Properties

Label 900.3.u.a.149.4
Level $900$
Weight $3$
Character 900.149
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
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] = [900,3,Mod(149,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 149.4
Root \(0.396143 + 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 900.149
Dual form 900.3.u.a.749.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.92048 + 0.686141i) q^{3} +(7.89542 - 4.55842i) q^{7} +(8.05842 + 4.00772i) q^{9} +O(q^{10})\) \(q+(2.92048 + 0.686141i) q^{3} +(7.89542 - 4.55842i) q^{7} +(8.05842 + 4.00772i) q^{9} +(-0.383156 + 0.221215i) q^{11} +(-9.62747 - 5.55842i) q^{13} +8.01544 q^{17} +8.11684 q^{19} +(26.1861 - 7.89542i) q^{21} +(11.8020 - 20.4416i) q^{23} +(20.7846 + 17.2337i) q^{27} +(45.9090 - 26.5055i) q^{29} +(-14.6753 + 25.4183i) q^{31} +(-1.27078 + 0.383156i) q^{33} +18.4674i q^{37} +(-24.3030 - 22.8391i) q^{39} +(-38.9674 - 22.4978i) q^{41} +(-19.9186 + 11.5000i) q^{43} +(-4.22894 - 7.32473i) q^{47} +(17.0584 - 29.5461i) q^{49} +(23.4090 + 5.49972i) q^{51} +60.5841 q^{53} +(23.7051 + 5.56930i) q^{57} +(-65.9674 - 38.0863i) q^{59} +(-2.67527 - 4.63370i) q^{61} +(81.8935 - 5.09105i) q^{63} +(95.0039 + 54.8505i) q^{67} +(48.4932 - 51.6014i) q^{69} +16.0309i q^{71} +4.35053i q^{73} +(-2.01678 + 3.49317i) q^{77} +(0.792110 + 1.37197i) q^{79} +(48.8763 + 64.5918i) q^{81} +(4.22894 + 7.32473i) q^{83} +(152.263 - 45.9090i) q^{87} -64.1236i q^{89} -101.351 q^{91} +(-60.2994 + 64.1644i) q^{93} +(99.7953 - 57.6168i) q^{97} +(-3.97420 + 0.247063i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 30 q^{9} - 72 q^{11} - 4 q^{19} + 198 q^{21} + 126 q^{29} - 14 q^{31} - 114 q^{39} - 36 q^{41} + 102 q^{49} - 54 q^{51} - 252 q^{59} + 82 q^{61} - 198 q^{69} - 166 q^{79} - 126 q^{81} - 604 q^{91} - 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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 0
\(3\) 2.92048 + 0.686141i 0.973494 + 0.228714i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 7.89542 4.55842i 1.12792 0.651203i 0.184507 0.982831i \(-0.440931\pi\)
0.943410 + 0.331628i \(0.107598\pi\)
\(8\) 0 0
\(9\) 8.05842 + 4.00772i 0.895380 + 0.445302i
\(10\) 0 0
\(11\) −0.383156 + 0.221215i −0.0348324 + 0.0201105i −0.517315 0.855795i \(-0.673068\pi\)
0.482483 + 0.875906i \(0.339735\pi\)
\(12\) 0 0
\(13\) −9.62747 5.55842i −0.740575 0.427571i 0.0817036 0.996657i \(-0.473964\pi\)
−0.822278 + 0.569086i \(0.807297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.01544 0.471497 0.235748 0.971814i \(-0.424246\pi\)
0.235748 + 0.971814i \(0.424246\pi\)
\(18\) 0 0
\(19\) 8.11684 0.427202 0.213601 0.976921i \(-0.431481\pi\)
0.213601 + 0.976921i \(0.431481\pi\)
\(20\) 0 0
\(21\) 26.1861 7.89542i 1.24696 0.375972i
\(22\) 0 0
\(23\) 11.8020 20.4416i 0.513128 0.888764i −0.486756 0.873538i \(-0.661820\pi\)
0.999884 0.0152262i \(-0.00484683\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 20.7846 + 17.2337i 0.769800 + 0.638285i
\(28\) 0 0
\(29\) 45.9090 26.5055i 1.58307 0.913984i 0.588659 0.808381i \(-0.299656\pi\)
0.994408 0.105603i \(-0.0336772\pi\)
\(30\) 0 0
\(31\) −14.6753 + 25.4183i −0.473396 + 0.819945i −0.999536 0.0304523i \(-0.990305\pi\)
0.526141 + 0.850398i \(0.323639\pi\)
\(32\) 0 0
\(33\) −1.27078 + 0.383156i −0.0385086 + 0.0116108i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 18.4674i 0.499118i 0.968360 + 0.249559i \(0.0802857\pi\)
−0.968360 + 0.249559i \(0.919714\pi\)
\(38\) 0 0
\(39\) −24.3030 22.8391i −0.623153 0.585617i
\(40\) 0 0
\(41\) −38.9674 22.4978i −0.950424 0.548727i −0.0572112 0.998362i \(-0.518221\pi\)
−0.893213 + 0.449635i \(0.851554\pi\)
\(42\) 0 0
\(43\) −19.9186 + 11.5000i −0.463223 + 0.267442i −0.713398 0.700759i \(-0.752845\pi\)
0.250176 + 0.968200i \(0.419512\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.22894 7.32473i −0.0899774 0.155845i 0.817524 0.575895i \(-0.195346\pi\)
−0.907501 + 0.420049i \(0.862013\pi\)
\(48\) 0 0
\(49\) 17.0584 29.5461i 0.348131 0.602981i
\(50\) 0 0
\(51\) 23.4090 + 5.49972i 0.458999 + 0.107838i
\(52\) 0 0
\(53\) 60.5841 1.14310 0.571548 0.820569i \(-0.306343\pi\)
0.571548 + 0.820569i \(0.306343\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 23.7051 + 5.56930i 0.415879 + 0.0977070i
\(58\) 0 0
\(59\) −65.9674 38.0863i −1.11809 0.645530i −0.177178 0.984179i \(-0.556697\pi\)
−0.940913 + 0.338649i \(0.890030\pi\)
\(60\) 0 0
\(61\) −2.67527 4.63370i −0.0438568 0.0759622i 0.843264 0.537500i \(-0.180631\pi\)
−0.887121 + 0.461538i \(0.847298\pi\)
\(62\) 0 0
\(63\) 81.8935 5.09105i 1.29990 0.0808103i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 95.0039 + 54.8505i 1.41797 + 0.818665i 0.996120 0.0880017i \(-0.0280481\pi\)
0.421848 + 0.906666i \(0.361381\pi\)
\(68\) 0 0
\(69\) 48.4932 51.6014i 0.702800 0.747847i
\(70\) 0 0
\(71\) 16.0309i 0.225787i 0.993607 + 0.112894i \(0.0360119\pi\)
−0.993607 + 0.112894i \(0.963988\pi\)
\(72\) 0 0
\(73\) 4.35053i 0.0595963i 0.999556 + 0.0297982i \(0.00948645\pi\)
−0.999556 + 0.0297982i \(0.990514\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.01678 + 3.49317i −0.0261920 + 0.0453659i
\(78\) 0 0
\(79\) 0.792110 + 1.37197i 0.0100267 + 0.0173668i 0.870995 0.491291i \(-0.163475\pi\)
−0.860969 + 0.508658i \(0.830142\pi\)
\(80\) 0 0
\(81\) 48.8763 + 64.5918i 0.603411 + 0.797430i
\(82\) 0 0
\(83\) 4.22894 + 7.32473i 0.0509511 + 0.0882498i 0.890376 0.455226i \(-0.150441\pi\)
−0.839425 + 0.543475i \(0.817108\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 152.263 45.9090i 1.75015 0.527689i
\(88\) 0 0
\(89\) 64.1236i 0.720489i −0.932858 0.360245i \(-0.882693\pi\)
0.932858 0.360245i \(-0.117307\pi\)
\(90\) 0 0
\(91\) −101.351 −1.11374
\(92\) 0 0
\(93\) −60.2994 + 64.1644i −0.648380 + 0.689940i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 99.7953 57.6168i 1.02882 0.593988i 0.112172 0.993689i \(-0.464219\pi\)
0.916646 + 0.399701i \(0.130886\pi\)
\(98\) 0 0
\(99\) −3.97420 + 0.247063i −0.0401435 + 0.00249558i
\(100\) 0 0
\(101\) 114.558 66.1403i 1.13424 0.654855i 0.189244 0.981930i \(-0.439396\pi\)
0.944998 + 0.327075i \(0.106063\pi\)
\(102\) 0 0
\(103\) 108.557 + 62.6753i 1.05395 + 0.608498i 0.923752 0.382990i \(-0.125106\pi\)
0.130197 + 0.991488i \(0.458439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −36.5378 −0.341475 −0.170737 0.985317i \(-0.554615\pi\)
−0.170737 + 0.985317i \(0.554615\pi\)
\(108\) 0 0
\(109\) −134.701 −1.23579 −0.617895 0.786261i \(-0.712014\pi\)
−0.617895 + 0.786261i \(0.712014\pi\)
\(110\) 0 0
\(111\) −12.6712 + 53.9336i −0.114155 + 0.485889i
\(112\) 0 0
\(113\) −95.1051 + 164.727i −0.841638 + 1.45776i 0.0468711 + 0.998901i \(0.485075\pi\)
−0.888509 + 0.458859i \(0.848258\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −55.3056 83.3763i −0.472697 0.712618i
\(118\) 0 0
\(119\) 63.2853 36.5378i 0.531809 0.307040i
\(120\) 0 0
\(121\) −60.4021 + 104.620i −0.499191 + 0.864624i
\(122\) 0 0
\(123\) −98.3668 92.4416i −0.799730 0.751558i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 184.103i 1.44963i 0.688943 + 0.724816i \(0.258075\pi\)
−0.688943 + 0.724816i \(0.741925\pi\)
\(128\) 0 0
\(129\) −66.0625 + 19.9186i −0.512112 + 0.154408i
\(130\) 0 0
\(131\) −109.194 63.0433i −0.833544 0.481247i 0.0215207 0.999768i \(-0.493149\pi\)
−0.855064 + 0.518522i \(0.826483\pi\)
\(132\) 0 0
\(133\) 64.0859 37.0000i 0.481849 0.278195i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 62.1326 + 107.617i 0.453523 + 0.785524i 0.998602 0.0528602i \(-0.0168338\pi\)
−0.545079 + 0.838385i \(0.683500\pi\)
\(138\) 0 0
\(139\) 13.3832 23.1803i 0.0962817 0.166765i −0.813861 0.581059i \(-0.802638\pi\)
0.910143 + 0.414295i \(0.135972\pi\)
\(140\) 0 0
\(141\) −7.32473 24.2934i −0.0519485 0.172294i
\(142\) 0 0
\(143\) 4.91843 0.0343946
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 70.0916 74.5842i 0.476813 0.507376i
\(148\) 0 0
\(149\) −69.8437 40.3243i −0.468750 0.270633i 0.246966 0.969024i \(-0.420566\pi\)
−0.715716 + 0.698391i \(0.753900\pi\)
\(150\) 0 0
\(151\) −49.9742 86.5579i −0.330955 0.573231i 0.651744 0.758439i \(-0.274037\pi\)
−0.982699 + 0.185208i \(0.940704\pi\)
\(152\) 0 0
\(153\) 64.5918 + 32.1237i 0.422169 + 0.209959i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 60.1487 + 34.7269i 0.383113 + 0.221190i 0.679172 0.733979i \(-0.262339\pi\)
−0.296059 + 0.955170i \(0.595673\pi\)
\(158\) 0 0
\(159\) 176.935 + 41.5692i 1.11280 + 0.261442i
\(160\) 0 0
\(161\) 215.193i 1.33660i
\(162\) 0 0
\(163\) 162.467i 0.996732i −0.866967 0.498366i \(-0.833934\pi\)
0.866967 0.498366i \(-0.166066\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −48.3397 + 83.7269i −0.289459 + 0.501358i −0.973681 0.227916i \(-0.926809\pi\)
0.684221 + 0.729274i \(0.260142\pi\)
\(168\) 0 0
\(169\) −22.7079 39.3312i −0.134366 0.232729i
\(170\) 0 0
\(171\) 65.4090 + 32.5301i 0.382509 + 0.190234i
\(172\) 0 0
\(173\) 14.0141 + 24.2731i 0.0810064 + 0.140307i 0.903682 0.428203i \(-0.140853\pi\)
−0.822676 + 0.568510i \(0.807520\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −166.524 156.493i −0.940813 0.884142i
\(178\) 0 0
\(179\) 35.6012i 0.198889i 0.995043 + 0.0994447i \(0.0317067\pi\)
−0.995043 + 0.0994447i \(0.968293\pi\)
\(180\) 0 0
\(181\) −19.6358 −0.108485 −0.0542426 0.998528i \(-0.517274\pi\)
−0.0542426 + 0.998528i \(0.517274\pi\)
\(182\) 0 0
\(183\) −4.63370 15.3682i −0.0253207 0.0839794i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.07117 + 1.77314i −0.0164233 + 0.00948202i
\(188\) 0 0
\(189\) 242.662 + 41.3222i 1.28392 + 0.218636i
\(190\) 0 0
\(191\) 188.662 108.924i 0.987757 0.570282i 0.0831540 0.996537i \(-0.473501\pi\)
0.904603 + 0.426255i \(0.140167\pi\)
\(192\) 0 0
\(193\) −42.4352 24.5000i −0.219872 0.126943i 0.386019 0.922491i \(-0.373850\pi\)
−0.605891 + 0.795548i \(0.707183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −359.965 −1.82723 −0.913617 0.406575i \(-0.866723\pi\)
−0.913617 + 0.406575i \(0.866723\pi\)
\(198\) 0 0
\(199\) 61.0652 0.306861 0.153430 0.988159i \(-0.450968\pi\)
0.153430 + 0.988159i \(0.450968\pi\)
\(200\) 0 0
\(201\) 239.822 + 225.376i 1.19314 + 1.12127i
\(202\) 0 0
\(203\) 241.647 418.545i 1.19038 2.06180i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 177.029 117.428i 0.855214 0.567285i
\(208\) 0 0
\(209\) −3.11002 + 1.79557i −0.0148805 + 0.00859124i
\(210\) 0 0
\(211\) −193.493 + 335.140i −0.917029 + 1.58834i −0.113126 + 0.993581i \(0.536087\pi\)
−0.803903 + 0.594761i \(0.797247\pi\)
\(212\) 0 0
\(213\) −10.9994 + 46.8179i −0.0516406 + 0.219802i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 267.584i 1.23311i
\(218\) 0 0
\(219\) −2.98508 + 12.7056i −0.0136305 + 0.0580167i
\(220\) 0 0
\(221\) −77.1684 44.5532i −0.349178 0.201598i
\(222\) 0 0
\(223\) −88.9864 + 51.3763i −0.399042 + 0.230387i −0.686071 0.727535i \(-0.740666\pi\)
0.287028 + 0.957922i \(0.407333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −169.482 293.552i −0.746617 1.29318i −0.949435 0.313962i \(-0.898343\pi\)
0.202818 0.979216i \(-0.434990\pi\)
\(228\) 0 0
\(229\) −148.376 + 256.995i −0.647932 + 1.12225i 0.335685 + 0.941974i \(0.391032\pi\)
−0.983616 + 0.180276i \(0.942301\pi\)
\(230\) 0 0
\(231\) −8.28679 + 8.81795i −0.0358736 + 0.0381729i
\(232\) 0 0
\(233\) −346.537 −1.48728 −0.743642 0.668578i \(-0.766903\pi\)
−0.743642 + 0.668578i \(0.766903\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.37197 + 4.55033i 0.00578892 + 0.0191997i
\(238\) 0 0
\(239\) 140.026 + 80.8439i 0.585882 + 0.338259i 0.763468 0.645846i \(-0.223495\pi\)
−0.177586 + 0.984105i \(0.556829\pi\)
\(240\) 0 0
\(241\) 162.370 + 281.232i 0.673732 + 1.16694i 0.976838 + 0.213982i \(0.0686433\pi\)
−0.303105 + 0.952957i \(0.598023\pi\)
\(242\) 0 0
\(243\) 98.4233 + 222.175i 0.405034 + 0.914302i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −78.1447 45.1168i −0.316375 0.182659i
\(248\) 0 0
\(249\) 7.32473 + 24.2934i 0.0294166 + 0.0975638i
\(250\) 0 0
\(251\) 384.012i 1.52993i −0.644074 0.764963i \(-0.722757\pi\)
0.644074 0.764963i \(-0.277243\pi\)
\(252\) 0 0
\(253\) 10.4431i 0.0412770i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.46694 11.2011i 0.0251632 0.0435839i −0.853170 0.521634i \(-0.825323\pi\)
0.878333 + 0.478050i \(0.158656\pi\)
\(258\) 0 0
\(259\) 84.1821 + 145.808i 0.325027 + 0.562964i
\(260\) 0 0
\(261\) 476.181 29.6026i 1.82445 0.113420i
\(262\) 0 0
\(263\) 24.7358 + 42.8437i 0.0940526 + 0.162904i 0.909213 0.416332i \(-0.136684\pi\)
−0.815160 + 0.579236i \(0.803351\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 43.9978 187.272i 0.164786 0.701392i
\(268\) 0 0
\(269\) 21.4434i 0.0797154i −0.999205 0.0398577i \(-0.987310\pi\)
0.999205 0.0398577i \(-0.0126905\pi\)
\(270\) 0 0
\(271\) −326.907 −1.20630 −0.603150 0.797628i \(-0.706088\pi\)
−0.603150 + 0.797628i \(0.706088\pi\)
\(272\) 0 0
\(273\) −295.992 69.5407i −1.08422 0.254728i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −413.487 + 238.727i −1.49273 + 0.861830i −0.999965 0.00833105i \(-0.997348\pi\)
−0.492768 + 0.870161i \(0.664015\pi\)
\(278\) 0 0
\(279\) −220.129 + 146.017i −0.788993 + 0.523359i
\(280\) 0 0
\(281\) −103.064 + 59.5039i −0.366775 + 0.211758i −0.672049 0.740507i \(-0.734585\pi\)
0.305274 + 0.952265i \(0.401252\pi\)
\(282\) 0 0
\(283\) −338.920 195.675i −1.19760 0.691432i −0.237577 0.971369i \(-0.576353\pi\)
−0.960018 + 0.279937i \(0.909687\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −410.218 −1.42933
\(288\) 0 0
\(289\) −224.753 −0.777691
\(290\) 0 0
\(291\) 330.984 99.7953i 1.13740 0.342939i
\(292\) 0 0
\(293\) 35.8483 62.0910i 0.122349 0.211915i −0.798345 0.602201i \(-0.794291\pi\)
0.920694 + 0.390286i \(0.127624\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −11.7761 2.00532i −0.0396502 0.00675192i
\(298\) 0 0
\(299\) −227.246 + 131.200i −0.760020 + 0.438797i
\(300\) 0 0
\(301\) −104.844 + 181.595i −0.348318 + 0.603304i
\(302\) 0 0
\(303\) 379.947 114.558i 1.25395 0.378081i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 172.351i 0.561402i −0.959795 0.280701i \(-0.909433\pi\)
0.959795 0.280701i \(-0.0905670\pi\)
\(308\) 0 0
\(309\) 274.034 + 257.527i 0.886841 + 0.833421i
\(310\) 0 0
\(311\) −524.246 302.673i −1.68568 0.973227i −0.957763 0.287559i \(-0.907156\pi\)
−0.727915 0.685667i \(-0.759511\pi\)
\(312\) 0 0
\(313\) −283.595 + 163.734i −0.906055 + 0.523111i −0.879160 0.476527i \(-0.841895\pi\)
−0.0268949 + 0.999638i \(0.508562\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −34.5210 59.7921i −0.108899 0.188619i 0.806425 0.591336i \(-0.201399\pi\)
−0.915325 + 0.402717i \(0.868066\pi\)
\(318\) 0 0
\(319\) −11.7269 + 20.3115i −0.0367613 + 0.0636725i
\(320\) 0 0
\(321\) −106.708 25.0701i −0.332423 0.0780999i
\(322\) 0 0
\(323\) 65.0601 0.201424
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −393.392 92.4239i −1.20303 0.282642i
\(328\) 0 0
\(329\) −66.7785 38.5546i −0.202974 0.117187i
\(330\) 0 0
\(331\) −254.895 441.492i −0.770076 1.33381i −0.937521 0.347930i \(-0.886885\pi\)
0.167444 0.985882i \(-0.446449\pi\)
\(332\) 0 0
\(333\) −74.0121 + 148.818i −0.222259 + 0.446901i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 292.232 + 168.720i 0.867156 + 0.500653i 0.866402 0.499347i \(-0.166427\pi\)
0.000754096 1.00000i \(0.499760\pi\)
\(338\) 0 0
\(339\) −390.778 + 415.826i −1.15274 + 1.22663i
\(340\) 0 0
\(341\) 12.9856i 0.0380808i
\(342\) 0 0
\(343\) 135.687i 0.395590i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −107.622 + 186.407i −0.310151 + 0.537197i −0.978395 0.206745i \(-0.933713\pi\)
0.668244 + 0.743942i \(0.267046\pi\)
\(348\) 0 0
\(349\) 181.012 + 313.522i 0.518659 + 0.898345i 0.999765 + 0.0216818i \(0.00690207\pi\)
−0.481105 + 0.876663i \(0.659765\pi\)
\(350\) 0 0
\(351\) −104.311 281.446i −0.297183 0.801842i
\(352\) 0 0
\(353\) 292.420 + 506.486i 0.828385 + 1.43481i 0.899304 + 0.437323i \(0.144074\pi\)
−0.0709189 + 0.997482i \(0.522593\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 209.894 63.2853i 0.587937 0.177270i
\(358\) 0 0
\(359\) 393.693i 1.09664i −0.836269 0.548319i \(-0.815268\pi\)
0.836269 0.548319i \(-0.184732\pi\)
\(360\) 0 0
\(361\) −295.117 −0.817498
\(362\) 0 0
\(363\) −248.187 + 264.095i −0.683711 + 0.727535i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 329.831 190.428i 0.898722 0.518877i 0.0219364 0.999759i \(-0.493017\pi\)
0.876785 + 0.480882i \(0.159684\pi\)
\(368\) 0 0
\(369\) −223.851 337.467i −0.606641 0.914546i
\(370\) 0 0
\(371\) 478.337 276.168i 1.28932 0.744388i
\(372\) 0 0
\(373\) 115.080 + 66.4416i 0.308526 + 0.178128i 0.646267 0.763112i \(-0.276329\pi\)
−0.337741 + 0.941239i \(0.609663\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −589.316 −1.56317
\(378\) 0 0
\(379\) −507.622 −1.33937 −0.669686 0.742644i \(-0.733571\pi\)
−0.669686 + 0.742644i \(0.733571\pi\)
\(380\) 0 0
\(381\) −126.321 + 537.670i −0.331550 + 1.41121i
\(382\) 0 0
\(383\) 165.968 287.466i 0.433338 0.750564i −0.563820 0.825898i \(-0.690669\pi\)
0.997158 + 0.0753339i \(0.0240023\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −206.601 + 12.8437i −0.533853 + 0.0331879i
\(388\) 0 0
\(389\) −296.662 + 171.278i −0.762626 + 0.440302i −0.830238 0.557409i \(-0.811795\pi\)
0.0676116 + 0.997712i \(0.478462\pi\)
\(390\) 0 0
\(391\) 94.5979 163.848i 0.241938 0.419049i
\(392\) 0 0
\(393\) −275.643 259.039i −0.701382 0.659133i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.8043i 0.0624792i −0.999512 0.0312396i \(-0.990055\pi\)
0.999512 0.0312396i \(-0.00994550\pi\)
\(398\) 0 0
\(399\) 212.549 64.0859i 0.532704 0.160616i
\(400\) 0 0
\(401\) 52.0842 + 30.0708i 0.129886 + 0.0749896i 0.563535 0.826092i \(-0.309441\pi\)
−0.433649 + 0.901082i \(0.642774\pi\)
\(402\) 0 0
\(403\) 282.571 163.143i 0.701170 0.404820i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.08526 7.07589i −0.0100375 0.0173855i
\(408\) 0 0
\(409\) −240.720 + 416.939i −0.588558 + 1.01941i 0.405864 + 0.913933i \(0.366971\pi\)
−0.994422 + 0.105478i \(0.966363\pi\)
\(410\) 0 0
\(411\) 107.617 + 356.925i 0.261841 + 0.868430i
\(412\) 0 0
\(413\) −694.453 −1.68149
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 54.9902 58.5149i 0.131871 0.140324i
\(418\) 0 0
\(419\) 479.531 + 276.857i 1.14447 + 0.660758i 0.947533 0.319659i \(-0.103568\pi\)
0.196933 + 0.980417i \(0.436902\pi\)
\(420\) 0 0
\(421\) 190.947 + 330.730i 0.453556 + 0.785581i 0.998604 0.0528233i \(-0.0168220\pi\)
−0.545048 + 0.838405i \(0.683489\pi\)
\(422\) 0 0
\(423\) −4.72306 75.9742i −0.0111656 0.179608i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −42.2447 24.3900i −0.0989337 0.0571194i
\(428\) 0 0
\(429\) 14.3642 + 3.37474i 0.0334829 + 0.00786652i
\(430\) 0 0
\(431\) 821.321i 1.90562i 0.303570 + 0.952809i \(0.401821\pi\)
−0.303570 + 0.952809i \(0.598179\pi\)
\(432\) 0 0
\(433\) 199.155i 0.459942i 0.973198 + 0.229971i \(0.0738631\pi\)
−0.973198 + 0.229971i \(0.926137\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 95.7946 165.921i 0.219210 0.379682i
\(438\) 0 0
\(439\) −240.830 417.130i −0.548588 0.950182i −0.998372 0.0570445i \(-0.981832\pi\)
0.449784 0.893137i \(-0.351501\pi\)
\(440\) 0 0
\(441\) 255.876 169.729i 0.580218 0.384873i
\(442\) 0 0
\(443\) 270.143 + 467.902i 0.609805 + 1.05621i 0.991272 + 0.131830i \(0.0420854\pi\)
−0.381468 + 0.924382i \(0.624581\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −176.309 165.689i −0.394428 0.370669i
\(448\) 0 0
\(449\) 300.318i 0.668859i −0.942421 0.334429i \(-0.891456\pi\)
0.942421 0.334429i \(-0.108544\pi\)
\(450\) 0 0
\(451\) 19.9074 0.0441407
\(452\) 0 0
\(453\) −86.5579 287.080i −0.191077 0.633731i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 134.841 77.8505i 0.295057 0.170351i −0.345163 0.938543i \(-0.612176\pi\)
0.640220 + 0.768191i \(0.278843\pi\)
\(458\) 0 0
\(459\) 166.598 + 138.136i 0.362958 + 0.300949i
\(460\) 0 0
\(461\) 261.143 150.771i 0.566470 0.327052i −0.189268 0.981925i \(-0.560612\pi\)
0.755738 + 0.654874i \(0.227278\pi\)
\(462\) 0 0
\(463\) 206.790 + 119.390i 0.446630 + 0.257862i 0.706406 0.707807i \(-0.250315\pi\)
−0.259776 + 0.965669i \(0.583649\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 423.152 0.906107 0.453054 0.891483i \(-0.350335\pi\)
0.453054 + 0.891483i \(0.350335\pi\)
\(468\) 0 0
\(469\) 1000.13 2.13247
\(470\) 0 0
\(471\) 151.836 + 142.690i 0.322369 + 0.302950i
\(472\) 0 0
\(473\) 5.08795 8.81259i 0.0107568 0.0186313i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 488.212 + 242.804i 1.02351 + 0.509024i
\(478\) 0 0
\(479\) 379.284 218.980i 0.791824 0.457160i −0.0487802 0.998810i \(-0.515533\pi\)
0.840604 + 0.541650i \(0.182200\pi\)
\(480\) 0 0
\(481\) 102.649 177.794i 0.213408 0.369634i
\(482\) 0 0
\(483\) 147.653 628.467i 0.305699 1.30117i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 401.945i 0.825350i 0.910878 + 0.412675i \(0.135405\pi\)
−0.910878 + 0.412675i \(0.864595\pi\)
\(488\) 0 0
\(489\) 111.475 474.483i 0.227966 0.970313i
\(490\) 0 0
\(491\) −241.084 139.190i −0.491007 0.283483i 0.233985 0.972240i \(-0.424823\pi\)
−0.724992 + 0.688757i \(0.758157\pi\)
\(492\) 0 0
\(493\) 367.981 212.454i 0.746411 0.430941i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 73.0756 + 126.571i 0.147033 + 0.254669i
\(498\) 0 0
\(499\) 272.655 472.252i 0.546402 0.946397i −0.452115 0.891960i \(-0.649330\pi\)
0.998517 0.0544369i \(-0.0173364\pi\)
\(500\) 0 0
\(501\) −198.624 + 211.355i −0.396454 + 0.421866i
\(502\) 0 0
\(503\) −306.460 −0.609264 −0.304632 0.952470i \(-0.598534\pi\)
−0.304632 + 0.952470i \(0.598534\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −39.3312 130.447i −0.0775764 0.257292i
\(508\) 0 0
\(509\) −480.208 277.248i −0.943434 0.544692i −0.0523989 0.998626i \(-0.516687\pi\)
−0.891035 + 0.453934i \(0.850020\pi\)
\(510\) 0 0
\(511\) 19.8316 + 34.3493i 0.0388093 + 0.0672197i
\(512\) 0 0
\(513\) 168.705 + 139.883i 0.328860 + 0.272677i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.24069 + 1.87101i 0.00626825 + 0.00361898i
\(518\) 0 0
\(519\) 24.2731 + 80.5049i 0.0467691 + 0.155115i
\(520\) 0 0
\(521\) 154.167i 0.295905i 0.988994 + 0.147953i \(0.0472683\pi\)
−0.988994 + 0.147953i \(0.952732\pi\)
\(522\) 0 0
\(523\) 480.598i 0.918925i −0.888197 0.459463i \(-0.848042\pi\)
0.888197 0.459463i \(-0.151958\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −117.629 + 203.739i −0.223204 + 0.386602i
\(528\) 0 0
\(529\) −14.0721 24.3735i −0.0266013 0.0460748i
\(530\) 0 0
\(531\) −378.954 571.294i −0.713660 1.07588i
\(532\) 0 0
\(533\) 250.105 + 433.194i 0.469240 + 0.812747i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −24.4274 + 103.973i −0.0454887 + 0.193618i
\(538\) 0 0
\(539\) 15.0943i 0.0280043i
\(540\) 0 0
\(541\) −300.543 −0.555533 −0.277766 0.960649i \(-0.589594\pi\)
−0.277766 + 0.960649i \(0.589594\pi\)
\(542\) 0 0
\(543\) −57.3460 13.4729i −0.105610 0.0248120i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −87.6473 + 50.6032i −0.160233 + 0.0925104i −0.577972 0.816056i \(-0.696156\pi\)
0.417739 + 0.908567i \(0.362822\pi\)
\(548\) 0 0
\(549\) −2.98785 48.0620i −0.00544236 0.0875446i
\(550\) 0 0
\(551\) 372.636 215.141i 0.676290 0.390456i
\(552\) 0 0
\(553\) 12.5081 + 7.22154i 0.0226186 + 0.0130588i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 433.041 0.777452 0.388726 0.921353i \(-0.372915\pi\)
0.388726 + 0.921353i \(0.372915\pi\)
\(558\) 0 0
\(559\) 255.687 0.457401
\(560\) 0 0
\(561\) −10.1859 + 3.07117i −0.0181567 + 0.00547445i
\(562\) 0 0
\(563\) 520.886 902.201i 0.925197 1.60249i 0.133954 0.990988i \(-0.457233\pi\)
0.791244 0.611501i \(-0.209434\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 680.336 + 287.181i 1.19989 + 0.506491i
\(568\) 0 0
\(569\) −257.445 + 148.636i −0.452452 + 0.261223i −0.708865 0.705344i \(-0.750793\pi\)
0.256413 + 0.966567i \(0.417459\pi\)
\(570\) 0 0
\(571\) 339.524 588.073i 0.594613 1.02990i −0.398988 0.916956i \(-0.630638\pi\)
0.993601 0.112945i \(-0.0360282\pi\)
\(572\) 0 0
\(573\) 625.720 188.662i 1.09201 0.329252i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 148.351i 0.257107i −0.991703 0.128553i \(-0.958967\pi\)
0.991703 0.128553i \(-0.0410333\pi\)
\(578\) 0 0
\(579\) −107.121 100.668i −0.185010 0.173866i
\(580\) 0 0
\(581\) 66.7785 + 38.5546i 0.114937 + 0.0663590i
\(582\) 0 0
\(583\) −23.2132 + 13.4021i −0.0398168 + 0.0229882i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 263.559 + 456.497i 0.448993 + 0.777678i 0.998321 0.0579287i \(-0.0184496\pi\)
−0.549328 + 0.835607i \(0.685116\pi\)
\(588\) 0 0
\(589\) −119.117 + 206.316i −0.202236 + 0.350283i
\(590\) 0 0
\(591\) −1051.27 246.987i −1.77880 0.417913i
\(592\) 0 0
\(593\) −473.848 −0.799069 −0.399534 0.916718i \(-0.630828\pi\)
−0.399534 + 0.916718i \(0.630828\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 178.340 + 41.8993i 0.298727 + 0.0701832i
\(598\) 0 0
\(599\) 601.414 + 347.227i 1.00403 + 0.579677i 0.909438 0.415839i \(-0.136512\pi\)
0.0945922 + 0.995516i \(0.469845\pi\)
\(600\) 0 0
\(601\) −93.3559 161.697i −0.155334 0.269047i 0.777846 0.628454i \(-0.216312\pi\)
−0.933181 + 0.359408i \(0.882979\pi\)
\(602\) 0 0
\(603\) 545.756 + 822.758i 0.905068 + 1.36444i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 712.209 + 411.194i 1.17333 + 0.677420i 0.954461 0.298335i \(-0.0964311\pi\)
0.218865 + 0.975755i \(0.429764\pi\)
\(608\) 0 0
\(609\) 992.906 1056.55i 1.63039 1.73489i
\(610\) 0 0
\(611\) 94.0249i 0.153887i
\(612\) 0 0
\(613\) 482.206i 0.786634i 0.919403 + 0.393317i \(0.128672\pi\)
−0.919403 + 0.393317i \(0.871328\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 344.052 595.916i 0.557621 0.965828i −0.440073 0.897962i \(-0.645048\pi\)
0.997694 0.0678661i \(-0.0216191\pi\)
\(618\) 0 0
\(619\) −380.253 658.617i −0.614302 1.06400i −0.990507 0.137465i \(-0.956105\pi\)
0.376205 0.926536i \(-0.377229\pi\)
\(620\) 0 0
\(621\) 597.583 221.479i 0.962291 0.356649i
\(622\) 0 0
\(623\) −292.302 506.282i −0.469185 0.812652i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10.3148 + 3.11002i −0.0164510 + 0.00496016i
\(628\) 0 0
\(629\) 148.024i 0.235333i
\(630\) 0 0
\(631\) 1008.08 1.59758 0.798792 0.601607i \(-0.205473\pi\)
0.798792 + 0.601607i \(0.205473\pi\)
\(632\) 0 0
\(633\) −795.046 + 846.007i −1.25600 + 1.33650i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −328.459 + 189.636i −0.515634 + 0.297701i
\(638\) 0 0
\(639\) −64.2473 + 129.184i −0.100544 + 0.202165i
\(640\) 0 0
\(641\) −488.095 + 281.802i −0.761458 + 0.439628i −0.829819 0.558032i \(-0.811556\pi\)
0.0683607 + 0.997661i \(0.478223\pi\)
\(642\) 0 0
\(643\) −499.697 288.500i −0.777133 0.448678i 0.0582801 0.998300i \(-0.481438\pi\)
−0.835413 + 0.549622i \(0.814772\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1024.52 1.58349 0.791744 0.610853i \(-0.209173\pi\)
0.791744 + 0.610853i \(0.209173\pi\)
\(648\) 0 0
\(649\) 33.7011 0.0519277
\(650\) 0 0
\(651\) −183.600 + 781.475i −0.282028 + 1.20042i
\(652\) 0 0
\(653\) 199.697 345.885i 0.305814 0.529686i −0.671628 0.740888i \(-0.734405\pi\)
0.977442 + 0.211203i \(0.0677381\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −17.4357 + 35.0584i −0.0265384 + 0.0533614i
\(658\) 0 0
\(659\) 646.308 373.146i 0.980741 0.566231i 0.0782470 0.996934i \(-0.475068\pi\)
0.902494 + 0.430703i \(0.141734\pi\)
\(660\) 0 0
\(661\) −475.624 + 823.804i −0.719552 + 1.24630i 0.241626 + 0.970369i \(0.422319\pi\)
−0.961178 + 0.275931i \(0.911014\pi\)
\(662\) 0 0
\(663\) −194.799 183.065i −0.293815 0.276117i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1251.27i 1.87596i
\(668\) 0 0
\(669\) −295.135 + 88.9864i −0.441158 + 0.133014i
\(670\) 0 0
\(671\) 2.05009 + 1.18362i 0.00305527 + 0.00176396i
\(672\) 0 0
\(673\) 298.113 172.115i 0.442961 0.255743i −0.261892 0.965097i \(-0.584346\pi\)
0.704853 + 0.709354i \(0.251013\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 492.832 + 853.610i 0.727965 + 1.26087i 0.957742 + 0.287628i \(0.0928668\pi\)
−0.229778 + 0.973243i \(0.573800\pi\)
\(678\) 0 0
\(679\) 525.284 909.818i 0.773614 1.33994i
\(680\) 0 0
\(681\) −293.552 973.600i −0.431060 1.42966i
\(682\) 0 0
\(683\) 166.658 0.244009 0.122004 0.992530i \(-0.461068\pi\)
0.122004 + 0.992530i \(0.461068\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −609.665 + 648.743i −0.887431 + 0.944313i
\(688\) 0 0
\(689\) −583.272 336.752i −0.846548 0.488755i
\(690\) 0 0
\(691\) 449.077 + 777.825i 0.649895 + 1.12565i 0.983148 + 0.182813i \(0.0585204\pi\)
−0.333253 + 0.942838i \(0.608146\pi\)
\(692\) 0 0
\(693\) −30.2518 + 20.0668i −0.0436534 + 0.0289564i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −312.341 180.330i −0.448122 0.258723i
\(698\) 0 0
\(699\) −1012.06 237.773i −1.44786 0.340162i
\(700\) 0 0
\(701\) 730.549i 1.04215i 0.853510 + 0.521076i \(0.174469\pi\)
−0.853510 + 0.521076i \(0.825531\pi\)
\(702\) 0 0
\(703\) 149.897i 0.213224i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 602.991 1044.41i 0.852887 1.47724i
\(708\) 0 0
\(709\) −114.961 199.118i −0.162145 0.280843i 0.773493 0.633805i \(-0.218508\pi\)
−0.935638 + 0.352962i \(0.885174\pi\)
\(710\) 0 0
\(711\) 0.884663 + 14.2305i 0.00124425 + 0.0200148i
\(712\) 0 0
\(713\) 346.394 + 599.971i 0.485825 + 0.841474i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 353.472 + 332.181i 0.492988 + 0.463292i
\(718\) 0 0
\(719\) 907.095i 1.26161i 0.775943 + 0.630803i \(0.217275\pi\)
−0.775943 + 0.630803i \(0.782725\pi\)
\(720\) 0 0
\(721\) 1142.80 1.58502
\(722\) 0 0
\(723\) 281.232 + 932.742i 0.388980 + 1.29010i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 186.838 107.871i 0.256999 0.148378i −0.365966 0.930628i \(-0.619261\pi\)
0.622965 + 0.782250i \(0.285928\pi\)
\(728\) 0 0
\(729\) 135.000 + 716.391i 0.185185 + 0.982704i
\(730\) 0 0
\(731\) −159.656 + 92.1776i −0.218408 + 0.126098i
\(732\) 0 0
\(733\) 544.963 + 314.634i 0.743469 + 0.429242i 0.823329 0.567564i \(-0.192114\pi\)
−0.0798604 + 0.996806i \(0.525447\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −48.5351 −0.0658549
\(738\) 0 0
\(739\) −547.649 −0.741068 −0.370534 0.928819i \(-0.620825\pi\)
−0.370534 + 0.928819i \(0.620825\pi\)
\(740\) 0 0
\(741\) −197.264 185.381i −0.266213 0.250177i
\(742\) 0 0
\(743\) −267.072 + 462.583i −0.359451 + 0.622588i −0.987869 0.155288i \(-0.950369\pi\)
0.628418 + 0.777876i \(0.283703\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.72306 + 75.9742i 0.00632271 + 0.101706i
\(748\) 0 0
\(749\) −288.481 + 166.555i −0.385155 + 0.222369i
\(750\) 0 0
\(751\) 225.545 390.655i 0.300326 0.520180i −0.675884 0.737008i \(-0.736238\pi\)
0.976210 + 0.216828i \(0.0695712\pi\)
\(752\) 0 0
\(753\) 263.486 1121.50i 0.349915 1.48937i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 352.391i 0.465511i −0.972535 0.232755i \(-0.925226\pi\)
0.972535 0.232755i \(-0.0747741\pi\)
\(758\) 0 0
\(759\) −7.16543 + 30.4988i −0.00944061 + 0.0401829i
\(760\) 0 0
\(761\) −929.923 536.891i −1.22197 0.705507i −0.256636 0.966508i \(-0.582614\pi\)
−0.965339 + 0.261001i \(0.915947\pi\)
\(762\) 0 0
\(763\) −1063.52 + 614.024i −1.39387 + 0.804750i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 423.399 + 733.349i 0.552020 + 0.956126i
\(768\) 0 0
\(769\) 177.988 308.284i 0.231454 0.400889i −0.726782 0.686868i \(-0.758985\pi\)
0.958236 + 0.285978i \(0.0923185\pi\)
\(770\) 0 0
\(771\) 26.5721 28.2753i 0.0344644 0.0366735i
\(772\) 0 0
\(773\) 370.790 0.479677 0.239838 0.970813i \(-0.422906\pi\)
0.239838 + 0.970813i \(0.422906\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 145.808 + 483.589i 0.187655 + 0.622380i
\(778\) 0 0
\(779\) −316.292 182.611i −0.406023 0.234418i
\(780\) 0 0
\(781\) −3.54628 6.14233i −0.00454069 0.00786470i
\(782\) 0 0
\(783\) 1410.99 + 240.273i 1.80203 + 0.306862i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −998.581 576.531i −1.26885 0.732568i −0.294076 0.955782i \(-0.595012\pi\)
−0.974769 + 0.223214i \(0.928345\pi\)
\(788\) 0 0
\(789\) 42.8437 + 142.096i 0.0543013 + 0.180097i
\(790\) 0 0
\(791\) 1734.12i 2.19231i
\(792\) 0 0
\(793\) 59.4810i 0.0750076i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 439.873 761.882i 0.551910 0.955937i −0.446226 0.894920i \(-0.647232\pi\)
0.998137 0.0610167i \(-0.0194343\pi\)
\(798\) 0 0
\(799\) −33.8968 58.7110i −0.0424240 0.0734806i
\(800\) 0 0
\(801\) 256.989 516.735i 0.320836 0.645112i
\(802\) 0 0
\(803\) −0.962404 1.66693i −0.00119851 0.00207588i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.7132 62.6252i 0.0182320 0.0776025i
\(808\) 0 0
\(809\) 884.508i 1.09334i −0.837350 0.546668i \(-0.815896\pi\)
0.837350 0.546668i \(-0.184104\pi\)
\(810\) 0 0
\(811\) −961.464 −1.18553 −0.592765 0.805376i \(-0.701964\pi\)
−0.592765 + 0.805376i \(0.701964\pi\)
\(812\) 0 0
\(813\) −954.727 224.304i −1.17433 0.275897i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −161.676 + 93.3437i −0.197890 + 0.114252i
\(818\) 0 0
\(819\) −816.725 406.185i −0.997223 0.495952i
\(820\) 0 0
\(821\) 778.064 449.215i 0.947702 0.547156i 0.0553360 0.998468i \(-0.482377\pi\)
0.892366 + 0.451312i \(0.149044\pi\)
\(822\) 0 0
\(823\) 187.219 + 108.091i 0.227484 + 0.131338i 0.609411 0.792855i \(-0.291406\pi\)
−0.381927 + 0.924193i \(0.624739\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −113.883 −0.137706 −0.0688528 0.997627i \(-0.521934\pi\)
−0.0688528 + 0.997627i \(0.521934\pi\)
\(828\) 0 0
\(829\) −101.326 −0.122227 −0.0611135 0.998131i \(-0.519465\pi\)
−0.0611135 + 0.998131i \(0.519465\pi\)
\(830\) 0 0
\(831\) −1371.38 + 413.487i −1.65028 + 0.497578i
\(832\) 0 0
\(833\) 136.731 236.825i 0.164143 0.284303i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −743.071 + 275.401i −0.887779 + 0.329033i
\(838\) 0 0
\(839\) −60.4689 + 34.9117i −0.0720726 + 0.0416111i −0.535603 0.844470i \(-0.679916\pi\)
0.463531 + 0.886081i \(0.346582\pi\)
\(840\) 0 0
\(841\) 984.588 1705.36i 1.17073 2.02777i
\(842\) 0 0
\(843\) −341.824 + 103.064i −0.405485 + 0.122258i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1101.35i 1.30030i
\(848\) 0 0
\(849\) −855.547 804.012i −1.00771 0.947011i
\(850\) 0 0
\(851\) 377.502 + 217.951i 0.443598 + 0.256112i
\(852\) 0 0
\(853\) 993.028 573.325i 1.16416 0.672127i 0.211862 0.977300i \(-0.432047\pi\)
0.952297 + 0.305172i \(0.0987140\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 125.788 + 217.871i 0.146777 + 0.254225i 0.930035 0.367472i \(-0.119777\pi\)
−0.783258 + 0.621697i \(0.786443\pi\)
\(858\) 0 0
\(859\) −244.266 + 423.082i −0.284361 + 0.492528i −0.972454 0.233095i \(-0.925115\pi\)
0.688093 + 0.725623i \(0.258448\pi\)
\(860\) 0 0
\(861\) −1198.03 281.467i −1.39145 0.326908i
\(862\) 0 0
\(863\) −596.889 −0.691644 −0.345822 0.938300i \(-0.612400\pi\)
−0.345822 + 0.938300i \(0.612400\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −656.386 154.212i −0.757077 0.177868i
\(868\) 0 0
\(869\) −0.607003 0.350454i −0.000698508 0.000403284i
\(870\) 0 0
\(871\) −609.765 1056.14i −0.700074 1.21256i
\(872\) 0 0
\(873\) 1035.10 64.3490i 1.18569 0.0737102i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −620.434 358.208i −0.707451 0.408447i 0.102666 0.994716i \(-0.467263\pi\)
−0.810116 + 0.586269i \(0.800596\pi\)
\(878\) 0 0
\(879\) 147.297 156.739i 0.167574 0.178315i
\(880\) 0 0
\(881\) 1200.86i 1.36306i 0.731790 + 0.681531i \(0.238685\pi\)
−0.731790 + 0.681531i \(0.761315\pi\)
\(882\) 0 0
\(883\) 22.8938i 0.0259273i −0.999916 0.0129636i \(-0.995873\pi\)
0.999916 0.0129636i \(-0.00412657\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −301.294 + 521.857i −0.339678 + 0.588340i −0.984372 0.176101i \(-0.943651\pi\)
0.644694 + 0.764441i \(0.276985\pi\)
\(888\) 0 0
\(889\) 839.220 + 1453.57i 0.944005 + 1.63506i
\(890\) 0 0
\(891\) −33.0160 13.9366i −0.0370549 0.0156415i
\(892\) 0 0
\(893\) −34.3256 59.4537i −0.0384385 0.0665775i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −753.689 + 227.246i −0.840233 + 0.253340i
\(898\) 0 0
\(899\) 1555.90i 1.73071i
\(900\) 0 0
\(901\) 485.609 0.538966
\(902\) 0 0
\(903\) −430.794 + 458.406i −0.477069 + 0.507648i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 366.281 211.473i 0.403838 0.233156i −0.284300 0.958735i \(-0.591761\pi\)
0.688139 + 0.725579i \(0.258428\pi\)
\(908\) 0 0
\(909\) 1188.23 73.8684i 1.30719 0.0812634i
\(910\) 0 0
\(911\) 125.376 72.3861i 0.137625 0.0794578i −0.429607 0.903016i \(-0.641348\pi\)
0.567232 + 0.823558i \(0.308014\pi\)
\(912\) 0 0
\(913\) −3.24069 1.87101i −0.00354949 0.00204930i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1149.51 −1.25356
\(918\) 0 0
\(919\) 869.093 0.945694 0.472847 0.881145i \(-0.343226\pi\)
0.472847 + 0.881145i \(0.343226\pi\)
\(920\) 0 0
\(921\) 118.257 503.347i 0.128400 0.546522i
\(922\) 0 0
\(923\) 89.1064 154.337i 0.0965400 0.167212i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 623.611 + 940.129i 0.672720 + 1.01416i
\(928\) 0 0
\(929\) −82.2838 + 47.5066i −0.0885724 + 0.0511373i −0.543632 0.839324i \(-0.682951\pi\)
0.455060 + 0.890461i \(0.349618\pi\)
\(930\) 0 0
\(931\) 138.461 239.821i 0.148722 0.257595i
\(932\) 0 0
\(933\) −1323.37 1243.66i −1.41841 1.33297i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1555.55i 1.66014i −0.557657 0.830071i \(-0.688300\pi\)
0.557657 0.830071i \(-0.311700\pi\)
\(938\) 0 0
\(939\) −940.578 + 283.595i −1.00168 + 0.302018i
\(940\) 0 0
\(941\) 335.246 + 193.554i 0.356265 + 0.205690i 0.667441 0.744662i \(-0.267389\pi\)
−0.311176 + 0.950352i \(0.600723\pi\)
\(942\) 0 0
\(943\) −919.782 + 531.036i −0.975379 + 0.563135i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −484.894 839.861i −0.512032 0.886865i −0.999903 0.0139492i \(-0.995560\pi\)
0.487871 0.872916i \(-0.337774\pi\)
\(948\) 0 0
\(949\) 24.1821 41.8846i 0.0254817 0.0441355i
\(950\) 0 0
\(951\) −59.7921 198.308i −0.0628729 0.208526i
\(952\) 0 0
\(953\) −1294.65 −1.35850 −0.679248 0.733909i \(-0.737694\pi\)
−0.679248 + 0.733909i \(0.737694\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −48.1846 + 51.2731i −0.0503497 + 0.0535769i
\(958\) 0 0
\(959\) 981.126 + 566.453i 1.02307 + 0.590671i
\(960\) 0 0
\(961\) 49.7731 + 86.2096i 0.0517931 + 0.0897082i
\(962\) 0 0
\(963\) −294.437 146.433i −0.305750 0.152059i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −713.668 412.036i −0.738023 0.426098i 0.0833272 0.996522i \(-0.473445\pi\)
−0.821350 + 0.570425i \(0.806779\pi\)
\(968\) 0 0
\(969\) 190.007 + 44.6404i 0.196085 + 0.0460685i
\(970\) 0 0
\(971\) 1518.35i 1.56370i 0.623469 + 0.781848i \(0.285723\pi\)
−0.623469 + 0.781848i \(0.714277\pi\)
\(972\) 0 0
\(973\) 244.024i 0.250796i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 185.616 321.497i 0.189986 0.329065i −0.755259 0.655426i \(-0.772489\pi\)
0.945245 + 0.326361i \(0.105822\pi\)
\(978\) 0 0
\(979\) 14.1851 + 24.5693i 0.0144894 + 0.0250963i
\(980\) 0 0
\(981\) −1085.48 539.844i −1.10650 0.550300i
\(982\) 0 0
\(983\) 466.522 + 808.039i 0.474590 + 0.822014i 0.999577 0.0290967i \(-0.00926307\pi\)
−0.524987 + 0.851110i \(0.675930\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −168.571 158.417i −0.170792 0.160504i
\(988\) 0 0
\(989\) 542.890i 0.548928i
\(990\) 0 0
\(991\) 1615.53 1.63020 0.815099 0.579321i \(-0.196682\pi\)
0.815099 + 0.579321i \(0.196682\pi\)
\(992\) 0 0
\(993\) −441.492 1464.26i −0.444604 1.47458i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 899.586 519.376i 0.902293 0.520939i 0.0243496 0.999704i \(-0.492249\pi\)
0.877943 + 0.478764i \(0.158915\pi\)
\(998\) 0 0
\(999\) −318.261 + 383.837i −0.318580 + 0.384221i
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 900.3.u.a.149.4 8
3.2 odd 2 2700.3.u.b.449.4 8
5.2 odd 4 900.3.p.a.401.2 4
5.3 odd 4 36.3.g.a.5.1 4
5.4 even 2 inner 900.3.u.a.149.1 8
9.2 odd 6 inner 900.3.u.a.749.1 8
9.7 even 3 2700.3.u.b.2249.1 8
15.2 even 4 2700.3.p.b.2501.2 4
15.8 even 4 108.3.g.a.17.1 4
15.14 odd 2 2700.3.u.b.449.1 8
20.3 even 4 144.3.q.b.113.2 4
40.3 even 4 576.3.q.g.257.1 4
40.13 odd 4 576.3.q.d.257.2 4
45.2 even 12 900.3.p.a.101.2 4
45.7 odd 12 2700.3.p.b.1601.2 4
45.13 odd 12 324.3.c.b.161.1 4
45.23 even 12 324.3.c.b.161.4 4
45.29 odd 6 inner 900.3.u.a.749.4 8
45.34 even 6 2700.3.u.b.2249.4 8
45.38 even 12 36.3.g.a.29.1 yes 4
45.43 odd 12 108.3.g.a.89.1 4
60.23 odd 4 432.3.q.b.17.1 4
120.53 even 4 1728.3.q.g.449.2 4
120.83 odd 4 1728.3.q.h.449.2 4
180.23 odd 12 1296.3.e.e.161.4 4
180.43 even 12 432.3.q.b.305.1 4
180.83 odd 12 144.3.q.b.65.2 4
180.103 even 12 1296.3.e.e.161.1 4
360.43 even 12 1728.3.q.h.1601.2 4
360.83 odd 12 576.3.q.g.65.1 4
360.133 odd 12 1728.3.q.g.1601.2 4
360.173 even 12 576.3.q.d.65.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.g.a.5.1 4 5.3 odd 4
36.3.g.a.29.1 yes 4 45.38 even 12
108.3.g.a.17.1 4 15.8 even 4
108.3.g.a.89.1 4 45.43 odd 12
144.3.q.b.65.2 4 180.83 odd 12
144.3.q.b.113.2 4 20.3 even 4
324.3.c.b.161.1 4 45.13 odd 12
324.3.c.b.161.4 4 45.23 even 12
432.3.q.b.17.1 4 60.23 odd 4
432.3.q.b.305.1 4 180.43 even 12
576.3.q.d.65.2 4 360.173 even 12
576.3.q.d.257.2 4 40.13 odd 4
576.3.q.g.65.1 4 360.83 odd 12
576.3.q.g.257.1 4 40.3 even 4
900.3.p.a.101.2 4 45.2 even 12
900.3.p.a.401.2 4 5.2 odd 4
900.3.u.a.149.1 8 5.4 even 2 inner
900.3.u.a.149.4 8 1.1 even 1 trivial
900.3.u.a.749.1 8 9.2 odd 6 inner
900.3.u.a.749.4 8 45.29 odd 6 inner
1296.3.e.e.161.1 4 180.103 even 12
1296.3.e.e.161.4 4 180.23 odd 12
1728.3.q.g.449.2 4 120.53 even 4
1728.3.q.g.1601.2 4 360.133 odd 12
1728.3.q.h.449.2 4 120.83 odd 4
1728.3.q.h.1601.2 4 360.43 even 12
2700.3.p.b.1601.2 4 45.7 odd 12
2700.3.p.b.2501.2 4 15.2 even 4
2700.3.u.b.449.1 8 15.14 odd 2
2700.3.u.b.449.4 8 3.2 odd 2
2700.3.u.b.2249.1 8 9.7 even 3
2700.3.u.b.2249.4 8 45.34 even 6