Properties

Label 495.2.d.a.494.4
Level $495$
Weight $2$
Character 495.494
Analytic conductor $3.953$
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] = [495,2,Mod(494,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.494");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.d (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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 494.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 495.494
Dual form 495.2.d.a.494.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +(2.12132 - 0.707107i) q^{5} +1.41421 q^{7} +3.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{4} +(2.12132 - 0.707107i) q^{5} +1.41421 q^{7} +3.00000i q^{8} +(0.707107 + 2.12132i) q^{10} +(-3.00000 - 1.41421i) q^{11} +2.82843 q^{13} +1.41421i q^{14} -1.00000 q^{16} +2.00000i q^{17} -4.24264i q^{19} +(2.12132 - 0.707107i) q^{20} +(1.41421 - 3.00000i) q^{22} +(4.00000 - 3.00000i) q^{25} +2.82843i q^{26} +1.41421 q^{28} -6.00000 q^{29} +5.00000i q^{32} -2.00000 q^{34} +(3.00000 - 1.00000i) q^{35} +6.00000i q^{37} +4.24264 q^{38} +(2.12132 + 6.36396i) q^{40} +6.00000 q^{41} -1.41421 q^{43} +(-3.00000 - 1.41421i) q^{44} -8.48528 q^{47} -5.00000 q^{49} +(3.00000 + 4.00000i) q^{50} +2.82843 q^{52} +4.24264 q^{53} +(-7.36396 - 0.878680i) q^{55} +4.24264i q^{56} -6.00000i q^{58} +8.48528i q^{59} +8.48528i q^{61} -7.00000 q^{64} +(6.00000 - 2.00000i) q^{65} +2.00000i q^{68} +(1.00000 + 3.00000i) q^{70} -8.48528i q^{71} -11.3137 q^{73} -6.00000 q^{74} -4.24264i q^{76} +(-4.24264 - 2.00000i) q^{77} -12.7279i q^{79} +(-2.12132 + 0.707107i) q^{80} +6.00000i q^{82} -14.0000i q^{83} +(1.41421 + 4.24264i) q^{85} -1.41421i q^{86} +(4.24264 - 9.00000i) q^{88} -7.07107i q^{89} +4.00000 q^{91} -8.48528i q^{94} +(-3.00000 - 9.00000i) q^{95} +12.0000i q^{97} -5.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 12 q^{11} - 4 q^{16} + 16 q^{25} - 24 q^{29} - 8 q^{34} + 12 q^{35} + 24 q^{41} - 12 q^{44} - 20 q^{49} + 12 q^{50} - 4 q^{55} - 28 q^{64} + 24 q^{65} + 4 q^{70} - 24 q^{74} + 16 q^{91} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.12132 0.707107i 0.948683 0.316228i
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 0 0
\(10\) 0.707107 + 2.12132i 0.223607 + 0.670820i
\(11\) −3.00000 1.41421i −0.904534 0.426401i
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 1.41421i 0.377964i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 4.24264i 0.973329i −0.873589 0.486664i \(-0.838214\pi\)
0.873589 0.486664i \(-0.161786\pi\)
\(20\) 2.12132 0.707107i 0.474342 0.158114i
\(21\) 0 0
\(22\) 1.41421 3.00000i 0.301511 0.639602i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 3.00000i 0.800000 0.600000i
\(26\) 2.82843i 0.554700i
\(27\) 0 0
\(28\) 1.41421 0.267261
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 3.00000 1.00000i 0.507093 0.169031i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 4.24264 0.688247
\(39\) 0 0
\(40\) 2.12132 + 6.36396i 0.335410 + 1.00623i
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.41421 −0.215666 −0.107833 0.994169i \(-0.534391\pi\)
−0.107833 + 0.994169i \(0.534391\pi\)
\(44\) −3.00000 1.41421i −0.452267 0.213201i
\(45\) 0 0
\(46\) 0 0
\(47\) −8.48528 −1.23771 −0.618853 0.785507i \(-0.712402\pi\)
−0.618853 + 0.785507i \(0.712402\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 3.00000 + 4.00000i 0.424264 + 0.565685i
\(51\) 0 0
\(52\) 2.82843 0.392232
\(53\) 4.24264 0.582772 0.291386 0.956606i \(-0.405884\pi\)
0.291386 + 0.956606i \(0.405884\pi\)
\(54\) 0 0
\(55\) −7.36396 0.878680i −0.992956 0.118481i
\(56\) 4.24264i 0.566947i
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 8.48528i 1.10469i 0.833616 + 0.552345i \(0.186267\pi\)
−0.833616 + 0.552345i \(0.813733\pi\)
\(60\) 0 0
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 6.00000 2.00000i 0.744208 0.248069i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) 1.00000 + 3.00000i 0.119523 + 0.358569i
\(71\) 8.48528i 1.00702i −0.863990 0.503509i \(-0.832042\pi\)
0.863990 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.24264i 0.486664i
\(77\) −4.24264 2.00000i −0.483494 0.227921i
\(78\) 0 0
\(79\) 12.7279i 1.43200i −0.698099 0.716002i \(-0.745970\pi\)
0.698099 0.716002i \(-0.254030\pi\)
\(80\) −2.12132 + 0.707107i −0.237171 + 0.0790569i
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) 14.0000i 1.53670i −0.640030 0.768350i \(-0.721078\pi\)
0.640030 0.768350i \(-0.278922\pi\)
\(84\) 0 0
\(85\) 1.41421 + 4.24264i 0.153393 + 0.460179i
\(86\) 1.41421i 0.152499i
\(87\) 0 0
\(88\) 4.24264 9.00000i 0.452267 0.959403i
\(89\) 7.07107i 0.749532i −0.927119 0.374766i \(-0.877723\pi\)
0.927119 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 8.48528i 0.875190i
\(95\) −3.00000 9.00000i −0.307794 0.923381i
\(96\) 0 0
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 5.00000i 0.505076i
\(99\) 0 0
\(100\) 4.00000 3.00000i 0.400000 0.300000i
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) 8.48528i 0.832050i
\(105\) 0 0
\(106\) 4.24264i 0.412082i
\(107\) 14.0000i 1.35343i −0.736245 0.676716i \(-0.763403\pi\)
0.736245 0.676716i \(-0.236597\pi\)
\(108\) 0 0
\(109\) 16.9706i 1.62549i −0.582623 0.812743i \(-0.697974\pi\)
0.582623 0.812743i \(-0.302026\pi\)
\(110\) 0.878680 7.36396i 0.0837788 0.702126i
\(111\) 0 0
\(112\) −1.41421 −0.133631
\(113\) −12.7279 −1.19734 −0.598671 0.800995i \(-0.704304\pi\)
−0.598671 + 0.800995i \(0.704304\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −8.48528 −0.781133
\(119\) 2.82843i 0.259281i
\(120\) 0 0
\(121\) 7.00000 + 8.48528i 0.636364 + 0.771389i
\(122\) −8.48528 −0.768221
\(123\) 0 0
\(124\) 0 0
\(125\) 6.36396 9.19239i 0.569210 0.822192i
\(126\) 0 0
\(127\) −18.3848 −1.63139 −0.815693 0.578486i \(-0.803644\pi\)
−0.815693 + 0.578486i \(0.803644\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 0 0
\(130\) 2.00000 + 6.00000i 0.175412 + 0.526235i
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 4.24264 0.362473 0.181237 0.983440i \(-0.441990\pi\)
0.181237 + 0.983440i \(0.441990\pi\)
\(138\) 0 0
\(139\) 4.24264i 0.359856i −0.983680 0.179928i \(-0.942414\pi\)
0.983680 0.179928i \(-0.0575865\pi\)
\(140\) 3.00000 1.00000i 0.253546 0.0845154i
\(141\) 0 0
\(142\) 8.48528 0.712069
\(143\) −8.48528 4.00000i −0.709575 0.334497i
\(144\) 0 0
\(145\) −12.7279 + 4.24264i −1.05700 + 0.352332i
\(146\) 11.3137i 0.936329i
\(147\) 0 0
\(148\) 6.00000i 0.493197i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 4.24264i 0.345261i 0.984987 + 0.172631i \(0.0552267\pi\)
−0.984987 + 0.172631i \(0.944773\pi\)
\(152\) 12.7279 1.03237
\(153\) 0 0
\(154\) 2.00000 4.24264i 0.161165 0.341882i
\(155\) 0 0
\(156\) 0 0
\(157\) 24.0000i 1.91541i −0.287754 0.957704i \(-0.592909\pi\)
0.287754 0.957704i \(-0.407091\pi\)
\(158\) 12.7279 1.01258
\(159\) 0 0
\(160\) 3.53553 + 10.6066i 0.279508 + 0.838525i
\(161\) 0 0
\(162\) 0 0
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 8.00000i 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) −4.24264 + 1.41421i −0.325396 + 0.108465i
\(171\) 0 0
\(172\) −1.41421 −0.107833
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 0 0
\(175\) 5.65685 4.24264i 0.427618 0.320713i
\(176\) 3.00000 + 1.41421i 0.226134 + 0.106600i
\(177\) 0 0
\(178\) 7.07107 0.529999
\(179\) 5.65685i 0.422813i −0.977398 0.211407i \(-0.932196\pi\)
0.977398 0.211407i \(-0.0678044\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 4.00000i 0.296500i
\(183\) 0 0
\(184\) 0 0
\(185\) 4.24264 + 12.7279i 0.311925 + 0.935775i
\(186\) 0 0
\(187\) 2.82843 6.00000i 0.206835 0.438763i
\(188\) −8.48528 −0.618853
\(189\) 0 0
\(190\) 9.00000 3.00000i 0.652929 0.217643i
\(191\) 2.82843i 0.204658i −0.994751 0.102329i \(-0.967371\pi\)
0.994751 0.102329i \(-0.0326294\pi\)
\(192\) 0 0
\(193\) 19.7990 1.42516 0.712581 0.701590i \(-0.247526\pi\)
0.712581 + 0.701590i \(0.247526\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −5.00000 −0.357143
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 9.00000 + 12.0000i 0.636396 + 0.848528i
\(201\) 0 0
\(202\) 6.00000i 0.422159i
\(203\) −8.48528 −0.595550
\(204\) 0 0
\(205\) 12.7279 4.24264i 0.888957 0.296319i
\(206\) −12.0000 −0.836080
\(207\) 0 0
\(208\) −2.82843 −0.196116
\(209\) −6.00000 + 12.7279i −0.415029 + 0.880409i
\(210\) 0 0
\(211\) 4.24264i 0.292075i 0.989279 + 0.146038i \(0.0466521\pi\)
−0.989279 + 0.146038i \(0.953348\pi\)
\(212\) 4.24264 0.291386
\(213\) 0 0
\(214\) 14.0000 0.957020
\(215\) −3.00000 + 1.00000i −0.204598 + 0.0681994i
\(216\) 0 0
\(217\) 0 0
\(218\) 16.9706 1.14939
\(219\) 0 0
\(220\) −7.36396 0.878680i −0.496478 0.0592406i
\(221\) 5.65685i 0.380521i
\(222\) 0 0
\(223\) 24.0000i 1.60716i −0.595198 0.803579i \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) 7.07107i 0.472456i
\(225\) 0 0
\(226\) 12.7279i 0.846649i
\(227\) 2.00000i 0.132745i 0.997795 + 0.0663723i \(0.0211425\pi\)
−0.997795 + 0.0663723i \(0.978857\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000i 1.18176i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) −18.0000 + 6.00000i −1.17419 + 0.391397i
\(236\) 8.48528i 0.552345i
\(237\) 0 0
\(238\) −2.82843 −0.183340
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 8.48528i 0.546585i 0.961931 + 0.273293i \(0.0881127\pi\)
−0.961931 + 0.273293i \(0.911887\pi\)
\(242\) −8.48528 + 7.00000i −0.545455 + 0.449977i
\(243\) 0 0
\(244\) 8.48528i 0.543214i
\(245\) −10.6066 + 3.53553i −0.677631 + 0.225877i
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 9.19239 + 6.36396i 0.581378 + 0.402492i
\(251\) 16.9706i 1.07117i 0.844481 + 0.535586i \(0.179909\pi\)
−0.844481 + 0.535586i \(0.820091\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 18.3848i 1.15356i
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 29.6985 1.85254 0.926270 0.376860i \(-0.122996\pi\)
0.926270 + 0.376860i \(0.122996\pi\)
\(258\) 0 0
\(259\) 8.48528i 0.527250i
\(260\) 6.00000 2.00000i 0.372104 0.124035i
\(261\) 0 0
\(262\) 18.0000i 1.11204i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 9.00000 3.00000i 0.552866 0.184289i
\(266\) 6.00000 0.367884
\(267\) 0 0
\(268\) 0 0
\(269\) 12.7279i 0.776035i 0.921652 + 0.388018i \(0.126840\pi\)
−0.921652 + 0.388018i \(0.873160\pi\)
\(270\) 0 0
\(271\) 12.7279i 0.773166i 0.922255 + 0.386583i \(0.126345\pi\)
−0.922255 + 0.386583i \(0.873655\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) 4.24264i 0.256307i
\(275\) −16.2426 + 3.34315i −0.979468 + 0.201599i
\(276\) 0 0
\(277\) 31.1127 1.86938 0.934690 0.355463i \(-0.115677\pi\)
0.934690 + 0.355463i \(0.115677\pi\)
\(278\) 4.24264 0.254457
\(279\) 0 0
\(280\) 3.00000 + 9.00000i 0.179284 + 0.537853i
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 1.41421 0.0840663 0.0420331 0.999116i \(-0.486616\pi\)
0.0420331 + 0.999116i \(0.486616\pi\)
\(284\) 8.48528i 0.503509i
\(285\) 0 0
\(286\) 4.00000 8.48528i 0.236525 0.501745i
\(287\) 8.48528 0.500870
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) −4.24264 12.7279i −0.249136 0.747409i
\(291\) 0 0
\(292\) −11.3137 −0.662085
\(293\) 10.0000i 0.584206i 0.956387 + 0.292103i \(0.0943550\pi\)
−0.956387 + 0.292103i \(0.905645\pi\)
\(294\) 0 0
\(295\) 6.00000 + 18.0000i 0.349334 + 1.04800i
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) 18.0000i 1.04271i
\(299\) 0 0
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) −4.24264 −0.244137
\(303\) 0 0
\(304\) 4.24264i 0.243332i
\(305\) 6.00000 + 18.0000i 0.343559 + 1.03068i
\(306\) 0 0
\(307\) −24.0416 −1.37213 −0.686064 0.727541i \(-0.740663\pi\)
−0.686064 + 0.727541i \(0.740663\pi\)
\(308\) −4.24264 2.00000i −0.241747 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) 16.9706i 0.962312i −0.876635 0.481156i \(-0.840217\pi\)
0.876635 0.481156i \(-0.159783\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 24.0000 1.35440
\(315\) 0 0
\(316\) 12.7279i 0.716002i
\(317\) 21.2132 1.19145 0.595726 0.803188i \(-0.296864\pi\)
0.595726 + 0.803188i \(0.296864\pi\)
\(318\) 0 0
\(319\) 18.0000 + 8.48528i 1.00781 + 0.475085i
\(320\) −14.8492 + 4.94975i −0.830098 + 0.276699i
\(321\) 0 0
\(322\) 0 0
\(323\) 8.48528 0.472134
\(324\) 0 0
\(325\) 11.3137 8.48528i 0.627572 0.470679i
\(326\) −24.0000 −1.32924
\(327\) 0 0
\(328\) 18.0000i 0.993884i
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 14.0000i 0.768350i
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 11.3137 0.616297 0.308148 0.951338i \(-0.400291\pi\)
0.308148 + 0.951338i \(0.400291\pi\)
\(338\) 5.00000i 0.271964i
\(339\) 0 0
\(340\) 1.41421 + 4.24264i 0.0766965 + 0.230089i
\(341\) 0 0
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 4.24264i 0.228748i
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 14.0000i 0.751559i 0.926709 + 0.375780i \(0.122625\pi\)
−0.926709 + 0.375780i \(0.877375\pi\)
\(348\) 0 0
\(349\) 8.48528i 0.454207i −0.973871 0.227103i \(-0.927074\pi\)
0.973871 0.227103i \(-0.0729255\pi\)
\(350\) 4.24264 + 5.65685i 0.226779 + 0.302372i
\(351\) 0 0
\(352\) 7.07107 15.0000i 0.376889 0.799503i
\(353\) −4.24264 −0.225813 −0.112906 0.993606i \(-0.536016\pi\)
−0.112906 + 0.993606i \(0.536016\pi\)
\(354\) 0 0
\(355\) −6.00000 18.0000i −0.318447 0.955341i
\(356\) 7.07107i 0.374766i
\(357\) 0 0
\(358\) 5.65685 0.298974
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −24.0000 + 8.00000i −1.25622 + 0.418739i
\(366\) 0 0
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −12.7279 + 4.24264i −0.661693 + 0.220564i
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 22.6274 1.17160 0.585802 0.810454i \(-0.300780\pi\)
0.585802 + 0.810454i \(0.300780\pi\)
\(374\) 6.00000 + 2.82843i 0.310253 + 0.146254i
\(375\) 0 0
\(376\) 25.4558i 1.31278i
\(377\) −16.9706 −0.874028
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −3.00000 9.00000i −0.153897 0.461690i
\(381\) 0 0
\(382\) 2.82843 0.144715
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −10.4142 1.24264i −0.530757 0.0633308i
\(386\) 19.7990i 1.00774i
\(387\) 0 0
\(388\) 12.0000i 0.609208i
\(389\) 21.2132i 1.07555i 0.843088 + 0.537776i \(0.180735\pi\)
−0.843088 + 0.537776i \(0.819265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 15.0000i 0.757614i
\(393\) 0 0
\(394\) −14.0000 −0.705310
\(395\) −9.00000 27.0000i −0.452839 1.35852i
\(396\) 0 0
\(397\) 6.00000i 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 20.0000i 1.00251i
\(399\) 0 0
\(400\) −4.00000 + 3.00000i −0.200000 + 0.150000i
\(401\) 26.8701i 1.34183i −0.741536 0.670913i \(-0.765902\pi\)
0.741536 0.670913i \(-0.234098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 8.48528i 0.421117i
\(407\) 8.48528 18.0000i 0.420600 0.892227i
\(408\) 0 0
\(409\) 8.48528i 0.419570i 0.977748 + 0.209785i \(0.0672764\pi\)
−0.977748 + 0.209785i \(0.932724\pi\)
\(410\) 4.24264 + 12.7279i 0.209529 + 0.628587i
\(411\) 0 0
\(412\) 12.0000i 0.591198i
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) −9.89949 29.6985i −0.485947 1.45784i
\(416\) 14.1421i 0.693375i
\(417\) 0 0
\(418\) −12.7279 6.00000i −0.622543 0.293470i
\(419\) 39.5980i 1.93449i −0.253849 0.967244i \(-0.581697\pi\)
0.253849 0.967244i \(-0.418303\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) −4.24264 −0.206529
\(423\) 0 0
\(424\) 12.7279i 0.618123i
\(425\) 6.00000 + 8.00000i 0.291043 + 0.388057i
\(426\) 0 0
\(427\) 12.0000i 0.580721i
\(428\) 14.0000i 0.676716i
\(429\) 0 0
\(430\) −1.00000 3.00000i −0.0482243 0.144673i
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) 18.0000i 0.865025i 0.901628 + 0.432512i \(0.142373\pi\)
−0.901628 + 0.432512i \(0.857627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 16.9706i 0.812743i
\(437\) 0 0
\(438\) 0 0
\(439\) 38.1838i 1.82241i 0.411951 + 0.911206i \(0.364847\pi\)
−0.411951 + 0.911206i \(0.635153\pi\)
\(440\) 2.63604 22.0919i 0.125668 1.05319i
\(441\) 0 0
\(442\) −5.65685 −0.269069
\(443\) −25.4558 −1.20944 −0.604722 0.796437i \(-0.706716\pi\)
−0.604722 + 0.796437i \(0.706716\pi\)
\(444\) 0 0
\(445\) −5.00000 15.0000i −0.237023 0.711068i
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) −9.89949 −0.467707
\(449\) 32.5269i 1.53504i −0.641025 0.767520i \(-0.721491\pi\)
0.641025 0.767520i \(-0.278509\pi\)
\(450\) 0 0
\(451\) −18.0000 8.48528i −0.847587 0.399556i
\(452\) −12.7279 −0.598671
\(453\) 0 0
\(454\) −2.00000 −0.0938647
\(455\) 8.48528 2.82843i 0.397796 0.132599i
\(456\) 0 0
\(457\) −28.2843 −1.32308 −0.661541 0.749909i \(-0.730097\pi\)
−0.661541 + 0.749909i \(0.730097\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −8.48528 −0.392652 −0.196326 0.980539i \(-0.562901\pi\)
−0.196326 + 0.980539i \(0.562901\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −6.00000 18.0000i −0.276759 0.830278i
\(471\) 0 0
\(472\) −25.4558 −1.17170
\(473\) 4.24264 + 2.00000i 0.195077 + 0.0919601i
\(474\) 0 0
\(475\) −12.7279 16.9706i −0.583997 0.778663i
\(476\) 2.82843i 0.129641i
\(477\) 0 0
\(478\) 6.00000i 0.274434i
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 16.9706i 0.773791i
\(482\) −8.48528 −0.386494
\(483\) 0 0
\(484\) 7.00000 + 8.48528i 0.318182 + 0.385695i
\(485\) 8.48528 + 25.4558i 0.385297 + 1.15589i
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) −25.4558 −1.15233
\(489\) 0 0
\(490\) −3.53553 10.6066i −0.159719 0.479157i
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 6.36396 9.19239i 0.284605 0.411096i
\(501\) 0 0
\(502\) −16.9706 −0.757433
\(503\) 16.0000i 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 12.7279 4.24264i 0.566385 0.188795i
\(506\) 0 0
\(507\) 0 0
\(508\) −18.3848 −0.815693
\(509\) 4.24264i 0.188052i 0.995570 + 0.0940259i \(0.0299736\pi\)
−0.995570 + 0.0940259i \(0.970026\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 11.0000i 0.486136i
\(513\) 0 0
\(514\) 29.6985i 1.30994i
\(515\) 8.48528 + 25.4558i 0.373906 + 1.12172i
\(516\) 0 0
\(517\) 25.4558 + 12.0000i 1.11955 + 0.527759i
\(518\) −8.48528 −0.372822
\(519\) 0 0
\(520\) 6.00000 + 18.0000i 0.263117 + 0.789352i
\(521\) 7.07107i 0.309789i 0.987931 + 0.154895i \(0.0495038\pi\)
−0.987931 + 0.154895i \(0.950496\pi\)
\(522\) 0 0
\(523\) −7.07107 −0.309196 −0.154598 0.987977i \(-0.549408\pi\)
−0.154598 + 0.987977i \(0.549408\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 3.00000 + 9.00000i 0.130312 + 0.390935i
\(531\) 0 0
\(532\) 6.00000i 0.260133i
\(533\) 16.9706 0.735077
\(534\) 0 0
\(535\) −9.89949 29.6985i −0.427992 1.28398i
\(536\) 0 0
\(537\) 0 0
\(538\) −12.7279 −0.548740
\(539\) 15.0000 + 7.07107i 0.646096 + 0.304572i
\(540\) 0 0
\(541\) 33.9411i 1.45924i 0.683851 + 0.729621i \(0.260304\pi\)
−0.683851 + 0.729621i \(0.739696\pi\)
\(542\) −12.7279 −0.546711
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) −12.0000 36.0000i −0.514024 1.54207i
\(546\) 0 0
\(547\) 9.89949 0.423272 0.211636 0.977349i \(-0.432121\pi\)
0.211636 + 0.977349i \(0.432121\pi\)
\(548\) 4.24264 0.181237
\(549\) 0 0
\(550\) −3.34315 16.2426i −0.142552 0.692589i
\(551\) 25.4558i 1.08446i
\(552\) 0 0
\(553\) 18.0000i 0.765438i
\(554\) 31.1127i 1.32185i
\(555\) 0 0
\(556\) 4.24264i 0.179928i
\(557\) 34.0000i 1.44063i 0.693649 + 0.720313i \(0.256002\pi\)
−0.693649 + 0.720313i \(0.743998\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) −3.00000 + 1.00000i −0.126773 + 0.0422577i
\(561\) 0 0
\(562\) 6.00000i 0.253095i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) −27.0000 + 9.00000i −1.13590 + 0.378633i
\(566\) 1.41421i 0.0594438i
\(567\) 0 0
\(568\) 25.4558 1.06810
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.24264i 0.177549i 0.996052 + 0.0887745i \(0.0282950\pi\)
−0.996052 + 0.0887745i \(0.971705\pi\)
\(572\) −8.48528 4.00000i −0.354787 0.167248i
\(573\) 0 0
\(574\) 8.48528i 0.354169i
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) −12.7279 + 4.24264i −0.528498 + 0.176166i
\(581\) 19.7990i 0.821401i
\(582\) 0 0
\(583\) −12.7279 6.00000i −0.527137 0.248495i
\(584\) 33.9411i 1.40449i
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) −42.4264 −1.75113 −0.875563 0.483105i \(-0.839509\pi\)
−0.875563 + 0.483105i \(0.839509\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −18.0000 + 6.00000i −0.741048 + 0.247016i
\(591\) 0 0
\(592\) 6.00000i 0.246598i
\(593\) 2.00000i 0.0821302i −0.999156 0.0410651i \(-0.986925\pi\)
0.999156 0.0410651i \(-0.0130751\pi\)
\(594\) 0 0
\(595\) 2.00000 + 6.00000i 0.0819920 + 0.245976i
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) 11.3137i 0.462266i 0.972922 + 0.231133i \(0.0742432\pi\)
−0.972922 + 0.231133i \(0.925757\pi\)
\(600\) 0 0
\(601\) 33.9411i 1.38449i −0.721664 0.692244i \(-0.756622\pi\)
0.721664 0.692244i \(-0.243378\pi\)
\(602\) 2.00000i 0.0815139i
\(603\) 0 0
\(604\) 4.24264i 0.172631i
\(605\) 20.8492 + 13.0503i 0.847642 + 0.530568i
\(606\) 0 0
\(607\) 32.5269 1.32023 0.660113 0.751166i \(-0.270508\pi\)
0.660113 + 0.751166i \(0.270508\pi\)
\(608\) 21.2132 0.860309
\(609\) 0 0
\(610\) −18.0000 + 6.00000i −0.728799 + 0.242933i
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 28.2843 1.14239 0.571195 0.820814i \(-0.306480\pi\)
0.571195 + 0.820814i \(0.306480\pi\)
\(614\) 24.0416i 0.970241i
\(615\) 0 0
\(616\) 6.00000 12.7279i 0.241747 0.512823i
\(617\) −4.24264 −0.170802 −0.0854011 0.996347i \(-0.527217\pi\)
−0.0854011 + 0.996347i \(0.527217\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.9706 0.680458
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 7.00000 24.0000i 0.280000 0.960000i
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 24.0000i 0.957704i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 38.1838 1.51887
\(633\) 0 0
\(634\) 21.2132i 0.842484i
\(635\) −39.0000 + 13.0000i −1.54767 + 0.515889i
\(636\) 0 0
\(637\) −14.1421 −0.560332
\(638\) −8.48528 + 18.0000i −0.335936 + 0.712627i
\(639\) 0 0
\(640\) 2.12132 + 6.36396i 0.0838525 + 0.251558i
\(641\) 15.5563i 0.614439i 0.951639 + 0.307219i \(0.0993986\pi\)
−0.951639 + 0.307219i \(0.900601\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i −0.971603 0.236617i \(-0.923961\pi\)
0.971603 0.236617i \(-0.0760386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.48528i 0.333849i
\(647\) −16.9706 −0.667182 −0.333591 0.942718i \(-0.608260\pi\)
−0.333591 + 0.942718i \(0.608260\pi\)
\(648\) 0 0
\(649\) 12.0000 25.4558i 0.471041 0.999229i
\(650\) 8.48528 + 11.3137i 0.332820 + 0.443760i
\(651\) 0 0
\(652\) 24.0000i 0.939913i
\(653\) 29.6985 1.16219 0.581096 0.813835i \(-0.302624\pi\)
0.581096 + 0.813835i \(0.302624\pi\)
\(654\) 0 0
\(655\) 38.1838 12.7279i 1.49196 0.497321i
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 12.0000i 0.467809i
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 0 0
\(664\) 42.0000 1.62992
\(665\) −4.24264 12.7279i −0.164523 0.493568i
\(666\) 0 0
\(667\) 0 0
\(668\) 8.00000i 0.309529i
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 25.4558i 0.463255 0.982712i
\(672\) 0 0
\(673\) −36.7696 −1.41736 −0.708681 0.705529i \(-0.750709\pi\)
−0.708681 + 0.705529i \(0.750709\pi\)
\(674\) 11.3137i 0.435788i
\(675\) 0 0
\(676\) −5.00000 −0.192308
\(677\) 46.0000i 1.76792i 0.467559 + 0.883962i \(0.345134\pi\)
−0.467559 + 0.883962i \(0.654866\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) −12.7279 + 4.24264i −0.488094 + 0.162698i
\(681\) 0 0
\(682\) 0 0
\(683\) 33.9411 1.29872 0.649361 0.760481i \(-0.275037\pi\)
0.649361 + 0.760481i \(0.275037\pi\)
\(684\) 0 0
\(685\) 9.00000 3.00000i 0.343872 0.114624i
\(686\) 16.9706i 0.647939i
\(687\) 0 0
\(688\) 1.41421 0.0539164
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 0 0
\(694\) −14.0000 −0.531433
\(695\) −3.00000 9.00000i −0.113796 0.341389i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 8.48528 0.321173
\(699\) 0 0
\(700\) 5.65685 4.24264i 0.213809 0.160357i
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 25.4558 0.960085
\(704\) 21.0000 + 9.89949i 0.791467 + 0.373101i
\(705\) 0 0
\(706\) 4.24264i 0.159674i
\(707\) 8.48528 0.319122
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 18.0000 6.00000i 0.675528 0.225176i
\(711\) 0 0
\(712\) 21.2132 0.794998
\(713\) 0 0
\(714\) 0 0
\(715\) −20.8284 2.48528i −0.778939 0.0929443i
\(716\) 5.65685i 0.211407i
\(717\) 0 0
\(718\) 24.0000i 0.895672i
\(719\) 31.1127i 1.16031i 0.814507 + 0.580154i \(0.197008\pi\)
−0.814507 + 0.580154i \(0.802992\pi\)
\(720\) 0 0
\(721\) 16.9706i 0.632017i
\(722\) 1.00000i 0.0372161i
\(723\) 0 0
\(724\) 0 0
\(725\) −24.0000 + 18.0000i −0.891338 + 0.668503i
\(726\) 0 0
\(727\) 48.0000i 1.78022i 0.455744 + 0.890111i \(0.349373\pi\)
−0.455744 + 0.890111i \(0.650627\pi\)
\(728\) 12.0000i 0.444750i
\(729\) 0 0
\(730\) −8.00000 24.0000i −0.296093 0.888280i
\(731\) 2.82843i 0.104613i
\(732\) 0 0
\(733\) 31.1127 1.14917 0.574587 0.818444i \(-0.305163\pi\)
0.574587 + 0.818444i \(0.305163\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 46.6690i 1.71675i 0.513024 + 0.858374i \(0.328525\pi\)
−0.513024 + 0.858374i \(0.671475\pi\)
\(740\) 4.24264 + 12.7279i 0.155963 + 0.467888i
\(741\) 0 0
\(742\) 6.00000i 0.220267i
\(743\) 10.0000i 0.366864i −0.983032 0.183432i \(-0.941279\pi\)
0.983032 0.183432i \(-0.0587208\pi\)
\(744\) 0 0
\(745\) −38.1838 + 12.7279i −1.39894 + 0.466315i
\(746\) 22.6274i 0.828449i
\(747\) 0 0
\(748\) 2.82843 6.00000i 0.103418 0.219382i
\(749\) 19.7990i 0.723439i
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 8.48528 0.309426
\(753\) 0 0
\(754\) 16.9706i 0.618031i
\(755\) 3.00000 + 9.00000i 0.109181 + 0.327544i
\(756\) 0 0
\(757\) 42.0000i 1.52652i −0.646094 0.763258i \(-0.723599\pi\)
0.646094 0.763258i \(-0.276401\pi\)
\(758\) 16.0000i 0.581146i
\(759\) 0 0
\(760\) 27.0000 9.00000i 0.979393 0.326464i
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 24.0000i 0.868858i
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 1.24264 10.4142i 0.0447817 0.375302i
\(771\) 0 0
\(772\) 19.7990 0.712581
\(773\) 4.24264 0.152597 0.0762986 0.997085i \(-0.475690\pi\)
0.0762986 + 0.997085i \(0.475690\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −36.0000 −1.29232
\(777\) 0 0
\(778\) −21.2132 −0.760530
\(779\) 25.4558i 0.912050i
\(780\) 0 0
\(781\) −12.0000 + 25.4558i −0.429394 + 0.910882i
\(782\) 0 0
\(783\) 0 0
\(784\) 5.00000 0.178571
\(785\) −16.9706 50.9117i −0.605705 1.81712i
\(786\) 0 0
\(787\) −9.89949 −0.352879 −0.176439 0.984311i \(-0.556458\pi\)
−0.176439 + 0.984311i \(0.556458\pi\)
\(788\) 14.0000i 0.498729i
\(789\) 0 0
\(790\) 27.0000 9.00000i 0.960617 0.320206i
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 24.0000i 0.852265i
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −29.6985 −1.05197 −0.525987 0.850493i \(-0.676304\pi\)
−0.525987 + 0.850493i \(0.676304\pi\)
\(798\) 0 0
\(799\) 16.9706i 0.600375i
\(800\) 15.0000 + 20.0000i 0.530330 + 0.707107i
\(801\) 0 0
\(802\) 26.8701 0.948815
\(803\) 33.9411 + 16.0000i 1.19776 + 0.564628i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 18.0000i 0.633238i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 46.6690i 1.63877i −0.573242 0.819386i \(-0.694315\pi\)
0.573242 0.819386i \(-0.305685\pi\)
\(812\) −8.48528 −0.297775
\(813\) 0 0
\(814\) 18.0000 + 8.48528i 0.630900 + 0.297409i
\(815\) 16.9706 + 50.9117i 0.594453 + 1.78336i
\(816\) 0 0
\(817\) 6.00000i 0.209913i
\(818\) −8.48528 −0.296681
\(819\) 0 0
\(820\) 12.7279 4.24264i 0.444478 0.148159i
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 24.0000i 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) −36.0000 −1.25412
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 34.0000i 1.18230i −0.806563 0.591148i \(-0.798675\pi\)
0.806563 0.591148i \(-0.201325\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 29.6985 9.89949i 1.03085 0.343616i
\(831\) 0 0
\(832\) −19.7990 −0.686406
\(833\) 10.0000i 0.346479i
\(834\) 0 0
\(835\) −5.65685 16.9706i −0.195764 0.587291i
\(836\) −6.00000 + 12.7279i −0.207514 + 0.440204i
\(837\) 0 0
\(838\) 39.5980 1.36789
\(839\) 50.9117i 1.75767i −0.477129 0.878833i \(-0.658323\pi\)
0.477129 0.878833i \(-0.341677\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 24.0000i 0.827095i
\(843\) 0 0
\(844\) 4.24264i 0.146038i
\(845\) −10.6066 + 3.53553i −0.364878 + 0.121626i
\(846\) 0 0
\(847\) 9.89949 + 12.0000i 0.340151 + 0.412325i
\(848\) −4.24264 −0.145693
\(849\) 0 0
\(850\) −8.00000 + 6.00000i −0.274398 + 0.205798i
\(851\) 0 0
\(852\) 0 0
\(853\) −28.2843 −0.968435 −0.484218 0.874948i \(-0.660896\pi\)
−0.484218 + 0.874948i \(0.660896\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) 42.0000 1.43553
\(857\) 26.0000i 0.888143i −0.895991 0.444072i \(-0.853534\pi\)
0.895991 0.444072i \(-0.146466\pi\)
\(858\) 0 0
\(859\) −48.0000 −1.63774 −0.818869 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) −3.00000 + 1.00000i −0.102299 + 0.0340997i
\(861\) 0 0
\(862\) 18.0000i 0.613082i
\(863\) −33.9411 −1.15537 −0.577685 0.816260i \(-0.696044\pi\)
−0.577685 + 0.816260i \(0.696044\pi\)
\(864\) 0 0
\(865\) 9.89949 + 29.6985i 0.336593 + 1.00978i
\(866\) −18.0000 −0.611665
\(867\) 0 0
\(868\) 0 0
\(869\) −18.0000 + 38.1838i −0.610608 + 1.29530i
\(870\) 0 0
\(871\) 0 0
\(872\) 50.9117 1.72409
\(873\) 0 0
\(874\) 0 0
\(875\) 9.00000 13.0000i 0.304256 0.439480i
\(876\) 0 0
\(877\) 11.3137 0.382037 0.191018 0.981586i \(-0.438821\pi\)
0.191018 + 0.981586i \(0.438821\pi\)
\(878\) −38.1838 −1.28864
\(879\) 0 0
\(880\) 7.36396 + 0.878680i 0.248239 + 0.0296203i
\(881\) 21.2132i 0.714691i 0.933972 + 0.357345i \(0.116318\pi\)
−0.933972 + 0.357345i \(0.883682\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 5.65685i 0.190261i
\(885\) 0 0
\(886\) 25.4558i 0.855206i
\(887\) 2.00000i 0.0671534i 0.999436 + 0.0335767i \(0.0106898\pi\)
−0.999436 + 0.0335767i \(0.989310\pi\)
\(888\) 0 0
\(889\) −26.0000 −0.872012
\(890\) 15.0000 5.00000i 0.502801 0.167600i
\(891\) 0 0
\(892\) 24.0000i 0.803579i
\(893\) 36.0000i 1.20469i
\(894\) 0 0
\(895\) −4.00000 12.0000i −0.133705 0.401116i
\(896\) 4.24264i 0.141737i
\(897\) 0 0
\(898\) 32.5269 1.08544
\(899\) 0 0
\(900\) 0 0
\(901\) 8.48528i 0.282686i
\(902\) 8.48528 18.0000i 0.282529 0.599334i
\(903\) 0 0
\(904\) 38.1838i 1.26997i
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000i 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 2.00000i 0.0663723i
\(909\) 0 0
\(910\) 2.82843 + 8.48528i 0.0937614 + 0.281284i
\(911\) 2.82843i 0.0937100i 0.998902 + 0.0468550i \(0.0149199\pi\)
−0.998902 + 0.0468550i \(0.985080\pi\)
\(912\) 0 0
\(913\) −19.7990 + 42.0000i −0.655251 + 1.39000i
\(914\) 28.2843i 0.935561i
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 25.4558 0.840626
\(918\) 0 0
\(919\) 12.7279i 0.419855i 0.977717 + 0.209928i \(0.0673229\pi\)
−0.977717 + 0.209928i \(0.932677\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 18.0000i 0.592798i
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) 18.0000 + 24.0000i 0.591836 + 0.789115i
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 30.0000i 0.984798i
\(929\) 55.1543i 1.80955i 0.425885 + 0.904777i \(0.359963\pi\)
−0.425885 + 0.904777i \(0.640037\pi\)
\(930\) 0 0
\(931\) 21.2132i 0.695235i
\(932\) 10.0000i 0.327561i
\(933\) 0 0
\(934\) 8.48528i 0.277647i
\(935\) 1.75736 14.7279i 0.0574718 0.481655i
\(936\) 0 0
\(937\) 2.82843 0.0924007 0.0462003 0.998932i \(-0.485289\pi\)
0.0462003 + 0.998932i \(0.485289\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −18.0000 + 6.00000i −0.587095 + 0.195698i
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 8.48528i 0.276172i
\(945\) 0 0
\(946\) −2.00000 + 4.24264i −0.0650256 + 0.137940i
\(947\) 33.9411 1.10294 0.551469 0.834195i \(-0.314067\pi\)
0.551469 + 0.834195i \(0.314067\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 16.9706 12.7279i 0.550598 0.412948i
\(951\) 0 0
\(952\) −8.48528 −0.275010
\(953\) 46.0000i 1.49009i −0.667016 0.745043i \(-0.732429\pi\)
0.667016 0.745043i \(-0.267571\pi\)
\(954\) 0 0
\(955\) −2.00000 6.00000i −0.0647185 0.194155i
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 6.00000i 0.193851i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −16.9706 −0.547153
\(963\) 0 0
\(964\) 8.48528i 0.273293i
\(965\) 42.0000 14.0000i 1.35203 0.450676i
\(966\) 0 0
\(967\) −52.3259 −1.68269 −0.841344 0.540500i \(-0.818235\pi\)
−0.841344 + 0.540500i \(0.818235\pi\)
\(968\) −25.4558 + 21.0000i −0.818182 + 0.674966i
\(969\) 0 0
\(970\) −25.4558 + 8.48528i −0.817338 + 0.272446i
\(971\) 19.7990i 0.635380i 0.948195 + 0.317690i \(0.102907\pi\)
−0.948195 + 0.317690i \(0.897093\pi\)
\(972\) 0 0
\(973\) 6.00000i 0.192351i
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 8.48528i 0.271607i
\(977\) −38.1838 −1.22161 −0.610803 0.791782i \(-0.709153\pi\)
−0.610803 + 0.791782i \(0.709153\pi\)
\(978\) 0 0
\(979\) −10.0000 + 21.2132i −0.319601 + 0.677977i
\(980\) −10.6066 + 3.53553i −0.338815 + 0.112938i
\(981\) 0 0
\(982\) 36.0000i 1.14881i
\(983\) −50.9117 −1.62383 −0.811915 0.583775i \(-0.801575\pi\)
−0.811915 + 0.583775i \(0.801575\pi\)
\(984\) 0 0
\(985\) 9.89949 + 29.6985i 0.315424 + 0.946272i
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 12.0000i 0.381771i
\(989\) 0 0
\(990\) 0 0
\(991\) −12.0000 −0.381193 −0.190596 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) −42.4264 + 14.1421i −1.34501 + 0.448336i
\(996\) 0 0
\(997\) −39.5980 −1.25408 −0.627040 0.778987i \(-0.715734\pi\)
−0.627040 + 0.778987i \(0.715734\pi\)
\(998\) 32.0000i 1.01294i
\(999\) 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 495.2.d.a.494.4 yes 4
3.2 odd 2 495.2.d.b.494.1 yes 4
5.2 odd 4 2475.2.f.a.2276.2 2
5.3 odd 4 2475.2.f.c.2276.1 2
5.4 even 2 inner 495.2.d.a.494.1 4
11.10 odd 2 495.2.d.b.494.2 yes 4
15.2 even 4 2475.2.f.d.2276.2 2
15.8 even 4 2475.2.f.b.2276.1 2
15.14 odd 2 495.2.d.b.494.4 yes 4
33.32 even 2 inner 495.2.d.a.494.3 yes 4
55.32 even 4 2475.2.f.d.2276.1 2
55.43 even 4 2475.2.f.b.2276.2 2
55.54 odd 2 495.2.d.b.494.3 yes 4
165.32 odd 4 2475.2.f.a.2276.1 2
165.98 odd 4 2475.2.f.c.2276.2 2
165.164 even 2 inner 495.2.d.a.494.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.d.a.494.1 4 5.4 even 2 inner
495.2.d.a.494.2 yes 4 165.164 even 2 inner
495.2.d.a.494.3 yes 4 33.32 even 2 inner
495.2.d.a.494.4 yes 4 1.1 even 1 trivial
495.2.d.b.494.1 yes 4 3.2 odd 2
495.2.d.b.494.2 yes 4 11.10 odd 2
495.2.d.b.494.3 yes 4 55.54 odd 2
495.2.d.b.494.4 yes 4 15.14 odd 2
2475.2.f.a.2276.1 2 165.32 odd 4
2475.2.f.a.2276.2 2 5.2 odd 4
2475.2.f.b.2276.1 2 15.8 even 4
2475.2.f.b.2276.2 2 55.43 even 4
2475.2.f.c.2276.1 2 5.3 odd 4
2475.2.f.c.2276.2 2 165.98 odd 4
2475.2.f.d.2276.1 2 55.32 even 4
2475.2.f.d.2276.2 2 15.2 even 4