
1764.2.w.a.1109.8 16 7.4 even 3
1764.2.bm.b.1685.4 16 1.1 even 1 trivial
1764.2.bm.b.1685.5 16 7.6 odd 2 inner
1764.2.bm.b.1697.4 16 63.59 even 6 inner
1764.2.bm.b.1697.5 16 63.32 odd 6 inner
2268.2.f.b.1133.7 16 63.61 odd 6
2268.2.f.b.1133.8 16 63.2 odd 6
2268.2.f.b.1133.9 16 63.47 even 6
2268.2.f.b.1133.10 16 63.16 even 3
3024.2.cc.c.881.4 16 252.247 odd 6
3024.2.cc.c.881.5 16 252.103 even 6
3024.2.cc.c.2897.4 16 84.47 odd 6
3024.2.cc.c.2897.5 16 84.23 even 6
5292.2.w.a.521.4 16 21.17 even 6
5292.2.w.a.521.5 16 21.11 odd 6
5292.2.w.a.1097.4 16 9.4 even 3
5292.2.w.a.1097.5 16 63.13 odd 6
5292.2.bm.b.2285.4 16 63.31 odd 6
5292.2.bm.b.2285.5 16 63.4 even 3
5292.2.bm.b.4625.4 16 3.2 odd 2
5292.2.bm.b.4625.5 16 21.20 even 2