Properties

Label 1110.2.h.b.961.2
Level $1110$
Weight $2$
Character 1110.961
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(961,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.h (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: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1110.961
Dual form 1110.2.h.b.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +2.00000 q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +2.00000 q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +2.00000i q^{13} +2.00000i q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +6.00000i q^{19} -1.00000i q^{20} -2.00000 q^{21} -4.00000i q^{22} +1.00000i q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -2.00000i q^{29} +1.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} +4.00000 q^{33} +2.00000 q^{34} +2.00000i q^{35} -1.00000 q^{36} +(-6.00000 + 1.00000i) q^{37} -6.00000 q^{38} -2.00000i q^{39} +1.00000 q^{40} -2.00000 q^{41} -2.00000i q^{42} +12.0000i q^{43} +4.00000 q^{44} +1.00000i q^{45} -6.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000i q^{50} +2.00000i q^{51} -2.00000i q^{52} -4.00000 q^{53} -1.00000i q^{54} -4.00000i q^{55} -2.00000i q^{56} -6.00000i q^{57} +2.00000 q^{58} -10.0000i q^{59} +1.00000i q^{60} -2.00000i q^{61} -4.00000 q^{62} +2.00000 q^{63} -1.00000 q^{64} -2.00000 q^{65} +4.00000i q^{66} -12.0000 q^{67} +2.00000i q^{68} -2.00000 q^{70} +12.0000 q^{71} -1.00000i q^{72} -10.0000 q^{73} +(-1.00000 - 6.00000i) q^{74} +1.00000 q^{75} -6.00000i q^{76} -8.00000 q^{77} +2.00000 q^{78} +4.00000i q^{79} +1.00000i q^{80} +1.00000 q^{81} -2.00000i q^{82} +2.00000 q^{84} +2.00000 q^{85} -12.0000 q^{86} +2.00000i q^{87} +4.00000i q^{88} -12.0000i q^{89} -1.00000 q^{90} +4.00000i q^{91} -4.00000i q^{93} -6.00000i q^{94} -6.00000 q^{95} -1.00000i q^{96} -6.00000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 4 q^{7} + 2 q^{9} - 2 q^{10} - 8 q^{11} + 2 q^{12} + 2 q^{16} - 4 q^{21} - 2 q^{25} - 4 q^{26} - 2 q^{27} - 4 q^{28} + 2 q^{30} + 8 q^{33} + 4 q^{34} - 2 q^{36} - 12 q^{37} - 12 q^{38} + 2 q^{40} - 4 q^{41} + 8 q^{44} - 12 q^{47} - 2 q^{48} - 6 q^{49} - 8 q^{53} + 4 q^{58} - 8 q^{62} + 4 q^{63} - 2 q^{64} - 4 q^{65} - 24 q^{67} - 4 q^{70} + 24 q^{71} - 20 q^{73} - 2 q^{74} + 2 q^{75} - 16 q^{77} + 4 q^{78} + 2 q^{81} + 4 q^{84} + 4 q^{85} - 24 q^{86} - 2 q^{90} - 12 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 1.00000i 0.223607i
\(21\) −2.00000 −0.436436
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 2.00000i 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) 2.00000i 0.338062i
\(36\) −1.00000 −0.166667
\(37\) −6.00000 + 1.00000i −0.986394 + 0.164399i
\(38\) −6.00000 −0.973329
\(39\) 2.00000i 0.320256i
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000i 0.141421i
\(51\) 2.00000i 0.280056i
\(52\) 2.00000i 0.277350i
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 4.00000i 0.539360i
\(56\) 2.00000i 0.267261i
\(57\) 6.00000i 0.794719i
\(58\) 2.00000 0.262613
\(59\) 10.0000i 1.30189i −0.759125 0.650945i \(-0.774373\pi\)
0.759125 0.650945i \(-0.225627\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 2.00000i 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) −4.00000 −0.508001
\(63\) 2.00000 0.251976
\(64\) −1.00000 −0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000i 0.492366i
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000i 0.117851i
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −1.00000 6.00000i −0.116248 0.697486i
\(75\) 1.00000 0.115470
\(76\) 6.00000i 0.688247i
\(77\) −8.00000 −0.911685
\(78\) 2.00000 0.226455
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 2.00000 0.216930
\(86\) −12.0000 −1.29399
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 12.0000i 1.27200i −0.771690 0.635999i \(-0.780588\pi\)
0.771690 0.635999i \(-0.219412\pi\)
\(90\) −1.00000 −0.105409
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 6.00000i 0.618853i
\(95\) −6.00000 −0.615587
\(96\) 1.00000i 0.102062i
\(97\) 6.00000i 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 3.00000i 0.303046i
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.00000i 0.195180i
\(106\) 4.00000i 0.388514i
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000i 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 1.00000i 0.569495 0.0949158i
\(112\) 2.00000 0.188982
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) 2.00000i 0.185695i
\(117\) 2.00000i 0.184900i
\(118\) 10.0000 0.920575
\(119\) 4.00000i 0.366679i
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 2.00000 0.180334
\(124\) 4.00000i 0.359211i
\(125\) 1.00000i 0.0894427i
\(126\) 2.00000i 0.178174i
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 12.0000i 1.05654i
\(130\) 2.00000i 0.175412i
\(131\) 22.0000i 1.92215i 0.276289 + 0.961074i \(0.410895\pi\)
−0.276289 + 0.961074i \(0.589105\pi\)
\(132\) −4.00000 −0.348155
\(133\) 12.0000i 1.04053i
\(134\) 12.0000i 1.03664i
\(135\) 1.00000i 0.0860663i
\(136\) −2.00000 −0.171499
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 2.00000i 0.169031i
\(141\) 6.00000 0.505291
\(142\) 12.0000i 1.00702i
\(143\) 8.00000i 0.668994i
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) 10.0000i 0.827606i
\(147\) 3.00000 0.247436
\(148\) 6.00000 1.00000i 0.493197 0.0821995i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 6.00000 0.486664
\(153\) 2.00000i 0.161690i
\(154\) 8.00000i 0.644658i
\(155\) −4.00000 −0.321288
\(156\) 2.00000i 0.160128i
\(157\) 24.0000 1.91541 0.957704 0.287754i \(-0.0929087\pi\)
0.957704 + 0.287754i \(0.0929087\pi\)
\(158\) −4.00000 −0.318223
\(159\) 4.00000 0.317221
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 2.00000 0.156174
\(165\) 4.00000i 0.311400i
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000i 0.154303i
\(169\) 9.00000 0.692308
\(170\) 2.00000i 0.153393i
\(171\) 6.00000i 0.458831i
\(172\) 12.0000i 0.914991i
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) −2.00000 −0.151620
\(175\) −2.00000 −0.151186
\(176\) −4.00000 −0.301511
\(177\) 10.0000i 0.751646i
\(178\) 12.0000 0.899438
\(179\) 18.0000i 1.34538i 0.739923 + 0.672692i \(0.234862\pi\)
−0.739923 + 0.672692i \(0.765138\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.00000 −0.296500
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) −1.00000 6.00000i −0.0735215 0.441129i
\(186\) 4.00000 0.293294
\(187\) 8.00000i 0.585018i
\(188\) 6.00000 0.437595
\(189\) −2.00000 −0.145479
\(190\) 6.00000i 0.435286i
\(191\) 12.0000i 0.868290i −0.900843 0.434145i \(-0.857051\pi\)
0.900843 0.434145i \(-0.142949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 6.00000 0.430775
\(195\) 2.00000 0.143223
\(196\) 3.00000 0.214286
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 4.00000i 0.283552i −0.989899 0.141776i \(-0.954719\pi\)
0.989899 0.141776i \(-0.0452813\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 12.0000 0.846415
\(202\) 14.0000i 0.985037i
\(203\) 4.00000i 0.280745i
\(204\) 2.00000i 0.140028i
\(205\) 2.00000i 0.139686i
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) 24.0000i 1.66011i
\(210\) 2.00000 0.138013
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 4.00000 0.274721
\(213\) −12.0000 −0.822226
\(214\) 12.0000i 0.820303i
\(215\) −12.0000 −0.818393
\(216\) 1.00000i 0.0680414i
\(217\) 8.00000i 0.543075i
\(218\) 2.00000 0.135457
\(219\) 10.0000 0.675737
\(220\) 4.00000i 0.269680i
\(221\) 4.00000 0.269069
\(222\) 1.00000 + 6.00000i 0.0671156 + 0.402694i
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 2.00000i 0.133631i
\(225\) −1.00000 −0.0666667
\(226\) −18.0000 −1.19734
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 6.00000i 0.397360i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 6.00000i 0.391397i
\(236\) 10.0000i 0.650945i
\(237\) 4.00000i 0.259828i
\(238\) 4.00000 0.259281
\(239\) 4.00000i 0.258738i −0.991596 0.129369i \(-0.958705\pi\)
0.991596 0.129369i \(-0.0412952\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 8.00000i 0.515325i −0.966235 0.257663i \(-0.917048\pi\)
0.966235 0.257663i \(-0.0829523\pi\)
\(242\) 5.00000i 0.321412i
\(243\) −1.00000 −0.0641500
\(244\) 2.00000i 0.128037i
\(245\) 3.00000i 0.191663i
\(246\) 2.00000i 0.127515i
\(247\) −12.0000 −0.763542
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 10.0000i 0.631194i 0.948893 + 0.315597i \(0.102205\pi\)
−0.948893 + 0.315597i \(0.897795\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 2.00000i 0.125491i
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 12.0000 0.747087
\(259\) −12.0000 + 2.00000i −0.745644 + 0.124274i
\(260\) 2.00000 0.124035
\(261\) 2.00000i 0.123797i
\(262\) −22.0000 −1.35916
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 4.00000i 0.246183i
\(265\) 4.00000i 0.245718i
\(266\) −12.0000 −0.735767
\(267\) 12.0000i 0.734388i
\(268\) 12.0000 0.733017
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 4.00000i 0.242091i
\(274\) 18.0000i 1.08742i
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 22.0000i 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 4.00000i 0.239474i
\(280\) 2.00000 0.119523
\(281\) 4.00000i 0.238620i −0.992857 0.119310i \(-0.961932\pi\)
0.992857 0.119310i \(-0.0380682\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 8.00000i 0.475551i 0.971320 + 0.237775i \(0.0764182\pi\)
−0.971320 + 0.237775i \(0.923582\pi\)
\(284\) −12.0000 −0.712069
\(285\) 6.00000 0.355409
\(286\) 8.00000 0.473050
\(287\) −4.00000 −0.236113
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 2.00000i 0.117444i
\(291\) 6.00000i 0.351726i
\(292\) 10.0000 0.585206
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 10.0000 0.582223
\(296\) 1.00000 + 6.00000i 0.0581238 + 0.348743i
\(297\) 4.00000 0.232104
\(298\) 6.00000i 0.347571i
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 24.0000i 1.38334i
\(302\) 4.00000i 0.230174i
\(303\) 14.0000 0.804279
\(304\) 6.00000i 0.344124i
\(305\) 2.00000 0.114520
\(306\) 2.00000 0.114332
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 8.00000 0.455842
\(309\) 8.00000i 0.455104i
\(310\) 4.00000i 0.227185i
\(311\) 8.00000i 0.453638i −0.973937 0.226819i \(-0.927167\pi\)
0.973937 0.226819i \(-0.0728326\pi\)
\(312\) −2.00000 −0.113228
\(313\) 26.0000i 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) 24.0000i 1.35440i
\(315\) 2.00000i 0.112687i
\(316\) 4.00000i 0.225018i
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 4.00000i 0.224309i
\(319\) 8.00000i 0.447914i
\(320\) 1.00000i 0.0559017i
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) −1.00000 −0.0555556
\(325\) 2.00000i 0.110940i
\(326\) −8.00000 −0.443079
\(327\) 2.00000i 0.110600i
\(328\) 2.00000i 0.110432i
\(329\) −12.0000 −0.661581
\(330\) −4.00000 −0.220193
\(331\) 10.0000i 0.549650i 0.961494 + 0.274825i \(0.0886199\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(332\) 0 0
\(333\) −6.00000 + 1.00000i −0.328798 + 0.0547997i
\(334\) 0 0
\(335\) 12.0000i 0.655630i
\(336\) −2.00000 −0.109109
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 18.0000i 0.977626i
\(340\) −2.00000 −0.108465
\(341\) 16.0000i 0.866449i
\(342\) −6.00000 −0.324443
\(343\) −20.0000 −1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 12.0000i 0.645124i
\(347\) 4.00000i 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 2.00000i 0.107211i
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 2.00000i 0.106904i
\(351\) 2.00000i 0.106752i
\(352\) 4.00000i 0.213201i
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) −10.0000 −0.531494
\(355\) 12.0000i 0.636894i
\(356\) 12.0000i 0.635999i
\(357\) 4.00000i 0.211702i
\(358\) −18.0000 −0.951330
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 1.00000 0.0527046
\(361\) −17.0000 −0.894737
\(362\) 2.00000i 0.105118i
\(363\) −5.00000 −0.262432
\(364\) 4.00000i 0.209657i
\(365\) 10.0000i 0.523424i
\(366\) −2.00000 −0.104542
\(367\) −6.00000 −0.313197 −0.156599 0.987662i \(-0.550053\pi\)
−0.156599 + 0.987662i \(0.550053\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 6.00000 1.00000i 0.311925 0.0519875i
\(371\) −8.00000 −0.415339
\(372\) 4.00000i 0.207390i
\(373\) −24.0000 −1.24267 −0.621336 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) −8.00000 −0.413670
\(375\) 1.00000i 0.0516398i
\(376\) 6.00000i 0.309426i
\(377\) 4.00000 0.206010
\(378\) 2.00000i 0.102869i
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 6.00000 0.307794
\(381\) −2.00000 −0.102463
\(382\) 12.0000 0.613973
\(383\) 32.0000i 1.63512i 0.575841 + 0.817562i \(0.304675\pi\)
−0.575841 + 0.817562i \(0.695325\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 8.00000i 0.407718i
\(386\) −14.0000 −0.712581
\(387\) 12.0000i 0.609994i
\(388\) 6.00000i 0.304604i
\(389\) 22.0000i 1.11544i −0.830028 0.557722i \(-0.811675\pi\)
0.830028 0.557722i \(-0.188325\pi\)
\(390\) 2.00000i 0.101274i
\(391\) 0 0
\(392\) 3.00000i 0.151523i
\(393\) 22.0000i 1.10975i
\(394\) 8.00000i 0.403034i
\(395\) −4.00000 −0.201262
\(396\) 4.00000 0.201008
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 4.00000 0.200502
\(399\) 12.0000i 0.600751i
\(400\) −1.00000 −0.0500000
\(401\) 12.0000i 0.599251i −0.954057 0.299626i \(-0.903138\pi\)
0.954057 0.299626i \(-0.0968618\pi\)
\(402\) 12.0000i 0.598506i
\(403\) −8.00000 −0.398508
\(404\) 14.0000 0.696526
\(405\) 1.00000i 0.0496904i
\(406\) 4.00000 0.198517
\(407\) 24.0000 4.00000i 1.18964 0.198273i
\(408\) 2.00000 0.0990148
\(409\) 4.00000i 0.197787i 0.995098 + 0.0988936i \(0.0315304\pi\)
−0.995098 + 0.0988936i \(0.968470\pi\)
\(410\) 2.00000 0.0987730
\(411\) −18.0000 −0.887875
\(412\) 8.00000i 0.394132i
\(413\) 20.0000i 0.984136i
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 24.0000 1.17388
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 2.00000i 0.0975900i
\(421\) 18.0000i 0.877266i 0.898666 + 0.438633i \(0.144537\pi\)
−0.898666 + 0.438633i \(0.855463\pi\)
\(422\) 20.0000i 0.973585i
\(423\) −6.00000 −0.291730
\(424\) 4.00000i 0.194257i
\(425\) 2.00000i 0.0970143i
\(426\) 12.0000i 0.581402i
\(427\) 4.00000i 0.193574i
\(428\) 12.0000 0.580042
\(429\) 8.00000i 0.386244i
\(430\) 12.0000i 0.578691i
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) −8.00000 −0.384012
\(435\) −2.00000 −0.0958927
\(436\) 2.00000i 0.0957826i
\(437\) 0 0
\(438\) 10.0000i 0.477818i
\(439\) 4.00000i 0.190910i 0.995434 + 0.0954548i \(0.0304305\pi\)
−0.995434 + 0.0954548i \(0.969569\pi\)
\(440\) −4.00000 −0.190693
\(441\) −3.00000 −0.142857
\(442\) 4.00000i 0.190261i
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −6.00000 + 1.00000i −0.284747 + 0.0474579i
\(445\) 12.0000 0.568855
\(446\) 2.00000i 0.0947027i
\(447\) −6.00000 −0.283790
\(448\) −2.00000 −0.0944911
\(449\) 20.0000i 0.943858i −0.881636 0.471929i \(-0.843558\pi\)
0.881636 0.471929i \(-0.156442\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 8.00000 0.376705
\(452\) 18.0000i 0.846649i
\(453\) −4.00000 −0.187936
\(454\) −20.0000 −0.938647
\(455\) −4.00000 −0.187523
\(456\) −6.00000 −0.280976
\(457\) 14.0000i 0.654892i −0.944870 0.327446i \(-0.893812\pi\)
0.944870 0.327446i \(-0.106188\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 10.0000i 0.465746i 0.972507 + 0.232873i \(0.0748127\pi\)
−0.972507 + 0.232873i \(0.925187\pi\)
\(462\) 8.00000i 0.372194i
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 2.00000i 0.0928477i
\(465\) 4.00000 0.185496
\(466\) 6.00000i 0.277945i
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −24.0000 −1.10822
\(470\) 6.00000 0.276759
\(471\) −24.0000 −1.10586
\(472\) −10.0000 −0.460287
\(473\) 48.0000i 2.20704i
\(474\) 4.00000 0.183726
\(475\) 6.00000i 0.275299i
\(476\) 4.00000i 0.183340i
\(477\) −4.00000 −0.183147
\(478\) 4.00000 0.182956
\(479\) 12.0000i 0.548294i 0.961688 + 0.274147i \(0.0883955\pi\)
−0.961688 + 0.274147i \(0.911605\pi\)
\(480\) 1.00000 0.0456435
\(481\) −2.00000 12.0000i −0.0911922 0.547153i
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 6.00000 0.272446
\(486\) 1.00000i 0.0453609i
\(487\) 32.0000i 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 8.00000i 0.361773i
\(490\) 3.00000 0.135526
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −4.00000 −0.180151
\(494\) 12.0000i 0.539906i
\(495\) 4.00000i 0.179787i
\(496\) 4.00000i 0.179605i
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 34.0000i 1.52205i −0.648723 0.761025i \(-0.724697\pi\)
0.648723 0.761025i \(-0.275303\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 0 0
\(502\) −10.0000 −0.446322
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) 2.00000i 0.0890871i
\(505\) 14.0000i 0.622992i
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −2.00000 −0.0887357
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 2.00000i 0.0885615i
\(511\) −20.0000 −0.884748
\(512\) 1.00000i 0.0441942i
\(513\) 6.00000i 0.264906i
\(514\) −6.00000 −0.264649
\(515\) −8.00000 −0.352522
\(516\) 12.0000i 0.528271i
\(517\) 24.0000 1.05552
\(518\) −2.00000 12.0000i −0.0878750 0.527250i
\(519\) 12.0000 0.526742
\(520\) 2.00000i 0.0877058i
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 2.00000 0.0875376
\(523\) 16.0000i 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 22.0000i 0.961074i
\(525\) 2.00000 0.0872872
\(526\) 26.0000i 1.13365i
\(527\) 8.00000 0.348485
\(528\) 4.00000 0.174078
\(529\) 23.0000 1.00000
\(530\) 4.00000 0.173749
\(531\) 10.0000i 0.433963i
\(532\) 12.0000i 0.520266i
\(533\) 4.00000i 0.173259i
\(534\) −12.0000 −0.519291
\(535\) 12.0000i 0.518805i
\(536\) 12.0000i 0.518321i
\(537\) 18.0000i 0.776757i
\(538\) 2.00000i 0.0862261i
\(539\) 12.0000 0.516877
\(540\) 1.00000i 0.0430331i
\(541\) 10.0000i 0.429934i 0.976621 + 0.214967i \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) 20.0000i 0.859074i
\(543\) −2.00000 −0.0858282
\(544\) 2.00000 0.0857493
\(545\) 2.00000 0.0856706
\(546\) 4.00000 0.171184
\(547\) 32.0000i 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) −18.0000 −0.768922
\(549\) 2.00000i 0.0853579i
\(550\) 4.00000i 0.170561i
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 22.0000 0.934690
\(555\) 1.00000 + 6.00000i 0.0424476 + 0.254686i
\(556\) −12.0000 −0.508913
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) −4.00000 −0.169334
\(559\) −24.0000 −1.01509
\(560\) 2.00000i 0.0845154i
\(561\) 8.00000i 0.337760i
\(562\) 4.00000 0.168730
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) −6.00000 −0.252646
\(565\) −18.0000 −0.757266
\(566\) −8.00000 −0.336265
\(567\) 2.00000 0.0839921
\(568\) 12.0000i 0.503509i
\(569\) 12.0000i 0.503066i −0.967849 0.251533i \(-0.919065\pi\)
0.967849 0.251533i \(-0.0809347\pi\)
\(570\) 6.00000i 0.251312i
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 12.0000i 0.501307i
\(574\) 4.00000i 0.166957i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 14.0000i 0.582828i 0.956597 + 0.291414i \(0.0941257\pi\)
−0.956597 + 0.291414i \(0.905874\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 14.0000i 0.581820i
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) 16.0000 0.662652
\(584\) 10.0000i 0.413803i
\(585\) −2.00000 −0.0826898
\(586\) 24.0000i 0.991431i
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) −3.00000 −0.123718
\(589\) −24.0000 −0.988903
\(590\) 10.0000i 0.411693i
\(591\) 8.00000 0.329076
\(592\) −6.00000 + 1.00000i −0.246598 + 0.0410997i
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 4.00000i 0.164122i
\(595\) 4.00000 0.163984
\(596\) −6.00000 −0.245770
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −24.0000 −0.978167
\(603\) −12.0000 −0.488678
\(604\) −4.00000 −0.162758
\(605\) 5.00000i 0.203279i
\(606\) 14.0000i 0.568711i
\(607\) 48.0000i 1.94826i −0.225989 0.974130i \(-0.572561\pi\)
0.225989 0.974130i \(-0.427439\pi\)
\(608\) −6.00000 −0.243332
\(609\) 4.00000i 0.162088i
\(610\) 2.00000i 0.0809776i
\(611\) 12.0000i 0.485468i
\(612\) 2.00000i 0.0808452i
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 16.0000i 0.645707i
\(615\) 2.00000i 0.0806478i
\(616\) 8.00000i 0.322329i
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 8.00000 0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) 24.0000i 0.961540i
\(624\) 2.00000i 0.0800641i
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 24.0000i 0.958468i
\(628\) −24.0000 −0.957704
\(629\) 2.00000 + 12.0000i 0.0797452 + 0.478471i
\(630\) −2.00000 −0.0796819
\(631\) 16.0000i 0.636950i 0.947931 + 0.318475i \(0.103171\pi\)
−0.947931 + 0.318475i \(0.896829\pi\)
\(632\) 4.00000 0.159111
\(633\) −20.0000 −0.794929
\(634\) 12.0000i 0.476581i
\(635\) 2.00000i 0.0793676i
\(636\) −4.00000 −0.158610
\(637\) 6.00000i 0.237729i
\(638\) −8.00000 −0.316723
\(639\) 12.0000 0.474713
\(640\) 1.00000 0.0395285
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 12.0000i 0.472134i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 40.0000i 1.57014i
\(650\) 2.00000 0.0784465
\(651\) 8.00000i 0.313545i
\(652\) 8.00000i 0.313304i
\(653\) 14.0000i 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −22.0000 −0.859611
\(656\) −2.00000 −0.0780869
\(657\) −10.0000 −0.390137
\(658\) 12.0000i 0.467809i
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 4.00000i 0.155700i
\(661\) 2.00000i 0.0777910i 0.999243 + 0.0388955i \(0.0123839\pi\)
−0.999243 + 0.0388955i \(0.987616\pi\)
\(662\) −10.0000 −0.388661
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) −1.00000 6.00000i −0.0387492 0.232495i
\(667\) 0 0
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 12.0000 0.463600
\(671\) 8.00000i 0.308837i
\(672\) 2.00000i 0.0771517i
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 22.0000i 0.847408i
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 16.0000 0.614930 0.307465 0.951559i \(-0.400519\pi\)
0.307465 + 0.951559i \(0.400519\pi\)
\(678\) 18.0000 0.691286
\(679\) 12.0000i 0.460518i
\(680\) 2.00000i 0.0766965i
\(681\) 20.0000i 0.766402i
\(682\) 16.0000 0.612672
\(683\) 28.0000i 1.07139i −0.844411 0.535695i \(-0.820050\pi\)
0.844411 0.535695i \(-0.179950\pi\)
\(684\) 6.00000i 0.229416i
\(685\) 18.0000i 0.687745i
\(686\) 20.0000i 0.763604i
\(687\) −10.0000 −0.381524
\(688\) 12.0000i 0.457496i
\(689\) 8.00000i 0.304776i
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 12.0000 0.456172
\(693\) −8.00000 −0.303895
\(694\) 4.00000 0.151838
\(695\) 12.0000i 0.455186i
\(696\) 2.00000 0.0758098
\(697\) 4.00000i 0.151511i
\(698\) 18.0000i 0.681310i
\(699\) −6.00000 −0.226941
\(700\) 2.00000 0.0755929
\(701\) 18.0000i 0.679851i −0.940452 0.339925i \(-0.889598\pi\)
0.940452 0.339925i \(-0.110402\pi\)
\(702\) 2.00000 0.0754851
\(703\) −6.00000 36.0000i −0.226294 1.35777i
\(704\) 4.00000 0.150756
\(705\) 6.00000i 0.225973i
\(706\) −14.0000 −0.526897
\(707\) −28.0000 −1.05305
\(708\) 10.0000i 0.375823i
\(709\) 42.0000i 1.57734i 0.614815 + 0.788672i \(0.289231\pi\)
−0.614815 + 0.788672i \(0.710769\pi\)
\(710\) −12.0000 −0.450352
\(711\) 4.00000i 0.150012i
\(712\) −12.0000 −0.449719
\(713\) 0 0
\(714\) −4.00000 −0.149696
\(715\) 8.00000 0.299183
\(716\) 18.0000i 0.672692i
\(717\) 4.00000i 0.149383i
\(718\) 16.0000i 0.597115i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 16.0000i 0.595871i
\(722\) 17.0000i 0.632674i
\(723\) 8.00000i 0.297523i
\(724\) −2.00000 −0.0743294
\(725\) 2.00000i 0.0742781i
\(726\) 5.00000i 0.185567i
\(727\) 16.0000i 0.593407i 0.954970 + 0.296704i \(0.0958873\pi\)
−0.954970 + 0.296704i \(0.904113\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 24.0000 0.887672
\(732\) 2.00000i 0.0739221i
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 6.00000i 0.221464i
\(735\) 3.00000i 0.110657i
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 2.00000i 0.0736210i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 1.00000 + 6.00000i 0.0367607 + 0.220564i
\(741\) 12.0000 0.440831
\(742\) 8.00000i 0.293689i
\(743\) −46.0000 −1.68758 −0.843788 0.536676i \(-0.819680\pi\)
−0.843788 + 0.536676i \(0.819680\pi\)
\(744\) −4.00000 −0.146647
\(745\) 6.00000i 0.219823i
\(746\) 24.0000i 0.878702i
\(747\) 0 0
\(748\) 8.00000i 0.292509i
\(749\) −24.0000 −0.876941
\(750\) −1.00000 −0.0365148
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) −6.00000 −0.218797
\(753\) 10.0000i 0.364420i
\(754\) 4.00000i 0.145671i
\(755\) 4.00000i 0.145575i
\(756\) 2.00000 0.0727393
\(757\) 10.0000i 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 28.0000i 1.01701i
\(759\) 0 0
\(760\) 6.00000i 0.217643i
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 2.00000i 0.0724524i
\(763\) 4.00000i 0.144810i
\(764\) 12.0000i 0.434145i
\(765\) 2.00000 0.0723102
\(766\) −32.0000 −1.15621
\(767\) 20.0000 0.722158
\(768\) −1.00000 −0.0360844
\(769\) 16.0000i 0.576975i 0.957484 + 0.288487i \(0.0931523\pi\)
−0.957484 + 0.288487i \(0.906848\pi\)
\(770\) 8.00000 0.288300
\(771\) 6.00000i 0.216085i
\(772\) 14.0000i 0.503871i
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −12.0000 −0.431331
\(775\) 4.00000i 0.143684i
\(776\) −6.00000 −0.215387
\(777\) 12.0000 2.00000i 0.430498 0.0717496i
\(778\) 22.0000 0.788738
\(779\) 12.0000i 0.429945i
\(780\) −2.00000 −0.0716115
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) −3.00000 −0.107143
\(785\) 24.0000i 0.856597i
\(786\) 22.0000 0.784714
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 8.00000 0.284988
\(789\) 26.0000 0.925625
\(790\) 4.00000i 0.142314i
\(791\) 36.0000i 1.28001i
\(792\) 4.00000i 0.142134i
\(793\) 4.00000 0.142044
\(794\) 4.00000i 0.141955i
\(795\) 4.00000i 0.141865i
\(796\) 4.00000i 0.141776i
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 12.0000 0.424795
\(799\) 12.0000i 0.424529i
\(800\) 1.00000i 0.0353553i
\(801\) 12.0000i 0.423999i
\(802\) 12.0000 0.423735
\(803\) 40.0000 1.41157
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 8.00000i 0.281788i
\(807\) 2.00000 0.0704033
\(808\) 14.0000i 0.492518i
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 4.00000i 0.140372i
\(813\) −20.0000 −0.701431
\(814\) 4.00000 + 24.0000i 0.140200 + 0.841200i
\(815\) −8.00000 −0.280228
\(816\) 2.00000i 0.0700140i
\(817\) −72.0000 −2.51896
\(818\) −4.00000 −0.139857
\(819\) 4.00000i 0.139771i
\(820\) 2.00000i 0.0698430i
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 18.0000i 0.627822i
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 8.00000 0.278693
\(825\) −4.00000 −0.139262
\(826\) 20.0000 0.695889
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) 34.0000i 1.18087i 0.807086 + 0.590434i \(0.201044\pi\)
−0.807086 + 0.590434i \(0.798956\pi\)
\(830\) 0 0
\(831\) 22.0000i 0.763172i
\(832\) 2.00000i 0.0693375i
\(833\) 6.00000i 0.207888i
\(834\) 12.0000i 0.415526i
\(835\) 0 0
\(836\) 24.0000i 0.830057i
\(837\) 4.00000i 0.138260i
\(838\) 20.0000i 0.690889i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 25.0000 0.862069
\(842\) −18.0000 −0.620321
\(843\) 4.00000i 0.137767i
\(844\) −20.0000 −0.688428
\(845\) 9.00000i 0.309609i
\(846\) 6.00000i 0.206284i
\(847\) 10.0000 0.343604
\(848\) −4.00000 −0.137361
\(849\) 8.00000i 0.274559i
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 12.0000 0.411113
\(853\) 42.0000i 1.43805i 0.694983 + 0.719026i \(0.255412\pi\)
−0.694983 + 0.719026i \(0.744588\pi\)
\(854\) 4.00000 0.136877
\(855\) −6.00000 −0.205196
\(856\) 12.0000i 0.410152i
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) −8.00000 −0.273115
\(859\) 6.00000i 0.204717i 0.994748 + 0.102359i \(0.0326389\pi\)
−0.994748 + 0.102359i \(0.967361\pi\)
\(860\) 12.0000 0.409197
\(861\) 4.00000 0.136320
\(862\) −12.0000 −0.408722
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 12.0000i 0.408012i
\(866\) 30.0000i 1.01944i
\(867\) −13.0000 −0.441503
\(868\) 8.00000i 0.271538i
\(869\) 16.0000i 0.542763i
\(870\) 2.00000i 0.0678064i
\(871\) 24.0000i 0.813209i
\(872\) −2.00000 −0.0677285
\(873\) 6.00000i 0.203069i
\(874\) 0 0
\(875\) 2.00000i 0.0676123i
\(876\) −10.0000 −0.337869
\(877\) 40.0000 1.35070 0.675352 0.737496i \(-0.263992\pi\)
0.675352 + 0.737496i \(0.263992\pi\)
\(878\) −4.00000 −0.134993
\(879\) −24.0000 −0.809500
\(880\) 4.00000i 0.134840i
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 12.0000i 0.403832i −0.979403 0.201916i \(-0.935283\pi\)
0.979403 0.201916i \(-0.0647168\pi\)
\(884\) −4.00000 −0.134535
\(885\) −10.0000 −0.336146
\(886\) 4.00000i 0.134383i
\(887\) 30.0000 1.00730 0.503651 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(888\) −1.00000 6.00000i −0.0335578 0.201347i
\(889\) 4.00000 0.134156
\(890\) 12.0000i 0.402241i
\(891\) −4.00000 −0.134005
\(892\) −2.00000 −0.0669650
\(893\) 36.0000i 1.20469i
\(894\) 6.00000i 0.200670i
\(895\) −18.0000 −0.601674
\(896\) 2.00000i 0.0668153i
\(897\) 0 0
\(898\) 20.0000 0.667409
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) 8.00000i 0.266519i
\(902\) 8.00000i 0.266371i
\(903\) 24.0000i 0.798670i
\(904\) 18.0000 0.598671
\(905\) 2.00000i 0.0664822i
\(906\) 4.00000i 0.132891i
\(907\) 28.0000i 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 20.0000i 0.663723i
\(909\) −14.0000 −0.464351
\(910\) 4.00000i 0.132599i
\(911\) 48.0000i 1.59031i −0.606406 0.795155i \(-0.707389\pi\)
0.606406 0.795155i \(-0.292611\pi\)
\(912\) 6.00000i 0.198680i
\(913\) 0 0
\(914\) 14.0000 0.463079
\(915\) −2.00000 −0.0661180
\(916\) −10.0000 −0.330409
\(917\) 44.0000i 1.45301i
\(918\) −2.00000 −0.0660098
\(919\) 32.0000i 1.05558i 0.849374 + 0.527791i \(0.176980\pi\)
−0.849374 + 0.527791i \(0.823020\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −10.0000 −0.329332
\(923\) 24.0000i 0.789970i
\(924\) −8.00000 −0.263181
\(925\) 6.00000 1.00000i 0.197279 0.0328798i
\(926\) −32.0000 −1.05159
\(927\) 8.00000i 0.262754i
\(928\) 2.00000 0.0656532
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 4.00000i 0.131165i
\(931\) 18.0000i 0.589926i
\(932\) −6.00000 −0.196537
\(933\) 8.00000i 0.261908i
\(934\) −8.00000 −0.261768
\(935\) −8.00000 −0.261628
\(936\) 2.00000 0.0653720
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 24.0000i 0.783628i
\(939\) 26.0000i 0.848478i
\(940\) 6.00000i 0.195698i
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 24.0000i 0.781962i
\(943\) 0 0
\(944\) 10.0000i 0.325472i
\(945\) 2.00000i 0.0650600i
\(946\) 48.0000 1.56061
\(947\) 24.0000i 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 4.00000i 0.129914i
\(949\) 20.0000i 0.649227i
\(950\) 6.00000 0.194666
\(951\) −12.0000 −0.389127
\(952\) −4.00000 −0.129641
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 4.00000i 0.129505i
\(955\) 12.0000 0.388311
\(956\) 4.00000i 0.129369i
\(957\) 8.00000i 0.258603i
\(958\) −12.0000 −0.387702
\(959\) 36.0000 1.16250
\(960\) 1.00000i 0.0322749i
\(961\) 15.0000 0.483871
\(962\) 12.0000 2.00000i 0.386896 0.0644826i
\(963\) −12.0000 −0.386695
\(964\) 8.00000i 0.257663i
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −12.0000 −0.385496
\(970\) 6.00000i 0.192648i
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000 0.0320750
\(973\) 24.0000 0.769405
\(974\) 32.0000 1.02535
\(975\) 2.00000i 0.0640513i
\(976\) 2.00000i 0.0640184i
\(977\) 30.0000i 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 8.00000 0.255812
\(979\) 48.0000i 1.53409i
\(980\) 3.00000i 0.0958315i
\(981\) 2.00000i 0.0638551i
\(982\) 20.0000i 0.638226i
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 2.00000i 0.0637577i
\(985\) 8.00000i 0.254901i
\(986\) 4.00000i 0.127386i
\(987\) 12.0000 0.381964
\(988\) 12.0000 0.381771
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 16.0000i 0.508257i 0.967170 + 0.254128i \(0.0817886\pi\)
−0.967170 + 0.254128i \(0.918211\pi\)
\(992\) −4.00000 −0.127000
\(993\) 10.0000i 0.317340i
\(994\) 24.0000i 0.761234i
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 10.0000i 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 34.0000 1.07625
\(999\) 6.00000 1.00000i 0.189832 0.0316386i
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 1110.2.h.b.961.2 yes 2
3.2 odd 2 3330.2.h.h.2071.1 2
37.36 even 2 inner 1110.2.h.b.961.1 2
111.110 odd 2 3330.2.h.h.2071.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.h.b.961.1 2 37.36 even 2 inner
1110.2.h.b.961.2 yes 2 1.1 even 1 trivial
3330.2.h.h.2071.1 2 3.2 odd 2
3330.2.h.h.2071.2 2 111.110 odd 2