Properties

Label 630.2.g.g.379.4
Level $630$
Weight $2$
Character 630.379
Analytic conductor $5.031$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(379,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.g (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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 379.4
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 630.379
Dual form 630.2.g.g.379.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(0.224745 - 2.22474i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(0.224745 - 2.22474i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(2.22474 + 0.224745i) q^{10} -4.89898 q^{11} +0.449490i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -6.44949 q^{19} +(-0.224745 + 2.22474i) q^{20} -4.89898i q^{22} -6.89898i q^{23} +(-4.89898 - 1.00000i) q^{25} -0.449490 q^{26} +1.00000i q^{28} -2.89898 q^{29} -0.898979 q^{31} +1.00000i q^{32} +2.00000 q^{34} +(-2.22474 - 0.224745i) q^{35} +2.00000i q^{37} -6.44949i q^{38} +(-2.22474 - 0.224745i) q^{40} +10.8990 q^{41} -8.89898i q^{43} +4.89898 q^{44} +6.89898 q^{46} +0.898979i q^{47} -1.00000 q^{49} +(1.00000 - 4.89898i) q^{50} -0.449490i q^{52} -1.10102i q^{53} +(-1.10102 + 10.8990i) q^{55} -1.00000 q^{56} -2.89898i q^{58} -6.44949 q^{59} +8.44949 q^{61} -0.898979i q^{62} -1.00000 q^{64} +(1.00000 + 0.101021i) q^{65} -8.00000i q^{67} +2.00000i q^{68} +(0.224745 - 2.22474i) q^{70} +10.8990 q^{71} +6.89898i q^{73} -2.00000 q^{74} +6.44949 q^{76} +4.89898i q^{77} +2.89898 q^{79} +(0.224745 - 2.22474i) q^{80} +10.8990i q^{82} +2.44949i q^{83} +(-4.44949 - 0.449490i) q^{85} +8.89898 q^{86} +4.89898i q^{88} -10.0000 q^{89} +0.449490 q^{91} +6.89898i q^{92} -0.898979 q^{94} +(-1.44949 + 14.3485i) q^{95} -3.79796i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{5} + 4 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{19} + 4 q^{20} + 8 q^{26} + 8 q^{29} + 16 q^{31} + 8 q^{34} - 4 q^{35} - 4 q^{40} + 24 q^{41} + 8 q^{46} - 4 q^{49} + 4 q^{50} - 24 q^{55} - 4 q^{56} - 16 q^{59} + 24 q^{61} - 4 q^{64} + 4 q^{65} - 4 q^{70} + 24 q^{71} - 8 q^{74} + 16 q^{76} - 8 q^{79} - 4 q^{80} - 8 q^{85} + 16 q^{86} - 40 q^{89} - 8 q^{91} + 16 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0.224745 2.22474i 0.100509 0.994936i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.22474 + 0.224745i 0.703526 + 0.0710706i
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 0.449490i 0.124666i 0.998055 + 0.0623330i \(0.0198541\pi\)
−0.998055 + 0.0623330i \(0.980146\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −6.44949 −1.47961 −0.739807 0.672819i \(-0.765083\pi\)
−0.739807 + 0.672819i \(0.765083\pi\)
\(20\) −0.224745 + 2.22474i −0.0502545 + 0.497468i
\(21\) 0 0
\(22\) 4.89898i 1.04447i
\(23\) 6.89898i 1.43854i −0.694732 0.719268i \(-0.744477\pi\)
0.694732 0.719268i \(-0.255523\pi\)
\(24\) 0 0
\(25\) −4.89898 1.00000i −0.979796 0.200000i
\(26\) −0.449490 −0.0881522
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −2.89898 −0.538327 −0.269163 0.963095i \(-0.586747\pi\)
−0.269163 + 0.963095i \(0.586747\pi\)
\(30\) 0 0
\(31\) −0.898979 −0.161461 −0.0807307 0.996736i \(-0.525725\pi\)
−0.0807307 + 0.996736i \(0.525725\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −2.22474 0.224745i −0.376051 0.0379888i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 6.44949i 1.04625i
\(39\) 0 0
\(40\) −2.22474 0.224745i −0.351763 0.0355353i
\(41\) 10.8990 1.70213 0.851067 0.525057i \(-0.175956\pi\)
0.851067 + 0.525057i \(0.175956\pi\)
\(42\) 0 0
\(43\) 8.89898i 1.35708i −0.734563 0.678541i \(-0.762613\pi\)
0.734563 0.678541i \(-0.237387\pi\)
\(44\) 4.89898 0.738549
\(45\) 0 0
\(46\) 6.89898 1.01720
\(47\) 0.898979i 0.131130i 0.997848 + 0.0655648i \(0.0208849\pi\)
−0.997848 + 0.0655648i \(0.979115\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 1.00000 4.89898i 0.141421 0.692820i
\(51\) 0 0
\(52\) 0.449490i 0.0623330i
\(53\) 1.10102i 0.151237i −0.997137 0.0756184i \(-0.975907\pi\)
0.997137 0.0756184i \(-0.0240931\pi\)
\(54\) 0 0
\(55\) −1.10102 + 10.8990i −0.148462 + 1.46962i
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.89898i 0.380655i
\(59\) −6.44949 −0.839652 −0.419826 0.907605i \(-0.637909\pi\)
−0.419826 + 0.907605i \(0.637909\pi\)
\(60\) 0 0
\(61\) 8.44949 1.08185 0.540923 0.841072i \(-0.318075\pi\)
0.540923 + 0.841072i \(0.318075\pi\)
\(62\) 0.898979i 0.114171i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.00000 + 0.101021i 0.124035 + 0.0125301i
\(66\) 0 0
\(67\) 8.00000i 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) 0.224745 2.22474i 0.0268622 0.265908i
\(71\) 10.8990 1.29347 0.646735 0.762714i \(-0.276134\pi\)
0.646735 + 0.762714i \(0.276134\pi\)
\(72\) 0 0
\(73\) 6.89898i 0.807464i 0.914877 + 0.403732i \(0.132287\pi\)
−0.914877 + 0.403732i \(0.867713\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 6.44949 0.739807
\(77\) 4.89898i 0.558291i
\(78\) 0 0
\(79\) 2.89898 0.326161 0.163080 0.986613i \(-0.447857\pi\)
0.163080 + 0.986613i \(0.447857\pi\)
\(80\) 0.224745 2.22474i 0.0251272 0.248734i
\(81\) 0 0
\(82\) 10.8990i 1.20359i
\(83\) 2.44949i 0.268866i 0.990923 + 0.134433i \(0.0429214\pi\)
−0.990923 + 0.134433i \(0.957079\pi\)
\(84\) 0 0
\(85\) −4.44949 0.449490i −0.482615 0.0487540i
\(86\) 8.89898 0.959602
\(87\) 0 0
\(88\) 4.89898i 0.522233i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0.449490 0.0471193
\(92\) 6.89898i 0.719268i
\(93\) 0 0
\(94\) −0.898979 −0.0927227
\(95\) −1.44949 + 14.3485i −0.148715 + 1.47212i
\(96\) 0 0
\(97\) 3.79796i 0.385624i −0.981236 0.192812i \(-0.938239\pi\)
0.981236 0.192812i \(-0.0617608\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 4.89898 + 1.00000i 0.489898 + 0.100000i
\(101\) −8.44949 −0.840756 −0.420378 0.907349i \(-0.638102\pi\)
−0.420378 + 0.907349i \(0.638102\pi\)
\(102\) 0 0
\(103\) 3.10102i 0.305553i −0.988261 0.152776i \(-0.951179\pi\)
0.988261 0.152776i \(-0.0488214\pi\)
\(104\) 0.449490 0.0440761
\(105\) 0 0
\(106\) 1.10102 0.106941
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) 2.89898 0.277672 0.138836 0.990315i \(-0.455664\pi\)
0.138836 + 0.990315i \(0.455664\pi\)
\(110\) −10.8990 1.10102i −1.03918 0.104978i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 0.202041i 0.0190064i 0.999955 + 0.00950321i \(0.00302501\pi\)
−0.999955 + 0.00950321i \(0.996975\pi\)
\(114\) 0 0
\(115\) −15.3485 1.55051i −1.43125 0.144586i
\(116\) 2.89898 0.269163
\(117\) 0 0
\(118\) 6.44949i 0.593724i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 8.44949i 0.764981i
\(123\) 0 0
\(124\) 0.898979 0.0807307
\(125\) −3.32577 + 10.6742i −0.297465 + 0.954733i
\(126\) 0 0
\(127\) 5.10102i 0.452642i −0.974053 0.226321i \(-0.927330\pi\)
0.974053 0.226321i \(-0.0726699\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −0.101021 + 1.00000i −0.00886009 + 0.0877058i
\(131\) 1.55051 0.135469 0.0677344 0.997703i \(-0.478423\pi\)
0.0677344 + 0.997703i \(0.478423\pi\)
\(132\) 0 0
\(133\) 6.44949i 0.559242i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 17.7980i 1.52058i −0.649582 0.760291i \(-0.725056\pi\)
0.649582 0.760291i \(-0.274944\pi\)
\(138\) 0 0
\(139\) −6.44949 −0.547039 −0.273519 0.961867i \(-0.588188\pi\)
−0.273519 + 0.961867i \(0.588188\pi\)
\(140\) 2.22474 + 0.224745i 0.188025 + 0.0189944i
\(141\) 0 0
\(142\) 10.8990i 0.914622i
\(143\) 2.20204i 0.184144i
\(144\) 0 0
\(145\) −0.651531 + 6.44949i −0.0541067 + 0.535601i
\(146\) −6.89898 −0.570964
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) 15.7980 1.29422 0.647110 0.762397i \(-0.275978\pi\)
0.647110 + 0.762397i \(0.275978\pi\)
\(150\) 0 0
\(151\) −19.5959 −1.59469 −0.797347 0.603522i \(-0.793764\pi\)
−0.797347 + 0.603522i \(0.793764\pi\)
\(152\) 6.44949i 0.523123i
\(153\) 0 0
\(154\) −4.89898 −0.394771
\(155\) −0.202041 + 2.00000i −0.0162283 + 0.160644i
\(156\) 0 0
\(157\) 8.44949i 0.674343i 0.941443 + 0.337171i \(0.109470\pi\)
−0.941443 + 0.337171i \(0.890530\pi\)
\(158\) 2.89898i 0.230630i
\(159\) 0 0
\(160\) 2.22474 + 0.224745i 0.175882 + 0.0177676i
\(161\) −6.89898 −0.543716
\(162\) 0 0
\(163\) 16.8990i 1.32363i 0.749667 + 0.661815i \(0.230214\pi\)
−0.749667 + 0.661815i \(0.769786\pi\)
\(164\) −10.8990 −0.851067
\(165\) 0 0
\(166\) −2.44949 −0.190117
\(167\) 4.89898i 0.379094i −0.981872 0.189547i \(-0.939298\pi\)
0.981872 0.189547i \(-0.0607020\pi\)
\(168\) 0 0
\(169\) 12.7980 0.984458
\(170\) 0.449490 4.44949i 0.0344743 0.341260i
\(171\) 0 0
\(172\) 8.89898i 0.678541i
\(173\) 18.2474i 1.38733i 0.720299 + 0.693664i \(0.244005\pi\)
−0.720299 + 0.693664i \(0.755995\pi\)
\(174\) 0 0
\(175\) −1.00000 + 4.89898i −0.0755929 + 0.370328i
\(176\) −4.89898 −0.369274
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) −5.79796 −0.433360 −0.216680 0.976243i \(-0.569523\pi\)
−0.216680 + 0.976243i \(0.569523\pi\)
\(180\) 0 0
\(181\) 14.2474 1.05900 0.529502 0.848309i \(-0.322379\pi\)
0.529502 + 0.848309i \(0.322379\pi\)
\(182\) 0.449490i 0.0333184i
\(183\) 0 0
\(184\) −6.89898 −0.508600
\(185\) 4.44949 + 0.449490i 0.327133 + 0.0330471i
\(186\) 0 0
\(187\) 9.79796i 0.716498i
\(188\) 0.898979i 0.0655648i
\(189\) 0 0
\(190\) −14.3485 1.44949i −1.04095 0.105157i
\(191\) 16.6969 1.20815 0.604074 0.796928i \(-0.293543\pi\)
0.604074 + 0.796928i \(0.293543\pi\)
\(192\) 0 0
\(193\) 17.5959i 1.26658i −0.773914 0.633291i \(-0.781704\pi\)
0.773914 0.633291i \(-0.218296\pi\)
\(194\) 3.79796 0.272678
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 9.10102i 0.648421i −0.945985 0.324210i \(-0.894901\pi\)
0.945985 0.324210i \(-0.105099\pi\)
\(198\) 0 0
\(199\) −7.10102 −0.503378 −0.251689 0.967808i \(-0.580986\pi\)
−0.251689 + 0.967808i \(0.580986\pi\)
\(200\) −1.00000 + 4.89898i −0.0707107 + 0.346410i
\(201\) 0 0
\(202\) 8.44949i 0.594504i
\(203\) 2.89898i 0.203468i
\(204\) 0 0
\(205\) 2.44949 24.2474i 0.171080 1.69352i
\(206\) 3.10102 0.216058
\(207\) 0 0
\(208\) 0.449490i 0.0311665i
\(209\) 31.5959 2.18554
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 1.10102i 0.0756184i
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −19.7980 2.00000i −1.35021 0.136399i
\(216\) 0 0
\(217\) 0.898979i 0.0610267i
\(218\) 2.89898i 0.196344i
\(219\) 0 0
\(220\) 1.10102 10.8990i 0.0742308 0.734809i
\(221\) 0.898979 0.0604719
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −0.202041 −0.0134396
\(227\) 7.34847i 0.487735i 0.969809 + 0.243868i \(0.0784162\pi\)
−0.969809 + 0.243868i \(0.921584\pi\)
\(228\) 0 0
\(229\) −15.1464 −1.00090 −0.500452 0.865764i \(-0.666833\pi\)
−0.500452 + 0.865764i \(0.666833\pi\)
\(230\) 1.55051 15.3485i 0.102238 1.01205i
\(231\) 0 0
\(232\) 2.89898i 0.190327i
\(233\) 10.2020i 0.668358i 0.942510 + 0.334179i \(0.108459\pi\)
−0.942510 + 0.334179i \(0.891541\pi\)
\(234\) 0 0
\(235\) 2.00000 + 0.202041i 0.130466 + 0.0131797i
\(236\) 6.44949 0.419826
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) −25.7980 −1.66873 −0.834366 0.551211i \(-0.814166\pi\)
−0.834366 + 0.551211i \(0.814166\pi\)
\(240\) 0 0
\(241\) 20.6969 1.33321 0.666604 0.745412i \(-0.267747\pi\)
0.666604 + 0.745412i \(0.267747\pi\)
\(242\) 13.0000i 0.835672i
\(243\) 0 0
\(244\) −8.44949 −0.540923
\(245\) −0.224745 + 2.22474i −0.0143584 + 0.142134i
\(246\) 0 0
\(247\) 2.89898i 0.184458i
\(248\) 0.898979i 0.0570853i
\(249\) 0 0
\(250\) −10.6742 3.32577i −0.675098 0.210340i
\(251\) 1.55051 0.0978673 0.0489337 0.998802i \(-0.484418\pi\)
0.0489337 + 0.998802i \(0.484418\pi\)
\(252\) 0 0
\(253\) 33.7980i 2.12486i
\(254\) 5.10102 0.320066
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.6969i 1.29104i −0.763744 0.645520i \(-0.776641\pi\)
0.763744 0.645520i \(-0.223359\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −1.00000 0.101021i −0.0620174 0.00626503i
\(261\) 0 0
\(262\) 1.55051i 0.0957908i
\(263\) 9.79796i 0.604168i −0.953281 0.302084i \(-0.902318\pi\)
0.953281 0.302084i \(-0.0976823\pi\)
\(264\) 0 0
\(265\) −2.44949 0.247449i −0.150471 0.0152007i
\(266\) −6.44949 −0.395444
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) −15.1464 −0.923494 −0.461747 0.887012i \(-0.652777\pi\)
−0.461747 + 0.887012i \(0.652777\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) 17.7980 1.07521
\(275\) 24.0000 + 4.89898i 1.44725 + 0.295420i
\(276\) 0 0
\(277\) 5.10102i 0.306491i −0.988188 0.153245i \(-0.951028\pi\)
0.988188 0.153245i \(-0.0489725\pi\)
\(278\) 6.44949i 0.386815i
\(279\) 0 0
\(280\) −0.224745 + 2.22474i −0.0134311 + 0.132954i
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 28.2474i 1.67914i −0.543254 0.839568i \(-0.682808\pi\)
0.543254 0.839568i \(-0.317192\pi\)
\(284\) −10.8990 −0.646735
\(285\) 0 0
\(286\) 2.20204 0.130209
\(287\) 10.8990i 0.643346i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) −6.44949 0.651531i −0.378727 0.0382592i
\(291\) 0 0
\(292\) 6.89898i 0.403732i
\(293\) 6.24745i 0.364980i −0.983208 0.182490i \(-0.941584\pi\)
0.983208 0.182490i \(-0.0584157\pi\)
\(294\) 0 0
\(295\) −1.44949 + 14.3485i −0.0843926 + 0.835400i
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 15.7980i 0.915151i
\(299\) 3.10102 0.179337
\(300\) 0 0
\(301\) −8.89898 −0.512929
\(302\) 19.5959i 1.12762i
\(303\) 0 0
\(304\) −6.44949 −0.369904
\(305\) 1.89898 18.7980i 0.108735 1.07637i
\(306\) 0 0
\(307\) 4.24745i 0.242415i 0.992627 + 0.121207i \(0.0386766\pi\)
−0.992627 + 0.121207i \(0.961323\pi\)
\(308\) 4.89898i 0.279145i
\(309\) 0 0
\(310\) −2.00000 0.202041i −0.113592 0.0114752i
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 17.5959i 0.994580i −0.867584 0.497290i \(-0.834328\pi\)
0.867584 0.497290i \(-0.165672\pi\)
\(314\) −8.44949 −0.476832
\(315\) 0 0
\(316\) −2.89898 −0.163080
\(317\) 26.4949i 1.48810i −0.668123 0.744051i \(-0.732902\pi\)
0.668123 0.744051i \(-0.267098\pi\)
\(318\) 0 0
\(319\) 14.2020 0.795162
\(320\) −0.224745 + 2.22474i −0.0125636 + 0.124367i
\(321\) 0 0
\(322\) 6.89898i 0.384465i
\(323\) 12.8990i 0.717718i
\(324\) 0 0
\(325\) 0.449490 2.20204i 0.0249332 0.122147i
\(326\) −16.8990 −0.935948
\(327\) 0 0
\(328\) 10.8990i 0.601795i
\(329\) 0.898979 0.0495623
\(330\) 0 0
\(331\) 10.6969 0.587957 0.293978 0.955812i \(-0.405021\pi\)
0.293978 + 0.955812i \(0.405021\pi\)
\(332\) 2.44949i 0.134433i
\(333\) 0 0
\(334\) 4.89898 0.268060
\(335\) −17.7980 1.79796i −0.972406 0.0982330i
\(336\) 0 0
\(337\) 29.5959i 1.61219i −0.591785 0.806096i \(-0.701576\pi\)
0.591785 0.806096i \(-0.298424\pi\)
\(338\) 12.7980i 0.696117i
\(339\) 0 0
\(340\) 4.44949 + 0.449490i 0.241307 + 0.0243770i
\(341\) 4.40408 0.238494
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −8.89898 −0.479801
\(345\) 0 0
\(346\) −18.2474 −0.980989
\(347\) 19.1010i 1.02540i −0.858569 0.512698i \(-0.828646\pi\)
0.858569 0.512698i \(-0.171354\pi\)
\(348\) 0 0
\(349\) 3.55051 0.190054 0.0950272 0.995475i \(-0.469706\pi\)
0.0950272 + 0.995475i \(0.469706\pi\)
\(350\) −4.89898 1.00000i −0.261861 0.0534522i
\(351\) 0 0
\(352\) 4.89898i 0.261116i
\(353\) 13.1010i 0.697297i 0.937254 + 0.348648i \(0.113359\pi\)
−0.937254 + 0.348648i \(0.886641\pi\)
\(354\) 0 0
\(355\) 2.44949 24.2474i 0.130005 1.28692i
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 5.79796i 0.306432i
\(359\) 11.5959 0.612009 0.306005 0.952030i \(-0.401008\pi\)
0.306005 + 0.952030i \(0.401008\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) 14.2474i 0.748829i
\(363\) 0 0
\(364\) −0.449490 −0.0235597
\(365\) 15.3485 + 1.55051i 0.803376 + 0.0811574i
\(366\) 0 0
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) 6.89898i 0.359634i
\(369\) 0 0
\(370\) −0.449490 + 4.44949i −0.0233679 + 0.231318i
\(371\) −1.10102 −0.0571621
\(372\) 0 0
\(373\) 24.6969i 1.27876i −0.768891 0.639380i \(-0.779191\pi\)
0.768891 0.639380i \(-0.220809\pi\)
\(374\) −9.79796 −0.506640
\(375\) 0 0
\(376\) 0.898979 0.0463613
\(377\) 1.30306i 0.0671111i
\(378\) 0 0
\(379\) −1.30306 −0.0669338 −0.0334669 0.999440i \(-0.510655\pi\)
−0.0334669 + 0.999440i \(0.510655\pi\)
\(380\) 1.44949 14.3485i 0.0743573 0.736061i
\(381\) 0 0
\(382\) 16.6969i 0.854290i
\(383\) 16.8990i 0.863498i −0.901994 0.431749i \(-0.857897\pi\)
0.901994 0.431749i \(-0.142103\pi\)
\(384\) 0 0
\(385\) 10.8990 + 1.10102i 0.555463 + 0.0561132i
\(386\) 17.5959 0.895609
\(387\) 0 0
\(388\) 3.79796i 0.192812i
\(389\) 22.8990 1.16102 0.580512 0.814252i \(-0.302852\pi\)
0.580512 + 0.814252i \(0.302852\pi\)
\(390\) 0 0
\(391\) −13.7980 −0.697793
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 9.10102 0.458503
\(395\) 0.651531 6.44949i 0.0327821 0.324509i
\(396\) 0 0
\(397\) 17.3485i 0.870695i −0.900263 0.435347i \(-0.856626\pi\)
0.900263 0.435347i \(-0.143374\pi\)
\(398\) 7.10102i 0.355942i
\(399\) 0 0
\(400\) −4.89898 1.00000i −0.244949 0.0500000i
\(401\) −29.3939 −1.46786 −0.733930 0.679225i \(-0.762316\pi\)
−0.733930 + 0.679225i \(0.762316\pi\)
\(402\) 0 0
\(403\) 0.404082i 0.0201288i
\(404\) 8.44949 0.420378
\(405\) 0 0
\(406\) −2.89898 −0.143874
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) 14.4949 0.716727 0.358363 0.933582i \(-0.383335\pi\)
0.358363 + 0.933582i \(0.383335\pi\)
\(410\) 24.2474 + 2.44949i 1.19750 + 0.120972i
\(411\) 0 0
\(412\) 3.10102i 0.152776i
\(413\) 6.44949i 0.317359i
\(414\) 0 0
\(415\) 5.44949 + 0.550510i 0.267505 + 0.0270235i
\(416\) −0.449490 −0.0220380
\(417\) 0 0
\(418\) 31.5959i 1.54541i
\(419\) −6.44949 −0.315078 −0.157539 0.987513i \(-0.550356\pi\)
−0.157539 + 0.987513i \(0.550356\pi\)
\(420\) 0 0
\(421\) −23.7980 −1.15984 −0.579921 0.814673i \(-0.696917\pi\)
−0.579921 + 0.814673i \(0.696917\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) −1.10102 −0.0534703
\(425\) −2.00000 + 9.79796i −0.0970143 + 0.475271i
\(426\) 0 0
\(427\) 8.44949i 0.408899i
\(428\) 8.00000i 0.386695i
\(429\) 0 0
\(430\) 2.00000 19.7980i 0.0964486 0.954742i
\(431\) −17.7980 −0.857298 −0.428649 0.903471i \(-0.641010\pi\)
−0.428649 + 0.903471i \(0.641010\pi\)
\(432\) 0 0
\(433\) 19.7980i 0.951429i 0.879600 + 0.475715i \(0.157810\pi\)
−0.879600 + 0.475715i \(0.842190\pi\)
\(434\) −0.898979 −0.0431524
\(435\) 0 0
\(436\) −2.89898 −0.138836
\(437\) 44.4949i 2.12848i
\(438\) 0 0
\(439\) −37.3939 −1.78471 −0.892356 0.451332i \(-0.850949\pi\)
−0.892356 + 0.451332i \(0.850949\pi\)
\(440\) 10.8990 + 1.10102i 0.519588 + 0.0524891i
\(441\) 0 0
\(442\) 0.898979i 0.0427601i
\(443\) 9.79796i 0.465515i −0.972535 0.232758i \(-0.925225\pi\)
0.972535 0.232758i \(-0.0747749\pi\)
\(444\) 0 0
\(445\) −2.24745 + 22.2474i −0.106539 + 1.05463i
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −53.3939 −2.51422
\(452\) 0.202041i 0.00950321i
\(453\) 0 0
\(454\) −7.34847 −0.344881
\(455\) 0.101021 1.00000i 0.00473591 0.0468807i
\(456\) 0 0
\(457\) 9.59592i 0.448878i −0.974488 0.224439i \(-0.927945\pi\)
0.974488 0.224439i \(-0.0720550\pi\)
\(458\) 15.1464i 0.707746i
\(459\) 0 0
\(460\) 15.3485 + 1.55051i 0.715626 + 0.0722929i
\(461\) −2.65153 −0.123494 −0.0617470 0.998092i \(-0.519667\pi\)
−0.0617470 + 0.998092i \(0.519667\pi\)
\(462\) 0 0
\(463\) 35.5959i 1.65428i 0.561994 + 0.827141i \(0.310034\pi\)
−0.561994 + 0.827141i \(0.689966\pi\)
\(464\) −2.89898 −0.134582
\(465\) 0 0
\(466\) −10.2020 −0.472600
\(467\) 5.55051i 0.256847i −0.991719 0.128423i \(-0.959008\pi\)
0.991719 0.128423i \(-0.0409917\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −0.202041 + 2.00000i −0.00931946 + 0.0922531i
\(471\) 0 0
\(472\) 6.44949i 0.296862i
\(473\) 43.5959i 2.00454i
\(474\) 0 0
\(475\) 31.5959 + 6.44949i 1.44972 + 0.295923i
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 25.7980i 1.17997i
\(479\) 38.6969 1.76811 0.884054 0.467385i \(-0.154804\pi\)
0.884054 + 0.467385i \(0.154804\pi\)
\(480\) 0 0
\(481\) −0.898979 −0.0409899
\(482\) 20.6969i 0.942720i
\(483\) 0 0
\(484\) −13.0000 −0.590909
\(485\) −8.44949 0.853572i −0.383672 0.0387587i
\(486\) 0 0
\(487\) 36.6969i 1.66290i −0.555602 0.831449i \(-0.687512\pi\)
0.555602 0.831449i \(-0.312488\pi\)
\(488\) 8.44949i 0.382490i
\(489\) 0 0
\(490\) −2.22474 0.224745i −0.100504 0.0101529i
\(491\) 19.5959 0.884351 0.442176 0.896928i \(-0.354207\pi\)
0.442176 + 0.896928i \(0.354207\pi\)
\(492\) 0 0
\(493\) 5.79796i 0.261127i
\(494\) 2.89898 0.130431
\(495\) 0 0
\(496\) −0.898979 −0.0403654
\(497\) 10.8990i 0.488886i
\(498\) 0 0
\(499\) −25.7980 −1.15488 −0.577438 0.816435i \(-0.695947\pi\)
−0.577438 + 0.816435i \(0.695947\pi\)
\(500\) 3.32577 10.6742i 0.148733 0.477366i
\(501\) 0 0
\(502\) 1.55051i 0.0692027i
\(503\) 4.00000i 0.178351i −0.996016 0.0891756i \(-0.971577\pi\)
0.996016 0.0891756i \(-0.0284232\pi\)
\(504\) 0 0
\(505\) −1.89898 + 18.7980i −0.0845035 + 0.836498i
\(506\) −33.7980 −1.50250
\(507\) 0 0
\(508\) 5.10102i 0.226321i
\(509\) −36.4495 −1.61560 −0.807798 0.589460i \(-0.799341\pi\)
−0.807798 + 0.589460i \(0.799341\pi\)
\(510\) 0 0
\(511\) 6.89898 0.305193
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 20.6969 0.912903
\(515\) −6.89898 0.696938i −0.304005 0.0307108i
\(516\) 0 0
\(517\) 4.40408i 0.193691i
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 0.101021 1.00000i 0.00443004 0.0438529i
\(521\) −3.30306 −0.144710 −0.0723549 0.997379i \(-0.523051\pi\)
−0.0723549 + 0.997379i \(0.523051\pi\)
\(522\) 0 0
\(523\) 1.14643i 0.0501298i −0.999686 0.0250649i \(-0.992021\pi\)
0.999686 0.0250649i \(-0.00797924\pi\)
\(524\) −1.55051 −0.0677344
\(525\) 0 0
\(526\) 9.79796 0.427211
\(527\) 1.79796i 0.0783203i
\(528\) 0 0
\(529\) −24.5959 −1.06939
\(530\) 0.247449 2.44949i 0.0107485 0.106399i
\(531\) 0 0
\(532\) 6.44949i 0.279621i
\(533\) 4.89898i 0.212198i
\(534\) 0 0
\(535\) 17.7980 + 1.79796i 0.769473 + 0.0777325i
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 15.1464i 0.653009i
\(539\) 4.89898 0.211014
\(540\) 0 0
\(541\) −29.5959 −1.27243 −0.636214 0.771513i \(-0.719500\pi\)
−0.636214 + 0.771513i \(0.719500\pi\)
\(542\) 12.0000i 0.515444i
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 0.651531 6.44949i 0.0279085 0.276266i
\(546\) 0 0
\(547\) 10.6969i 0.457368i 0.973501 + 0.228684i \(0.0734423\pi\)
−0.973501 + 0.228684i \(0.926558\pi\)
\(548\) 17.7980i 0.760291i
\(549\) 0 0
\(550\) −4.89898 + 24.0000i −0.208893 + 1.02336i
\(551\) 18.6969 0.796516
\(552\) 0 0
\(553\) 2.89898i 0.123277i
\(554\) 5.10102 0.216722
\(555\) 0 0
\(556\) 6.44949 0.273519
\(557\) 16.6969i 0.707472i 0.935345 + 0.353736i \(0.115089\pi\)
−0.935345 + 0.353736i \(0.884911\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −2.22474 0.224745i −0.0940126 0.00949720i
\(561\) 0 0
\(562\) 18.0000i 0.759284i
\(563\) 14.0454i 0.591943i 0.955197 + 0.295972i \(0.0956434\pi\)
−0.955197 + 0.295972i \(0.904357\pi\)
\(564\) 0 0
\(565\) 0.449490 + 0.0454077i 0.0189102 + 0.00191032i
\(566\) 28.2474 1.18733
\(567\) 0 0
\(568\) 10.8990i 0.457311i
\(569\) 14.2020 0.595381 0.297690 0.954663i \(-0.403784\pi\)
0.297690 + 0.954663i \(0.403784\pi\)
\(570\) 0 0
\(571\) −20.8990 −0.874595 −0.437298 0.899317i \(-0.644064\pi\)
−0.437298 + 0.899317i \(0.644064\pi\)
\(572\) 2.20204i 0.0920720i
\(573\) 0 0
\(574\) 10.8990 0.454915
\(575\) −6.89898 + 33.7980i −0.287707 + 1.40947i
\(576\) 0 0
\(577\) 46.4949i 1.93561i 0.251705 + 0.967804i \(0.419009\pi\)
−0.251705 + 0.967804i \(0.580991\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 0.651531 6.44949i 0.0270533 0.267800i
\(581\) 2.44949 0.101622
\(582\) 0 0
\(583\) 5.39388i 0.223392i
\(584\) 6.89898 0.285482
\(585\) 0 0
\(586\) 6.24745 0.258080
\(587\) 33.1464i 1.36810i 0.729435 + 0.684050i \(0.239783\pi\)
−0.729435 + 0.684050i \(0.760217\pi\)
\(588\) 0 0
\(589\) 5.79796 0.238901
\(590\) −14.3485 1.44949i −0.590717 0.0596745i
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 1.10102i 0.0452135i −0.999744 0.0226067i \(-0.992803\pi\)
0.999744 0.0226067i \(-0.00719656\pi\)
\(594\) 0 0
\(595\) −0.449490 + 4.44949i −0.0184273 + 0.182411i
\(596\) −15.7980 −0.647110
\(597\) 0 0
\(598\) 3.10102i 0.126810i
\(599\) −22.8990 −0.935627 −0.467813 0.883827i \(-0.654958\pi\)
−0.467813 + 0.883827i \(0.654958\pi\)
\(600\) 0 0
\(601\) 19.3939 0.791093 0.395546 0.918446i \(-0.370555\pi\)
0.395546 + 0.918446i \(0.370555\pi\)
\(602\) 8.89898i 0.362695i
\(603\) 0 0
\(604\) 19.5959 0.797347
\(605\) 2.92168 28.9217i 0.118783 1.17583i
\(606\) 0 0
\(607\) 25.3939i 1.03071i −0.856978 0.515353i \(-0.827661\pi\)
0.856978 0.515353i \(-0.172339\pi\)
\(608\) 6.44949i 0.261561i
\(609\) 0 0
\(610\) 18.7980 + 1.89898i 0.761107 + 0.0768874i
\(611\) −0.404082 −0.0163474
\(612\) 0 0
\(613\) 8.20204i 0.331277i 0.986187 + 0.165639i \(0.0529685\pi\)
−0.986187 + 0.165639i \(0.947031\pi\)
\(614\) −4.24745 −0.171413
\(615\) 0 0
\(616\) 4.89898 0.197386
\(617\) 9.59592i 0.386317i 0.981168 + 0.193159i \(0.0618732\pi\)
−0.981168 + 0.193159i \(0.938127\pi\)
\(618\) 0 0
\(619\) 46.4495 1.86696 0.933481 0.358626i \(-0.116755\pi\)
0.933481 + 0.358626i \(0.116755\pi\)
\(620\) 0.202041 2.00000i 0.00811416 0.0803219i
\(621\) 0 0
\(622\) 12.0000i 0.481156i
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 23.0000 + 9.79796i 0.920000 + 0.391918i
\(626\) 17.5959 0.703274
\(627\) 0 0
\(628\) 8.44949i 0.337171i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 6.49490 0.258558 0.129279 0.991608i \(-0.458734\pi\)
0.129279 + 0.991608i \(0.458734\pi\)
\(632\) 2.89898i 0.115315i
\(633\) 0 0
\(634\) 26.4949 1.05225
\(635\) −11.3485 1.14643i −0.450350 0.0454946i
\(636\) 0 0
\(637\) 0.449490i 0.0178094i
\(638\) 14.2020i 0.562264i
\(639\) 0 0
\(640\) −2.22474 0.224745i −0.0879408 0.00888382i
\(641\) −6.20204 −0.244966 −0.122483 0.992471i \(-0.539086\pi\)
−0.122483 + 0.992471i \(0.539086\pi\)
\(642\) 0 0
\(643\) 9.14643i 0.360700i 0.983603 + 0.180350i \(0.0577230\pi\)
−0.983603 + 0.180350i \(0.942277\pi\)
\(644\) 6.89898 0.271858
\(645\) 0 0
\(646\) −12.8990 −0.507504
\(647\) 22.2929i 0.876423i −0.898872 0.438211i \(-0.855612\pi\)
0.898872 0.438211i \(-0.144388\pi\)
\(648\) 0 0
\(649\) 31.5959 1.24025
\(650\) 2.20204 + 0.449490i 0.0863712 + 0.0176304i
\(651\) 0 0
\(652\) 16.8990i 0.661815i
\(653\) 39.7980i 1.55741i −0.627387 0.778707i \(-0.715876\pi\)
0.627387 0.778707i \(-0.284124\pi\)
\(654\) 0 0
\(655\) 0.348469 3.44949i 0.0136158 0.134783i
\(656\) 10.8990 0.425534
\(657\) 0 0
\(658\) 0.898979i 0.0350459i
\(659\) 7.10102 0.276616 0.138308 0.990389i \(-0.455834\pi\)
0.138308 + 0.990389i \(0.455834\pi\)
\(660\) 0 0
\(661\) 12.9444 0.503478 0.251739 0.967795i \(-0.418997\pi\)
0.251739 + 0.967795i \(0.418997\pi\)
\(662\) 10.6969i 0.415748i
\(663\) 0 0
\(664\) 2.44949 0.0950586
\(665\) 14.3485 + 1.44949i 0.556410 + 0.0562088i
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 4.89898i 0.189547i
\(669\) 0 0
\(670\) 1.79796 17.7980i 0.0694612 0.687595i
\(671\) −41.3939 −1.59799
\(672\) 0 0
\(673\) 1.79796i 0.0693062i −0.999399 0.0346531i \(-0.988967\pi\)
0.999399 0.0346531i \(-0.0110326\pi\)
\(674\) 29.5959 1.13999
\(675\) 0 0
\(676\) −12.7980 −0.492229
\(677\) 31.5505i 1.21258i 0.795242 + 0.606292i \(0.207344\pi\)
−0.795242 + 0.606292i \(0.792656\pi\)
\(678\) 0 0
\(679\) −3.79796 −0.145752
\(680\) −0.449490 + 4.44949i −0.0172371 + 0.170630i
\(681\) 0 0
\(682\) 4.40408i 0.168641i
\(683\) 35.5959i 1.36204i −0.732265 0.681020i \(-0.761537\pi\)
0.732265 0.681020i \(-0.238463\pi\)
\(684\) 0 0
\(685\) −39.5959 4.00000i −1.51288 0.152832i
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 8.89898i 0.339270i
\(689\) 0.494897 0.0188541
\(690\) 0 0
\(691\) −13.1464 −0.500114 −0.250057 0.968231i \(-0.580449\pi\)
−0.250057 + 0.968231i \(0.580449\pi\)
\(692\) 18.2474i 0.693664i
\(693\) 0 0
\(694\) 19.1010 0.725065
\(695\) −1.44949 + 14.3485i −0.0549823 + 0.544268i
\(696\) 0 0
\(697\) 21.7980i 0.825657i
\(698\) 3.55051i 0.134389i
\(699\) 0 0
\(700\) 1.00000 4.89898i 0.0377964 0.185164i
\(701\) −40.6969 −1.53710 −0.768551 0.639788i \(-0.779022\pi\)
−0.768551 + 0.639788i \(0.779022\pi\)
\(702\) 0 0
\(703\) 12.8990i 0.486494i
\(704\) 4.89898 0.184637
\(705\) 0 0
\(706\) −13.1010 −0.493063
\(707\) 8.44949i 0.317776i
\(708\) 0 0
\(709\) −40.2929 −1.51323 −0.756615 0.653861i \(-0.773148\pi\)
−0.756615 + 0.653861i \(0.773148\pi\)
\(710\) 24.2474 + 2.44949i 0.909991 + 0.0919277i
\(711\) 0 0
\(712\) 10.0000i 0.374766i
\(713\) 6.20204i 0.232268i
\(714\) 0 0
\(715\) −4.89898 0.494897i −0.183211 0.0185081i
\(716\) 5.79796 0.216680
\(717\) 0 0
\(718\) 11.5959i 0.432756i
\(719\) 44.4949 1.65938 0.829690 0.558225i \(-0.188517\pi\)
0.829690 + 0.558225i \(0.188517\pi\)
\(720\) 0 0
\(721\) −3.10102 −0.115488
\(722\) 22.5959i 0.840933i
\(723\) 0 0
\(724\) −14.2474 −0.529502
\(725\) 14.2020 + 2.89898i 0.527451 + 0.107665i
\(726\) 0 0
\(727\) 6.69694i 0.248376i −0.992259 0.124188i \(-0.960367\pi\)
0.992259 0.124188i \(-0.0396325\pi\)
\(728\) 0.449490i 0.0166592i
\(729\) 0 0
\(730\) −1.55051 + 15.3485i −0.0573870 + 0.568072i
\(731\) −17.7980 −0.658281
\(732\) 0 0
\(733\) 43.6413i 1.61193i 0.591964 + 0.805965i \(0.298353\pi\)
−0.591964 + 0.805965i \(0.701647\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 6.89898 0.254300
\(737\) 39.1918i 1.44365i
\(738\) 0 0
\(739\) 44.4949 1.63677 0.818386 0.574669i \(-0.194869\pi\)
0.818386 + 0.574669i \(0.194869\pi\)
\(740\) −4.44949 0.449490i −0.163566 0.0165236i
\(741\) 0 0
\(742\) 1.10102i 0.0404197i
\(743\) 15.3031i 0.561415i −0.959793 0.280707i \(-0.909431\pi\)
0.959793 0.280707i \(-0.0905691\pi\)
\(744\) 0 0
\(745\) 3.55051 35.1464i 0.130081 1.28767i
\(746\) 24.6969 0.904219
\(747\) 0 0
\(748\) 9.79796i 0.358249i
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −22.2020 −0.810164 −0.405082 0.914280i \(-0.632757\pi\)
−0.405082 + 0.914280i \(0.632757\pi\)
\(752\) 0.898979i 0.0327824i
\(753\) 0 0
\(754\) 1.30306 0.0474547
\(755\) −4.40408 + 43.5959i −0.160281 + 1.58662i
\(756\) 0 0
\(757\) 32.2020i 1.17040i −0.810888 0.585202i \(-0.801015\pi\)
0.810888 0.585202i \(-0.198985\pi\)
\(758\) 1.30306i 0.0473293i
\(759\) 0 0
\(760\) 14.3485 + 1.44949i 0.520474 + 0.0525785i
\(761\) 30.8990 1.12009 0.560044 0.828463i \(-0.310784\pi\)
0.560044 + 0.828463i \(0.310784\pi\)
\(762\) 0 0
\(763\) 2.89898i 0.104950i
\(764\) −16.6969 −0.604074
\(765\) 0 0
\(766\) 16.8990 0.610585
\(767\) 2.89898i 0.104676i
\(768\) 0 0
\(769\) 11.3031 0.407599 0.203799 0.979013i \(-0.434671\pi\)
0.203799 + 0.979013i \(0.434671\pi\)
\(770\) −1.10102 + 10.8990i −0.0396780 + 0.392772i
\(771\) 0 0
\(772\) 17.5959i 0.633291i
\(773\) 13.3485i 0.480111i −0.970759 0.240056i \(-0.922834\pi\)
0.970759 0.240056i \(-0.0771657\pi\)
\(774\) 0 0
\(775\) 4.40408 + 0.898979i 0.158199 + 0.0322923i
\(776\) −3.79796 −0.136339
\(777\) 0 0
\(778\) 22.8990i 0.820968i
\(779\) −70.2929 −2.51850
\(780\) 0 0
\(781\) −53.3939 −1.91058
\(782\) 13.7980i 0.493414i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 18.7980 + 1.89898i 0.670928 + 0.0677775i
\(786\) 0 0
\(787\) 45.5505i 1.62370i 0.583866 + 0.811850i \(0.301539\pi\)
−0.583866 + 0.811850i \(0.698461\pi\)
\(788\) 9.10102i 0.324210i
\(789\) 0 0
\(790\) 6.44949 + 0.651531i 0.229463 + 0.0231804i
\(791\) 0.202041 0.00718375
\(792\) 0 0
\(793\) 3.79796i 0.134869i
\(794\) 17.3485 0.615674
\(795\) 0 0
\(796\) 7.10102 0.251689
\(797\) 52.9444i 1.87539i −0.347464 0.937693i \(-0.612957\pi\)
0.347464 0.937693i \(-0.387043\pi\)
\(798\) 0 0
\(799\) 1.79796 0.0636072
\(800\) 1.00000 4.89898i 0.0353553 0.173205i
\(801\) 0 0
\(802\) 29.3939i 1.03793i
\(803\) 33.7980i 1.19270i
\(804\) 0 0
\(805\) −1.55051 + 15.3485i −0.0546483 + 0.540962i
\(806\) 0.404082 0.0142332
\(807\) 0 0
\(808\) 8.44949i 0.297252i
\(809\) 8.40408 0.295472 0.147736 0.989027i \(-0.452801\pi\)
0.147736 + 0.989027i \(0.452801\pi\)
\(810\) 0 0
\(811\) −38.9444 −1.36752 −0.683761 0.729706i \(-0.739657\pi\)
−0.683761 + 0.729706i \(0.739657\pi\)
\(812\) 2.89898i 0.101734i
\(813\) 0 0
\(814\) 9.79796 0.343418
\(815\) 37.5959 + 3.79796i 1.31693 + 0.133037i
\(816\) 0 0
\(817\) 57.3939i 2.00796i
\(818\) 14.4949i 0.506802i
\(819\) 0 0
\(820\) −2.44949 + 24.2474i −0.0855399 + 0.846758i
\(821\) −27.7980 −0.970155 −0.485078 0.874471i \(-0.661209\pi\)
−0.485078 + 0.874471i \(0.661209\pi\)
\(822\) 0 0
\(823\) 39.1918i 1.36614i −0.730352 0.683071i \(-0.760644\pi\)
0.730352 0.683071i \(-0.239356\pi\)
\(824\) −3.10102 −0.108029
\(825\) 0 0
\(826\) −6.44949 −0.224406
\(827\) 23.5959i 0.820510i −0.911971 0.410255i \(-0.865440\pi\)
0.911971 0.410255i \(-0.134560\pi\)
\(828\) 0 0
\(829\) −39.6413 −1.37680 −0.688400 0.725331i \(-0.741687\pi\)
−0.688400 + 0.725331i \(0.741687\pi\)
\(830\) −0.550510 + 5.44949i −0.0191085 + 0.189155i
\(831\) 0 0
\(832\) 0.449490i 0.0155833i
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) −10.8990 1.10102i −0.377175 0.0381024i
\(836\) −31.5959 −1.09277
\(837\) 0 0
\(838\) 6.44949i 0.222794i
\(839\) 27.1010 0.935631 0.467816 0.883826i \(-0.345041\pi\)
0.467816 + 0.883826i \(0.345041\pi\)
\(840\) 0 0
\(841\) −20.5959 −0.710204
\(842\) 23.7980i 0.820132i
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 2.87628 28.4722i 0.0989469 0.979473i
\(846\) 0 0
\(847\) 13.0000i 0.446685i
\(848\) 1.10102i 0.0378092i
\(849\) 0 0
\(850\) −9.79796 2.00000i −0.336067 0.0685994i
\(851\) 13.7980 0.472988
\(852\) 0 0
\(853\) 29.8434i 1.02182i −0.859635 0.510909i \(-0.829309\pi\)
0.859635 0.510909i \(-0.170691\pi\)
\(854\) 8.44949 0.289136
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 25.1918i 0.860537i −0.902701 0.430268i \(-0.858419\pi\)
0.902701 0.430268i \(-0.141581\pi\)
\(858\) 0 0
\(859\) 29.6413 1.01135 0.505674 0.862724i \(-0.331244\pi\)
0.505674 + 0.862724i \(0.331244\pi\)
\(860\) 19.7980 + 2.00000i 0.675105 + 0.0681994i
\(861\) 0 0
\(862\) 17.7980i 0.606201i
\(863\) 13.3939i 0.455933i 0.973669 + 0.227966i \(0.0732076\pi\)
−0.973669 + 0.227966i \(0.926792\pi\)
\(864\) 0 0
\(865\) 40.5959 + 4.10102i 1.38030 + 0.139439i
\(866\) −19.7980 −0.672762
\(867\) 0 0
\(868\) 0.898979i 0.0305134i
\(869\) −14.2020 −0.481771
\(870\) 0 0
\(871\) 3.59592 0.121843
\(872\) 2.89898i 0.0981718i
\(873\) 0 0
\(874\) −44.4949 −1.50506
\(875\) 10.6742 + 3.32577i 0.360855 + 0.112431i
\(876\) 0 0
\(877\) 19.3939i 0.654885i 0.944871 + 0.327442i \(0.106187\pi\)
−0.944871 + 0.327442i \(0.893813\pi\)
\(878\) 37.3939i 1.26198i
\(879\) 0 0
\(880\) −1.10102 + 10.8990i −0.0371154 + 0.367405i
\(881\) −27.7980 −0.936537 −0.468269 0.883586i \(-0.655122\pi\)
−0.468269 + 0.883586i \(0.655122\pi\)
\(882\) 0 0
\(883\) 41.7980i 1.40661i −0.710887 0.703307i \(-0.751706\pi\)
0.710887 0.703307i \(-0.248294\pi\)
\(884\) −0.898979 −0.0302360
\(885\) 0 0
\(886\) 9.79796 0.329169
\(887\) 26.6969i 0.896395i 0.893934 + 0.448198i \(0.147934\pi\)
−0.893934 + 0.448198i \(0.852066\pi\)
\(888\) 0 0
\(889\) −5.10102 −0.171083
\(890\) −22.2474 2.24745i −0.745736 0.0753347i
\(891\) 0 0
\(892\) 4.00000i 0.133930i
\(893\) 5.79796i 0.194021i
\(894\) 0 0
\(895\) −1.30306 + 12.8990i −0.0435565 + 0.431165i
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.0000i 0.333704i
\(899\) 2.60612 0.0869191
\(900\) 0 0
\(901\) −2.20204 −0.0733606
\(902\) 53.3939i 1.77782i
\(903\) 0 0
\(904\) 0.202041 0.00671978
\(905\) 3.20204 31.6969i 0.106439 1.05364i
\(906\) 0 0
\(907\) 22.2020i 0.737207i −0.929587 0.368603i \(-0.879836\pi\)
0.929587 0.368603i \(-0.120164\pi\)
\(908\) 7.34847i 0.243868i
\(909\) 0 0
\(910\) 1.00000 + 0.101021i 0.0331497 + 0.00334880i
\(911\) −3.59592 −0.119138 −0.0595690 0.998224i \(-0.518973\pi\)
−0.0595690 + 0.998224i \(0.518973\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 9.59592 0.317405
\(915\) 0 0
\(916\) 15.1464 0.500452
\(917\) 1.55051i 0.0512024i
\(918\) 0 0
\(919\) 17.1010 0.564111 0.282055 0.959398i \(-0.408984\pi\)
0.282055 + 0.959398i \(0.408984\pi\)
\(920\) −1.55051 + 15.3485i −0.0511188 + 0.506024i
\(921\) 0 0
\(922\) 2.65153i 0.0873235i
\(923\) 4.89898i 0.161252i
\(924\) 0 0
\(925\) 2.00000 9.79796i 0.0657596 0.322155i
\(926\) −35.5959 −1.16975
\(927\) 0 0
\(928\) 2.89898i 0.0951637i
\(929\) 40.2929 1.32197 0.660983 0.750401i \(-0.270140\pi\)
0.660983 + 0.750401i \(0.270140\pi\)
\(930\) 0 0
\(931\) 6.44949 0.211373
\(932\) 10.2020i 0.334179i
\(933\) 0 0
\(934\) 5.55051 0.181618
\(935\) 21.7980 + 2.20204i 0.712869 + 0.0720144i
\(936\) 0 0
\(937\) 50.8990i 1.66280i −0.555677 0.831399i \(-0.687541\pi\)
0.555677 0.831399i \(-0.312459\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 0 0
\(940\) −2.00000 0.202041i −0.0652328 0.00658985i
\(941\) 24.4495 0.797031 0.398515 0.917162i \(-0.369526\pi\)
0.398515 + 0.917162i \(0.369526\pi\)
\(942\) 0 0
\(943\) 75.1918i 2.44858i
\(944\) −6.44949 −0.209913
\(945\) 0 0
\(946\) −43.5959 −1.41743
\(947\) 44.0908i 1.43276i 0.697711 + 0.716379i \(0.254202\pi\)
−0.697711 + 0.716379i \(0.745798\pi\)
\(948\) 0 0
\(949\) −3.10102 −0.100663
\(950\) −6.44949 + 31.5959i −0.209249 + 1.02511i
\(951\) 0 0
\(952\) 2.00000i 0.0648204i
\(953\) 21.7980i 0.706105i 0.935603 + 0.353053i \(0.114856\pi\)
−0.935603 + 0.353053i \(0.885144\pi\)
\(954\) 0 0
\(955\) 3.75255 37.1464i 0.121430 1.20203i
\(956\) 25.7980 0.834366
\(957\) 0 0
\(958\) 38.6969i 1.25024i
\(959\) −17.7980 −0.574726
\(960\) 0 0
\(961\) −30.1918 −0.973930
\(962\) 0.898979i 0.0289843i
\(963\) 0 0
\(964\) −20.6969 −0.666604
\(965\) −39.1464 3.95459i −1.26017 0.127303i
\(966\) 0 0
\(967\) 32.2929i 1.03847i 0.854632 + 0.519234i \(0.173783\pi\)
−0.854632 + 0.519234i \(0.826217\pi\)
\(968\) 13.0000i 0.417836i
\(969\) 0 0
\(970\) 0.853572 8.44949i 0.0274065 0.271297i
\(971\) 14.4495 0.463706 0.231853 0.972751i \(-0.425521\pi\)
0.231853 + 0.972751i \(0.425521\pi\)
\(972\) 0 0
\(973\) 6.44949i 0.206761i
\(974\) 36.6969 1.17585
\(975\) 0 0
\(976\) 8.44949 0.270462
\(977\) 29.3939i 0.940393i −0.882562 0.470197i \(-0.844183\pi\)
0.882562 0.470197i \(-0.155817\pi\)
\(978\) 0 0
\(979\) 48.9898 1.56572
\(980\) 0.224745 2.22474i 0.00717921 0.0710669i
\(981\) 0 0
\(982\) 19.5959i 0.625331i
\(983\) 42.6969i 1.36182i −0.732367 0.680910i \(-0.761584\pi\)
0.732367 0.680910i \(-0.238416\pi\)
\(984\) 0 0
\(985\) −20.2474 2.04541i −0.645137 0.0651721i
\(986\) −5.79796 −0.184645
\(987\) 0 0
\(988\) 2.89898i 0.0922288i
\(989\) −61.3939 −1.95221
\(990\) 0 0
\(991\) 60.6969 1.92810 0.964051 0.265718i \(-0.0856089\pi\)
0.964051 + 0.265718i \(0.0856089\pi\)
\(992\) 0.898979i 0.0285426i
\(993\) 0 0
\(994\) 10.8990 0.345695
\(995\) −1.59592 + 15.7980i −0.0505940 + 0.500829i
\(996\) 0 0
\(997\) 42.6515i 1.35079i 0.737457 + 0.675394i \(0.236026\pi\)
−0.737457 + 0.675394i \(0.763974\pi\)
\(998\) 25.7980i 0.816620i
\(999\) 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 630.2.g.g.379.4 4
3.2 odd 2 70.2.c.a.29.1 4
4.3 odd 2 5040.2.t.t.1009.3 4
5.2 odd 4 3150.2.a.bs.1.1 2
5.3 odd 4 3150.2.a.bt.1.1 2
5.4 even 2 inner 630.2.g.g.379.2 4
12.11 even 2 560.2.g.e.449.3 4
15.2 even 4 350.2.a.h.1.1 2
15.8 even 4 350.2.a.g.1.2 2
15.14 odd 2 70.2.c.a.29.4 yes 4
20.19 odd 2 5040.2.t.t.1009.4 4
21.2 odd 6 490.2.i.c.459.2 8
21.5 even 6 490.2.i.f.459.1 8
21.11 odd 6 490.2.i.c.79.3 8
21.17 even 6 490.2.i.f.79.4 8
21.20 even 2 490.2.c.e.99.2 4
24.5 odd 2 2240.2.g.j.449.4 4
24.11 even 2 2240.2.g.i.449.2 4
60.23 odd 4 2800.2.a.bm.1.1 2
60.47 odd 4 2800.2.a.bl.1.2 2
60.59 even 2 560.2.g.e.449.1 4
105.44 odd 6 490.2.i.c.459.3 8
105.59 even 6 490.2.i.f.79.1 8
105.62 odd 4 2450.2.a.bq.1.2 2
105.74 odd 6 490.2.i.c.79.2 8
105.83 odd 4 2450.2.a.bl.1.1 2
105.89 even 6 490.2.i.f.459.4 8
105.104 even 2 490.2.c.e.99.3 4
120.29 odd 2 2240.2.g.j.449.2 4
120.59 even 2 2240.2.g.i.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.1 4 3.2 odd 2
70.2.c.a.29.4 yes 4 15.14 odd 2
350.2.a.g.1.2 2 15.8 even 4
350.2.a.h.1.1 2 15.2 even 4
490.2.c.e.99.2 4 21.20 even 2
490.2.c.e.99.3 4 105.104 even 2
490.2.i.c.79.2 8 105.74 odd 6
490.2.i.c.79.3 8 21.11 odd 6
490.2.i.c.459.2 8 21.2 odd 6
490.2.i.c.459.3 8 105.44 odd 6
490.2.i.f.79.1 8 105.59 even 6
490.2.i.f.79.4 8 21.17 even 6
490.2.i.f.459.1 8 21.5 even 6
490.2.i.f.459.4 8 105.89 even 6
560.2.g.e.449.1 4 60.59 even 2
560.2.g.e.449.3 4 12.11 even 2
630.2.g.g.379.2 4 5.4 even 2 inner
630.2.g.g.379.4 4 1.1 even 1 trivial
2240.2.g.i.449.2 4 24.11 even 2
2240.2.g.i.449.4 4 120.59 even 2
2240.2.g.j.449.2 4 120.29 odd 2
2240.2.g.j.449.4 4 24.5 odd 2
2450.2.a.bl.1.1 2 105.83 odd 4
2450.2.a.bq.1.2 2 105.62 odd 4
2800.2.a.bl.1.2 2 60.47 odd 4
2800.2.a.bm.1.1 2 60.23 odd 4
3150.2.a.bs.1.1 2 5.2 odd 4
3150.2.a.bt.1.1 2 5.3 odd 4
5040.2.t.t.1009.3 4 4.3 odd 2
5040.2.t.t.1009.4 4 20.19 odd 2