Properties

Label 810.3.b.b.809.6
Level $810$
Weight $3$
Character 810.809
Analytic conductor $22.071$
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] = [810,3,Mod(809,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.809");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.b (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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 809.6
Root \(0.173499 + 2.99498i\) of defining polynomial
Character \(\chi\) \(=\) 810.809
Dual form 810.3.b.b.809.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +(3.56108 + 3.50980i) q^{5} +3.17976i q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +(3.56108 + 3.50980i) q^{5} +3.17976i q^{7} -2.82843 q^{8} +(-5.03613 - 4.96360i) q^{10} -15.6543i q^{11} -1.33596i q^{13} -4.49686i q^{14} +4.00000 q^{16} -23.4761 q^{17} -23.3447 q^{19} +(7.12217 + 7.01960i) q^{20} +22.1386i q^{22} -28.6849 q^{23} +(0.362627 + 24.9974i) q^{25} +1.88933i q^{26} +6.35952i q^{28} -33.1593i q^{29} +18.5472 q^{31} -5.65685 q^{32} +33.2002 q^{34} +(-11.1603 + 11.3234i) q^{35} -3.95544i q^{37} +33.0144 q^{38} +(-10.0723 - 9.92721i) q^{40} -60.3301i q^{41} -2.83257i q^{43} -31.3087i q^{44} +40.5665 q^{46} -71.6735 q^{47} +38.8891 q^{49} +(-0.512832 - 35.3516i) q^{50} -2.67192i q^{52} +14.3465 q^{53} +(54.9436 - 55.7464i) q^{55} -8.99372i q^{56} +46.8943i q^{58} +72.3760i q^{59} +11.2385 q^{61} -26.2296 q^{62} +8.00000 q^{64} +(4.68894 - 4.75746i) q^{65} +3.35443i q^{67} -46.9521 q^{68} +(15.7831 - 16.0137i) q^{70} -40.0554i q^{71} +86.6579i q^{73} +5.59384i q^{74} -46.6894 q^{76} +49.7770 q^{77} -90.5804 q^{79} +(14.2443 + 14.0392i) q^{80} +85.3196i q^{82} -102.892 q^{83} +(-83.6002 - 82.3962i) q^{85} +4.00586i q^{86} +44.2772i q^{88} -144.459i q^{89} +4.24802 q^{91} -57.3697 q^{92} +101.362 q^{94} +(-83.1323 - 81.9351i) q^{95} -74.8437i q^{97} -54.9975 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 16 q^{10} + 64 q^{16} + 144 q^{19} - 12 q^{25} - 56 q^{31} + 272 q^{34} - 32 q^{40} - 56 q^{46} - 24 q^{49} + 20 q^{55} - 136 q^{61} + 128 q^{64} + 224 q^{70} + 288 q^{76} - 840 q^{79} + 272 q^{85} - 168 q^{91} + 328 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 3.56108 + 3.50980i 0.712217 + 0.701960i
\(6\) 0 0
\(7\) 3.17976i 0.454251i 0.973865 + 0.227126i \(0.0729328\pi\)
−0.973865 + 0.227126i \(0.927067\pi\)
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) −5.03613 4.96360i −0.503613 0.496360i
\(11\) 15.6543i 1.42312i −0.702624 0.711561i \(-0.747989\pi\)
0.702624 0.711561i \(-0.252011\pi\)
\(12\) 0 0
\(13\) 1.33596i 0.102766i −0.998679 0.0513830i \(-0.983637\pi\)
0.998679 0.0513830i \(-0.0163629\pi\)
\(14\) 4.49686i 0.321204i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) −23.4761 −1.38094 −0.690472 0.723359i \(-0.742597\pi\)
−0.690472 + 0.723359i \(0.742597\pi\)
\(18\) 0 0
\(19\) −23.3447 −1.22867 −0.614334 0.789046i \(-0.710575\pi\)
−0.614334 + 0.789046i \(0.710575\pi\)
\(20\) 7.12217 + 7.01960i 0.356108 + 0.350980i
\(21\) 0 0
\(22\) 22.1386i 1.00630i
\(23\) −28.6849 −1.24717 −0.623584 0.781756i \(-0.714324\pi\)
−0.623584 + 0.781756i \(0.714324\pi\)
\(24\) 0 0
\(25\) 0.362627 + 24.9974i 0.0145051 + 0.999895i
\(26\) 1.88933i 0.0726665i
\(27\) 0 0
\(28\) 6.35952i 0.227126i
\(29\) 33.1593i 1.14342i −0.820454 0.571712i \(-0.806279\pi\)
0.820454 0.571712i \(-0.193721\pi\)
\(30\) 0 0
\(31\) 18.5472 0.598296 0.299148 0.954207i \(-0.403298\pi\)
0.299148 + 0.954207i \(0.403298\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) 33.2002 0.976475
\(35\) −11.1603 + 11.3234i −0.318866 + 0.323525i
\(36\) 0 0
\(37\) 3.95544i 0.106904i −0.998570 0.0534519i \(-0.982978\pi\)
0.998570 0.0534519i \(-0.0170224\pi\)
\(38\) 33.0144 0.868799
\(39\) 0 0
\(40\) −10.0723 9.92721i −0.251807 0.248180i
\(41\) 60.3301i 1.47147i −0.677272 0.735733i \(-0.736838\pi\)
0.677272 0.735733i \(-0.263162\pi\)
\(42\) 0 0
\(43\) 2.83257i 0.0658738i −0.999457 0.0329369i \(-0.989514\pi\)
0.999457 0.0329369i \(-0.0104860\pi\)
\(44\) 31.3087i 0.711561i
\(45\) 0 0
\(46\) 40.5665 0.881881
\(47\) −71.6735 −1.52497 −0.762485 0.647006i \(-0.776021\pi\)
−0.762485 + 0.647006i \(0.776021\pi\)
\(48\) 0 0
\(49\) 38.8891 0.793656
\(50\) −0.512832 35.3516i −0.0102566 0.707032i
\(51\) 0 0
\(52\) 2.67192i 0.0513830i
\(53\) 14.3465 0.270689 0.135345 0.990799i \(-0.456786\pi\)
0.135345 + 0.990799i \(0.456786\pi\)
\(54\) 0 0
\(55\) 54.9436 55.7464i 0.998974 1.01357i
\(56\) 8.99372i 0.160602i
\(57\) 0 0
\(58\) 46.8943i 0.808523i
\(59\) 72.3760i 1.22671i 0.789807 + 0.613356i \(0.210181\pi\)
−0.789807 + 0.613356i \(0.789819\pi\)
\(60\) 0 0
\(61\) 11.2385 0.184237 0.0921186 0.995748i \(-0.470636\pi\)
0.0921186 + 0.995748i \(0.470636\pi\)
\(62\) −26.2296 −0.423059
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 4.68894 4.75746i 0.0721376 0.0731917i
\(66\) 0 0
\(67\) 3.35443i 0.0500661i 0.999687 + 0.0250330i \(0.00796909\pi\)
−0.999687 + 0.0250330i \(0.992031\pi\)
\(68\) −46.9521 −0.690472
\(69\) 0 0
\(70\) 15.7831 16.0137i 0.225472 0.228767i
\(71\) 40.0554i 0.564160i −0.959391 0.282080i \(-0.908976\pi\)
0.959391 0.282080i \(-0.0910244\pi\)
\(72\) 0 0
\(73\) 86.6579i 1.18709i 0.804799 + 0.593547i \(0.202273\pi\)
−0.804799 + 0.593547i \(0.797727\pi\)
\(74\) 5.59384i 0.0755925i
\(75\) 0 0
\(76\) −46.6894 −0.614334
\(77\) 49.7770 0.646455
\(78\) 0 0
\(79\) −90.5804 −1.14659 −0.573294 0.819350i \(-0.694335\pi\)
−0.573294 + 0.819350i \(0.694335\pi\)
\(80\) 14.2443 + 14.0392i 0.178054 + 0.175490i
\(81\) 0 0
\(82\) 85.3196i 1.04048i
\(83\) −102.892 −1.23967 −0.619833 0.784734i \(-0.712800\pi\)
−0.619833 + 0.784734i \(0.712800\pi\)
\(84\) 0 0
\(85\) −83.6002 82.3962i −0.983531 0.969367i
\(86\) 4.00586i 0.0465798i
\(87\) 0 0
\(88\) 44.2772i 0.503150i
\(89\) 144.459i 1.62313i −0.584259 0.811567i \(-0.698615\pi\)
0.584259 0.811567i \(-0.301385\pi\)
\(90\) 0 0
\(91\) 4.24802 0.0466816
\(92\) −57.3697 −0.623584
\(93\) 0 0
\(94\) 101.362 1.07832
\(95\) −83.1323 81.9351i −0.875077 0.862475i
\(96\) 0 0
\(97\) 74.8437i 0.771584i −0.922586 0.385792i \(-0.873928\pi\)
0.922586 0.385792i \(-0.126072\pi\)
\(98\) −54.9975 −0.561199
\(99\) 0 0
\(100\) 0.725253 + 49.9947i 0.00725253 + 0.499947i
\(101\) 16.9570i 0.167891i 0.996470 + 0.0839455i \(0.0267522\pi\)
−0.996470 + 0.0839455i \(0.973248\pi\)
\(102\) 0 0
\(103\) 105.653i 1.02576i −0.858461 0.512879i \(-0.828579\pi\)
0.858461 0.512879i \(-0.171421\pi\)
\(104\) 3.77866i 0.0363333i
\(105\) 0 0
\(106\) −20.2891 −0.191406
\(107\) 58.8159 0.549681 0.274840 0.961490i \(-0.411375\pi\)
0.274840 + 0.961490i \(0.411375\pi\)
\(108\) 0 0
\(109\) −184.365 −1.69142 −0.845710 0.533643i \(-0.820823\pi\)
−0.845710 + 0.533643i \(0.820823\pi\)
\(110\) −77.7020 + 78.8373i −0.706382 + 0.716703i
\(111\) 0 0
\(112\) 12.7190i 0.113563i
\(113\) 17.1551 0.151815 0.0759073 0.997115i \(-0.475815\pi\)
0.0759073 + 0.997115i \(0.475815\pi\)
\(114\) 0 0
\(115\) −102.149 100.678i −0.888254 0.875461i
\(116\) 66.3186i 0.571712i
\(117\) 0 0
\(118\) 102.355i 0.867416i
\(119\) 74.6482i 0.627296i
\(120\) 0 0
\(121\) −124.058 −1.02528
\(122\) −15.8936 −0.130275
\(123\) 0 0
\(124\) 37.0943 0.299148
\(125\) −86.4444 + 90.2905i −0.691555 + 0.722324i
\(126\) 0 0
\(127\) 17.8557i 0.140596i 0.997526 + 0.0702979i \(0.0223950\pi\)
−0.997526 + 0.0702979i \(0.977605\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) −6.63117 + 6.72806i −0.0510090 + 0.0517543i
\(131\) 212.823i 1.62460i −0.583239 0.812301i \(-0.698215\pi\)
0.583239 0.812301i \(-0.301785\pi\)
\(132\) 0 0
\(133\) 74.2305i 0.558124i
\(134\) 4.74387i 0.0354020i
\(135\) 0 0
\(136\) 66.4003 0.488238
\(137\) 197.548 1.44195 0.720977 0.692959i \(-0.243693\pi\)
0.720977 + 0.692959i \(0.243693\pi\)
\(138\) 0 0
\(139\) −51.5742 −0.371037 −0.185519 0.982641i \(-0.559396\pi\)
−0.185519 + 0.982641i \(0.559396\pi\)
\(140\) −22.3206 + 22.6468i −0.159433 + 0.161763i
\(141\) 0 0
\(142\) 56.6469i 0.398922i
\(143\) −20.9135 −0.146249
\(144\) 0 0
\(145\) 116.382 118.083i 0.802638 0.814366i
\(146\) 122.553i 0.839403i
\(147\) 0 0
\(148\) 7.91089i 0.0534519i
\(149\) 225.983i 1.51666i 0.651868 + 0.758332i \(0.273985\pi\)
−0.651868 + 0.758332i \(0.726015\pi\)
\(150\) 0 0
\(151\) −53.6516 −0.355308 −0.177654 0.984093i \(-0.556851\pi\)
−0.177654 + 0.984093i \(0.556851\pi\)
\(152\) 66.0287 0.434400
\(153\) 0 0
\(154\) −70.3954 −0.457113
\(155\) 66.0480 + 65.0968i 0.426116 + 0.419979i
\(156\) 0 0
\(157\) 153.382i 0.976953i 0.872577 + 0.488477i \(0.162447\pi\)
−0.872577 + 0.488477i \(0.837553\pi\)
\(158\) 128.100 0.810760
\(159\) 0 0
\(160\) −20.1445 19.8544i −0.125903 0.124090i
\(161\) 91.2109i 0.566528i
\(162\) 0 0
\(163\) 272.064i 1.66910i −0.550929 0.834552i \(-0.685727\pi\)
0.550929 0.834552i \(-0.314273\pi\)
\(164\) 120.660i 0.735733i
\(165\) 0 0
\(166\) 145.512 0.876576
\(167\) −160.173 −0.959122 −0.479561 0.877508i \(-0.659204\pi\)
−0.479561 + 0.877508i \(0.659204\pi\)
\(168\) 0 0
\(169\) 167.215 0.989439
\(170\) 118.228 + 116.526i 0.695462 + 0.685446i
\(171\) 0 0
\(172\) 5.66514i 0.0329369i
\(173\) −30.4179 −0.175826 −0.0879130 0.996128i \(-0.528020\pi\)
−0.0879130 + 0.996128i \(0.528020\pi\)
\(174\) 0 0
\(175\) −79.4856 + 1.15307i −0.454204 + 0.00658895i
\(176\) 62.6174i 0.355781i
\(177\) 0 0
\(178\) 204.296i 1.14773i
\(179\) 261.709i 1.46206i 0.682344 + 0.731031i \(0.260961\pi\)
−0.682344 + 0.731031i \(0.739039\pi\)
\(180\) 0 0
\(181\) −158.159 −0.873806 −0.436903 0.899509i \(-0.643925\pi\)
−0.436903 + 0.899509i \(0.643925\pi\)
\(182\) −6.00761 −0.0330089
\(183\) 0 0
\(184\) 81.1330 0.440940
\(185\) 13.8828 14.0857i 0.0750422 0.0761387i
\(186\) 0 0
\(187\) 367.502i 1.96525i
\(188\) −143.347 −0.762485
\(189\) 0 0
\(190\) 117.567 + 115.874i 0.618773 + 0.609862i
\(191\) 123.681i 0.647542i −0.946135 0.323771i \(-0.895049\pi\)
0.946135 0.323771i \(-0.104951\pi\)
\(192\) 0 0
\(193\) 44.7888i 0.232066i 0.993245 + 0.116033i \(0.0370179\pi\)
−0.993245 + 0.116033i \(0.962982\pi\)
\(194\) 105.845i 0.545592i
\(195\) 0 0
\(196\) 77.7783 0.396828
\(197\) −203.271 −1.03183 −0.515917 0.856638i \(-0.672549\pi\)
−0.515917 + 0.856638i \(0.672549\pi\)
\(198\) 0 0
\(199\) −77.5958 −0.389929 −0.194964 0.980810i \(-0.562459\pi\)
−0.194964 + 0.980810i \(0.562459\pi\)
\(200\) −1.02566 70.7032i −0.00512832 0.353516i
\(201\) 0 0
\(202\) 23.9808i 0.118717i
\(203\) 105.439 0.519402
\(204\) 0 0
\(205\) 211.746 214.840i 1.03291 1.04800i
\(206\) 149.416i 0.725320i
\(207\) 0 0
\(208\) 5.34383i 0.0256915i
\(209\) 365.446i 1.74854i
\(210\) 0 0
\(211\) −132.330 −0.627157 −0.313578 0.949562i \(-0.601528\pi\)
−0.313578 + 0.949562i \(0.601528\pi\)
\(212\) 28.6931 0.135345
\(213\) 0 0
\(214\) −83.1782 −0.388683
\(215\) 9.94176 10.0870i 0.0462407 0.0469164i
\(216\) 0 0
\(217\) 58.9755i 0.271777i
\(218\) 260.731 1.19601
\(219\) 0 0
\(220\) 109.887 111.493i 0.499487 0.506786i
\(221\) 31.3630i 0.141914i
\(222\) 0 0
\(223\) 349.183i 1.56584i −0.622122 0.782921i \(-0.713729\pi\)
0.622122 0.782921i \(-0.286271\pi\)
\(224\) 17.9874i 0.0803010i
\(225\) 0 0
\(226\) −24.2609 −0.107349
\(227\) 218.334 0.961825 0.480912 0.876769i \(-0.340306\pi\)
0.480912 + 0.876769i \(0.340306\pi\)
\(228\) 0 0
\(229\) 380.456 1.66138 0.830690 0.556735i \(-0.187946\pi\)
0.830690 + 0.556735i \(0.187946\pi\)
\(230\) 144.461 + 142.380i 0.628090 + 0.619045i
\(231\) 0 0
\(232\) 93.7887i 0.404261i
\(233\) 232.991 0.999963 0.499981 0.866036i \(-0.333340\pi\)
0.499981 + 0.866036i \(0.333340\pi\)
\(234\) 0 0
\(235\) −255.235 251.560i −1.08611 1.07047i
\(236\) 144.752i 0.613356i
\(237\) 0 0
\(238\) 105.568i 0.443565i
\(239\) 157.493i 0.658965i 0.944162 + 0.329482i \(0.106874\pi\)
−0.944162 + 0.329482i \(0.893126\pi\)
\(240\) 0 0
\(241\) −115.151 −0.477806 −0.238903 0.971043i \(-0.576788\pi\)
−0.238903 + 0.971043i \(0.576788\pi\)
\(242\) 175.445 0.724980
\(243\) 0 0
\(244\) 22.4769 0.0921186
\(245\) 138.487 + 136.493i 0.565255 + 0.557114i
\(246\) 0 0
\(247\) 31.1875i 0.126265i
\(248\) −52.4593 −0.211529
\(249\) 0 0
\(250\) 122.251 127.690i 0.489003 0.510760i
\(251\) 157.563i 0.627740i 0.949466 + 0.313870i \(0.101626\pi\)
−0.949466 + 0.313870i \(0.898374\pi\)
\(252\) 0 0
\(253\) 449.043i 1.77487i
\(254\) 25.2517i 0.0994162i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 20.1945 0.0785777 0.0392889 0.999228i \(-0.487491\pi\)
0.0392889 + 0.999228i \(0.487491\pi\)
\(258\) 0 0
\(259\) 12.5774 0.0485612
\(260\) 9.37789 9.51491i 0.0360688 0.0365958i
\(261\) 0 0
\(262\) 300.977i 1.14877i
\(263\) −101.679 −0.386614 −0.193307 0.981138i \(-0.561921\pi\)
−0.193307 + 0.981138i \(0.561921\pi\)
\(264\) 0 0
\(265\) 51.0892 + 50.3534i 0.192789 + 0.190013i
\(266\) 104.978i 0.394653i
\(267\) 0 0
\(268\) 6.70885i 0.0250330i
\(269\) 83.9984i 0.312262i −0.987736 0.156131i \(-0.950098\pi\)
0.987736 0.156131i \(-0.0499022\pi\)
\(270\) 0 0
\(271\) 349.591 1.29000 0.645001 0.764182i \(-0.276857\pi\)
0.645001 + 0.764182i \(0.276857\pi\)
\(272\) −93.9042 −0.345236
\(273\) 0 0
\(274\) −279.375 −1.01962
\(275\) 391.317 5.67668i 1.42297 0.0206425i
\(276\) 0 0
\(277\) 326.401i 1.17834i 0.808008 + 0.589172i \(0.200546\pi\)
−0.808008 + 0.589172i \(0.799454\pi\)
\(278\) 72.9369 0.262363
\(279\) 0 0
\(280\) 31.5661 32.0274i 0.112736 0.114383i
\(281\) 108.743i 0.386986i −0.981102 0.193493i \(-0.938018\pi\)
0.981102 0.193493i \(-0.0619816\pi\)
\(282\) 0 0
\(283\) 380.702i 1.34524i −0.739990 0.672618i \(-0.765170\pi\)
0.739990 0.672618i \(-0.234830\pi\)
\(284\) 80.1108i 0.282080i
\(285\) 0 0
\(286\) 29.5762 0.103413
\(287\) 191.835 0.668415
\(288\) 0 0
\(289\) 262.125 0.907007
\(290\) −164.590 + 166.995i −0.567550 + 0.575843i
\(291\) 0 0
\(292\) 173.316i 0.593547i
\(293\) 101.317 0.345790 0.172895 0.984940i \(-0.444688\pi\)
0.172895 + 0.984940i \(0.444688\pi\)
\(294\) 0 0
\(295\) −254.025 + 257.737i −0.861102 + 0.873685i
\(296\) 11.1877i 0.0377962i
\(297\) 0 0
\(298\) 319.588i 1.07244i
\(299\) 38.3218i 0.128166i
\(300\) 0 0
\(301\) 9.00690 0.0299232
\(302\) 75.8748 0.251241
\(303\) 0 0
\(304\) −93.3787 −0.307167
\(305\) 40.0211 + 39.4448i 0.131217 + 0.129327i
\(306\) 0 0
\(307\) 115.566i 0.376436i 0.982127 + 0.188218i \(0.0602711\pi\)
−0.982127 + 0.188218i \(0.939729\pi\)
\(308\) 99.5541 0.323228
\(309\) 0 0
\(310\) −93.4060 92.0608i −0.301310 0.296970i
\(311\) 338.675i 1.08899i 0.838765 + 0.544494i \(0.183278\pi\)
−0.838765 + 0.544494i \(0.816722\pi\)
\(312\) 0 0
\(313\) 267.167i 0.853569i −0.904353 0.426785i \(-0.859646\pi\)
0.904353 0.426785i \(-0.140354\pi\)
\(314\) 216.914i 0.690810i
\(315\) 0 0
\(316\) −181.161 −0.573294
\(317\) −293.696 −0.926485 −0.463243 0.886231i \(-0.653314\pi\)
−0.463243 + 0.886231i \(0.653314\pi\)
\(318\) 0 0
\(319\) −519.087 −1.62723
\(320\) 28.4887 + 28.0784i 0.0890271 + 0.0877450i
\(321\) 0 0
\(322\) 128.992i 0.400595i
\(323\) 548.041 1.69672
\(324\) 0 0
\(325\) 33.3954 0.484454i 0.102755 0.00149063i
\(326\) 384.757i 1.18023i
\(327\) 0 0
\(328\) 170.639i 0.520242i
\(329\) 227.905i 0.692719i
\(330\) 0 0
\(331\) −47.2025 −0.142606 −0.0713028 0.997455i \(-0.522716\pi\)
−0.0713028 + 0.997455i \(0.522716\pi\)
\(332\) −205.785 −0.619833
\(333\) 0 0
\(334\) 226.519 0.678202
\(335\) −11.7734 + 11.9454i −0.0351444 + 0.0356579i
\(336\) 0 0
\(337\) 468.253i 1.38947i 0.719264 + 0.694737i \(0.244479\pi\)
−0.719264 + 0.694737i \(0.755521\pi\)
\(338\) −236.478 −0.699639
\(339\) 0 0
\(340\) −167.200 164.792i −0.491766 0.484684i
\(341\) 290.344i 0.851448i
\(342\) 0 0
\(343\) 279.466i 0.814770i
\(344\) 8.01172i 0.0232899i
\(345\) 0 0
\(346\) 43.0174 0.124328
\(347\) 57.0054 0.164281 0.0821404 0.996621i \(-0.473824\pi\)
0.0821404 + 0.996621i \(0.473824\pi\)
\(348\) 0 0
\(349\) −179.131 −0.513268 −0.256634 0.966509i \(-0.582613\pi\)
−0.256634 + 0.966509i \(0.582613\pi\)
\(350\) 112.410 1.63068i 0.321170 0.00465909i
\(351\) 0 0
\(352\) 88.5543i 0.251575i
\(353\) 22.8170 0.0646375 0.0323188 0.999478i \(-0.489711\pi\)
0.0323188 + 0.999478i \(0.489711\pi\)
\(354\) 0 0
\(355\) 140.586 142.641i 0.396018 0.401804i
\(356\) 288.918i 0.811567i
\(357\) 0 0
\(358\) 370.113i 1.03383i
\(359\) 598.539i 1.66724i 0.552339 + 0.833619i \(0.313735\pi\)
−0.552339 + 0.833619i \(0.686265\pi\)
\(360\) 0 0
\(361\) 183.974 0.509623
\(362\) 223.670 0.617874
\(363\) 0 0
\(364\) 8.49605 0.0233408
\(365\) −304.152 + 308.596i −0.833293 + 0.845469i
\(366\) 0 0
\(367\) 18.7000i 0.0509536i 0.999675 + 0.0254768i \(0.00811039\pi\)
−0.999675 + 0.0254768i \(0.991890\pi\)
\(368\) −114.739 −0.311792
\(369\) 0 0
\(370\) −19.6333 + 19.9201i −0.0530629 + 0.0538382i
\(371\) 45.6185i 0.122961i
\(372\) 0 0
\(373\) 288.883i 0.774485i −0.921978 0.387243i \(-0.873428\pi\)
0.921978 0.387243i \(-0.126572\pi\)
\(374\) 519.727i 1.38964i
\(375\) 0 0
\(376\) 202.723 0.539158
\(377\) −44.2994 −0.117505
\(378\) 0 0
\(379\) 427.427 1.12778 0.563888 0.825852i \(-0.309305\pi\)
0.563888 + 0.825852i \(0.309305\pi\)
\(380\) −166.265 163.870i −0.437539 0.431237i
\(381\) 0 0
\(382\) 174.911i 0.457881i
\(383\) −178.280 −0.465484 −0.232742 0.972539i \(-0.574770\pi\)
−0.232742 + 0.972539i \(0.574770\pi\)
\(384\) 0 0
\(385\) 177.260 + 174.707i 0.460416 + 0.453785i
\(386\) 63.3409i 0.164096i
\(387\) 0 0
\(388\) 149.687i 0.385792i
\(389\) 368.910i 0.948356i 0.880429 + 0.474178i \(0.157255\pi\)
−0.880429 + 0.474178i \(0.842745\pi\)
\(390\) 0 0
\(391\) 673.407 1.72227
\(392\) −109.995 −0.280600
\(393\) 0 0
\(394\) 287.469 0.729617
\(395\) −322.564 317.919i −0.816619 0.804859i
\(396\) 0 0
\(397\) 520.661i 1.31149i 0.754983 + 0.655745i \(0.227645\pi\)
−0.754983 + 0.655745i \(0.772355\pi\)
\(398\) 109.737 0.275721
\(399\) 0 0
\(400\) 1.45051 + 99.9895i 0.00362627 + 0.249974i
\(401\) 0.323245i 0.000806096i 1.00000 0.000403048i \(0.000128294\pi\)
−1.00000 0.000403048i \(0.999872\pi\)
\(402\) 0 0
\(403\) 24.7782i 0.0614844i
\(404\) 33.9140i 0.0839455i
\(405\) 0 0
\(406\) −149.113 −0.367273
\(407\) −61.9199 −0.152137
\(408\) 0 0
\(409\) 9.52823 0.0232964 0.0116482 0.999932i \(-0.496292\pi\)
0.0116482 + 0.999932i \(0.496292\pi\)
\(410\) −299.455 + 303.830i −0.730377 + 0.741050i
\(411\) 0 0
\(412\) 211.306i 0.512879i
\(413\) −230.138 −0.557236
\(414\) 0 0
\(415\) −366.408 361.131i −0.882911 0.870196i
\(416\) 7.55732i 0.0181666i
\(417\) 0 0
\(418\) 516.818i 1.23641i
\(419\) 390.835i 0.932780i −0.884579 0.466390i \(-0.845554\pi\)
0.884579 0.466390i \(-0.154446\pi\)
\(420\) 0 0
\(421\) 427.682 1.01587 0.507935 0.861395i \(-0.330409\pi\)
0.507935 + 0.861395i \(0.330409\pi\)
\(422\) 187.143 0.443467
\(423\) 0 0
\(424\) −40.5781 −0.0957031
\(425\) −8.51304 586.840i −0.0200307 1.38080i
\(426\) 0 0
\(427\) 35.7356i 0.0836900i
\(428\) 117.632 0.274840
\(429\) 0 0
\(430\) −14.0598 + 14.2652i −0.0326971 + 0.0331749i
\(431\) 396.306i 0.919503i −0.888048 0.459752i \(-0.847938\pi\)
0.888048 0.459752i \(-0.152062\pi\)
\(432\) 0 0
\(433\) 700.722i 1.61830i −0.587606 0.809148i \(-0.699929\pi\)
0.587606 0.809148i \(-0.300071\pi\)
\(434\) 83.4040i 0.192175i
\(435\) 0 0
\(436\) −368.730 −0.845710
\(437\) 669.639 1.53235
\(438\) 0 0
\(439\) −95.1798 −0.216811 −0.108405 0.994107i \(-0.534574\pi\)
−0.108405 + 0.994107i \(0.534574\pi\)
\(440\) −155.404 + 157.675i −0.353191 + 0.358352i
\(441\) 0 0
\(442\) 44.3540i 0.100348i
\(443\) −237.685 −0.536536 −0.268268 0.963344i \(-0.586451\pi\)
−0.268268 + 0.963344i \(0.586451\pi\)
\(444\) 0 0
\(445\) 507.022 514.430i 1.13937 1.15602i
\(446\) 493.819i 1.10722i
\(447\) 0 0
\(448\) 25.4381i 0.0567814i
\(449\) 574.978i 1.28057i 0.768136 + 0.640287i \(0.221185\pi\)
−0.768136 + 0.640287i \(0.778815\pi\)
\(450\) 0 0
\(451\) −944.428 −2.09408
\(452\) 34.3101 0.0759073
\(453\) 0 0
\(454\) −308.771 −0.680113
\(455\) 15.1276 + 14.9097i 0.0332474 + 0.0327686i
\(456\) 0 0
\(457\) 552.639i 1.20927i 0.796501 + 0.604637i \(0.206682\pi\)
−0.796501 + 0.604637i \(0.793318\pi\)
\(458\) −538.046 −1.17477
\(459\) 0 0
\(460\) −204.298 201.356i −0.444127 0.437731i
\(461\) 753.521i 1.63454i −0.576258 0.817268i \(-0.695488\pi\)
0.576258 0.817268i \(-0.304512\pi\)
\(462\) 0 0
\(463\) 393.957i 0.850878i 0.904987 + 0.425439i \(0.139880\pi\)
−0.904987 + 0.425439i \(0.860120\pi\)
\(464\) 132.637i 0.285856i
\(465\) 0 0
\(466\) −329.499 −0.707080
\(467\) 144.091 0.308546 0.154273 0.988028i \(-0.450696\pi\)
0.154273 + 0.988028i \(0.450696\pi\)
\(468\) 0 0
\(469\) −10.6663 −0.0227426
\(470\) 360.957 + 355.759i 0.767995 + 0.756934i
\(471\) 0 0
\(472\) 204.710i 0.433708i
\(473\) −44.3421 −0.0937464
\(474\) 0 0
\(475\) −8.46540 583.556i −0.0178219 1.22854i
\(476\) 149.296i 0.313648i
\(477\) 0 0
\(478\) 222.728i 0.465959i
\(479\) 160.218i 0.334484i −0.985916 0.167242i \(-0.946514\pi\)
0.985916 0.167242i \(-0.0534861\pi\)
\(480\) 0 0
\(481\) −5.28431 −0.0109861
\(482\) 162.848 0.337860
\(483\) 0 0
\(484\) −248.117 −0.512638
\(485\) 262.686 266.524i 0.541621 0.549535i
\(486\) 0 0
\(487\) 184.489i 0.378827i 0.981897 + 0.189414i \(0.0606587\pi\)
−0.981897 + 0.189414i \(0.939341\pi\)
\(488\) −31.7872 −0.0651377
\(489\) 0 0
\(490\) −195.851 193.030i −0.399696 0.393939i
\(491\) 83.3423i 0.169740i 0.996392 + 0.0848700i \(0.0270475\pi\)
−0.996392 + 0.0848700i \(0.972953\pi\)
\(492\) 0 0
\(493\) 778.449i 1.57900i
\(494\) 44.1058i 0.0892830i
\(495\) 0 0
\(496\) 74.1886 0.149574
\(497\) 127.366 0.256271
\(498\) 0 0
\(499\) −402.812 −0.807238 −0.403619 0.914927i \(-0.632248\pi\)
−0.403619 + 0.914927i \(0.632248\pi\)
\(500\) −172.889 + 180.581i −0.345778 + 0.361162i
\(501\) 0 0
\(502\) 222.827i 0.443879i
\(503\) −743.290 −1.47771 −0.738857 0.673862i \(-0.764634\pi\)
−0.738857 + 0.673862i \(0.764634\pi\)
\(504\) 0 0
\(505\) −59.5156 + 60.3852i −0.117853 + 0.119575i
\(506\) 635.042i 1.25502i
\(507\) 0 0
\(508\) 35.7113i 0.0702979i
\(509\) 97.6240i 0.191796i 0.995391 + 0.0958978i \(0.0305722\pi\)
−0.995391 + 0.0958978i \(0.969428\pi\)
\(510\) 0 0
\(511\) −275.551 −0.539239
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) −28.5593 −0.0555628
\(515\) 370.821 376.239i 0.720040 0.730562i
\(516\) 0 0
\(517\) 1122.00i 2.17022i
\(518\) −17.7871 −0.0343380
\(519\) 0 0
\(520\) −13.2623 + 13.4561i −0.0255045 + 0.0258772i
\(521\) 630.912i 1.21096i 0.795859 + 0.605482i \(0.207019\pi\)
−0.795859 + 0.605482i \(0.792981\pi\)
\(522\) 0 0
\(523\) 441.750i 0.844647i −0.906445 0.422323i \(-0.861215\pi\)
0.906445 0.422323i \(-0.138785\pi\)
\(524\) 425.646i 0.812301i
\(525\) 0 0
\(526\) 143.797 0.273377
\(527\) −435.414 −0.826213
\(528\) 0 0
\(529\) 293.821 0.555427
\(530\) −72.2510 71.2105i −0.136323 0.134359i
\(531\) 0 0
\(532\) 148.461i 0.279062i
\(533\) −80.5985 −0.151217
\(534\) 0 0
\(535\) 209.448 + 206.432i 0.391492 + 0.385854i
\(536\) 9.48775i 0.0177010i
\(537\) 0 0
\(538\) 118.792i 0.220802i
\(539\) 608.784i 1.12947i
\(540\) 0 0
\(541\) 365.131 0.674919 0.337459 0.941340i \(-0.390432\pi\)
0.337459 + 0.941340i \(0.390432\pi\)
\(542\) −494.396 −0.912169
\(543\) 0 0
\(544\) 132.801 0.244119
\(545\) −656.538 647.083i −1.20466 1.18731i
\(546\) 0 0
\(547\) 364.069i 0.665574i −0.943002 0.332787i \(-0.892011\pi\)
0.943002 0.332787i \(-0.107989\pi\)
\(548\) 395.095 0.720977
\(549\) 0 0
\(550\) −553.406 + 8.02804i −1.00619 + 0.0145964i
\(551\) 774.093i 1.40489i
\(552\) 0 0
\(553\) 288.024i 0.520839i
\(554\) 461.601i 0.833215i
\(555\) 0 0
\(556\) −103.148 −0.185519
\(557\) −901.292 −1.61812 −0.809059 0.587728i \(-0.800023\pi\)
−0.809059 + 0.587728i \(0.800023\pi\)
\(558\) 0 0
\(559\) −3.78420 −0.00676958
\(560\) −44.6413 + 45.2935i −0.0797165 + 0.0808813i
\(561\) 0 0
\(562\) 153.786i 0.273640i
\(563\) 81.6432 0.145015 0.0725073 0.997368i \(-0.476900\pi\)
0.0725073 + 0.997368i \(0.476900\pi\)
\(564\) 0 0
\(565\) 61.0906 + 60.2108i 0.108125 + 0.106568i
\(566\) 538.394i 0.951225i
\(567\) 0 0
\(568\) 113.294i 0.199461i
\(569\) 99.5184i 0.174900i 0.996169 + 0.0874502i \(0.0278719\pi\)
−0.996169 + 0.0874502i \(0.972128\pi\)
\(570\) 0 0
\(571\) −402.277 −0.704514 −0.352257 0.935903i \(-0.614586\pi\)
−0.352257 + 0.935903i \(0.614586\pi\)
\(572\) −41.8271 −0.0731243
\(573\) 0 0
\(574\) −271.296 −0.472641
\(575\) −10.4019 717.046i −0.0180903 1.24704i
\(576\) 0 0
\(577\) 278.698i 0.483012i 0.970399 + 0.241506i \(0.0776414\pi\)
−0.970399 + 0.241506i \(0.922359\pi\)
\(578\) −370.701 −0.641351
\(579\) 0 0
\(580\) 232.765 236.166i 0.401319 0.407183i
\(581\) 327.173i 0.563120i
\(582\) 0 0
\(583\) 224.585i 0.385224i
\(584\) 245.106i 0.419701i
\(585\) 0 0
\(586\) −143.283 −0.244511
\(587\) −325.927 −0.555243 −0.277621 0.960691i \(-0.589546\pi\)
−0.277621 + 0.960691i \(0.589546\pi\)
\(588\) 0 0
\(589\) −432.978 −0.735106
\(590\) 359.246 364.495i 0.608891 0.617788i
\(591\) 0 0
\(592\) 15.8218i 0.0267260i
\(593\) 113.167 0.190838 0.0954190 0.995437i \(-0.469581\pi\)
0.0954190 + 0.995437i \(0.469581\pi\)
\(594\) 0 0
\(595\) 262.000 265.828i 0.440336 0.446770i
\(596\) 451.966i 0.758332i
\(597\) 0 0
\(598\) 54.1952i 0.0906273i
\(599\) 283.109i 0.472636i −0.971676 0.236318i \(-0.924059\pi\)
0.971676 0.236318i \(-0.0759407\pi\)
\(600\) 0 0
\(601\) 523.593 0.871203 0.435601 0.900140i \(-0.356536\pi\)
0.435601 + 0.900140i \(0.356536\pi\)
\(602\) −12.7377 −0.0211589
\(603\) 0 0
\(604\) −107.303 −0.177654
\(605\) −441.782 435.420i −0.730219 0.719703i
\(606\) 0 0
\(607\) 1018.05i 1.67718i −0.544761 0.838591i \(-0.683380\pi\)
0.544761 0.838591i \(-0.316620\pi\)
\(608\) 132.057 0.217200
\(609\) 0 0
\(610\) −56.5984 55.7833i −0.0927843 0.0914481i
\(611\) 95.7528i 0.156715i
\(612\) 0 0
\(613\) 537.130i 0.876231i 0.898919 + 0.438116i \(0.144354\pi\)
−0.898919 + 0.438116i \(0.855646\pi\)
\(614\) 163.435i 0.266180i
\(615\) 0 0
\(616\) −140.791 −0.228556
\(617\) −1136.98 −1.84275 −0.921375 0.388676i \(-0.872933\pi\)
−0.921375 + 0.388676i \(0.872933\pi\)
\(618\) 0 0
\(619\) −234.742 −0.379228 −0.189614 0.981859i \(-0.560724\pi\)
−0.189614 + 0.981859i \(0.560724\pi\)
\(620\) 132.096 + 130.194i 0.213058 + 0.209990i
\(621\) 0 0
\(622\) 478.959i 0.770031i
\(623\) 459.345 0.737311
\(624\) 0 0
\(625\) −624.737 + 18.1294i −0.999579 + 0.0290071i
\(626\) 377.832i 0.603565i
\(627\) 0 0
\(628\) 306.763i 0.488477i
\(629\) 92.8582i 0.147628i
\(630\) 0 0
\(631\) 131.866 0.208980 0.104490 0.994526i \(-0.466679\pi\)
0.104490 + 0.994526i \(0.466679\pi\)
\(632\) 256.200 0.405380
\(633\) 0 0
\(634\) 415.349 0.655124
\(635\) −62.6698 + 63.5855i −0.0986925 + 0.100135i
\(636\) 0 0
\(637\) 51.9542i 0.0815608i
\(638\) 734.100 1.15063
\(639\) 0 0
\(640\) −40.2891 39.7088i −0.0629517 0.0620451i
\(641\) 536.826i 0.837483i 0.908106 + 0.418741i \(0.137529\pi\)
−0.908106 + 0.418741i \(0.862471\pi\)
\(642\) 0 0
\(643\) 585.189i 0.910092i 0.890468 + 0.455046i \(0.150377\pi\)
−0.890468 + 0.455046i \(0.849623\pi\)
\(644\) 182.422i 0.283264i
\(645\) 0 0
\(646\) −775.047 −1.19976
\(647\) 375.007 0.579609 0.289805 0.957086i \(-0.406410\pi\)
0.289805 + 0.957086i \(0.406410\pi\)
\(648\) 0 0
\(649\) 1133.00 1.74576
\(650\) −47.2283 + 0.685121i −0.0726589 + 0.00105403i
\(651\) 0 0
\(652\) 544.128i 0.834552i
\(653\) −453.112 −0.693892 −0.346946 0.937885i \(-0.612781\pi\)
−0.346946 + 0.937885i \(0.612781\pi\)
\(654\) 0 0
\(655\) 746.965 757.880i 1.14040 1.15707i
\(656\) 241.320i 0.367866i
\(657\) 0 0
\(658\) 322.306i 0.489826i
\(659\) 347.194i 0.526850i 0.964680 + 0.263425i \(0.0848521\pi\)
−0.964680 + 0.263425i \(0.915148\pi\)
\(660\) 0 0
\(661\) 946.578 1.43204 0.716020 0.698080i \(-0.245962\pi\)
0.716020 + 0.698080i \(0.245962\pi\)
\(662\) 66.7544 0.100837
\(663\) 0 0
\(664\) 291.023 0.438288
\(665\) 260.534 264.341i 0.391780 0.397505i
\(666\) 0 0
\(667\) 951.170i 1.42604i
\(668\) −320.347 −0.479561
\(669\) 0 0
\(670\) 16.6500 16.8933i 0.0248508 0.0252139i
\(671\) 175.931i 0.262192i
\(672\) 0 0
\(673\) 208.778i 0.310220i 0.987897 + 0.155110i \(0.0495732\pi\)
−0.987897 + 0.155110i \(0.950427\pi\)
\(674\) 662.210i 0.982507i
\(675\) 0 0
\(676\) 334.430 0.494720
\(677\) −416.186 −0.614751 −0.307375 0.951588i \(-0.599451\pi\)
−0.307375 + 0.951588i \(0.599451\pi\)
\(678\) 0 0
\(679\) 237.985 0.350493
\(680\) 236.457 + 233.052i 0.347731 + 0.342723i
\(681\) 0 0
\(682\) 410.608i 0.602064i
\(683\) −883.982 −1.29426 −0.647132 0.762378i \(-0.724032\pi\)
−0.647132 + 0.762378i \(0.724032\pi\)
\(684\) 0 0
\(685\) 703.484 + 693.352i 1.02698 + 1.01219i
\(686\) 395.225i 0.576130i
\(687\) 0 0
\(688\) 11.3303i 0.0164684i
\(689\) 19.1664i 0.0278176i
\(690\) 0 0
\(691\) 370.080 0.535571 0.267786 0.963479i \(-0.413708\pi\)
0.267786 + 0.963479i \(0.413708\pi\)
\(692\) −60.8358 −0.0879130
\(693\) 0 0
\(694\) −80.6178 −0.116164
\(695\) −183.660 181.015i −0.264259 0.260453i
\(696\) 0 0
\(697\) 1416.31i 2.03201i
\(698\) 253.329 0.362935
\(699\) 0 0
\(700\) −158.971 + 2.30613i −0.227102 + 0.00329447i
\(701\) 820.094i 1.16989i −0.811073 0.584946i \(-0.801116\pi\)
0.811073 0.584946i \(-0.198884\pi\)
\(702\) 0 0
\(703\) 92.3386i 0.131349i
\(704\) 125.235i 0.177890i
\(705\) 0 0
\(706\) −32.2682 −0.0457056
\(707\) −53.9191 −0.0762647
\(708\) 0 0
\(709\) −280.410 −0.395500 −0.197750 0.980252i \(-0.563363\pi\)
−0.197750 + 0.980252i \(0.563363\pi\)
\(710\) −198.819 + 201.724i −0.280027 + 0.284119i
\(711\) 0 0
\(712\) 408.592i 0.573865i
\(713\) −532.023 −0.746175
\(714\) 0 0
\(715\) −74.4749 73.4023i −0.104161 0.102661i
\(716\) 523.418i 0.731031i
\(717\) 0 0
\(718\) 846.462i 1.17892i
\(719\) 144.017i 0.200302i −0.994972 0.100151i \(-0.968067\pi\)
0.994972 0.100151i \(-0.0319327\pi\)
\(720\) 0 0
\(721\) 335.951 0.465952
\(722\) −260.179 −0.360358
\(723\) 0 0
\(724\) −316.318 −0.436903
\(725\) 828.895 12.0244i 1.14330 0.0165854i
\(726\) 0 0
\(727\) 567.714i 0.780900i 0.920624 + 0.390450i \(0.127681\pi\)
−0.920624 + 0.390450i \(0.872319\pi\)
\(728\) −12.0152 −0.0165044
\(729\) 0 0
\(730\) 430.136 436.421i 0.589227 0.597837i
\(731\) 66.4976i 0.0909680i
\(732\) 0 0
\(733\) 135.891i 0.185390i 0.995695 + 0.0926949i \(0.0295481\pi\)
−0.995695 + 0.0926949i \(0.970452\pi\)
\(734\) 26.4457i 0.0360296i
\(735\) 0 0
\(736\) 162.266 0.220470
\(737\) 52.5113 0.0712501
\(738\) 0 0
\(739\) −799.789 −1.08226 −0.541129 0.840939i \(-0.682003\pi\)
−0.541129 + 0.840939i \(0.682003\pi\)
\(740\) 27.7656 28.1713i 0.0375211 0.0380694i
\(741\) 0 0
\(742\) 64.5143i 0.0869465i
\(743\) 642.938 0.865327 0.432663 0.901556i \(-0.357574\pi\)
0.432663 + 0.901556i \(0.357574\pi\)
\(744\) 0 0
\(745\) −793.155 + 804.744i −1.06464 + 1.08019i
\(746\) 408.542i 0.547644i
\(747\) 0 0
\(748\) 735.004i 0.982626i
\(749\) 187.020i 0.249693i
\(750\) 0 0
\(751\) −804.131 −1.07075 −0.535374 0.844615i \(-0.679829\pi\)
−0.535374 + 0.844615i \(0.679829\pi\)
\(752\) −286.694 −0.381242
\(753\) 0 0
\(754\) 62.6488 0.0830887
\(755\) −191.058 188.306i −0.253057 0.249412i
\(756\) 0 0
\(757\) 998.545i 1.31908i −0.751668 0.659541i \(-0.770751\pi\)
0.751668 0.659541i \(-0.229249\pi\)
\(758\) −604.473 −0.797458
\(759\) 0 0
\(760\) 235.134 + 231.748i 0.309387 + 0.304931i
\(761\) 436.073i 0.573027i −0.958076 0.286513i \(-0.907504\pi\)
0.958076 0.286513i \(-0.0924963\pi\)
\(762\) 0 0
\(763\) 586.236i 0.768330i
\(764\) 247.361i 0.323771i
\(765\) 0 0
\(766\) 252.126 0.329147
\(767\) 96.6913 0.126064
\(768\) 0 0
\(769\) 1432.72 1.86310 0.931551 0.363612i \(-0.118456\pi\)
0.931551 + 0.363612i \(0.118456\pi\)
\(770\) −250.684 247.074i −0.325563 0.320875i
\(771\) 0 0
\(772\) 89.5776i 0.116033i
\(773\) 1341.08 1.73490 0.867449 0.497526i \(-0.165758\pi\)
0.867449 + 0.497526i \(0.165758\pi\)
\(774\) 0 0
\(775\) 6.72570 + 463.630i 0.00867832 + 0.598233i
\(776\) 211.690i 0.272796i
\(777\) 0 0
\(778\) 521.718i 0.670589i
\(779\) 1408.39i 1.80794i
\(780\) 0 0
\(781\) −627.041 −0.802869
\(782\) −952.342 −1.21783
\(783\) 0 0
\(784\) 155.557 0.198414
\(785\) −538.339 + 546.205i −0.685782 + 0.695803i
\(786\) 0 0
\(787\) 1048.34i 1.33207i −0.745921 0.666034i \(-0.767991\pi\)
0.745921 0.666034i \(-0.232009\pi\)
\(788\) −406.543 −0.515917
\(789\) 0 0
\(790\) 456.175 + 449.606i 0.577437 + 0.569121i
\(791\) 54.5489i 0.0689620i
\(792\) 0 0
\(793\) 15.0141i 0.0189333i
\(794\) 736.326i 0.927363i
\(795\) 0 0
\(796\) −155.192 −0.194964
\(797\) 1327.16 1.66519 0.832594 0.553883i \(-0.186855\pi\)
0.832594 + 0.553883i \(0.186855\pi\)
\(798\) 0 0
\(799\) 1682.61 2.10590
\(800\) −2.05133 141.406i −0.00256416 0.176758i
\(801\) 0 0
\(802\) 0.457137i 0.000569996i
\(803\) 1356.57 1.68938
\(804\) 0 0
\(805\) 320.132 324.810i 0.397680 0.403490i
\(806\) 35.0417i 0.0434761i
\(807\) 0 0
\(808\) 47.9616i 0.0593584i
\(809\) 976.178i 1.20665i 0.797496 + 0.603324i \(0.206157\pi\)
−0.797496 + 0.603324i \(0.793843\pi\)
\(810\) 0 0
\(811\) −289.065 −0.356431 −0.178215 0.983991i \(-0.557032\pi\)
−0.178215 + 0.983991i \(0.557032\pi\)
\(812\) 210.877 0.259701
\(813\) 0 0
\(814\) 87.5679 0.107577
\(815\) 954.890 968.842i 1.17164 1.18876i
\(816\) 0 0
\(817\) 66.1255i 0.0809369i
\(818\) −13.4750 −0.0164731
\(819\) 0 0
\(820\) 423.493 429.681i 0.516455 0.524001i
\(821\) 238.422i 0.290405i −0.989402 0.145202i \(-0.953617\pi\)
0.989402 0.145202i \(-0.0463833\pi\)
\(822\) 0 0
\(823\) 421.581i 0.512249i 0.966644 + 0.256125i \(0.0824458\pi\)
−0.966644 + 0.256125i \(0.917554\pi\)
\(824\) 298.832i 0.362660i
\(825\) 0 0
\(826\) 325.465 0.394025
\(827\) 944.973 1.14265 0.571326 0.820723i \(-0.306429\pi\)
0.571326 + 0.820723i \(0.306429\pi\)
\(828\) 0 0
\(829\) −609.872 −0.735672 −0.367836 0.929891i \(-0.619901\pi\)
−0.367836 + 0.929891i \(0.619901\pi\)
\(830\) 518.179 + 510.717i 0.624312 + 0.615321i
\(831\) 0 0
\(832\) 10.6877i 0.0128457i
\(833\) −912.963 −1.09599
\(834\) 0 0
\(835\) −570.391 562.176i −0.683103 0.673265i
\(836\) 730.891i 0.874272i
\(837\) 0 0
\(838\) 552.724i 0.659575i
\(839\) 66.6016i 0.0793821i 0.999212 + 0.0396911i \(0.0126374\pi\)
−0.999212 + 0.0396911i \(0.987363\pi\)
\(840\) 0 0
\(841\) −258.539 −0.307418
\(842\) −604.833 −0.718329
\(843\) 0 0
\(844\) −264.660 −0.313578
\(845\) 595.467 + 586.892i 0.704695 + 0.694546i
\(846\) 0 0
\(847\) 394.476i 0.465733i
\(848\) 57.3861 0.0676723
\(849\) 0 0
\(850\) 12.0393 + 829.916i 0.0141638 + 0.976372i
\(851\) 113.461i 0.133327i
\(852\) 0 0
\(853\) 361.920i 0.424291i 0.977238 + 0.212145i \(0.0680451\pi\)
−0.977238 + 0.212145i \(0.931955\pi\)
\(854\) 50.5378i 0.0591778i
\(855\) 0 0
\(856\) −166.356 −0.194342
\(857\) 1035.43 1.20821 0.604103 0.796906i \(-0.293532\pi\)
0.604103 + 0.796906i \(0.293532\pi\)
\(858\) 0 0
\(859\) 251.841 0.293180 0.146590 0.989197i \(-0.453170\pi\)
0.146590 + 0.989197i \(0.453170\pi\)
\(860\) 19.8835 20.1740i 0.0231204 0.0234582i
\(861\) 0 0
\(862\) 560.461i 0.650187i
\(863\) −872.334 −1.01082 −0.505408 0.862881i \(-0.668658\pi\)
−0.505408 + 0.862881i \(0.668658\pi\)
\(864\) 0 0
\(865\) −108.321 106.761i −0.125226 0.123423i
\(866\) 990.970i 1.14431i
\(867\) 0 0
\(868\) 117.951i 0.135888i
\(869\) 1417.98i 1.63173i
\(870\) 0 0
\(871\) 4.48137 0.00514509
\(872\) 521.462 0.598007
\(873\) 0 0
\(874\) −947.012 −1.08354
\(875\) −287.102 274.872i −0.328116 0.314140i
\(876\) 0 0
\(877\) 1468.28i 1.67421i −0.547044 0.837104i \(-0.684247\pi\)
0.547044 0.837104i \(-0.315753\pi\)
\(878\) 134.605 0.153308
\(879\) 0 0
\(880\) 219.774 222.986i 0.249744 0.253393i
\(881\) 727.513i 0.825781i 0.910781 + 0.412891i \(0.135481\pi\)
−0.910781 + 0.412891i \(0.864519\pi\)
\(882\) 0 0
\(883\) 384.077i 0.434968i −0.976064 0.217484i \(-0.930215\pi\)
0.976064 0.217484i \(-0.0697850\pi\)
\(884\) 62.7260i 0.0709571i
\(885\) 0 0
\(886\) 336.138 0.379388
\(887\) −464.833 −0.524051 −0.262026 0.965061i \(-0.584391\pi\)
−0.262026 + 0.965061i \(0.584391\pi\)
\(888\) 0 0
\(889\) −56.7767 −0.0638658
\(890\) −717.037 + 727.514i −0.805660 + 0.817432i
\(891\) 0 0
\(892\) 698.365i 0.782921i
\(893\) 1673.20 1.87368
\(894\) 0 0
\(895\) −918.546 + 931.968i −1.02631 + 1.04131i
\(896\) 35.9749i 0.0401505i
\(897\) 0 0
\(898\) 813.141i 0.905503i
\(899\) 615.011i 0.684105i
\(900\) 0 0
\(901\) −336.800 −0.373807
\(902\) 1335.62 1.48073
\(903\) 0 0
\(904\) −48.5218 −0.0536746
\(905\) −563.217 555.106i −0.622339 0.613376i
\(906\) 0 0
\(907\) 246.427i 0.271695i −0.990730 0.135847i \(-0.956624\pi\)
0.990730 0.135847i \(-0.0433757\pi\)
\(908\) 436.668 0.480912
\(909\) 0 0
\(910\) −21.3936 21.0855i −0.0235095 0.0231709i
\(911\) 144.890i 0.159045i −0.996833 0.0795224i \(-0.974660\pi\)
0.996833 0.0795224i \(-0.0253395\pi\)
\(912\) 0 0
\(913\) 1610.71i 1.76420i
\(914\) 781.549i 0.855086i
\(915\) 0 0
\(916\) 760.912 0.830690
\(917\) 676.725 0.737977
\(918\) 0 0
\(919\) −1421.82 −1.54714 −0.773569 0.633712i \(-0.781530\pi\)
−0.773569 + 0.633712i \(0.781530\pi\)
\(920\) 288.921 + 284.761i 0.314045 + 0.309522i
\(921\) 0 0
\(922\) 1065.64i 1.15579i
\(923\) −53.5123 −0.0579765
\(924\) 0 0
\(925\) 98.8757 1.43435i 0.106893 0.00155065i
\(926\) 557.139i 0.601662i
\(927\) 0 0
\(928\) 187.577i 0.202131i
\(929\) 547.295i 0.589123i −0.955632 0.294562i \(-0.904826\pi\)
0.955632 0.294562i \(-0.0951736\pi\)
\(930\) 0 0
\(931\) −907.854 −0.975139
\(932\) 465.983 0.499981
\(933\) 0 0
\(934\) −203.776 −0.218175
\(935\) −1289.86 + 1308.71i −1.37953 + 1.39969i
\(936\) 0 0
\(937\) 1384.47i 1.47756i −0.673948 0.738779i \(-0.735403\pi\)
0.673948 0.738779i \(-0.264597\pi\)
\(938\) 15.0844 0.0160814
\(939\) 0 0
\(940\) −510.471 503.119i −0.543054 0.535233i
\(941\) 701.042i 0.744996i 0.928033 + 0.372498i \(0.121499\pi\)
−0.928033 + 0.372498i \(0.878501\pi\)
\(942\) 0 0
\(943\) 1730.56i 1.83516i
\(944\) 289.504i 0.306678i
\(945\) 0 0
\(946\) 62.7091 0.0662887
\(947\) 655.997 0.692711 0.346355 0.938103i \(-0.387419\pi\)
0.346355 + 0.938103i \(0.387419\pi\)
\(948\) 0 0
\(949\) 115.771 0.121993
\(950\) 11.9719 + 825.272i 0.0126020 + 0.868708i
\(951\) 0 0
\(952\) 211.137i 0.221783i
\(953\) 290.181 0.304492 0.152246 0.988343i \(-0.451349\pi\)
0.152246 + 0.988343i \(0.451349\pi\)
\(954\) 0 0
\(955\) 434.094 440.437i 0.454548 0.461190i
\(956\) 314.985i 0.329482i
\(957\) 0 0
\(958\) 226.582i 0.236516i
\(959\) 628.154i 0.655009i
\(960\) 0 0
\(961\) −617.003 −0.642042
\(962\) 7.47314 0.00776834
\(963\) 0 0
\(964\) −230.302 −0.238903
\(965\) −157.200 + 159.497i −0.162901 + 0.165281i
\(966\) 0 0
\(967\) 1037.15i 1.07254i −0.844047 0.536270i \(-0.819833\pi\)
0.844047 0.536270i \(-0.180167\pi\)
\(968\) 350.890 0.362490
\(969\) 0 0
\(970\) −371.494 + 376.923i −0.382984 + 0.388580i
\(971\) 1762.50i 1.81514i 0.419901 + 0.907570i \(0.362065\pi\)
−0.419901 + 0.907570i \(0.637935\pi\)
\(972\) 0 0
\(973\) 163.993i 0.168544i
\(974\) 260.907i 0.267871i
\(975\) 0 0
\(976\) 44.9539 0.0460593
\(977\) 1771.34 1.81304 0.906519 0.422165i \(-0.138730\pi\)
0.906519 + 0.422165i \(0.138730\pi\)
\(978\) 0 0
\(979\) −2261.41 −2.30992
\(980\) 276.975 + 272.986i 0.282627 + 0.278557i
\(981\) 0 0
\(982\) 117.864i 0.120024i
\(983\) −1525.11 −1.55149 −0.775745 0.631047i \(-0.782626\pi\)
−0.775745 + 0.631047i \(0.782626\pi\)
\(984\) 0 0
\(985\) −723.867 713.442i −0.734890 0.724306i
\(986\) 1100.89i 1.11652i
\(987\) 0 0
\(988\) 62.3750i 0.0631326i
\(989\) 81.2519i 0.0821556i
\(990\) 0 0
\(991\) −296.858 −0.299554 −0.149777 0.988720i \(-0.547856\pi\)
−0.149777 + 0.988720i \(0.547856\pi\)
\(992\) −104.919 −0.105765
\(993\) 0 0
\(994\) −180.123 −0.181211
\(995\) −276.325 272.346i −0.277714 0.273714i
\(996\) 0 0
\(997\) 1318.45i 1.32242i −0.750203 0.661208i \(-0.770044\pi\)
0.750203 0.661208i \(-0.229956\pi\)
\(998\) 569.662 0.570803
\(999\) 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 810.3.b.b.809.6 16
3.2 odd 2 inner 810.3.b.b.809.11 16
5.4 even 2 inner 810.3.b.b.809.12 16
9.2 odd 6 90.3.j.b.59.1 yes 16
9.4 even 3 90.3.j.b.29.8 yes 16
9.5 odd 6 270.3.j.b.89.3 16
9.7 even 3 270.3.j.b.179.5 16
15.14 odd 2 inner 810.3.b.b.809.5 16
45.2 even 12 450.3.i.e.401.7 16
45.4 even 6 90.3.j.b.29.1 16
45.7 odd 12 1350.3.i.e.1151.3 16
45.13 odd 12 450.3.i.e.101.2 16
45.14 odd 6 270.3.j.b.89.5 16
45.22 odd 12 450.3.i.e.101.7 16
45.23 even 12 1350.3.i.e.251.6 16
45.29 odd 6 90.3.j.b.59.8 yes 16
45.32 even 12 1350.3.i.e.251.3 16
45.34 even 6 270.3.j.b.179.3 16
45.38 even 12 450.3.i.e.401.2 16
45.43 odd 12 1350.3.i.e.1151.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.3.j.b.29.1 16 45.4 even 6
90.3.j.b.29.8 yes 16 9.4 even 3
90.3.j.b.59.1 yes 16 9.2 odd 6
90.3.j.b.59.8 yes 16 45.29 odd 6
270.3.j.b.89.3 16 9.5 odd 6
270.3.j.b.89.5 16 45.14 odd 6
270.3.j.b.179.3 16 45.34 even 6
270.3.j.b.179.5 16 9.7 even 3
450.3.i.e.101.2 16 45.13 odd 12
450.3.i.e.101.7 16 45.22 odd 12
450.3.i.e.401.2 16 45.38 even 12
450.3.i.e.401.7 16 45.2 even 12
810.3.b.b.809.5 16 15.14 odd 2 inner
810.3.b.b.809.6 16 1.1 even 1 trivial
810.3.b.b.809.11 16 3.2 odd 2 inner
810.3.b.b.809.12 16 5.4 even 2 inner
1350.3.i.e.251.3 16 45.32 even 12
1350.3.i.e.251.6 16 45.23 even 12
1350.3.i.e.1151.3 16 45.7 odd 12
1350.3.i.e.1151.6 16 45.43 odd 12