Properties

Label 504.2.c.c.253.3
Level $504$
Weight $2$
Character 504.253
Analytic conductor $4.024$
Analytic rank $0$
Dimension $4$
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] = [504,2,Mod(253,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, 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("504.253");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.c (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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 253.3
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 504.253
Dual form 504.2.c.c.253.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -1.41421i q^{5} +1.00000 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -1.41421i q^{5} +1.00000 q^{7} -2.82843i q^{8} +(-1.00000 - 1.73205i) q^{10} -1.41421i q^{11} +(1.22474 - 0.707107i) q^{14} +(-2.00000 - 3.46410i) q^{16} -2.44949 q^{17} +3.46410i q^{19} +(-2.44949 - 1.41421i) q^{20} +(-1.00000 - 1.73205i) q^{22} +2.44949 q^{23} +3.00000 q^{25} +(1.00000 - 1.73205i) q^{28} -5.65685i q^{29} +2.00000 q^{31} +(-4.89898 - 2.82843i) q^{32} +(-3.00000 + 1.73205i) q^{34} -1.41421i q^{35} +3.46410i q^{37} +(2.44949 + 4.24264i) q^{38} -4.00000 q^{40} -7.34847 q^{41} +10.3923i q^{43} +(-2.44949 - 1.41421i) q^{44} +(3.00000 - 1.73205i) q^{46} +9.79796 q^{47} +1.00000 q^{49} +(3.67423 - 2.12132i) q^{50} -5.65685i q^{53} -2.00000 q^{55} -2.82843i q^{56} +(-4.00000 - 6.92820i) q^{58} +11.3137i q^{59} +13.8564i q^{61} +(2.44949 - 1.41421i) q^{62} -8.00000 q^{64} -3.46410i q^{67} +(-2.44949 + 4.24264i) q^{68} +(-1.00000 - 1.73205i) q^{70} +7.34847 q^{71} +2.00000 q^{73} +(2.44949 + 4.24264i) q^{74} +(6.00000 + 3.46410i) q^{76} -1.41421i q^{77} +8.00000 q^{79} +(-4.89898 + 2.82843i) q^{80} +(-9.00000 + 5.19615i) q^{82} +2.82843i q^{83} +3.46410i q^{85} +(7.34847 + 12.7279i) q^{86} -4.00000 q^{88} -12.2474 q^{89} +(2.44949 - 4.24264i) q^{92} +(12.0000 - 6.92820i) q^{94} +4.89898 q^{95} -10.0000 q^{97} +(1.22474 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{7} - 4 q^{10} - 8 q^{16} - 4 q^{22} + 12 q^{25} + 4 q^{28} + 8 q^{31} - 12 q^{34} - 16 q^{40} + 12 q^{46} + 4 q^{49} - 8 q^{55} - 16 q^{58} - 32 q^{64} - 4 q^{70} + 8 q^{73} + 24 q^{76} + 32 q^{79} - 36 q^{82} - 16 q^{88} + 48 q^{94} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22474 0.707107i 0.866025 0.500000i
\(3\) 0 0
\(4\) 1.00000 1.73205i 0.500000 0.866025i
\(5\) 1.41421i 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) −1.00000 1.73205i −0.316228 0.547723i
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.22474 0.707107i 0.327327 0.188982i
\(15\) 0 0
\(16\) −2.00000 3.46410i −0.500000 0.866025i
\(17\) −2.44949 −0.594089 −0.297044 0.954864i \(-0.596001\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −2.44949 1.41421i −0.547723 0.316228i
\(21\) 0 0
\(22\) −1.00000 1.73205i −0.213201 0.369274i
\(23\) 2.44949 0.510754 0.255377 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.00000 1.73205i 0.188982 0.327327i
\(29\) 5.65685i 1.05045i −0.850963 0.525226i \(-0.823981\pi\)
0.850963 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −4.89898 2.82843i −0.866025 0.500000i
\(33\) 0 0
\(34\) −3.00000 + 1.73205i −0.514496 + 0.297044i
\(35\) 1.41421i 0.239046i
\(36\) 0 0
\(37\) 3.46410i 0.569495i 0.958603 + 0.284747i \(0.0919097\pi\)
−0.958603 + 0.284747i \(0.908090\pi\)
\(38\) 2.44949 + 4.24264i 0.397360 + 0.688247i
\(39\) 0 0
\(40\) −4.00000 −0.632456
\(41\) −7.34847 −1.14764 −0.573819 0.818982i \(-0.694539\pi\)
−0.573819 + 0.818982i \(0.694539\pi\)
\(42\) 0 0
\(43\) 10.3923i 1.58481i 0.609994 + 0.792406i \(0.291172\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −2.44949 1.41421i −0.369274 0.213201i
\(45\) 0 0
\(46\) 3.00000 1.73205i 0.442326 0.255377i
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.67423 2.12132i 0.519615 0.300000i
\(51\) 0 0
\(52\) 0 0
\(53\) 5.65685i 0.777029i −0.921443 0.388514i \(-0.872988\pi\)
0.921443 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 2.82843i 0.377964i
\(57\) 0 0
\(58\) −4.00000 6.92820i −0.525226 0.909718i
\(59\) 11.3137i 1.47292i 0.676481 + 0.736460i \(0.263504\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 2.44949 1.41421i 0.311086 0.179605i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410i 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) −2.44949 + 4.24264i −0.297044 + 0.514496i
\(69\) 0 0
\(70\) −1.00000 1.73205i −0.119523 0.207020i
\(71\) 7.34847 0.872103 0.436051 0.899922i \(-0.356377\pi\)
0.436051 + 0.899922i \(0.356377\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.44949 + 4.24264i 0.284747 + 0.493197i
\(75\) 0 0
\(76\) 6.00000 + 3.46410i 0.688247 + 0.397360i
\(77\) 1.41421i 0.161165i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −4.89898 + 2.82843i −0.547723 + 0.316228i
\(81\) 0 0
\(82\) −9.00000 + 5.19615i −0.993884 + 0.573819i
\(83\) 2.82843i 0.310460i 0.987878 + 0.155230i \(0.0496119\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) 3.46410i 0.375735i
\(86\) 7.34847 + 12.7279i 0.792406 + 1.37249i
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −12.2474 −1.29823 −0.649113 0.760692i \(-0.724860\pi\)
−0.649113 + 0.760692i \(0.724860\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.44949 4.24264i 0.255377 0.442326i
\(93\) 0 0
\(94\) 12.0000 6.92820i 1.23771 0.714590i
\(95\) 4.89898 0.502625
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.22474 0.707107i 0.123718 0.0714286i
\(99\) 0 0
\(100\) 3.00000 5.19615i 0.300000 0.519615i
\(101\) 15.5563i 1.54791i 0.633238 + 0.773957i \(0.281726\pi\)
−0.633238 + 0.773957i \(0.718274\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −4.00000 6.92820i −0.388514 0.672927i
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i −0.748086 0.663602i \(-0.769027\pi\)
0.748086 0.663602i \(-0.230973\pi\)
\(110\) −2.44949 + 1.41421i −0.233550 + 0.134840i
\(111\) 0 0
\(112\) −2.00000 3.46410i −0.188982 0.327327i
\(113\) −14.6969 −1.38257 −0.691286 0.722581i \(-0.742955\pi\)
−0.691286 + 0.722581i \(0.742955\pi\)
\(114\) 0 0
\(115\) 3.46410i 0.323029i
\(116\) −9.79796 5.65685i −0.909718 0.525226i
\(117\) 0 0
\(118\) 8.00000 + 13.8564i 0.736460 + 1.27559i
\(119\) −2.44949 −0.224544
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 9.79796 + 16.9706i 0.887066 + 1.53644i
\(123\) 0 0
\(124\) 2.00000 3.46410i 0.179605 0.311086i
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −9.79796 + 5.65685i −0.866025 + 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 19.7990i 1.72985i 0.501905 + 0.864923i \(0.332633\pi\)
−0.501905 + 0.864923i \(0.667367\pi\)
\(132\) 0 0
\(133\) 3.46410i 0.300376i
\(134\) −2.44949 4.24264i −0.211604 0.366508i
\(135\) 0 0
\(136\) 6.92820i 0.594089i
\(137\) 4.89898 0.418548 0.209274 0.977857i \(-0.432890\pi\)
0.209274 + 0.977857i \(0.432890\pi\)
\(138\) 0 0
\(139\) 20.7846i 1.76293i −0.472252 0.881464i \(-0.656559\pi\)
0.472252 0.881464i \(-0.343441\pi\)
\(140\) −2.44949 1.41421i −0.207020 0.119523i
\(141\) 0 0
\(142\) 9.00000 5.19615i 0.755263 0.436051i
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 2.44949 1.41421i 0.202721 0.117041i
\(147\) 0 0
\(148\) 6.00000 + 3.46410i 0.493197 + 0.284747i
\(149\) 11.3137i 0.926855i 0.886135 + 0.463428i \(0.153381\pi\)
−0.886135 + 0.463428i \(0.846619\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 9.79796 0.794719
\(153\) 0 0
\(154\) −1.00000 1.73205i −0.0805823 0.139573i
\(155\) 2.82843i 0.227185i
\(156\) 0 0
\(157\) 13.8564i 1.10586i −0.833227 0.552931i \(-0.813509\pi\)
0.833227 0.552931i \(-0.186491\pi\)
\(158\) 9.79796 5.65685i 0.779484 0.450035i
\(159\) 0 0
\(160\) −4.00000 + 6.92820i −0.316228 + 0.547723i
\(161\) 2.44949 0.193047
\(162\) 0 0
\(163\) 3.46410i 0.271329i 0.990755 + 0.135665i \(0.0433170\pi\)
−0.990755 + 0.135665i \(0.956683\pi\)
\(164\) −7.34847 + 12.7279i −0.573819 + 0.993884i
\(165\) 0 0
\(166\) 2.00000 + 3.46410i 0.155230 + 0.268866i
\(167\) 14.6969 1.13728 0.568642 0.822585i \(-0.307469\pi\)
0.568642 + 0.822585i \(0.307469\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 2.44949 + 4.24264i 0.187867 + 0.325396i
\(171\) 0 0
\(172\) 18.0000 + 10.3923i 1.37249 + 0.792406i
\(173\) 9.89949i 0.752645i −0.926489 0.376322i \(-0.877189\pi\)
0.926489 0.376322i \(-0.122811\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) −4.89898 + 2.82843i −0.369274 + 0.213201i
\(177\) 0 0
\(178\) −15.0000 + 8.66025i −1.12430 + 0.649113i
\(179\) 18.3848i 1.37414i −0.726590 0.687071i \(-0.758896\pi\)
0.726590 0.687071i \(-0.241104\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i −0.635071 0.772454i \(-0.719029\pi\)
0.635071 0.772454i \(-0.280971\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.92820i 0.510754i
\(185\) 4.89898 0.360180
\(186\) 0 0
\(187\) 3.46410i 0.253320i
\(188\) 9.79796 16.9706i 0.714590 1.23771i
\(189\) 0 0
\(190\) 6.00000 3.46410i 0.435286 0.251312i
\(191\) −12.2474 −0.886194 −0.443097 0.896474i \(-0.646120\pi\)
−0.443097 + 0.896474i \(0.646120\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −12.2474 + 7.07107i −0.879316 + 0.507673i
\(195\) 0 0
\(196\) 1.00000 1.73205i 0.0714286 0.123718i
\(197\) 14.1421i 1.00759i −0.863825 0.503793i \(-0.831938\pi\)
0.863825 0.503793i \(-0.168062\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 8.48528i 0.600000i
\(201\) 0 0
\(202\) 11.0000 + 19.0526i 0.773957 + 1.34053i
\(203\) 5.65685i 0.397033i
\(204\) 0 0
\(205\) 10.3923i 0.725830i
\(206\) 2.44949 1.41421i 0.170664 0.0985329i
\(207\) 0 0
\(208\) 0 0
\(209\) 4.89898 0.338869
\(210\) 0 0
\(211\) 10.3923i 0.715436i 0.933830 + 0.357718i \(0.116445\pi\)
−0.933830 + 0.357718i \(0.883555\pi\)
\(212\) −9.79796 5.65685i −0.672927 0.388514i
\(213\) 0 0
\(214\) 5.00000 + 8.66025i 0.341793 + 0.592003i
\(215\) 14.6969 1.00232
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −9.79796 16.9706i −0.663602 1.14939i
\(219\) 0 0
\(220\) −2.00000 + 3.46410i −0.134840 + 0.233550i
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −4.89898 2.82843i −0.327327 0.188982i
\(225\) 0 0
\(226\) −18.0000 + 10.3923i −1.19734 + 0.691286i
\(227\) 2.82843i 0.187729i 0.995585 + 0.0938647i \(0.0299221\pi\)
−0.995585 + 0.0938647i \(0.970078\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i −0.973447 0.228914i \(-0.926482\pi\)
0.973447 0.228914i \(-0.0735176\pi\)
\(230\) −2.44949 4.24264i −0.161515 0.279751i
\(231\) 0 0
\(232\) −16.0000 −1.05045
\(233\) 9.79796 0.641886 0.320943 0.947099i \(-0.396000\pi\)
0.320943 + 0.947099i \(0.396000\pi\)
\(234\) 0 0
\(235\) 13.8564i 0.903892i
\(236\) 19.5959 + 11.3137i 1.27559 + 0.736460i
\(237\) 0 0
\(238\) −3.00000 + 1.73205i −0.194461 + 0.112272i
\(239\) 7.34847 0.475333 0.237666 0.971347i \(-0.423617\pi\)
0.237666 + 0.971347i \(0.423617\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 11.0227 6.36396i 0.708566 0.409091i
\(243\) 0 0
\(244\) 24.0000 + 13.8564i 1.53644 + 0.887066i
\(245\) 1.41421i 0.0903508i
\(246\) 0 0
\(247\) 0 0
\(248\) 5.65685i 0.359211i
\(249\) 0 0
\(250\) −8.00000 13.8564i −0.505964 0.876356i
\(251\) 14.1421i 0.892644i −0.894873 0.446322i \(-0.852734\pi\)
0.894873 0.446322i \(-0.147266\pi\)
\(252\) 0 0
\(253\) 3.46410i 0.217786i
\(254\) −4.89898 + 2.82843i −0.307389 + 0.177471i
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) −26.9444 −1.68074 −0.840372 0.542010i \(-0.817664\pi\)
−0.840372 + 0.542010i \(0.817664\pi\)
\(258\) 0 0
\(259\) 3.46410i 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 14.0000 + 24.2487i 0.864923 + 1.49809i
\(263\) −31.8434 −1.96355 −0.981773 0.190057i \(-0.939133\pi\)
−0.981773 + 0.190057i \(0.939133\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 2.44949 + 4.24264i 0.150188 + 0.260133i
\(267\) 0 0
\(268\) −6.00000 3.46410i −0.366508 0.211604i
\(269\) 7.07107i 0.431131i 0.976489 + 0.215565i \(0.0691594\pi\)
−0.976489 + 0.215565i \(0.930841\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 4.89898 + 8.48528i 0.297044 + 0.514496i
\(273\) 0 0
\(274\) 6.00000 3.46410i 0.362473 0.209274i
\(275\) 4.24264i 0.255841i
\(276\) 0 0
\(277\) 3.46410i 0.208138i −0.994570 0.104069i \(-0.966814\pi\)
0.994570 0.104069i \(-0.0331862\pi\)
\(278\) −14.6969 25.4558i −0.881464 1.52674i
\(279\) 0 0
\(280\) −4.00000 −0.239046
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 3.46410i 0.205919i −0.994686 0.102960i \(-0.967169\pi\)
0.994686 0.102960i \(-0.0328313\pi\)
\(284\) 7.34847 12.7279i 0.436051 0.755263i
\(285\) 0 0
\(286\) 0 0
\(287\) −7.34847 −0.433766
\(288\) 0 0
\(289\) −11.0000 −0.647059
\(290\) −9.79796 + 5.65685i −0.575356 + 0.332182i
\(291\) 0 0
\(292\) 2.00000 3.46410i 0.117041 0.202721i
\(293\) 9.89949i 0.578335i −0.957279 0.289167i \(-0.906622\pi\)
0.957279 0.289167i \(-0.0933784\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 9.79796 0.569495
\(297\) 0 0
\(298\) 8.00000 + 13.8564i 0.463428 + 0.802680i
\(299\) 0 0
\(300\) 0 0
\(301\) 10.3923i 0.599002i
\(302\) −4.89898 + 2.82843i −0.281905 + 0.162758i
\(303\) 0 0
\(304\) 12.0000 6.92820i 0.688247 0.397360i
\(305\) 19.5959 1.12206
\(306\) 0 0
\(307\) 31.1769i 1.77936i −0.456584 0.889680i \(-0.650927\pi\)
0.456584 0.889680i \(-0.349073\pi\)
\(308\) −2.44949 1.41421i −0.139573 0.0805823i
\(309\) 0 0
\(310\) −2.00000 3.46410i −0.113592 0.196748i
\(311\) −9.79796 −0.555591 −0.277796 0.960640i \(-0.589604\pi\)
−0.277796 + 0.960640i \(0.589604\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −9.79796 16.9706i −0.552931 0.957704i
\(315\) 0 0
\(316\) 8.00000 13.8564i 0.450035 0.779484i
\(317\) 11.3137i 0.635441i 0.948184 + 0.317721i \(0.102917\pi\)
−0.948184 + 0.317721i \(0.897083\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 11.3137i 0.632456i
\(321\) 0 0
\(322\) 3.00000 1.73205i 0.167183 0.0965234i
\(323\) 8.48528i 0.472134i
\(324\) 0 0
\(325\) 0 0
\(326\) 2.44949 + 4.24264i 0.135665 + 0.234978i
\(327\) 0 0
\(328\) 20.7846i 1.14764i
\(329\) 9.79796 0.540179
\(330\) 0 0
\(331\) 24.2487i 1.33283i 0.745581 + 0.666415i \(0.232172\pi\)
−0.745581 + 0.666415i \(0.767828\pi\)
\(332\) 4.89898 + 2.82843i 0.268866 + 0.155230i
\(333\) 0 0
\(334\) 18.0000 10.3923i 0.984916 0.568642i
\(335\) −4.89898 −0.267660
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 15.9217 9.19239i 0.866025 0.500000i
\(339\) 0 0
\(340\) 6.00000 + 3.46410i 0.325396 + 0.187867i
\(341\) 2.82843i 0.153168i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 29.3939 1.58481
\(345\) 0 0
\(346\) −7.00000 12.1244i −0.376322 0.651809i
\(347\) 1.41421i 0.0759190i −0.999279 0.0379595i \(-0.987914\pi\)
0.999279 0.0379595i \(-0.0120858\pi\)
\(348\) 0 0
\(349\) 20.7846i 1.11257i −0.830990 0.556287i \(-0.812225\pi\)
0.830990 0.556287i \(-0.187775\pi\)
\(350\) 3.67423 2.12132i 0.196396 0.113389i
\(351\) 0 0
\(352\) −4.00000 + 6.92820i −0.213201 + 0.369274i
\(353\) 26.9444 1.43411 0.717053 0.697019i \(-0.245491\pi\)
0.717053 + 0.697019i \(0.245491\pi\)
\(354\) 0 0
\(355\) 10.3923i 0.551566i
\(356\) −12.2474 + 21.2132i −0.649113 + 1.12430i
\(357\) 0 0
\(358\) −13.0000 22.5167i −0.687071 1.19004i
\(359\) −26.9444 −1.42207 −0.711035 0.703156i \(-0.751773\pi\)
−0.711035 + 0.703156i \(0.751773\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) −14.6969 25.4558i −0.772454 1.33793i
\(363\) 0 0
\(364\) 0 0
\(365\) 2.82843i 0.148047i
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −4.89898 8.48528i −0.255377 0.442326i
\(369\) 0 0
\(370\) 6.00000 3.46410i 0.311925 0.180090i
\(371\) 5.65685i 0.293689i
\(372\) 0 0
\(373\) 13.8564i 0.717458i 0.933442 + 0.358729i \(0.116790\pi\)
−0.933442 + 0.358729i \(0.883210\pi\)
\(374\) 2.44949 + 4.24264i 0.126660 + 0.219382i
\(375\) 0 0
\(376\) 27.7128i 1.42918i
\(377\) 0 0
\(378\) 0 0
\(379\) 10.3923i 0.533817i −0.963722 0.266908i \(-0.913998\pi\)
0.963722 0.266908i \(-0.0860021\pi\)
\(380\) 4.89898 8.48528i 0.251312 0.435286i
\(381\) 0 0
\(382\) −15.0000 + 8.66025i −0.767467 + 0.443097i
\(383\) 24.4949 1.25163 0.625815 0.779971i \(-0.284766\pi\)
0.625815 + 0.779971i \(0.284766\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) −19.5959 + 11.3137i −0.997406 + 0.575853i
\(387\) 0 0
\(388\) −10.0000 + 17.3205i −0.507673 + 0.879316i
\(389\) 28.2843i 1.43407i 0.697037 + 0.717035i \(0.254501\pi\)
−0.697037 + 0.717035i \(0.745499\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 2.82843i 0.142857i
\(393\) 0 0
\(394\) −10.0000 17.3205i −0.503793 0.872595i
\(395\) 11.3137i 0.569254i
\(396\) 0 0
\(397\) 6.92820i 0.347717i −0.984771 0.173858i \(-0.944377\pi\)
0.984771 0.173858i \(-0.0556235\pi\)
\(398\) −19.5959 + 11.3137i −0.982255 + 0.567105i
\(399\) 0 0
\(400\) −6.00000 10.3923i −0.300000 0.519615i
\(401\) 24.4949 1.22322 0.611608 0.791161i \(-0.290523\pi\)
0.611608 + 0.791161i \(0.290523\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 26.9444 + 15.5563i 1.34053 + 0.773957i
\(405\) 0 0
\(406\) −4.00000 6.92820i −0.198517 0.343841i
\(407\) 4.89898 0.242833
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 7.34847 + 12.7279i 0.362915 + 0.628587i
\(411\) 0 0
\(412\) 2.00000 3.46410i 0.0985329 0.170664i
\(413\) 11.3137i 0.556711i
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000 3.46410i 0.293470 0.169435i
\(419\) 11.3137i 0.552711i 0.961056 + 0.276355i \(0.0891267\pi\)
−0.961056 + 0.276355i \(0.910873\pi\)
\(420\) 0 0
\(421\) 31.1769i 1.51947i 0.650233 + 0.759735i \(0.274671\pi\)
−0.650233 + 0.759735i \(0.725329\pi\)
\(422\) 7.34847 + 12.7279i 0.357718 + 0.619586i
\(423\) 0 0
\(424\) −16.0000 −0.777029
\(425\) −7.34847 −0.356453
\(426\) 0 0
\(427\) 13.8564i 0.670559i
\(428\) 12.2474 + 7.07107i 0.592003 + 0.341793i
\(429\) 0 0
\(430\) 18.0000 10.3923i 0.868037 0.501161i
\(431\) −31.8434 −1.53384 −0.766921 0.641742i \(-0.778212\pi\)
−0.766921 + 0.641742i \(0.778212\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 2.44949 1.41421i 0.117579 0.0678844i
\(435\) 0 0
\(436\) −24.0000 13.8564i −1.14939 0.663602i
\(437\) 8.48528i 0.405906i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 5.65685i 0.269680i
\(441\) 0 0
\(442\) 0 0
\(443\) 35.3553i 1.67978i −0.542754 0.839891i \(-0.682619\pi\)
0.542754 0.839891i \(-0.317381\pi\)
\(444\) 0 0
\(445\) 17.3205i 0.821071i
\(446\) −19.5959 + 11.3137i −0.927894 + 0.535720i
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) 14.6969 0.693591 0.346796 0.937941i \(-0.387270\pi\)
0.346796 + 0.937941i \(0.387270\pi\)
\(450\) 0 0
\(451\) 10.3923i 0.489355i
\(452\) −14.6969 + 25.4558i −0.691286 + 1.19734i
\(453\) 0 0
\(454\) 2.00000 + 3.46410i 0.0938647 + 0.162578i
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −4.89898 8.48528i −0.228914 0.396491i
\(459\) 0 0
\(460\) −6.00000 3.46410i −0.279751 0.161515i
\(461\) 32.5269i 1.51493i 0.652876 + 0.757465i \(0.273562\pi\)
−0.652876 + 0.757465i \(0.726438\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −19.5959 + 11.3137i −0.909718 + 0.525226i
\(465\) 0 0
\(466\) 12.0000 6.92820i 0.555889 0.320943i
\(467\) 19.7990i 0.916188i 0.888904 + 0.458094i \(0.151468\pi\)
−0.888904 + 0.458094i \(0.848532\pi\)
\(468\) 0 0
\(469\) 3.46410i 0.159957i
\(470\) −9.79796 16.9706i −0.451946 0.782794i
\(471\) 0 0
\(472\) 32.0000 1.47292
\(473\) 14.6969 0.675766
\(474\) 0 0
\(475\) 10.3923i 0.476832i
\(476\) −2.44949 + 4.24264i −0.112272 + 0.194461i
\(477\) 0 0
\(478\) 9.00000 5.19615i 0.411650 0.237666i
\(479\) −39.1918 −1.79072 −0.895360 0.445342i \(-0.853082\pi\)
−0.895360 + 0.445342i \(0.853082\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −12.2474 + 7.07107i −0.557856 + 0.322078i
\(483\) 0 0
\(484\) 9.00000 15.5885i 0.409091 0.708566i
\(485\) 14.1421i 0.642161i
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 39.1918 1.77413
\(489\) 0 0
\(490\) −1.00000 1.73205i −0.0451754 0.0782461i
\(491\) 41.0122i 1.85085i 0.378925 + 0.925427i \(0.376294\pi\)
−0.378925 + 0.925427i \(0.623706\pi\)
\(492\) 0 0
\(493\) 13.8564i 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 6.92820i −0.179605 0.311086i
\(497\) 7.34847 0.329624
\(498\) 0 0
\(499\) 24.2487i 1.08552i 0.839887 + 0.542761i \(0.182621\pi\)
−0.839887 + 0.542761i \(0.817379\pi\)
\(500\) −19.5959 11.3137i −0.876356 0.505964i
\(501\) 0 0
\(502\) −10.0000 17.3205i −0.446322 0.773052i
\(503\) −14.6969 −0.655304 −0.327652 0.944798i \(-0.606257\pi\)
−0.327652 + 0.944798i \(0.606257\pi\)
\(504\) 0 0
\(505\) 22.0000 0.978987
\(506\) −2.44949 4.24264i −0.108893 0.188608i
\(507\) 0 0
\(508\) −4.00000 + 6.92820i −0.177471 + 0.307389i
\(509\) 18.3848i 0.814891i −0.913230 0.407445i \(-0.866420\pi\)
0.913230 0.407445i \(-0.133580\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) −33.0000 + 19.0526i −1.45557 + 0.840372i
\(515\) 2.82843i 0.124635i
\(516\) 0 0
\(517\) 13.8564i 0.609404i
\(518\) 2.44949 + 4.24264i 0.107624 + 0.186411i
\(519\) 0 0
\(520\) 0 0
\(521\) 12.2474 0.536570 0.268285 0.963340i \(-0.413543\pi\)
0.268285 + 0.963340i \(0.413543\pi\)
\(522\) 0 0
\(523\) 13.8564i 0.605898i 0.953007 + 0.302949i \(0.0979712\pi\)
−0.953007 + 0.302949i \(0.902029\pi\)
\(524\) 34.2929 + 19.7990i 1.49809 + 0.864923i
\(525\) 0 0
\(526\) −39.0000 + 22.5167i −1.70048 + 0.981773i
\(527\) −4.89898 −0.213403
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) −9.79796 + 5.65685i −0.425596 + 0.245718i
\(531\) 0 0
\(532\) 6.00000 + 3.46410i 0.260133 + 0.150188i
\(533\) 0 0
\(534\) 0 0
\(535\) 10.0000 0.432338
\(536\) −9.79796 −0.423207
\(537\) 0 0
\(538\) 5.00000 + 8.66025i 0.215565 + 0.373370i
\(539\) 1.41421i 0.0609145i
\(540\) 0 0
\(541\) 24.2487i 1.04253i 0.853394 + 0.521267i \(0.174540\pi\)
−0.853394 + 0.521267i \(0.825460\pi\)
\(542\) 2.44949 1.41421i 0.105215 0.0607457i
\(543\) 0 0
\(544\) 12.0000 + 6.92820i 0.514496 + 0.297044i
\(545\) −19.5959 −0.839397
\(546\) 0 0
\(547\) 10.3923i 0.444343i 0.975008 + 0.222171i \(0.0713145\pi\)
−0.975008 + 0.222171i \(0.928686\pi\)
\(548\) 4.89898 8.48528i 0.209274 0.362473i
\(549\) 0 0
\(550\) −3.00000 5.19615i −0.127920 0.221565i
\(551\) 19.5959 0.834814
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −2.44949 4.24264i −0.104069 0.180253i
\(555\) 0 0
\(556\) −36.0000 20.7846i −1.52674 0.881464i
\(557\) 28.2843i 1.19844i 0.800583 + 0.599222i \(0.204523\pi\)
−0.800583 + 0.599222i \(0.795477\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.89898 + 2.82843i −0.207020 + 0.119523i
\(561\) 0 0
\(562\) 0 0
\(563\) 39.5980i 1.66886i −0.551117 0.834428i \(-0.685798\pi\)
0.551117 0.834428i \(-0.314202\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) −2.44949 4.24264i −0.102960 0.178331i
\(567\) 0 0
\(568\) 20.7846i 0.872103i
\(569\) 39.1918 1.64301 0.821504 0.570203i \(-0.193136\pi\)
0.821504 + 0.570203i \(0.193136\pi\)
\(570\) 0 0
\(571\) 17.3205i 0.724841i 0.932015 + 0.362420i \(0.118050\pi\)
−0.932015 + 0.362420i \(0.881950\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −9.00000 + 5.19615i −0.375653 + 0.216883i
\(575\) 7.34847 0.306452
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −13.4722 + 7.77817i −0.560369 + 0.323529i
\(579\) 0 0
\(580\) −8.00000 + 13.8564i −0.332182 + 0.575356i
\(581\) 2.82843i 0.117343i
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 5.65685i 0.234082i
\(585\) 0 0
\(586\) −7.00000 12.1244i −0.289167 0.500853i
\(587\) 19.7990i 0.817192i 0.912715 + 0.408596i \(0.133981\pi\)
−0.912715 + 0.408596i \(0.866019\pi\)
\(588\) 0 0
\(589\) 6.92820i 0.285472i
\(590\) 19.5959 11.3137i 0.806751 0.465778i
\(591\) 0 0
\(592\) 12.0000 6.92820i 0.493197 0.284747i
\(593\) −26.9444 −1.10647 −0.553237 0.833024i \(-0.686607\pi\)
−0.553237 + 0.833024i \(0.686607\pi\)
\(594\) 0 0
\(595\) 3.46410i 0.142014i
\(596\) 19.5959 + 11.3137i 0.802680 + 0.463428i
\(597\) 0 0
\(598\) 0 0
\(599\) −2.44949 −0.100083 −0.0500417 0.998747i \(-0.515935\pi\)
−0.0500417 + 0.998747i \(0.515935\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 7.34847 + 12.7279i 0.299501 + 0.518751i
\(603\) 0 0
\(604\) −4.00000 + 6.92820i −0.162758 + 0.281905i
\(605\) 12.7279i 0.517464i
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 9.79796 16.9706i 0.397360 0.688247i
\(609\) 0 0
\(610\) 24.0000 13.8564i 0.971732 0.561029i
\(611\) 0 0
\(612\) 0 0
\(613\) 13.8564i 0.559655i −0.960050 0.279827i \(-0.909723\pi\)
0.960050 0.279827i \(-0.0902773\pi\)
\(614\) −22.0454 38.1838i −0.889680 1.54097i
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −44.0908 −1.77503 −0.887515 0.460779i \(-0.847570\pi\)
−0.887515 + 0.460779i \(0.847570\pi\)
\(618\) 0 0
\(619\) 13.8564i 0.556936i −0.960446 0.278468i \(-0.910173\pi\)
0.960446 0.278468i \(-0.0898266\pi\)
\(620\) −4.89898 2.82843i −0.196748 0.113592i
\(621\) 0 0
\(622\) −12.0000 + 6.92820i −0.481156 + 0.277796i
\(623\) −12.2474 −0.490684
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 31.8434 18.3848i 1.27272 0.734803i
\(627\) 0 0
\(628\) −24.0000 13.8564i −0.957704 0.552931i
\(629\) 8.48528i 0.338330i
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 22.6274i 0.900070i
\(633\) 0 0
\(634\) 8.00000 + 13.8564i 0.317721 + 0.550308i
\(635\) 5.65685i 0.224485i
\(636\) 0 0
\(637\) 0 0
\(638\) −9.79796 + 5.65685i −0.387905 + 0.223957i
\(639\) 0 0
\(640\) 8.00000 + 13.8564i 0.316228 + 0.547723i
\(641\) 34.2929 1.35449 0.677243 0.735759i \(-0.263174\pi\)
0.677243 + 0.735759i \(0.263174\pi\)
\(642\) 0 0
\(643\) 10.3923i 0.409832i 0.978780 + 0.204916i \(0.0656922\pi\)
−0.978780 + 0.204916i \(0.934308\pi\)
\(644\) 2.44949 4.24264i 0.0965234 0.167183i
\(645\) 0 0
\(646\) −6.00000 10.3923i −0.236067 0.408880i
\(647\) 4.89898 0.192599 0.0962994 0.995352i \(-0.469299\pi\)
0.0962994 + 0.995352i \(0.469299\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 6.00000 + 3.46410i 0.234978 + 0.135665i
\(653\) 14.1421i 0.553425i −0.960953 0.276712i \(-0.910755\pi\)
0.960953 0.276712i \(-0.0892449\pi\)
\(654\) 0 0
\(655\) 28.0000 1.09405
\(656\) 14.6969 + 25.4558i 0.573819 + 0.993884i
\(657\) 0 0
\(658\) 12.0000 6.92820i 0.467809 0.270089i
\(659\) 18.3848i 0.716169i −0.933689 0.358085i \(-0.883430\pi\)
0.933689 0.358085i \(-0.116570\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i 0.990881 + 0.134738i \(0.0430193\pi\)
−0.990881 + 0.134738i \(0.956981\pi\)
\(662\) 17.1464 + 29.6985i 0.666415 + 1.15426i
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 4.89898 0.189974
\(666\) 0 0
\(667\) 13.8564i 0.536522i
\(668\) 14.6969 25.4558i 0.568642 0.984916i
\(669\) 0 0
\(670\) −6.00000 + 3.46410i −0.231800 + 0.133830i
\(671\) 19.5959 0.756492
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) −12.2474 + 7.07107i −0.471754 + 0.272367i
\(675\) 0 0
\(676\) 13.0000 22.5167i 0.500000 0.866025i
\(677\) 35.3553i 1.35882i −0.733761 0.679408i \(-0.762237\pi\)
0.733761 0.679408i \(-0.237763\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 9.79796 0.375735
\(681\) 0 0
\(682\) −2.00000 3.46410i −0.0765840 0.132647i
\(683\) 18.3848i 0.703474i −0.936099 0.351737i \(-0.885591\pi\)
0.936099 0.351737i \(-0.114409\pi\)
\(684\) 0 0
\(685\) 6.92820i 0.264713i
\(686\) 1.22474 0.707107i 0.0467610 0.0269975i
\(687\) 0 0
\(688\) 36.0000 20.7846i 1.37249 0.792406i
\(689\) 0 0
\(690\) 0 0
\(691\) 34.6410i 1.31781i 0.752228 + 0.658903i \(0.228979\pi\)
−0.752228 + 0.658903i \(0.771021\pi\)
\(692\) −17.1464 9.89949i −0.651809 0.376322i
\(693\) 0 0
\(694\) −1.00000 1.73205i −0.0379595 0.0657477i
\(695\) −29.3939 −1.11497
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) −14.6969 25.4558i −0.556287 0.963518i
\(699\) 0 0
\(700\) 3.00000 5.19615i 0.113389 0.196396i
\(701\) 36.7696i 1.38877i 0.719605 + 0.694383i \(0.244323\pi\)
−0.719605 + 0.694383i \(0.755677\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 11.3137i 0.426401i
\(705\) 0 0
\(706\) 33.0000 19.0526i 1.24197 0.717053i
\(707\) 15.5563i 0.585057i
\(708\) 0 0
\(709\) 27.7128i 1.04078i −0.853930 0.520388i \(-0.825787\pi\)
0.853930 0.520388i \(-0.174213\pi\)
\(710\) −7.34847 12.7279i −0.275783 0.477670i
\(711\) 0 0
\(712\) 34.6410i 1.29823i
\(713\) 4.89898 0.183468
\(714\) 0 0
\(715\) 0 0
\(716\) −31.8434 18.3848i −1.19004 0.687071i
\(717\) 0 0
\(718\) −33.0000 + 19.0526i −1.23155 + 0.711035i
\(719\) 24.4949 0.913506 0.456753 0.889594i \(-0.349012\pi\)
0.456753 + 0.889594i \(0.349012\pi\)
\(720\) 0 0
\(721\) 2.00000 0.0744839
\(722\) 8.57321 4.94975i 0.319062 0.184211i
\(723\) 0 0
\(724\) −36.0000 20.7846i −1.33793 0.772454i
\(725\) 16.9706i 0.630271i
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.00000 3.46410i −0.0740233 0.128212i
\(731\) 25.4558i 0.941518i
\(732\) 0 0
\(733\) 27.7128i 1.02360i −0.859106 0.511798i \(-0.828980\pi\)
0.859106 0.511798i \(-0.171020\pi\)
\(734\) 9.79796 5.65685i 0.361649 0.208798i
\(735\) 0 0
\(736\) −12.0000 6.92820i −0.442326 0.255377i
\(737\) −4.89898 −0.180456
\(738\) 0 0
\(739\) 24.2487i 0.892003i −0.895032 0.446002i \(-0.852848\pi\)
0.895032 0.446002i \(-0.147152\pi\)
\(740\) 4.89898 8.48528i 0.180090 0.311925i
\(741\) 0 0
\(742\) −4.00000 6.92820i −0.146845 0.254342i
\(743\) 22.0454 0.808768 0.404384 0.914589i \(-0.367486\pi\)
0.404384 + 0.914589i \(0.367486\pi\)
\(744\) 0 0
\(745\) 16.0000 0.586195
\(746\) 9.79796 + 16.9706i 0.358729 + 0.621336i
\(747\) 0 0
\(748\) 6.00000 + 3.46410i 0.219382 + 0.126660i
\(749\) 7.07107i 0.258371i
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −19.5959 33.9411i −0.714590 1.23771i
\(753\) 0 0
\(754\) 0 0
\(755\) 5.65685i 0.205874i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −7.34847 12.7279i −0.266908 0.462299i
\(759\) 0 0
\(760\) 13.8564i 0.502625i
\(761\) −41.6413 −1.50950 −0.754748 0.656014i \(-0.772241\pi\)
−0.754748 + 0.656014i \(0.772241\pi\)
\(762\) 0 0
\(763\) 13.8564i 0.501636i
\(764\) −12.2474 + 21.2132i −0.443097 + 0.767467i
\(765\) 0 0
\(766\) 30.0000 17.3205i 1.08394 0.625815i
\(767\) 0 0
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −2.44949 + 1.41421i −0.0882735 + 0.0509647i
\(771\) 0 0
\(772\) −16.0000 + 27.7128i −0.575853 + 0.997406i
\(773\) 35.3553i 1.27164i −0.771836 0.635822i \(-0.780661\pi\)
0.771836 0.635822i \(-0.219339\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 28.2843i 1.01535i
\(777\) 0 0
\(778\) 20.0000 + 34.6410i 0.717035 + 1.24194i
\(779\) 25.4558i 0.912050i
\(780\) 0 0
\(781\) 10.3923i 0.371866i
\(782\) −7.34847 + 4.24264i −0.262781 + 0.151717i
\(783\) 0 0
\(784\) −2.00000 3.46410i −0.0714286 0.123718i
\(785\) −19.5959 −0.699408
\(786\) 0 0
\(787\) 34.6410i 1.23482i −0.786642 0.617409i \(-0.788182\pi\)
0.786642 0.617409i \(-0.211818\pi\)
\(788\) −24.4949 14.1421i −0.872595 0.503793i
\(789\) 0 0
\(790\) −8.00000 13.8564i −0.284627 0.492989i
\(791\) −14.6969 −0.522563
\(792\) 0 0
\(793\) 0 0
\(794\) −4.89898 8.48528i −0.173858 0.301131i
\(795\) 0 0
\(796\) −16.0000 + 27.7128i −0.567105 + 0.982255i
\(797\) 26.8701i 0.951786i −0.879503 0.475893i \(-0.842125\pi\)
0.879503 0.475893i \(-0.157875\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −14.6969 8.48528i −0.519615 0.300000i
\(801\) 0 0
\(802\) 30.0000 17.3205i 1.05934 0.611608i
\(803\) 2.82843i 0.0998130i
\(804\) 0 0
\(805\) 3.46410i 0.122094i
\(806\) 0 0
\(807\) 0 0
\(808\) 44.0000 1.54791
\(809\) 34.2929 1.20567 0.602836 0.797865i \(-0.294037\pi\)
0.602836 + 0.797865i \(0.294037\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −9.79796 5.65685i −0.343841 0.198517i
\(813\) 0 0
\(814\) 6.00000 3.46410i 0.210300 0.121417i
\(815\) 4.89898 0.171604
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 17.1464 9.89949i 0.599511 0.346128i
\(819\) 0 0
\(820\) 18.0000 + 10.3923i 0.628587 + 0.362915i
\(821\) 22.6274i 0.789702i −0.918745 0.394851i \(-0.870796\pi\)
0.918745 0.394851i \(-0.129204\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 5.65685i 0.197066i
\(825\) 0 0
\(826\) 8.00000 + 13.8564i 0.278356 + 0.482126i
\(827\) 49.4975i 1.72120i 0.509285 + 0.860598i \(0.329910\pi\)
−0.509285 + 0.860598i \(0.670090\pi\)
\(828\) 0 0
\(829\) 6.92820i 0.240626i 0.992736 + 0.120313i \(0.0383899\pi\)
−0.992736 + 0.120313i \(0.961610\pi\)
\(830\) 4.89898 2.82843i 0.170046 0.0981761i
\(831\) 0 0
\(832\) 0 0
\(833\) −2.44949 −0.0848698
\(834\) 0 0
\(835\) 20.7846i 0.719281i
\(836\) 4.89898 8.48528i 0.169435 0.293470i
\(837\) 0 0
\(838\) 8.00000 + 13.8564i 0.276355 + 0.478662i
\(839\) 14.6969 0.507395 0.253697 0.967284i \(-0.418353\pi\)
0.253697 + 0.967284i \(0.418353\pi\)
\(840\) 0 0
\(841\) −3.00000 −0.103448
\(842\) 22.0454 + 38.1838i 0.759735 + 1.31590i
\(843\) 0 0
\(844\) 18.0000 + 10.3923i 0.619586 + 0.357718i
\(845\) 18.3848i 0.632456i
\(846\) 0 0
\(847\) 9.00000 0.309244
\(848\) −19.5959 + 11.3137i −0.672927 + 0.388514i
\(849\) 0 0
\(850\) −9.00000 + 5.19615i −0.308697 + 0.178227i
\(851\) 8.48528i 0.290872i
\(852\) 0 0
\(853\) 20.7846i 0.711651i 0.934552 + 0.355826i \(0.115800\pi\)
−0.934552 + 0.355826i \(0.884200\pi\)
\(854\) 9.79796 + 16.9706i 0.335279 + 0.580721i
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) −2.44949 −0.0836730 −0.0418365 0.999124i \(-0.513321\pi\)
−0.0418365 + 0.999124i \(0.513321\pi\)
\(858\) 0 0
\(859\) 17.3205i 0.590968i −0.955348 0.295484i \(-0.904519\pi\)
0.955348 0.295484i \(-0.0954809\pi\)
\(860\) 14.6969 25.4558i 0.501161 0.868037i
\(861\) 0 0
\(862\) −39.0000 + 22.5167i −1.32835 + 0.766921i
\(863\) 2.44949 0.0833816 0.0416908 0.999131i \(-0.486726\pi\)
0.0416908 + 0.999131i \(0.486726\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) −26.9444 + 15.5563i −0.915608 + 0.528626i
\(867\) 0 0
\(868\) 2.00000 3.46410i 0.0678844 0.117579i
\(869\) 11.3137i 0.383791i
\(870\) 0 0
\(871\) 0 0
\(872\) −39.1918 −1.32720
\(873\) 0 0
\(874\) 6.00000 + 10.3923i 0.202953 + 0.351525i
\(875\) 11.3137i 0.382473i
\(876\) 0 0
\(877\) 27.7128i 0.935795i −0.883783 0.467898i \(-0.845012\pi\)
0.883783 0.467898i \(-0.154988\pi\)
\(878\) 39.1918 22.6274i 1.32266 0.763638i
\(879\) 0 0
\(880\) 4.00000 + 6.92820i 0.134840 + 0.233550i
\(881\) −7.34847 −0.247576 −0.123788 0.992309i \(-0.539504\pi\)
−0.123788 + 0.992309i \(0.539504\pi\)
\(882\) 0 0
\(883\) 31.1769i 1.04919i −0.851353 0.524593i \(-0.824217\pi\)
0.851353 0.524593i \(-0.175783\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −25.0000 43.3013i −0.839891 1.45473i
\(887\) 53.8888 1.80941 0.904704 0.426041i \(-0.140092\pi\)
0.904704 + 0.426041i \(0.140092\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 12.2474 + 21.2132i 0.410535 + 0.711068i
\(891\) 0 0
\(892\) −16.0000 + 27.7128i −0.535720 + 0.927894i
\(893\) 33.9411i 1.13580i
\(894\) 0 0
\(895\) −26.0000 −0.869084
\(896\) −9.79796 + 5.65685i −0.327327 + 0.188982i
\(897\) 0 0
\(898\) 18.0000 10.3923i 0.600668 0.346796i
\(899\) 11.3137i 0.377333i
\(900\) 0 0
\(901\) 13.8564i 0.461624i
\(902\) 7.34847 + 12.7279i 0.244677 + 0.423793i
\(903\) 0 0
\(904\) 41.5692i 1.38257i
\(905\) −29.3939 −0.977086
\(906\) 0 0
\(907\) 24.2487i 0.805165i −0.915384 0.402583i \(-0.868113\pi\)
0.915384 0.402583i \(-0.131887\pi\)
\(908\) 4.89898 + 2.82843i 0.162578 + 0.0938647i
\(909\) 0 0
\(910\) 0 0
\(911\) 7.34847 0.243466 0.121733 0.992563i \(-0.461155\pi\)
0.121733 + 0.992563i \(0.461155\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 31.8434 18.3848i 1.05328 0.608114i
\(915\) 0 0
\(916\) −12.0000 6.92820i −0.396491 0.228914i
\(917\) 19.7990i 0.653820i
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −9.79796 −0.323029
\(921\) 0 0
\(922\) 23.0000 + 39.8372i 0.757465 + 1.31197i
\(923\) 0 0
\(924\) 0 0
\(925\) 10.3923i 0.341697i
\(926\) 9.79796 5.65685i 0.321981 0.185896i
\(927\) 0 0
\(928\) −16.0000 + 27.7128i −0.525226 + 0.909718i
\(929\) −12.2474 −0.401826 −0.200913 0.979609i \(-0.564391\pi\)
−0.200913 + 0.979609i \(0.564391\pi\)
\(930\) 0 0
\(931\) 3.46410i 0.113531i
\(932\) 9.79796 16.9706i 0.320943 0.555889i
\(933\) 0 0
\(934\) 14.0000 + 24.2487i 0.458094 + 0.793442i
\(935\) 4.89898 0.160214
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −2.44949 4.24264i −0.0799787 0.138527i
\(939\) 0 0
\(940\) −24.0000 13.8564i −0.782794 0.451946i
\(941\) 18.3848i 0.599327i −0.954045 0.299663i \(-0.903126\pi\)
0.954045 0.299663i \(-0.0968743\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 39.1918 22.6274i 1.27559 0.736460i
\(945\) 0 0
\(946\) 18.0000 10.3923i 0.585230 0.337883i
\(947\) 9.89949i 0.321690i −0.986980 0.160845i \(-0.948578\pi\)
0.986980 0.160845i \(-0.0514220\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 7.34847 + 12.7279i 0.238416 + 0.412948i
\(951\) 0 0
\(952\) 6.92820i 0.224544i
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 17.3205i 0.560478i
\(956\) 7.34847 12.7279i 0.237666 0.411650i
\(957\) 0 0
\(958\) −48.0000 + 27.7128i −1.55081 + 0.895360i
\(959\) 4.89898 0.158196
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) −10.0000 + 17.3205i −0.322078 + 0.557856i
\(965\) 22.6274i 0.728402i
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 25.4558i 0.818182i
\(969\) 0 0
\(970\) 10.0000 + 17.3205i 0.321081 + 0.556128i
\(971\) 45.2548i 1.45230i 0.687538 + 0.726148i \(0.258691\pi\)
−0.687538 + 0.726148i \(0.741309\pi\)
\(972\) 0 0
\(973\) 20.7846i 0.666324i
\(974\) −19.5959 + 11.3137i −0.627894 + 0.362515i
\(975\) 0 0
\(976\) 48.0000 27.7128i 1.53644 0.887066i
\(977\) −39.1918 −1.25386 −0.626929 0.779076i \(-0.715688\pi\)
−0.626929 + 0.779076i \(0.715688\pi\)
\(978\) 0 0
\(979\) 17.3205i 0.553566i
\(980\) −2.44949 1.41421i −0.0782461 0.0451754i
\(981\) 0 0
\(982\) 29.0000 + 50.2295i 0.925427 + 1.60289i
\(983\) −9.79796 −0.312506 −0.156253 0.987717i \(-0.549942\pi\)
−0.156253 + 0.987717i \(0.549942\pi\)
\(984\) 0 0
\(985\) −20.0000 −0.637253
\(986\) 9.79796 + 16.9706i 0.312031 + 0.540453i
\(987\) 0 0
\(988\) 0 0
\(989\) 25.4558i 0.809449i
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −9.79796 5.65685i −0.311086 0.179605i
\(993\) 0 0
\(994\) 9.00000 5.19615i 0.285463 0.164812i
\(995\) 22.6274i 0.717337i
\(996\) 0 0
\(997\) 34.6410i 1.09709i −0.836120 0.548546i \(-0.815182\pi\)
0.836120 0.548546i \(-0.184818\pi\)
\(998\) 17.1464 + 29.6985i 0.542761 + 0.940089i
\(999\) 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 504.2.c.c.253.3 yes 4
3.2 odd 2 inner 504.2.c.c.253.2 yes 4
4.3 odd 2 2016.2.c.b.1009.1 4
8.3 odd 2 2016.2.c.b.1009.3 4
8.5 even 2 inner 504.2.c.c.253.4 yes 4
12.11 even 2 2016.2.c.b.1009.4 4
24.5 odd 2 inner 504.2.c.c.253.1 4
24.11 even 2 2016.2.c.b.1009.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.c.c.253.1 4 24.5 odd 2 inner
504.2.c.c.253.2 yes 4 3.2 odd 2 inner
504.2.c.c.253.3 yes 4 1.1 even 1 trivial
504.2.c.c.253.4 yes 4 8.5 even 2 inner
2016.2.c.b.1009.1 4 4.3 odd 2
2016.2.c.b.1009.2 4 24.11 even 2
2016.2.c.b.1009.3 4 8.3 odd 2
2016.2.c.b.1009.4 4 12.11 even 2