Properties

Label 1800.2.k.u.901.9
Level $1800$
Weight $2$
Character 1800.901
Analytic conductor $14.373$
Analytic rank $0$
Dimension $12$
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] = [1800,2,Mod(901,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1800.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.180227832610816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 901.9
Root \(0.806504 + 1.16170i\) of defining polynomial
Character \(\chi\) \(=\) 1800.901
Dual form 1800.2.k.u.901.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.806504 - 1.16170i) q^{2} +(-0.699104 - 1.87383i) q^{4} -0.746175 q^{7} +(-2.74067 - 0.699104i) q^{8} +O(q^{10})\) \(q+(0.806504 - 1.16170i) q^{2} +(-0.699104 - 1.87383i) q^{4} -0.746175 q^{7} +(-2.74067 - 0.699104i) q^{8} +5.36068i q^{11} +2.92520i q^{13} +(-0.601793 + 0.866833i) q^{14} +(-3.02251 + 2.62001i) q^{16} +2.13466 q^{17} -1.73367i q^{19} +(6.22751 + 4.32340i) q^{22} +7.49534 q^{23} +(3.39821 + 2.35918i) q^{26} +(0.521653 + 1.39821i) q^{28} +6.74916i q^{29} +2.64681 q^{31} +(0.606006 + 5.62430i) q^{32} +(1.72161 - 2.47984i) q^{34} -1.07480i q^{37} +(-2.01400 - 1.39821i) q^{38} +11.2936 q^{41} -7.44322i q^{43} +(10.0450 - 3.74767i) q^{44} +(6.04502 - 8.70735i) q^{46} +1.73367 q^{47} -6.44322 q^{49} +(5.48133 - 2.04502i) q^{52} +7.72161i q^{53} +(2.04502 + 0.521653i) q^{56} +(7.84052 + 5.44322i) q^{58} +6.85302i q^{59} +6.45203i q^{61} +(2.13466 - 3.07480i) q^{62} +(7.02251 + 3.83202i) q^{64} -7.44322i q^{67} +(-1.49235 - 4.00000i) q^{68} -13.2936 q^{71} +0.690358 q^{73} +(-1.24860 - 0.866833i) q^{74} +(-3.24860 + 1.21201i) q^{76} -4.00000i q^{77} +2.64681 q^{79} +(9.10834 - 13.1198i) q^{82} -5.85039i q^{83} +(-8.64681 - 6.00299i) q^{86} +(3.74767 - 14.6918i) q^{88} -7.59283 q^{89} -2.18271i q^{91} +(-5.24002 - 14.0450i) q^{92} +(1.39821 - 2.01400i) q^{94} +14.1887 q^{97} +(-5.19648 + 7.48511i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} - 20 q^{14} + 2 q^{16} + 28 q^{26} - 32 q^{31} - 24 q^{34} + 8 q^{41} + 44 q^{44} - 4 q^{46} + 12 q^{49} - 52 q^{56} + 46 q^{64} - 32 q^{71} + 36 q^{74} + 12 q^{76} - 32 q^{79} - 40 q^{86} - 40 q^{89} + 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.806504 1.16170i 0.570284 0.821447i
\(3\) 0 0
\(4\) −0.699104 1.87383i −0.349552 0.936917i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.746175 −0.282028 −0.141014 0.990008i \(-0.545036\pi\)
−0.141014 + 0.990008i \(0.545036\pi\)
\(8\) −2.74067 0.699104i −0.968972 0.247170i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.36068i 1.61630i 0.588974 + 0.808152i \(0.299532\pi\)
−0.588974 + 0.808152i \(0.700468\pi\)
\(12\) 0 0
\(13\) 2.92520i 0.811304i 0.914028 + 0.405652i \(0.132955\pi\)
−0.914028 + 0.405652i \(0.867045\pi\)
\(14\) −0.601793 + 0.866833i −0.160836 + 0.231671i
\(15\) 0 0
\(16\) −3.02251 + 2.62001i −0.755627 + 0.655002i
\(17\) 2.13466 0.517731 0.258866 0.965913i \(-0.416651\pi\)
0.258866 + 0.965913i \(0.416651\pi\)
\(18\) 0 0
\(19\) 1.73367i 0.397730i −0.980027 0.198865i \(-0.936274\pi\)
0.980027 0.198865i \(-0.0637255\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.22751 + 4.32340i 1.32771 + 0.921753i
\(23\) 7.49534 1.56289 0.781443 0.623977i \(-0.214484\pi\)
0.781443 + 0.623977i \(0.214484\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.39821 + 2.35918i 0.666443 + 0.462674i
\(27\) 0 0
\(28\) 0.521653 + 1.39821i 0.0985832 + 0.264236i
\(29\) 6.74916i 1.25329i 0.779306 + 0.626644i \(0.215572\pi\)
−0.779306 + 0.626644i \(0.784428\pi\)
\(30\) 0 0
\(31\) 2.64681 0.475381 0.237690 0.971341i \(-0.423610\pi\)
0.237690 + 0.971341i \(0.423610\pi\)
\(32\) 0.606006 + 5.62430i 0.107128 + 0.994245i
\(33\) 0 0
\(34\) 1.72161 2.47984i 0.295254 0.425289i
\(35\) 0 0
\(36\) 0 0
\(37\) 1.07480i 0.176697i −0.996090 0.0883483i \(-0.971841\pi\)
0.996090 0.0883483i \(-0.0281588\pi\)
\(38\) −2.01400 1.39821i −0.326714 0.226819i
\(39\) 0 0
\(40\) 0 0
\(41\) 11.2936 1.76377 0.881883 0.471468i \(-0.156276\pi\)
0.881883 + 0.471468i \(0.156276\pi\)
\(42\) 0 0
\(43\) 7.44322i 1.13508i −0.823346 0.567540i \(-0.807895\pi\)
0.823346 0.567540i \(-0.192105\pi\)
\(44\) 10.0450 3.74767i 1.51434 0.564982i
\(45\) 0 0
\(46\) 6.04502 8.70735i 0.891289 1.28383i
\(47\) 1.73367 0.252881 0.126441 0.991974i \(-0.459645\pi\)
0.126441 + 0.991974i \(0.459645\pi\)
\(48\) 0 0
\(49\) −6.44322 −0.920460
\(50\) 0 0
\(51\) 0 0
\(52\) 5.48133 2.04502i 0.760124 0.283593i
\(53\) 7.72161i 1.06064i 0.847796 + 0.530322i \(0.177929\pi\)
−0.847796 + 0.530322i \(0.822071\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.04502 + 0.521653i 0.273277 + 0.0697089i
\(57\) 0 0
\(58\) 7.84052 + 5.44322i 1.02951 + 0.714730i
\(59\) 6.85302i 0.892188i 0.894986 + 0.446094i \(0.147185\pi\)
−0.894986 + 0.446094i \(0.852815\pi\)
\(60\) 0 0
\(61\) 6.45203i 0.826098i 0.910709 + 0.413049i \(0.135536\pi\)
−0.910709 + 0.413049i \(0.864464\pi\)
\(62\) 2.13466 3.07480i 0.271102 0.390500i
\(63\) 0 0
\(64\) 7.02251 + 3.83202i 0.877813 + 0.479003i
\(65\) 0 0
\(66\) 0 0
\(67\) 7.44322i 0.909334i −0.890661 0.454667i \(-0.849758\pi\)
0.890661 0.454667i \(-0.150242\pi\)
\(68\) −1.49235 4.00000i −0.180974 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −13.2936 −1.57766 −0.788831 0.614610i \(-0.789313\pi\)
−0.788831 + 0.614610i \(0.789313\pi\)
\(72\) 0 0
\(73\) 0.690358 0.0808003 0.0404002 0.999184i \(-0.487137\pi\)
0.0404002 + 0.999184i \(0.487137\pi\)
\(74\) −1.24860 0.866833i −0.145147 0.100767i
\(75\) 0 0
\(76\) −3.24860 + 1.21201i −0.372640 + 0.139027i
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 2.64681 0.297789 0.148895 0.988853i \(-0.452428\pi\)
0.148895 + 0.988853i \(0.452428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.10834 13.1198i 1.00585 1.44884i
\(83\) 5.85039i 0.642164i −0.947051 0.321082i \(-0.895953\pi\)
0.947051 0.321082i \(-0.104047\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.64681 6.00299i −0.932409 0.647319i
\(87\) 0 0
\(88\) 3.74767 14.6918i 0.399503 1.56615i
\(89\) −7.59283 −0.804838 −0.402419 0.915456i \(-0.631831\pi\)
−0.402419 + 0.915456i \(0.631831\pi\)
\(90\) 0 0
\(91\) 2.18271i 0.228810i
\(92\) −5.24002 14.0450i −0.546310 1.46429i
\(93\) 0 0
\(94\) 1.39821 2.01400i 0.144214 0.207729i
\(95\) 0 0
\(96\) 0 0
\(97\) 14.1887 1.44064 0.720321 0.693641i \(-0.243994\pi\)
0.720321 + 0.693641i \(0.243994\pi\)
\(98\) −5.19648 + 7.48511i −0.524924 + 0.756110i
\(99\) 0 0
\(100\) 0 0
\(101\) 7.43952i 0.740260i −0.928980 0.370130i \(-0.879313\pi\)
0.928980 0.370130i \(-0.120687\pi\)
\(102\) 0 0
\(103\) −7.19820 −0.709260 −0.354630 0.935007i \(-0.615393\pi\)
−0.354630 + 0.935007i \(0.615393\pi\)
\(104\) 2.04502 8.01699i 0.200530 0.786131i
\(105\) 0 0
\(106\) 8.97021 + 6.22751i 0.871264 + 0.604869i
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 19.9504i 1.91090i 0.295158 + 0.955449i \(0.404628\pi\)
−0.295158 + 0.955449i \(0.595372\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.25532 1.95498i 0.213108 0.184729i
\(113\) 12.0540 1.13395 0.566973 0.823736i \(-0.308114\pi\)
0.566973 + 0.823736i \(0.308114\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.6468 4.71836i 1.17423 0.438089i
\(117\) 0 0
\(118\) 7.96117 + 5.52699i 0.732885 + 0.508801i
\(119\) −1.59283 −0.146014
\(120\) 0 0
\(121\) −17.7368 −1.61244
\(122\) 7.49534 + 5.20359i 0.678596 + 0.471110i
\(123\) 0 0
\(124\) −1.85039 4.95968i −0.166170 0.445392i
\(125\) 0 0
\(126\) 0 0
\(127\) 4.21351 0.373888 0.186944 0.982371i \(-0.440142\pi\)
0.186944 + 0.982371i \(0.440142\pi\)
\(128\) 10.1153 5.06752i 0.894079 0.447910i
\(129\) 0 0
\(130\) 0 0
\(131\) 10.3204i 0.901694i 0.892601 + 0.450847i \(0.148878\pi\)
−0.892601 + 0.450847i \(0.851122\pi\)
\(132\) 0 0
\(133\) 1.29362i 0.112171i
\(134\) −8.64681 6.00299i −0.746970 0.518579i
\(135\) 0 0
\(136\) −5.85039 1.49235i −0.501667 0.127968i
\(137\) 15.0387 1.28484 0.642422 0.766351i \(-0.277930\pi\)
0.642422 + 0.766351i \(0.277930\pi\)
\(138\) 0 0
\(139\) 9.47032i 0.803262i 0.915802 + 0.401631i \(0.131557\pi\)
−0.915802 + 0.401631i \(0.868443\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.7214 + 15.4432i −0.899716 + 1.29597i
\(143\) −15.6810 −1.31131
\(144\) 0 0
\(145\) 0 0
\(146\) 0.556777 0.801991i 0.0460792 0.0663732i
\(147\) 0 0
\(148\) −2.01400 + 0.751399i −0.165550 + 0.0617646i
\(149\) 1.78948i 0.146600i 0.997310 + 0.0733000i \(0.0233531\pi\)
−0.997310 + 0.0733000i \(0.976647\pi\)
\(150\) 0 0
\(151\) 10.6468 0.866425 0.433212 0.901292i \(-0.357380\pi\)
0.433212 + 0.901292i \(0.357380\pi\)
\(152\) −1.21201 + 4.75140i −0.0983071 + 0.385389i
\(153\) 0 0
\(154\) −4.64681 3.22601i −0.374451 0.259960i
\(155\) 0 0
\(156\) 0 0
\(157\) 6.92520i 0.552691i 0.961058 + 0.276345i \(0.0891234\pi\)
−0.961058 + 0.276345i \(0.910877\pi\)
\(158\) 2.13466 3.07480i 0.169824 0.244618i
\(159\) 0 0
\(160\) 0 0
\(161\) −5.59283 −0.440777
\(162\) 0 0
\(163\) 7.70079i 0.603172i −0.953439 0.301586i \(-0.902484\pi\)
0.953439 0.301586i \(-0.0975161\pi\)
\(164\) −7.89541 21.1624i −0.616528 1.65250i
\(165\) 0 0
\(166\) −6.79641 4.71836i −0.527504 0.366216i
\(167\) −3.22601 −0.249637 −0.124818 0.992180i \(-0.539835\pi\)
−0.124818 + 0.992180i \(0.539835\pi\)
\(168\) 0 0
\(169\) 4.44322 0.341786
\(170\) 0 0
\(171\) 0 0
\(172\) −13.9474 + 5.20359i −1.06348 + 0.396770i
\(173\) 6.42799i 0.488711i −0.969686 0.244356i \(-0.921424\pi\)
0.969686 0.244356i \(-0.0785764\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −14.0450 16.2027i −1.05868 1.22132i
\(177\) 0 0
\(178\) −6.12364 + 8.82061i −0.458987 + 0.661132i
\(179\) 8.13765i 0.608236i 0.952634 + 0.304118i \(0.0983618\pi\)
−0.952634 + 0.304118i \(0.901638\pi\)
\(180\) 0 0
\(181\) 1.49235i 0.110925i 0.998461 + 0.0554627i \(0.0176634\pi\)
−0.998461 + 0.0554627i \(0.982337\pi\)
\(182\) −2.53566 1.76036i −0.187955 0.130487i
\(183\) 0 0
\(184\) −20.5422 5.24002i −1.51439 0.386299i
\(185\) 0 0
\(186\) 0 0
\(187\) 11.4432i 0.836811i
\(188\) −1.21201 3.24860i −0.0883951 0.236929i
\(189\) 0 0
\(190\) 0 0
\(191\) −6.88645 −0.498286 −0.249143 0.968467i \(-0.580149\pi\)
−0.249143 + 0.968467i \(0.580149\pi\)
\(192\) 0 0
\(193\) 16.4830 1.18647 0.593237 0.805028i \(-0.297850\pi\)
0.593237 + 0.805028i \(0.297850\pi\)
\(194\) 11.4432 16.4830i 0.821576 1.18341i
\(195\) 0 0
\(196\) 4.50448 + 12.0735i 0.321749 + 0.862395i
\(197\) 13.5720i 0.966965i 0.875354 + 0.483483i \(0.160628\pi\)
−0.875354 + 0.483483i \(0.839372\pi\)
\(198\) 0 0
\(199\) 9.05398 0.641820 0.320910 0.947110i \(-0.396011\pi\)
0.320910 + 0.947110i \(0.396011\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.64251 6.00000i −0.608085 0.422159i
\(203\) 5.03605i 0.353462i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.80538 + 8.36217i −0.404480 + 0.582620i
\(207\) 0 0
\(208\) −7.66404 8.84143i −0.531406 0.613043i
\(209\) 9.29362 0.642853
\(210\) 0 0
\(211\) 2.53566i 0.174562i −0.996184 0.0872809i \(-0.972182\pi\)
0.996184 0.0872809i \(-0.0278178\pi\)
\(212\) 14.4690 5.39821i 0.993736 0.370750i
\(213\) 0 0
\(214\) 4.64681 + 3.22601i 0.317649 + 0.220526i
\(215\) 0 0
\(216\) 0 0
\(217\) −1.97498 −0.134070
\(218\) 23.1764 + 16.0900i 1.56970 + 1.08975i
\(219\) 0 0
\(220\) 0 0
\(221\) 6.24430i 0.420037i
\(222\) 0 0
\(223\) −12.1579 −0.814152 −0.407076 0.913394i \(-0.633452\pi\)
−0.407076 + 0.913394i \(0.633452\pi\)
\(224\) −0.452186 4.19671i −0.0302130 0.280405i
\(225\) 0 0
\(226\) 9.72161 14.0032i 0.646672 0.931478i
\(227\) 20.7368i 1.37635i −0.725544 0.688176i \(-0.758412\pi\)
0.725544 0.688176i \(-0.241588\pi\)
\(228\) 0 0
\(229\) 19.9504i 1.31836i −0.751987 0.659178i \(-0.770904\pi\)
0.751987 0.659178i \(-0.229096\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.71836 18.4972i 0.309776 1.21440i
\(233\) −13.3386 −0.873844 −0.436922 0.899499i \(-0.643931\pi\)
−0.436922 + 0.899499i \(0.643931\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.8414 4.79097i 0.835906 0.311866i
\(237\) 0 0
\(238\) −1.28462 + 1.85039i −0.0832697 + 0.119943i
\(239\) −22.8864 −1.48040 −0.740201 0.672386i \(-0.765269\pi\)
−0.740201 + 0.672386i \(0.765269\pi\)
\(240\) 0 0
\(241\) 3.59283 0.231435 0.115717 0.993282i \(-0.463083\pi\)
0.115717 + 0.993282i \(0.463083\pi\)
\(242\) −14.3048 + 20.6049i −0.919549 + 1.32453i
\(243\) 0 0
\(244\) 12.0900 4.51064i 0.773985 0.288764i
\(245\) 0 0
\(246\) 0 0
\(247\) 5.07131 0.322680
\(248\) −7.25402 1.85039i −0.460631 0.117500i
\(249\) 0 0
\(250\) 0 0
\(251\) 8.82801i 0.557219i −0.960404 0.278609i \(-0.910127\pi\)
0.960404 0.278609i \(-0.0898735\pi\)
\(252\) 0 0
\(253\) 40.1801i 2.52610i
\(254\) 3.39821 4.89484i 0.213222 0.307129i
\(255\) 0 0
\(256\) 2.27111 15.8380i 0.141944 0.989875i
\(257\) −22.2927 −1.39058 −0.695291 0.718728i \(-0.744725\pi\)
−0.695291 + 0.718728i \(0.744725\pi\)
\(258\) 0 0
\(259\) 0.801991i 0.0498333i
\(260\) 0 0
\(261\) 0 0
\(262\) 11.9892 + 8.32340i 0.740694 + 0.514222i
\(263\) −21.2014 −1.30733 −0.653667 0.756783i \(-0.726770\pi\)
−0.653667 + 0.756783i \(0.726770\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.50280 + 1.04331i 0.0921424 + 0.0639693i
\(267\) 0 0
\(268\) −13.9474 + 5.20359i −0.851971 + 0.317860i
\(269\) 14.6935i 0.895881i 0.894063 + 0.447940i \(0.147842\pi\)
−0.894063 + 0.447940i \(0.852158\pi\)
\(270\) 0 0
\(271\) −20.2396 −1.22947 −0.614735 0.788734i \(-0.710737\pi\)
−0.614735 + 0.788734i \(0.710737\pi\)
\(272\) −6.45203 + 5.59283i −0.391212 + 0.339115i
\(273\) 0 0
\(274\) 12.1288 17.4705i 0.732727 1.05543i
\(275\) 0 0
\(276\) 0 0
\(277\) 0.518027i 0.0311252i 0.999879 + 0.0155626i \(0.00495393\pi\)
−0.999879 + 0.0155626i \(0.995046\pi\)
\(278\) 11.0017 + 7.63785i 0.659837 + 0.458088i
\(279\) 0 0
\(280\) 0 0
\(281\) −13.7008 −0.817320 −0.408660 0.912687i \(-0.634004\pi\)
−0.408660 + 0.912687i \(0.634004\pi\)
\(282\) 0 0
\(283\) 18.0305i 1.07180i −0.844282 0.535900i \(-0.819973\pi\)
0.844282 0.535900i \(-0.180027\pi\)
\(284\) 9.29362 + 24.9100i 0.551475 + 1.47814i
\(285\) 0 0
\(286\) −12.6468 + 18.2167i −0.747821 + 1.07718i
\(287\) −8.42701 −0.497431
\(288\) 0 0
\(289\) −12.4432 −0.731954
\(290\) 0 0
\(291\) 0 0
\(292\) −0.482632 1.29362i −0.0282439 0.0757032i
\(293\) 15.9792i 0.933513i 0.884386 + 0.466757i \(0.154578\pi\)
−0.884386 + 0.466757i \(0.845422\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.751399 + 2.94568i −0.0436742 + 0.171214i
\(297\) 0 0
\(298\) 2.07884 + 1.44322i 0.120424 + 0.0836037i
\(299\) 21.9253i 1.26797i
\(300\) 0 0
\(301\) 5.55394i 0.320124i
\(302\) 8.58669 12.3684i 0.494108 0.711723i
\(303\) 0 0
\(304\) 4.54222 + 5.24002i 0.260514 + 0.300536i
\(305\) 0 0
\(306\) 0 0
\(307\) 22.5872i 1.28912i −0.764553 0.644561i \(-0.777040\pi\)
0.764553 0.644561i \(-0.222960\pi\)
\(308\) −7.49534 + 2.79641i −0.427086 + 0.159341i
\(309\) 0 0
\(310\) 0 0
\(311\) 18.5872 1.05399 0.526993 0.849870i \(-0.323320\pi\)
0.526993 + 0.849870i \(0.323320\pi\)
\(312\) 0 0
\(313\) 29.3871 1.66106 0.830528 0.556977i \(-0.188039\pi\)
0.830528 + 0.556977i \(0.188039\pi\)
\(314\) 8.04502 + 5.58520i 0.454007 + 0.315191i
\(315\) 0 0
\(316\) −1.85039 4.95968i −0.104093 0.279004i
\(317\) 5.57201i 0.312955i 0.987682 + 0.156478i \(0.0500139\pi\)
−0.987682 + 0.156478i \(0.949986\pi\)
\(318\) 0 0
\(319\) −36.1801 −2.02569
\(320\) 0 0
\(321\) 0 0
\(322\) −4.51064 + 6.49720i −0.251368 + 0.362075i
\(323\) 3.70079i 0.205917i
\(324\) 0 0
\(325\) 0 0
\(326\) −8.94602 6.21071i −0.495474 0.343980i
\(327\) 0 0
\(328\) −30.9520 7.89541i −1.70904 0.435951i
\(329\) −1.29362 −0.0713194
\(330\) 0 0
\(331\) 13.7396i 0.755199i −0.925969 0.377599i \(-0.876750\pi\)
0.925969 0.377599i \(-0.123250\pi\)
\(332\) −10.9627 + 4.09003i −0.601655 + 0.224470i
\(333\) 0 0
\(334\) −2.60179 + 3.74767i −0.142364 + 0.205063i
\(335\) 0 0
\(336\) 0 0
\(337\) −20.7523 −1.13045 −0.565226 0.824936i \(-0.691211\pi\)
−0.565226 + 0.824936i \(0.691211\pi\)
\(338\) 3.58348 5.16170i 0.194915 0.280760i
\(339\) 0 0
\(340\) 0 0
\(341\) 14.1887i 0.768360i
\(342\) 0 0
\(343\) 10.0310 0.541623
\(344\) −5.20359 + 20.3994i −0.280559 + 1.09986i
\(345\) 0 0
\(346\) −7.46742 5.18420i −0.401451 0.278704i
\(347\) 4.73684i 0.254287i 0.991884 + 0.127143i \(0.0405809\pi\)
−0.991884 + 0.127143i \(0.959419\pi\)
\(348\) 0 0
\(349\) 0.482632i 0.0258347i −0.999917 0.0129174i \(-0.995888\pi\)
0.999917 0.0129174i \(-0.00411184\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −30.1500 + 3.24860i −1.60700 + 0.173151i
\(353\) −2.13466 −0.113617 −0.0568083 0.998385i \(-0.518092\pi\)
−0.0568083 + 0.998385i \(0.518092\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.30818 + 14.2277i 0.281333 + 0.754067i
\(357\) 0 0
\(358\) 9.45352 + 6.56304i 0.499634 + 0.346868i
\(359\) 9.59283 0.506290 0.253145 0.967428i \(-0.418535\pi\)
0.253145 + 0.967428i \(0.418535\pi\)
\(360\) 0 0
\(361\) 15.9944 0.841811
\(362\) 1.73367 + 1.20359i 0.0911194 + 0.0632590i
\(363\) 0 0
\(364\) −4.09003 + 1.52594i −0.214376 + 0.0799809i
\(365\) 0 0
\(366\) 0 0
\(367\) −34.0832 −1.77913 −0.889565 0.456809i \(-0.848992\pi\)
−0.889565 + 0.456809i \(0.848992\pi\)
\(368\) −22.6547 + 19.6378i −1.18096 + 1.02369i
\(369\) 0 0
\(370\) 0 0
\(371\) 5.76167i 0.299131i
\(372\) 0 0
\(373\) 4.33796i 0.224611i −0.993674 0.112306i \(-0.964176\pi\)
0.993674 0.112306i \(-0.0358236\pi\)
\(374\) 13.2936 + 9.22900i 0.687397 + 0.477220i
\(375\) 0 0
\(376\) −4.75140 1.21201i −0.245035 0.0625047i
\(377\) −19.7426 −1.01680
\(378\) 0 0
\(379\) 6.90107i 0.354484i 0.984167 + 0.177242i \(0.0567176\pi\)
−0.984167 + 0.177242i \(0.943282\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5.55394 + 8.00000i −0.284165 + 0.409316i
\(383\) 22.3744 1.14328 0.571639 0.820506i \(-0.306308\pi\)
0.571639 + 0.820506i \(0.306308\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 13.2936 19.1484i 0.676627 0.974626i
\(387\) 0 0
\(388\) −9.91936 26.5872i −0.503579 1.34976i
\(389\) 11.0185i 0.558659i −0.960195 0.279330i \(-0.909888\pi\)
0.960195 0.279330i \(-0.0901122\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 17.6587 + 4.50448i 0.891900 + 0.227511i
\(393\) 0 0
\(394\) 15.7666 + 10.9459i 0.794311 + 0.551445i
\(395\) 0 0
\(396\) 0 0
\(397\) 25.2549i 1.26751i 0.773536 + 0.633753i \(0.218486\pi\)
−0.773536 + 0.633753i \(0.781514\pi\)
\(398\) 7.30207 10.5180i 0.366020 0.527221i
\(399\) 0 0
\(400\) 0 0
\(401\) −7.29362 −0.364226 −0.182113 0.983278i \(-0.558294\pi\)
−0.182113 + 0.983278i \(0.558294\pi\)
\(402\) 0 0
\(403\) 7.74244i 0.385678i
\(404\) −13.9404 + 5.20100i −0.693562 + 0.258759i
\(405\) 0 0
\(406\) −5.85039 4.06160i −0.290350 0.201574i
\(407\) 5.76167 0.285595
\(408\) 0 0
\(409\) 15.8504 0.783752 0.391876 0.920018i \(-0.371826\pi\)
0.391876 + 0.920018i \(0.371826\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5.03229 + 13.4882i 0.247923 + 0.664518i
\(413\) 5.11355i 0.251622i
\(414\) 0 0
\(415\) 0 0
\(416\) −16.4522 + 1.77269i −0.806635 + 0.0869131i
\(417\) 0 0
\(418\) 7.49534 10.7964i 0.366609 0.528070i
\(419\) 8.02602i 0.392097i −0.980594 0.196048i \(-0.937189\pi\)
0.980594 0.196048i \(-0.0628109\pi\)
\(420\) 0 0
\(421\) 22.9351i 1.11779i −0.829240 0.558893i \(-0.811226\pi\)
0.829240 0.558893i \(-0.188774\pi\)
\(422\) −2.94568 2.04502i −0.143393 0.0995498i
\(423\) 0 0
\(424\) 5.39821 21.1624i 0.262160 1.02774i
\(425\) 0 0
\(426\) 0 0
\(427\) 4.81434i 0.232982i
\(428\) 7.49534 2.79641i 0.362301 0.135170i
\(429\) 0 0
\(430\) 0 0
\(431\) 35.0665 1.68909 0.844547 0.535481i \(-0.179870\pi\)
0.844547 + 0.535481i \(0.179870\pi\)
\(432\) 0 0
\(433\) −17.0773 −0.820682 −0.410341 0.911932i \(-0.634590\pi\)
−0.410341 + 0.911932i \(0.634590\pi\)
\(434\) −1.59283 + 2.29434i −0.0764583 + 0.110132i
\(435\) 0 0
\(436\) 37.3836 13.9474i 1.79035 0.667958i
\(437\) 12.9944i 0.621607i
\(438\) 0 0
\(439\) 8.53885 0.407537 0.203769 0.979019i \(-0.434681\pi\)
0.203769 + 0.979019i \(0.434681\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.25402 + 5.03605i 0.345039 + 0.239541i
\(443\) 20.7368i 0.985237i −0.870245 0.492619i \(-0.836040\pi\)
0.870245 0.492619i \(-0.163960\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −9.80538 + 14.1238i −0.464298 + 0.668783i
\(447\) 0 0
\(448\) −5.24002 2.85936i −0.247568 0.135092i
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 60.5414i 2.85078i
\(452\) −8.42701 22.5872i −0.396373 1.06241i
\(453\) 0 0
\(454\) −24.0900 16.7243i −1.13060 0.784912i
\(455\) 0 0
\(456\) 0 0
\(457\) −1.28462 −0.0600921 −0.0300461 0.999549i \(-0.509565\pi\)
−0.0300461 + 0.999549i \(0.509565\pi\)
\(458\) −23.1764 16.0900i −1.08296 0.751838i
\(459\) 0 0
\(460\) 0 0
\(461\) 15.7033i 0.731374i 0.930738 + 0.365687i \(0.119166\pi\)
−0.930738 + 0.365687i \(0.880834\pi\)
\(462\) 0 0
\(463\) −18.7215 −0.870064 −0.435032 0.900415i \(-0.643263\pi\)
−0.435032 + 0.900415i \(0.643263\pi\)
\(464\) −17.6829 20.3994i −0.820906 0.947018i
\(465\) 0 0
\(466\) −10.7577 + 15.4955i −0.498339 + 0.717817i
\(467\) 2.14961i 0.0994719i 0.998762 + 0.0497360i \(0.0158380\pi\)
−0.998762 + 0.0497360i \(0.984162\pi\)
\(468\) 0 0
\(469\) 5.55394i 0.256457i
\(470\) 0 0
\(471\) 0 0
\(472\) 4.79097 18.7819i 0.220522 0.864505i
\(473\) 39.9007 1.83464
\(474\) 0 0
\(475\) 0 0
\(476\) 1.11355 + 2.98470i 0.0510396 + 0.136803i
\(477\) 0 0
\(478\) −18.4580 + 26.5872i −0.844249 + 1.21607i
\(479\) 12.1801 0.556521 0.278261 0.960506i \(-0.410242\pi\)
0.278261 + 0.960506i \(0.410242\pi\)
\(480\) 0 0
\(481\) 3.14401 0.143355
\(482\) 2.89763 4.17380i 0.131983 0.190111i
\(483\) 0 0
\(484\) 12.3999 + 33.2359i 0.563631 + 1.51072i
\(485\) 0 0
\(486\) 0 0
\(487\) −25.7678 −1.16765 −0.583826 0.811879i \(-0.698445\pi\)
−0.583826 + 0.811879i \(0.698445\pi\)
\(488\) 4.51064 17.6829i 0.204187 0.800466i
\(489\) 0 0
\(490\) 0 0
\(491\) 16.7724i 0.756927i −0.925616 0.378464i \(-0.876453\pi\)
0.925616 0.378464i \(-0.123547\pi\)
\(492\) 0 0
\(493\) 14.4072i 0.648866i
\(494\) 4.09003 5.89135i 0.184019 0.265065i
\(495\) 0 0
\(496\) −8.00000 + 6.93466i −0.359211 + 0.311375i
\(497\) 9.91936 0.444944
\(498\) 0 0
\(499\) 17.6224i 0.788888i −0.918920 0.394444i \(-0.870937\pi\)
0.918920 0.394444i \(-0.129063\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −10.2555 7.11982i −0.457726 0.317773i
\(503\) −27.1263 −1.20950 −0.604752 0.796414i \(-0.706728\pi\)
−0.604752 + 0.796414i \(0.706728\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 46.6773 + 32.4054i 2.07506 + 1.44059i
\(507\) 0 0
\(508\) −2.94568 7.89541i −0.130693 0.350302i
\(509\) 15.9782i 0.708220i −0.935204 0.354110i \(-0.884784\pi\)
0.935204 0.354110i \(-0.115216\pi\)
\(510\) 0 0
\(511\) −0.515128 −0.0227879
\(512\) −16.5674 15.4118i −0.732181 0.681110i
\(513\) 0 0
\(514\) −17.9792 + 25.8975i −0.793027 + 1.14229i
\(515\) 0 0
\(516\) 0 0
\(517\) 9.29362i 0.408733i
\(518\) 0.931674 + 0.646809i 0.0409354 + 0.0284191i
\(519\) 0 0
\(520\) 0 0
\(521\) −0.886447 −0.0388359 −0.0194180 0.999811i \(-0.506181\pi\)
−0.0194180 + 0.999811i \(0.506181\pi\)
\(522\) 0 0
\(523\) 41.7729i 1.82660i −0.407286 0.913301i \(-0.633525\pi\)
0.407286 0.913301i \(-0.366475\pi\)
\(524\) 19.3386 7.21500i 0.844812 0.315189i
\(525\) 0 0
\(526\) −17.0990 + 24.6297i −0.745552 + 1.07391i
\(527\) 5.65004 0.246120
\(528\) 0 0
\(529\) 33.1801 1.44261
\(530\) 0 0
\(531\) 0 0
\(532\) 2.42402 0.904373i 0.105095 0.0392095i
\(533\) 33.0361i 1.43095i
\(534\) 0 0
\(535\) 0 0
\(536\) −5.20359 + 20.3994i −0.224761 + 0.881120i
\(537\) 0 0
\(538\) 17.0695 + 11.8504i 0.735919 + 0.510907i
\(539\) 34.5400i 1.48774i
\(540\) 0 0
\(541\) 4.47705i 0.192483i 0.995358 + 0.0962417i \(0.0306822\pi\)
−0.995358 + 0.0962417i \(0.969318\pi\)
\(542\) −16.3233 + 23.5124i −0.701148 + 1.00995i
\(543\) 0 0
\(544\) 1.29362 + 12.0060i 0.0554634 + 0.514752i
\(545\) 0 0
\(546\) 0 0
\(547\) 14.3297i 0.612692i −0.951920 0.306346i \(-0.900893\pi\)
0.951920 0.306346i \(-0.0991065\pi\)
\(548\) −10.5136 28.1801i −0.449120 1.20379i
\(549\) 0 0
\(550\) 0 0
\(551\) 11.7008 0.498470
\(552\) 0 0
\(553\) −1.97498 −0.0839848
\(554\) 0.601793 + 0.417790i 0.0255677 + 0.0177502i
\(555\) 0 0
\(556\) 17.7458 6.62073i 0.752590 0.280782i
\(557\) 2.68556i 0.113791i −0.998380 0.0568954i \(-0.981880\pi\)
0.998380 0.0568954i \(-0.0181201\pi\)
\(558\) 0 0
\(559\) 21.7729 0.920895
\(560\) 0 0
\(561\) 0 0
\(562\) −11.0497 + 15.9162i −0.466105 + 0.671386i
\(563\) 20.7368i 0.873954i 0.899473 + 0.436977i \(0.143951\pi\)
−0.899473 + 0.436977i \(0.856049\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20.9460 14.5416i −0.880427 0.611230i
\(567\) 0 0
\(568\) 36.4334 + 9.29362i 1.52871 + 0.389952i
\(569\) −4.40717 −0.184758 −0.0923791 0.995724i \(-0.529447\pi\)
−0.0923791 + 0.995724i \(0.529447\pi\)
\(570\) 0 0
\(571\) 23.6590i 0.990098i −0.868865 0.495049i \(-0.835150\pi\)
0.868865 0.495049i \(-0.164850\pi\)
\(572\) 10.9627 + 29.3836i 0.458372 + 1.22859i
\(573\) 0 0
\(574\) −6.79641 + 9.78968i −0.283677 + 0.408613i
\(575\) 0 0
\(576\) 0 0
\(577\) 6.56366 0.273249 0.136624 0.990623i \(-0.456375\pi\)
0.136624 + 0.990623i \(0.456375\pi\)
\(578\) −10.0355 + 14.4553i −0.417422 + 0.601262i
\(579\) 0 0
\(580\) 0 0
\(581\) 4.36542i 0.181108i
\(582\) 0 0
\(583\) −41.3931 −1.71433
\(584\) −1.89204 0.482632i −0.0782933 0.0199715i
\(585\) 0 0
\(586\) 18.5630 + 12.8873i 0.766832 + 0.532368i
\(587\) 16.2992i 0.672741i 0.941730 + 0.336370i \(0.109199\pi\)
−0.941730 + 0.336370i \(0.890801\pi\)
\(588\) 0 0
\(589\) 4.58868i 0.189073i
\(590\) 0 0
\(591\) 0 0
\(592\) 2.81599 + 3.24860i 0.115737 + 0.133517i
\(593\) −16.3233 −0.670319 −0.335160 0.942161i \(-0.608790\pi\)
−0.335160 + 0.942161i \(0.608790\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.35319 1.25103i 0.137352 0.0512443i
\(597\) 0 0
\(598\) 25.4707 + 17.6829i 1.04157 + 0.723106i
\(599\) 25.5928 1.04569 0.522847 0.852426i \(-0.324870\pi\)
0.522847 + 0.852426i \(0.324870\pi\)
\(600\) 0 0
\(601\) 29.9225 1.22056 0.610282 0.792184i \(-0.291056\pi\)
0.610282 + 0.792184i \(0.291056\pi\)
\(602\) 6.45203 + 4.47928i 0.262965 + 0.182562i
\(603\) 0 0
\(604\) −7.44322 19.9504i −0.302860 0.811768i
\(605\) 0 0
\(606\) 0 0
\(607\) 20.6965 0.840046 0.420023 0.907513i \(-0.362022\pi\)
0.420023 + 0.907513i \(0.362022\pi\)
\(608\) 9.75065 1.05061i 0.395441 0.0426079i
\(609\) 0 0
\(610\) 0 0
\(611\) 5.07131i 0.205163i
\(612\) 0 0
\(613\) 22.6676i 0.915537i −0.889071 0.457769i \(-0.848649\pi\)
0.889071 0.457769i \(-0.151351\pi\)
\(614\) −26.2396 18.2167i −1.05895 0.735166i
\(615\) 0 0
\(616\) −2.79641 + 10.9627i −0.112671 + 0.441698i
\(617\) 22.1966 0.893603 0.446802 0.894633i \(-0.352563\pi\)
0.446802 + 0.894633i \(0.352563\pi\)
\(618\) 0 0
\(619\) 16.8204i 0.676070i 0.941133 + 0.338035i \(0.109762\pi\)
−0.941133 + 0.338035i \(0.890238\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 14.9907 21.5928i 0.601071 0.865794i
\(623\) 5.66558 0.226987
\(624\) 0 0
\(625\) 0 0
\(626\) 23.7008 34.1390i 0.947274 1.36447i
\(627\) 0 0
\(628\) 12.9767 4.84143i 0.517825 0.193194i
\(629\) 2.29434i 0.0914813i
\(630\) 0 0
\(631\) −44.1205 −1.75641 −0.878204 0.478285i \(-0.841258\pi\)
−0.878204 + 0.478285i \(0.841258\pi\)
\(632\) −7.25402 1.85039i −0.288549 0.0736047i
\(633\) 0 0
\(634\) 6.47301 + 4.49384i 0.257076 + 0.178473i
\(635\) 0 0
\(636\) 0 0
\(637\) 18.8477i 0.746773i
\(638\) −29.1794 + 42.0305i −1.15522 + 1.66400i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.18566 0.0468307 0.0234154 0.999726i \(-0.492546\pi\)
0.0234154 + 0.999726i \(0.492546\pi\)
\(642\) 0 0
\(643\) 22.5872i 0.890754i 0.895343 + 0.445377i \(0.146930\pi\)
−0.895343 + 0.445377i \(0.853070\pi\)
\(644\) 3.90997 + 10.4800i 0.154074 + 0.412971i
\(645\) 0 0
\(646\) −4.29921 2.98470i −0.169150 0.117431i
\(647\) −19.7090 −0.774842 −0.387421 0.921903i \(-0.626634\pi\)
−0.387421 + 0.921903i \(0.626634\pi\)
\(648\) 0 0
\(649\) −36.7368 −1.44205
\(650\) 0 0
\(651\) 0 0
\(652\) −14.4300 + 5.38365i −0.565122 + 0.210840i
\(653\) 44.4585i 1.73979i −0.493234 0.869897i \(-0.664185\pi\)
0.493234 0.869897i \(-0.335815\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −34.1350 + 29.5894i −1.33275 + 1.15527i
\(657\) 0 0
\(658\) −1.04331 + 1.50280i −0.0406723 + 0.0585852i
\(659\) 41.5863i 1.61997i −0.586448 0.809987i \(-0.699474\pi\)
0.586448 0.809987i \(-0.300526\pi\)
\(660\) 0 0
\(661\) 12.0060i 0.466978i 0.972359 + 0.233489i \(0.0750143\pi\)
−0.972359 + 0.233489i \(0.924986\pi\)
\(662\) −15.9614 11.0811i −0.620356 0.430678i
\(663\) 0 0
\(664\) −4.09003 + 16.0340i −0.158724 + 0.622239i
\(665\) 0 0
\(666\) 0 0
\(667\) 50.5872i 1.95875i
\(668\) 2.25532 + 6.04502i 0.0872609 + 0.233889i
\(669\) 0 0
\(670\) 0 0
\(671\) −34.5872 −1.33523
\(672\) 0 0
\(673\) −14.5080 −0.559244 −0.279622 0.960110i \(-0.590209\pi\)
−0.279622 + 0.960110i \(0.590209\pi\)
\(674\) −16.7368 + 24.1080i −0.644679 + 0.928607i
\(675\) 0 0
\(676\) −3.10627 8.32586i −0.119472 0.320226i
\(677\) 43.8600i 1.68568i 0.538166 + 0.842839i \(0.319117\pi\)
−0.538166 + 0.842839i \(0.680883\pi\)
\(678\) 0 0
\(679\) −10.5872 −0.406301
\(680\) 0 0
\(681\) 0 0
\(682\) 16.4830 + 11.4432i 0.631168 + 0.438184i
\(683\) 5.33527i 0.204148i 0.994777 + 0.102074i \(0.0325479\pi\)
−0.994777 + 0.102074i \(0.967452\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.09003 11.6530i 0.308879 0.444915i
\(687\) 0 0
\(688\) 19.5013 + 22.4972i 0.743480 + 0.857698i
\(689\) −22.5872 −0.860505
\(690\) 0 0
\(691\) 39.7710i 1.51296i −0.654016 0.756480i \(-0.726917\pi\)
0.654016 0.756480i \(-0.273083\pi\)
\(692\) −12.0450 + 4.49383i −0.457882 + 0.170830i
\(693\) 0 0
\(694\) 5.50280 + 3.82028i 0.208883 + 0.145016i
\(695\) 0 0
\(696\) 0 0
\(697\) 24.1080 0.913157
\(698\) −0.560675 0.389245i −0.0212219 0.0147331i
\(699\) 0 0
\(700\) 0 0
\(701\) 27.5015i 1.03872i −0.854556 0.519359i \(-0.826171\pi\)
0.854556 0.519359i \(-0.173829\pi\)
\(702\) 0 0
\(703\) −1.86335 −0.0702775
\(704\) −20.5422 + 37.6454i −0.774214 + 1.41881i
\(705\) 0 0
\(706\) −1.72161 + 2.47984i −0.0647937 + 0.0933300i
\(707\) 5.55118i 0.208774i
\(708\) 0 0
\(709\) 0.111632i 0.00419244i −0.999998 0.00209622i \(-0.999333\pi\)
0.999998 0.00209622i \(-0.000667249\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 20.8094 + 5.30818i 0.779866 + 0.198932i
\(713\) 19.8387 0.742966
\(714\) 0 0
\(715\) 0 0
\(716\) 15.2486 5.68906i 0.569867 0.212610i
\(717\) 0 0
\(718\) 7.73665 11.1440i 0.288729 0.415891i
\(719\) −10.7064 −0.399281 −0.199640 0.979869i \(-0.563977\pi\)
−0.199640 + 0.979869i \(0.563977\pi\)
\(720\) 0 0
\(721\) 5.37112 0.200031
\(722\) 12.8995 18.5807i 0.480071 0.691503i
\(723\) 0 0
\(724\) 2.79641 1.04331i 0.103928 0.0387742i
\(725\) 0 0
\(726\) 0 0
\(727\) 25.6562 0.951536 0.475768 0.879571i \(-0.342170\pi\)
0.475768 + 0.879571i \(0.342170\pi\)
\(728\) −1.52594 + 5.98207i −0.0565551 + 0.221710i
\(729\) 0 0
\(730\) 0 0
\(731\) 15.8888i 0.587667i
\(732\) 0 0
\(733\) 30.3684i 1.12168i 0.827923 + 0.560842i \(0.189522\pi\)
−0.827923 + 0.560842i \(0.810478\pi\)
\(734\) −27.4882 + 39.5945i −1.01461 + 1.46146i
\(735\) 0 0
\(736\) 4.54222 + 42.1560i 0.167428 + 1.55389i
\(737\) 39.9007 1.46976
\(738\) 0 0
\(739\) 20.1917i 0.742763i −0.928480 0.371381i \(-0.878884\pi\)
0.928480 0.371381i \(-0.121116\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.69335 4.64681i −0.245720 0.170590i
\(743\) 46.3863 1.70175 0.850875 0.525369i \(-0.176073\pi\)
0.850875 + 0.525369i \(0.176073\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.03942 3.49858i −0.184506 0.128092i
\(747\) 0 0
\(748\) 21.4427 8.00000i 0.784023 0.292509i
\(749\) 2.98470i 0.109059i
\(750\) 0 0
\(751\) −27.1261 −0.989845 −0.494922 0.868937i \(-0.664804\pi\)
−0.494922 + 0.868937i \(0.664804\pi\)
\(752\) −5.24002 + 4.54222i −0.191084 + 0.165638i
\(753\) 0 0
\(754\) −15.9225 + 22.9351i −0.579863 + 0.835245i
\(755\) 0 0
\(756\) 0 0
\(757\) 45.2549i 1.64482i −0.568898 0.822408i \(-0.692630\pi\)
0.568898 0.822408i \(-0.307370\pi\)
\(758\) 8.01699 + 5.56574i 0.291190 + 0.202157i
\(759\) 0 0
\(760\) 0 0
\(761\) −16.8864 −0.612133 −0.306067 0.952010i \(-0.599013\pi\)
−0.306067 + 0.952010i \(0.599013\pi\)
\(762\) 0 0
\(763\) 14.8864i 0.538926i
\(764\) 4.81434 + 12.9041i 0.174177 + 0.466852i
\(765\) 0 0
\(766\) 18.0450 25.9924i 0.651993 0.939142i
\(767\) −20.0464 −0.723835
\(768\) 0 0
\(769\) 16.3297 0.588863 0.294431 0.955673i \(-0.404870\pi\)
0.294431 + 0.955673i \(0.404870\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −11.5233 30.8864i −0.414734 1.11163i
\(773\) 41.3144i 1.48598i 0.669304 + 0.742989i \(0.266592\pi\)
−0.669304 + 0.742989i \(0.733408\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −38.8864 9.91936i −1.39594 0.356084i
\(777\) 0 0
\(778\) −12.8002 8.88645i −0.458909 0.318595i
\(779\) 19.5794i 0.701503i
\(780\) 0 0
\(781\) 71.2628i 2.54998i
\(782\) 12.9041 18.5872i 0.461448 0.664678i
\(783\) 0 0
\(784\) 19.4747 16.8813i 0.695525 0.602904i
\(785\) 0 0
\(786\) 0 0
\(787\) 11.4849i 0.409391i −0.978826 0.204696i \(-0.934380\pi\)
0.978826 0.204696i \(-0.0656205\pi\)
\(788\) 25.4317 9.48824i 0.905966 0.338005i
\(789\) 0 0
\(790\) 0 0
\(791\) −8.99440 −0.319804
\(792\) 0 0
\(793\) −18.8735 −0.670216
\(794\) 29.3386 + 20.3681i 1.04119 + 0.722838i
\(795\) 0 0
\(796\) −6.32967 16.9657i −0.224349 0.601332i
\(797\) 45.4945i 1.61150i −0.592257 0.805749i \(-0.701763\pi\)
0.592257 0.805749i \(-0.298237\pi\)
\(798\) 0 0
\(799\) 3.70079 0.130924
\(800\) 0 0
\(801\) 0 0
\(802\) −5.88233 + 8.47301i −0.207712 + 0.299192i
\(803\) 3.70079i 0.130598i
\(804\) 0 0
\(805\) 0 0
\(806\) 8.99440 + 6.24430i 0.316814 + 0.219946i
\(807\) 0 0
\(808\) −5.20100 + 20.3892i −0.182970 + 0.717291i
\(809\) 36.0721 1.26823 0.634114 0.773240i \(-0.281365\pi\)
0.634114 + 0.773240i \(0.281365\pi\)
\(810\) 0 0
\(811\) 44.5230i 1.56341i 0.623646 + 0.781707i \(0.285651\pi\)
−0.623646 + 0.781707i \(0.714349\pi\)
\(812\) −9.43673 + 3.52072i −0.331164 + 0.123553i
\(813\) 0 0
\(814\) 4.64681 6.69335i 0.162871 0.234602i
\(815\) 0 0
\(816\) 0 0
\(817\) −12.9041 −0.451456
\(818\) 12.7834 18.4134i 0.446961 0.643811i
\(819\) 0 0
\(820\) 0 0
\(821\) 34.1613i 1.19224i −0.802897 0.596118i \(-0.796709\pi\)
0.802897 0.596118i \(-0.203291\pi\)
\(822\) 0 0
\(823\) 2.12689 0.0741388 0.0370694 0.999313i \(-0.488198\pi\)
0.0370694 + 0.999313i \(0.488198\pi\)
\(824\) 19.7279 + 5.03229i 0.687253 + 0.175308i
\(825\) 0 0
\(826\) −5.94043 4.12410i −0.206694 0.143496i
\(827\) 38.5872i 1.34181i −0.741543 0.670905i \(-0.765906\pi\)
0.741543 0.670905i \(-0.234094\pi\)
\(828\) 0 0
\(829\) 34.2351i 1.18904i −0.804083 0.594518i \(-0.797343\pi\)
0.804083 0.594518i \(-0.202657\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −11.2094 + 20.5422i −0.388617 + 0.712173i
\(833\) −13.7541 −0.476551
\(834\) 0 0
\(835\) 0 0
\(836\) −6.49720 17.4147i −0.224710 0.602300i
\(837\) 0 0
\(838\) −9.32384 6.47301i −0.322087 0.223606i
\(839\) 41.5928 1.43594 0.717972 0.696072i \(-0.245071\pi\)
0.717972 + 0.696072i \(0.245071\pi\)
\(840\) 0 0
\(841\) −16.5512 −0.570730
\(842\) −26.6437 18.4972i −0.918202 0.637456i
\(843\) 0 0
\(844\) −4.75140 + 1.77269i −0.163550 + 0.0610184i
\(845\) 0 0
\(846\) 0 0
\(847\) 13.2348 0.454752
\(848\) −20.2307 23.3386i −0.694725 0.801452i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.05601i 0.276156i
\(852\) 0 0
\(853\) 23.1828i 0.793763i −0.917870 0.396881i \(-0.870092\pi\)
0.917870 0.396881i \(-0.129908\pi\)
\(854\) −5.59283 3.88278i −0.191383 0.132866i
\(855\) 0 0
\(856\) 2.79641 10.9627i 0.0955795 0.374696i
\(857\) 9.38868 0.320711 0.160356 0.987059i \(-0.448736\pi\)
0.160356 + 0.987059i \(0.448736\pi\)
\(858\) 0 0
\(859\) 8.98769i 0.306656i −0.988175 0.153328i \(-0.951001\pi\)
0.988175 0.153328i \(-0.0489991\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 28.2813 40.7368i 0.963264 1.38750i
\(863\) −12.2473 −0.416903 −0.208451 0.978033i \(-0.566842\pi\)
−0.208451 + 0.978033i \(0.566842\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −13.7729 + 19.8387i −0.468022 + 0.674147i
\(867\) 0 0
\(868\) 1.38072 + 3.70079i 0.0468646 + 0.125613i
\(869\) 14.1887i 0.481318i
\(870\) 0 0
\(871\) 21.7729 0.737746
\(872\) 13.9474 54.6773i 0.472317 1.85161i
\(873\) 0 0
\(874\) −15.0956 10.4800i −0.510617 0.354492i
\(875\) 0 0
\(876\) 0 0
\(877\) 26.1109i 0.881701i 0.897581 + 0.440850i \(0.145323\pi\)
−0.897581 + 0.440850i \(0.854677\pi\)
\(878\) 6.88661 9.91960i 0.232412 0.334770i
\(879\) 0 0
\(880\) 0 0
\(881\) 38.4793 1.29640 0.648200 0.761470i \(-0.275522\pi\)
0.648200 + 0.761470i \(0.275522\pi\)
\(882\) 0 0
\(883\) 6.58723i 0.221678i 0.993838 + 0.110839i \(0.0353538\pi\)
−0.993838 + 0.110839i \(0.964646\pi\)
\(884\) 11.7008 4.36542i 0.393540 0.146825i
\(885\) 0 0
\(886\) −24.0900 16.7243i −0.809320 0.561865i
\(887\) 50.9595 1.71105 0.855526 0.517760i \(-0.173234\pi\)
0.855526 + 0.517760i \(0.173234\pi\)
\(888\) 0 0
\(889\) −3.14401 −0.105447
\(890\) 0 0
\(891\) 0 0
\(892\) 8.49962 + 22.7819i 0.284588 + 0.762793i
\(893\) 3.00560i 0.100578i
\(894\) 0 0
\(895\) 0 0
\(896\) −7.54781 + 3.78126i −0.252155 + 0.126323i
\(897\) 0 0
\(898\) 1.61301 2.32340i 0.0538268 0.0775330i
\(899\) 17.8637i 0.595789i
\(900\) 0 0
\(901\) 16.4830i 0.549129i
\(902\) 70.3311 + 48.8269i 2.34177 + 1.62576i
\(903\) 0 0
\(904\) −33.0361 8.42701i −1.09876 0.280278i
\(905\) 0 0
\(906\) 0 0
\(907\) 24.5568i 0.815394i −0.913117 0.407697i \(-0.866332\pi\)
0.913117 0.407697i \(-0.133668\pi\)
\(908\) −38.8574 + 14.4972i −1.28953 + 0.481107i
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 31.3621 1.03793
\(914\) −1.03605 + 1.49235i −0.0342696 + 0.0493625i
\(915\) 0 0
\(916\) −37.3836 + 13.9474i −1.23519 + 0.460834i
\(917\) 7.70079i 0.254302i
\(918\) 0 0
\(919\) 28.7548 0.948532 0.474266 0.880382i \(-0.342713\pi\)
0.474266 + 0.880382i \(0.342713\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 18.2425 + 12.6647i 0.600785 + 0.417091i
\(923\) 38.8864i 1.27996i
\(924\) 0 0
\(925\) 0 0
\(926\) −15.0990 + 21.7489i −0.496184 + 0.714712i
\(927\) 0 0
\(928\) −37.9593 + 4.09003i −1.24608 + 0.134262i
\(929\) −28.2880 −0.928100 −0.464050 0.885809i \(-0.653604\pi\)
−0.464050 + 0.885809i \(0.653604\pi\)
\(930\) 0 0
\(931\) 11.1704i 0.366095i
\(932\) 9.32510 + 24.9944i 0.305454 + 0.818719i
\(933\) 0 0
\(934\) 2.49720 + 1.73367i 0.0817110 + 0.0567273i
\(935\) 0 0
\(936\) 0 0
\(937\) 33.9313 1.10849 0.554244 0.832354i \(-0.313008\pi\)
0.554244 + 0.832354i \(0.313008\pi\)
\(938\) 6.45203 + 4.47928i 0.210666 + 0.146254i
\(939\) 0 0
\(940\) 0 0
\(941\) 38.8016i 1.26490i −0.774603 0.632448i \(-0.782050\pi\)
0.774603 0.632448i \(-0.217950\pi\)
\(942\) 0 0
\(943\) 84.6495 2.75657
\(944\) −17.9550 20.7133i −0.584385 0.674161i
\(945\) 0 0
\(946\) 32.1801 46.3527i 1.04626 1.50706i
\(947\) 17.7729i 0.577541i −0.957398 0.288771i \(-0.906753\pi\)
0.957398 0.288771i \(-0.0932465\pi\)
\(948\) 0 0
\(949\) 2.01943i 0.0655536i
\(950\) 0 0
\(951\) 0 0
\(952\) 4.36542 + 1.11355i 0.141484 + 0.0360905i
\(953\) −46.3047 −1.49996 −0.749978 0.661463i \(-0.769936\pi\)
−0.749978 + 0.661463i \(0.769936\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 16.0000 + 42.8854i 0.517477 + 1.38701i
\(957\) 0 0
\(958\) 9.82327 14.1496i 0.317375 0.457153i
\(959\) −11.2215 −0.362361
\(960\) 0 0
\(961\) −23.9944 −0.774013
\(962\) 2.53566 3.65240i 0.0817528 0.117758i
\(963\) 0 0
\(964\) −2.51176 6.73237i −0.0808984 0.216835i
\(965\) 0 0
\(966\) 0 0
\(967\) −9.28482 −0.298580 −0.149290 0.988793i \(-0.547699\pi\)
−0.149290 + 0.988793i \(0.547699\pi\)
\(968\) 48.6108 + 12.3999i 1.56241 + 0.398548i
\(969\) 0 0
\(970\) 0 0
\(971\) 20.9301i 0.671678i 0.941919 + 0.335839i \(0.109020\pi\)
−0.941919 + 0.335839i \(0.890980\pi\)
\(972\) 0 0
\(973\) 7.06651i 0.226542i
\(974\) −20.7819 + 29.9346i −0.665894 + 0.959165i
\(975\) 0 0
\(976\) −16.9044 19.5013i −0.541096 0.624222i
\(977\) −30.8314 −0.986383 −0.493192 0.869921i \(-0.664170\pi\)
−0.493192 + 0.869921i \(0.664170\pi\)
\(978\) 0 0
\(979\) 40.7027i 1.30086i
\(980\) 0 0
\(981\) 0 0
\(982\) −19.4845 13.5270i −0.621776 0.431664i
\(983\) 31.3285 0.999223 0.499612 0.866250i \(-0.333476\pi\)
0.499612 + 0.866250i \(0.333476\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 16.7368 + 11.6194i 0.533010 + 0.370038i
\(987\) 0 0
\(988\) −3.54537 9.50280i −0.112793 0.302324i
\(989\) 55.7895i 1.77400i
\(990\) 0 0
\(991\) 31.3420 0.995611 0.497806 0.867289i \(-0.334139\pi\)
0.497806 + 0.867289i \(0.334139\pi\)
\(992\) 1.60398 + 14.8864i 0.0509265 + 0.472645i
\(993\) 0 0
\(994\) 8.00000 11.5233i 0.253745 0.365498i
\(995\) 0 0
\(996\) 0 0
\(997\) 20.9557i 0.663672i 0.943337 + 0.331836i \(0.107668\pi\)
−0.943337 + 0.331836i \(0.892332\pi\)
\(998\) −20.4720 14.2125i −0.648030 0.449890i
\(999\) 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 1800.2.k.u.901.9 12
3.2 odd 2 600.2.k.f.301.4 12
4.3 odd 2 7200.2.k.u.3601.7 12
5.2 odd 4 360.2.d.e.109.6 6
5.3 odd 4 360.2.d.f.109.1 6
5.4 even 2 inner 1800.2.k.u.901.4 12
8.3 odd 2 7200.2.k.u.3601.8 12
8.5 even 2 inner 1800.2.k.u.901.10 12
12.11 even 2 2400.2.k.f.1201.10 12
15.2 even 4 120.2.d.b.109.1 yes 6
15.8 even 4 120.2.d.a.109.6 yes 6
15.14 odd 2 600.2.k.f.301.9 12
20.3 even 4 1440.2.d.e.1009.6 6
20.7 even 4 1440.2.d.f.1009.2 6
20.19 odd 2 7200.2.k.u.3601.5 12
24.5 odd 2 600.2.k.f.301.3 12
24.11 even 2 2400.2.k.f.1201.4 12
40.3 even 4 1440.2.d.f.1009.1 6
40.13 odd 4 360.2.d.e.109.5 6
40.19 odd 2 7200.2.k.u.3601.6 12
40.27 even 4 1440.2.d.e.1009.5 6
40.29 even 2 inner 1800.2.k.u.901.3 12
40.37 odd 4 360.2.d.f.109.2 6
60.23 odd 4 480.2.d.a.49.1 6
60.47 odd 4 480.2.d.b.49.5 6
60.59 even 2 2400.2.k.f.1201.3 12
120.29 odd 2 600.2.k.f.301.10 12
120.53 even 4 120.2.d.b.109.2 yes 6
120.59 even 2 2400.2.k.f.1201.9 12
120.77 even 4 120.2.d.a.109.5 6
120.83 odd 4 480.2.d.b.49.6 6
120.107 odd 4 480.2.d.a.49.2 6
240.53 even 4 3840.2.f.l.769.5 12
240.77 even 4 3840.2.f.l.769.2 12
240.83 odd 4 3840.2.f.m.769.2 12
240.107 odd 4 3840.2.f.m.769.5 12
240.173 even 4 3840.2.f.l.769.8 12
240.197 even 4 3840.2.f.l.769.11 12
240.203 odd 4 3840.2.f.m.769.11 12
240.227 odd 4 3840.2.f.m.769.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.d.a.109.5 6 120.77 even 4
120.2.d.a.109.6 yes 6 15.8 even 4
120.2.d.b.109.1 yes 6 15.2 even 4
120.2.d.b.109.2 yes 6 120.53 even 4
360.2.d.e.109.5 6 40.13 odd 4
360.2.d.e.109.6 6 5.2 odd 4
360.2.d.f.109.1 6 5.3 odd 4
360.2.d.f.109.2 6 40.37 odd 4
480.2.d.a.49.1 6 60.23 odd 4
480.2.d.a.49.2 6 120.107 odd 4
480.2.d.b.49.5 6 60.47 odd 4
480.2.d.b.49.6 6 120.83 odd 4
600.2.k.f.301.3 12 24.5 odd 2
600.2.k.f.301.4 12 3.2 odd 2
600.2.k.f.301.9 12 15.14 odd 2
600.2.k.f.301.10 12 120.29 odd 2
1440.2.d.e.1009.5 6 40.27 even 4
1440.2.d.e.1009.6 6 20.3 even 4
1440.2.d.f.1009.1 6 40.3 even 4
1440.2.d.f.1009.2 6 20.7 even 4
1800.2.k.u.901.3 12 40.29 even 2 inner
1800.2.k.u.901.4 12 5.4 even 2 inner
1800.2.k.u.901.9 12 1.1 even 1 trivial
1800.2.k.u.901.10 12 8.5 even 2 inner
2400.2.k.f.1201.3 12 60.59 even 2
2400.2.k.f.1201.4 12 24.11 even 2
2400.2.k.f.1201.9 12 120.59 even 2
2400.2.k.f.1201.10 12 12.11 even 2
3840.2.f.l.769.2 12 240.77 even 4
3840.2.f.l.769.5 12 240.53 even 4
3840.2.f.l.769.8 12 240.173 even 4
3840.2.f.l.769.11 12 240.197 even 4
3840.2.f.m.769.2 12 240.83 odd 4
3840.2.f.m.769.5 12 240.107 odd 4
3840.2.f.m.769.8 12 240.227 odd 4
3840.2.f.m.769.11 12 240.203 odd 4
7200.2.k.u.3601.5 12 20.19 odd 2
7200.2.k.u.3601.6 12 40.19 odd 2
7200.2.k.u.3601.7 12 4.3 odd 2
7200.2.k.u.3601.8 12 8.3 odd 2