Properties

Label 1014.2.b.c.337.2
Level $1014$
Weight $2$
Character 1014.337
Analytic conductor $8.097$
Analytic rank $0$
Dimension $2$
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] = [1014,2,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1014.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.09683076496\)
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: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.2.b.c.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -3.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -3.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +3.00000 q^{10} +6.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} -3.00000i q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000i q^{18} -2.00000i q^{19} +3.00000i q^{20} +2.00000i q^{21} -6.00000 q^{22} +6.00000 q^{23} -1.00000i q^{24} -4.00000 q^{25} +1.00000 q^{27} -2.00000i q^{28} +3.00000 q^{29} +3.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} +6.00000i q^{33} +3.00000i q^{34} +6.00000 q^{35} -1.00000 q^{36} -7.00000i q^{37} +2.00000 q^{38} -3.00000 q^{40} +3.00000i q^{41} -2.00000 q^{42} +10.0000 q^{43} -6.00000i q^{44} -3.00000i q^{45} +6.00000i q^{46} +6.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} -4.00000i q^{50} +3.00000 q^{51} +3.00000 q^{53} +1.00000i q^{54} +18.0000 q^{55} +2.00000 q^{56} -2.00000i q^{57} +3.00000i q^{58} +3.00000i q^{60} -7.00000 q^{61} -4.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +10.0000i q^{67} -3.00000 q^{68} +6.00000 q^{69} +6.00000i q^{70} -6.00000i q^{71} -1.00000i q^{72} -13.0000i q^{73} +7.00000 q^{74} -4.00000 q^{75} +2.00000i q^{76} -12.0000 q^{77} -4.00000 q^{79} -3.00000i q^{80} +1.00000 q^{81} -3.00000 q^{82} +6.00000i q^{83} -2.00000i q^{84} -9.00000i q^{85} +10.0000i q^{86} +3.00000 q^{87} +6.00000 q^{88} +18.0000i q^{89} +3.00000 q^{90} -6.00000 q^{92} +4.00000i q^{93} -6.00000 q^{94} -6.00000 q^{95} +1.00000i q^{96} -14.0000i q^{97} +3.00000i q^{98} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 6 q^{17} - 12 q^{22} + 12 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 6 q^{30} + 12 q^{35} - 2 q^{36} + 4 q^{38} - 6 q^{40} - 4 q^{42} + 20 q^{43} + 2 q^{48} + 6 q^{49} + 6 q^{51} + 6 q^{53} + 36 q^{55} + 4 q^{56} - 14 q^{61} - 8 q^{62} - 2 q^{64} - 12 q^{66} - 6 q^{68} + 12 q^{69} + 14 q^{74} - 8 q^{75} - 24 q^{77} - 8 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{87} + 12 q^{88} + 6 q^{90} - 12 q^{92} - 12 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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
\(5\) − 3.00000i − 1.34164i −0.741620 0.670820i \(-0.765942\pi\)
0.741620 0.670820i \(-0.234058\pi\)
\(6\) 1.00000i 0.408248i
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) − 3.00000i − 0.774597i
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 3.00000i 0.670820i
\(21\) 2.00000i 0.436436i
\(22\) −6.00000 −1.27920
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 2.00000i − 0.377964i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 3.00000 0.547723
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 6.00000i 1.04447i
\(34\) 3.00000i 0.514496i
\(35\) 6.00000 1.01419
\(36\) −1.00000 −0.166667
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 3.00000i 0.468521i 0.972174 + 0.234261i \(0.0752669\pi\)
−0.972174 + 0.234261i \(0.924733\pi\)
\(42\) −2.00000 −0.308607
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) − 6.00000i − 0.904534i
\(45\) − 3.00000i − 0.447214i
\(46\) 6.00000i 0.884652i
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) − 4.00000i − 0.565685i
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 18.0000 2.42712
\(56\) 2.00000 0.267261
\(57\) − 2.00000i − 0.264906i
\(58\) 3.00000i 0.393919i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 3.00000i 0.387298i
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −4.00000 −0.508001
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) −3.00000 −0.363803
\(69\) 6.00000 0.722315
\(70\) 6.00000i 0.717137i
\(71\) − 6.00000i − 0.712069i −0.934473 0.356034i \(-0.884129\pi\)
0.934473 0.356034i \(-0.115871\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 13.0000i − 1.52153i −0.649025 0.760767i \(-0.724823\pi\)
0.649025 0.760767i \(-0.275177\pi\)
\(74\) 7.00000 0.813733
\(75\) −4.00000 −0.461880
\(76\) 2.00000i 0.229416i
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) − 3.00000i − 0.335410i
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) − 9.00000i − 0.976187i
\(86\) 10.0000i 1.07833i
\(87\) 3.00000 0.321634
\(88\) 6.00000 0.639602
\(89\) 18.0000i 1.90800i 0.299813 + 0.953998i \(0.403076\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 4.00000i 0.414781i
\(94\) −6.00000 −0.618853
\(95\) −6.00000 −0.615587
\(96\) 1.00000i 0.102062i
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 6.00000i 0.603023i
\(100\) 4.00000 0.400000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 3.00000i 0.297044i
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 3.00000i 0.291386i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 14.0000i − 1.34096i −0.741929 0.670478i \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 18.0000i 1.71623i
\(111\) − 7.00000i − 0.664411i
\(112\) 2.00000i 0.188982i
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 2.00000 0.187317
\(115\) − 18.0000i − 1.67851i
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000i 0.550019i
\(120\) −3.00000 −0.273861
\(121\) −25.0000 −2.27273
\(122\) − 7.00000i − 0.633750i
\(123\) 3.00000i 0.270501i
\(124\) − 4.00000i − 0.359211i
\(125\) − 3.00000i − 0.268328i
\(126\) −2.00000 −0.178174
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) 4.00000 0.346844
\(134\) −10.0000 −0.863868
\(135\) − 3.00000i − 0.258199i
\(136\) − 3.00000i − 0.257248i
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −6.00000 −0.507093
\(141\) 6.00000i 0.505291i
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 9.00000i − 0.747409i
\(146\) 13.0000 1.07589
\(147\) 3.00000 0.247436
\(148\) 7.00000i 0.575396i
\(149\) 9.00000i 0.737309i 0.929567 + 0.368654i \(0.120181\pi\)
−0.929567 + 0.368654i \(0.879819\pi\)
\(150\) − 4.00000i − 0.326599i
\(151\) − 10.0000i − 0.813788i −0.913475 0.406894i \(-0.866612\pi\)
0.913475 0.406894i \(-0.133388\pi\)
\(152\) −2.00000 −0.162221
\(153\) 3.00000 0.242536
\(154\) − 12.0000i − 0.966988i
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) − 4.00000i − 0.318223i
\(159\) 3.00000 0.237915
\(160\) 3.00000 0.237171
\(161\) 12.0000i 0.945732i
\(162\) 1.00000i 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) − 3.00000i − 0.234261i
\(165\) 18.0000 1.40130
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 9.00000 0.690268
\(171\) − 2.00000i − 0.152944i
\(172\) −10.0000 −0.762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 3.00000i 0.227429i
\(175\) − 8.00000i − 0.604743i
\(176\) 6.00000i 0.452267i
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 3.00000i 0.223607i
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) − 6.00000i − 0.442326i
\(185\) −21.0000 −1.54395
\(186\) −4.00000 −0.293294
\(187\) 18.0000i 1.31629i
\(188\) − 6.00000i − 0.437595i
\(189\) 2.00000i 0.145479i
\(190\) − 6.00000i − 0.435286i
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 23.0000i 1.65558i 0.561041 + 0.827788i \(0.310401\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) −6.00000 −0.426401
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 4.00000i 0.282843i
\(201\) 10.0000i 0.705346i
\(202\) − 15.0000i − 1.05540i
\(203\) 6.00000i 0.421117i
\(204\) −3.00000 −0.210042
\(205\) 9.00000 0.628587
\(206\) − 14.0000i − 0.975426i
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 6.00000i 0.414039i
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −3.00000 −0.206041
\(213\) − 6.00000i − 0.411113i
\(214\) − 6.00000i − 0.410152i
\(215\) − 30.0000i − 2.04598i
\(216\) − 1.00000i − 0.0680414i
\(217\) −8.00000 −0.543075
\(218\) 14.0000 0.948200
\(219\) − 13.0000i − 0.878459i
\(220\) −18.0000 −1.21356
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) −2.00000 −0.133631
\(225\) −4.00000 −0.266667
\(226\) − 3.00000i − 0.199557i
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 2.00000i 0.132453i
\(229\) − 22.0000i − 1.45380i −0.686743 0.726900i \(-0.740960\pi\)
0.686743 0.726900i \(-0.259040\pi\)
\(230\) 18.0000 1.18688
\(231\) −12.0000 −0.789542
\(232\) − 3.00000i − 0.196960i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) −6.00000 −0.388922
\(239\) − 6.00000i − 0.388108i −0.980991 0.194054i \(-0.937836\pi\)
0.980991 0.194054i \(-0.0621637\pi\)
\(240\) − 3.00000i − 0.193649i
\(241\) − 1.00000i − 0.0644157i −0.999481 0.0322078i \(-0.989746\pi\)
0.999481 0.0322078i \(-0.0102538\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) 1.00000 0.0641500
\(244\) 7.00000 0.448129
\(245\) − 9.00000i − 0.574989i
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 6.00000i 0.380235i
\(250\) 3.00000 0.189737
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 36.0000i 2.26330i
\(254\) 4.00000i 0.250982i
\(255\) − 9.00000i − 0.563602i
\(256\) 1.00000 0.0625000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 10.0000i 0.622573i
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 6.00000 0.369274
\(265\) − 9.00000i − 0.552866i
\(266\) 4.00000i 0.245256i
\(267\) 18.0000i 1.10158i
\(268\) − 10.0000i − 0.610847i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 3.00000 0.182574
\(271\) − 16.0000i − 0.971931i −0.873978 0.485965i \(-0.838468\pi\)
0.873978 0.485965i \(-0.161532\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) − 24.0000i − 1.44725i
\(276\) −6.00000 −0.361158
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 4.00000i 0.239474i
\(280\) − 6.00000i − 0.358569i
\(281\) 9.00000i 0.536895i 0.963294 + 0.268447i \(0.0865106\pi\)
−0.963294 + 0.268447i \(0.913489\pi\)
\(282\) −6.00000 −0.357295
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 6.00000i 0.356034i
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000i 0.0589256i
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) − 14.0000i − 0.820695i
\(292\) 13.0000i 0.760767i
\(293\) − 21.0000i − 1.22683i −0.789760 0.613417i \(-0.789795\pi\)
0.789760 0.613417i \(-0.210205\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 0 0
\(296\) −7.00000 −0.406867
\(297\) 6.00000i 0.348155i
\(298\) −9.00000 −0.521356
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) 20.0000i 1.15278i
\(302\) 10.0000 0.575435
\(303\) −15.0000 −0.861727
\(304\) − 2.00000i − 0.114708i
\(305\) 21.0000i 1.20246i
\(306\) 3.00000i 0.171499i
\(307\) − 10.0000i − 0.570730i −0.958419 0.285365i \(-0.907885\pi\)
0.958419 0.285365i \(-0.0921148\pi\)
\(308\) 12.0000 0.683763
\(309\) −14.0000 −0.796432
\(310\) 12.0000i 0.681554i
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 5.00000i 0.282166i
\(315\) 6.00000 0.338062
\(316\) 4.00000 0.225018
\(317\) − 3.00000i − 0.168497i −0.996445 0.0842484i \(-0.973151\pi\)
0.996445 0.0842484i \(-0.0268489\pi\)
\(318\) 3.00000i 0.168232i
\(319\) 18.0000i 1.00781i
\(320\) 3.00000i 0.167705i
\(321\) −6.00000 −0.334887
\(322\) −12.0000 −0.668734
\(323\) − 6.00000i − 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) − 14.0000i − 0.774202i
\(328\) 3.00000 0.165647
\(329\) −12.0000 −0.661581
\(330\) 18.0000i 0.990867i
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) − 7.00000i − 0.383598i
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 2.00000i 0.109109i
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) −3.00000 −0.162938
\(340\) 9.00000i 0.488094i
\(341\) −24.0000 −1.29967
\(342\) 2.00000 0.108148
\(343\) 20.0000i 1.07990i
\(344\) − 10.0000i − 0.539164i
\(345\) − 18.0000i − 0.969087i
\(346\) 6.00000i 0.322562i
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) −3.00000 −0.160817
\(349\) − 10.0000i − 0.535288i −0.963518 0.267644i \(-0.913755\pi\)
0.963518 0.267644i \(-0.0862451\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) 15.0000i 0.798369i 0.916871 + 0.399185i \(0.130707\pi\)
−0.916871 + 0.399185i \(0.869293\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) − 18.0000i − 0.953998i
\(357\) 6.00000i 0.317554i
\(358\) − 6.00000i − 0.317110i
\(359\) 6.00000i 0.316668i 0.987386 + 0.158334i \(0.0506123\pi\)
−0.987386 + 0.158334i \(0.949388\pi\)
\(360\) −3.00000 −0.158114
\(361\) 15.0000 0.789474
\(362\) 7.00000i 0.367912i
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −39.0000 −2.04135
\(366\) − 7.00000i − 0.365896i
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 6.00000 0.312772
\(369\) 3.00000i 0.156174i
\(370\) − 21.0000i − 1.09174i
\(371\) 6.00000i 0.311504i
\(372\) − 4.00000i − 0.207390i
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) −18.0000 −0.930758
\(375\) − 3.00000i − 0.154919i
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 6.00000 0.307794
\(381\) 4.00000 0.204926
\(382\) − 12.0000i − 0.613973i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 36.0000i 1.83473i
\(386\) −23.0000 −1.17067
\(387\) 10.0000 0.508329
\(388\) 14.0000i 0.710742i
\(389\) −39.0000 −1.97738 −0.988689 0.149979i \(-0.952080\pi\)
−0.988689 + 0.149979i \(0.952080\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) − 3.00000i − 0.151523i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 12.0000i 0.603786i
\(396\) − 6.00000i − 0.301511i
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 4.00000 0.200250
\(400\) −4.00000 −0.200000
\(401\) − 3.00000i − 0.149813i −0.997191 0.0749064i \(-0.976134\pi\)
0.997191 0.0749064i \(-0.0238658\pi\)
\(402\) −10.0000 −0.498755
\(403\) 0 0
\(404\) 15.0000 0.746278
\(405\) − 3.00000i − 0.149071i
\(406\) −6.00000 −0.297775
\(407\) 42.0000 2.08186
\(408\) − 3.00000i − 0.148522i
\(409\) 1.00000i 0.0494468i 0.999694 + 0.0247234i \(0.00787051\pi\)
−0.999694 + 0.0247234i \(0.992129\pi\)
\(410\) 9.00000i 0.444478i
\(411\) 9.00000i 0.443937i
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) 6.00000i 0.294884i
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 12.0000i 0.586939i
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) −6.00000 −0.292770
\(421\) − 29.0000i − 1.41337i −0.707527 0.706687i \(-0.750189\pi\)
0.707527 0.706687i \(-0.249811\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) 6.00000i 0.291730i
\(424\) − 3.00000i − 0.145693i
\(425\) −12.0000 −0.582086
\(426\) 6.00000 0.290701
\(427\) − 14.0000i − 0.677507i
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 30.0000 1.44673
\(431\) 6.00000i 0.289010i 0.989504 + 0.144505i \(0.0461589\pi\)
−0.989504 + 0.144505i \(0.953841\pi\)
\(432\) 1.00000 0.0481125
\(433\) 13.0000 0.624740 0.312370 0.949960i \(-0.398877\pi\)
0.312370 + 0.949960i \(0.398877\pi\)
\(434\) − 8.00000i − 0.384012i
\(435\) − 9.00000i − 0.431517i
\(436\) 14.0000i 0.670478i
\(437\) − 12.0000i − 0.574038i
\(438\) 13.0000 0.621164
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) − 18.0000i − 0.858116i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 7.00000i 0.332205i
\(445\) 54.0000 2.55985
\(446\) 8.00000 0.378811
\(447\) 9.00000i 0.425685i
\(448\) − 2.00000i − 0.0944911i
\(449\) 18.0000i 0.849473i 0.905317 + 0.424736i \(0.139633\pi\)
−0.905317 + 0.424736i \(0.860367\pi\)
\(450\) − 4.00000i − 0.188562i
\(451\) −18.0000 −0.847587
\(452\) 3.00000 0.141108
\(453\) − 10.0000i − 0.469841i
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) − 11.0000i − 0.514558i −0.966337 0.257279i \(-0.917174\pi\)
0.966337 0.257279i \(-0.0828260\pi\)
\(458\) 22.0000 1.02799
\(459\) 3.00000 0.140028
\(460\) 18.0000i 0.839254i
\(461\) − 15.0000i − 0.698620i −0.937007 0.349310i \(-0.886416\pi\)
0.937007 0.349310i \(-0.113584\pi\)
\(462\) − 12.0000i − 0.558291i
\(463\) 38.0000i 1.76601i 0.469364 + 0.883005i \(0.344483\pi\)
−0.469364 + 0.883005i \(0.655517\pi\)
\(464\) 3.00000 0.139272
\(465\) 12.0000 0.556487
\(466\) 6.00000i 0.277945i
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 18.0000i 0.830278i
\(471\) 5.00000 0.230388
\(472\) 0 0
\(473\) 60.0000i 2.75880i
\(474\) − 4.00000i − 0.183726i
\(475\) 8.00000i 0.367065i
\(476\) − 6.00000i − 0.275010i
\(477\) 3.00000 0.137361
\(478\) 6.00000 0.274434
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) 1.00000 0.0455488
\(483\) 12.0000i 0.546019i
\(484\) 25.0000 1.13636
\(485\) −42.0000 −1.90712
\(486\) 1.00000i 0.0453609i
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 7.00000i 0.316875i
\(489\) − 4.00000i − 0.180886i
\(490\) 9.00000 0.406579
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) − 3.00000i − 0.135250i
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) 4.00000i 0.179605i
\(497\) 12.0000 0.538274
\(498\) −6.00000 −0.268866
\(499\) − 32.0000i − 1.43252i −0.697835 0.716258i \(-0.745853\pi\)
0.697835 0.716258i \(-0.254147\pi\)
\(500\) 3.00000i 0.134164i
\(501\) 0 0
\(502\) 12.0000i 0.535586i
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 2.00000 0.0890871
\(505\) 45.0000i 2.00247i
\(506\) −36.0000 −1.60040
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) − 3.00000i − 0.132973i −0.997787 0.0664863i \(-0.978821\pi\)
0.997787 0.0664863i \(-0.0211789\pi\)
\(510\) 9.00000 0.398527
\(511\) 26.0000 1.15017
\(512\) 1.00000i 0.0441942i
\(513\) − 2.00000i − 0.0883022i
\(514\) 3.00000i 0.132324i
\(515\) 42.0000i 1.85074i
\(516\) −10.0000 −0.440225
\(517\) −36.0000 −1.58328
\(518\) 14.0000i 0.615125i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 3.00000i 0.131306i
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 0 0
\(525\) − 8.00000i − 0.349149i
\(526\) − 6.00000i − 0.261612i
\(527\) 12.0000i 0.522728i
\(528\) 6.00000i 0.261116i
\(529\) 13.0000 0.565217
\(530\) 9.00000 0.390935
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) 18.0000i 0.778208i
\(536\) 10.0000 0.431934
\(537\) −6.00000 −0.258919
\(538\) 18.0000i 0.776035i
\(539\) 18.0000i 0.775315i
\(540\) 3.00000i 0.129099i
\(541\) 29.0000i 1.24681i 0.781900 + 0.623404i \(0.214251\pi\)
−0.781900 + 0.623404i \(0.785749\pi\)
\(542\) 16.0000 0.687259
\(543\) 7.00000 0.300399
\(544\) 3.00000i 0.128624i
\(545\) −42.0000 −1.79908
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) − 9.00000i − 0.384461i
\(549\) −7.00000 −0.298753
\(550\) 24.0000 1.02336
\(551\) − 6.00000i − 0.255609i
\(552\) − 6.00000i − 0.255377i
\(553\) − 8.00000i − 0.340195i
\(554\) − 17.0000i − 0.722261i
\(555\) −21.0000 −0.891400
\(556\) 4.00000 0.169638
\(557\) 3.00000i 0.127114i 0.997978 + 0.0635570i \(0.0202445\pi\)
−0.997978 + 0.0635570i \(0.979756\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 6.00000 0.253546
\(561\) 18.0000i 0.759961i
\(562\) −9.00000 −0.379642
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) − 6.00000i − 0.252646i
\(565\) 9.00000i 0.378633i
\(566\) − 14.0000i − 0.588464i
\(567\) 2.00000i 0.0839921i
\(568\) −6.00000 −0.251754
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) − 6.00000i − 0.251312i
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) − 6.00000i − 0.250435i
\(575\) −24.0000 −1.00087
\(576\) −1.00000 −0.0416667
\(577\) − 11.0000i − 0.457936i −0.973434 0.228968i \(-0.926465\pi\)
0.973434 0.228968i \(-0.0735351\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 23.0000i 0.955847i
\(580\) 9.00000i 0.373705i
\(581\) −12.0000 −0.497844
\(582\) 14.0000 0.580319
\(583\) 18.0000i 0.745484i
\(584\) −13.0000 −0.537944
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) − 6.00000i − 0.246807i
\(592\) − 7.00000i − 0.287698i
\(593\) 9.00000i 0.369586i 0.982777 + 0.184793i \(0.0591614\pi\)
−0.982777 + 0.184793i \(0.940839\pi\)
\(594\) −6.00000 −0.246183
\(595\) 18.0000 0.737928
\(596\) − 9.00000i − 0.368654i
\(597\) 10.0000 0.409273
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 4.00000i 0.163299i
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) −20.0000 −0.815139
\(603\) 10.0000i 0.407231i
\(604\) 10.0000i 0.406894i
\(605\) 75.0000i 3.04918i
\(606\) − 15.0000i − 0.609333i
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 2.00000 0.0811107
\(609\) 6.00000i 0.243132i
\(610\) −21.0000 −0.850265
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 31.0000i 1.25208i 0.779792 + 0.626039i \(0.215325\pi\)
−0.779792 + 0.626039i \(0.784675\pi\)
\(614\) 10.0000 0.403567
\(615\) 9.00000 0.362915
\(616\) 12.0000i 0.483494i
\(617\) 15.0000i 0.603877i 0.953327 + 0.301939i \(0.0976338\pi\)
−0.953327 + 0.301939i \(0.902366\pi\)
\(618\) − 14.0000i − 0.563163i
\(619\) 8.00000i 0.321547i 0.986991 + 0.160774i \(0.0513989\pi\)
−0.986991 + 0.160774i \(0.948601\pi\)
\(620\) −12.0000 −0.481932
\(621\) 6.00000 0.240772
\(622\) 30.0000i 1.20289i
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) − 10.0000i − 0.399680i
\(627\) 12.0000 0.479234
\(628\) −5.00000 −0.199522
\(629\) − 21.0000i − 0.837325i
\(630\) 6.00000i 0.239046i
\(631\) 20.0000i 0.796187i 0.917345 + 0.398094i \(0.130328\pi\)
−0.917345 + 0.398094i \(0.869672\pi\)
\(632\) 4.00000i 0.159111i
\(633\) −16.0000 −0.635943
\(634\) 3.00000 0.119145
\(635\) − 12.0000i − 0.476205i
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) − 6.00000i − 0.237356i
\(640\) −3.00000 −0.118585
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) − 6.00000i − 0.236801i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) − 12.0000i − 0.472866i
\(645\) − 30.0000i − 1.18125i
\(646\) 6.00000 0.236067
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 4.00000i 0.156652i
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 3.00000i 0.117130i
\(657\) − 13.0000i − 0.507178i
\(658\) − 12.0000i − 0.467809i
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) −18.0000 −0.700649
\(661\) 5.00000i 0.194477i 0.995261 + 0.0972387i \(0.0310010\pi\)
−0.995261 + 0.0972387i \(0.968999\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) − 12.0000i − 0.465340i
\(666\) 7.00000 0.271244
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) − 8.00000i − 0.309298i
\(670\) 30.0000i 1.15900i
\(671\) − 42.0000i − 1.62139i
\(672\) −2.00000 −0.0771517
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) − 23.0000i − 0.885927i
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) − 3.00000i − 0.115214i
\(679\) 28.0000 1.07454
\(680\) −9.00000 −0.345134
\(681\) − 18.0000i − 0.689761i
\(682\) − 24.0000i − 0.919007i
\(683\) − 48.0000i − 1.83667i −0.395805 0.918334i \(-0.629534\pi\)
0.395805 0.918334i \(-0.370466\pi\)
\(684\) 2.00000i 0.0764719i
\(685\) 27.0000 1.03162
\(686\) −20.0000 −0.763604
\(687\) − 22.0000i − 0.839352i
\(688\) 10.0000 0.381246
\(689\) 0 0
\(690\) 18.0000 0.685248
\(691\) − 26.0000i − 0.989087i −0.869153 0.494543i \(-0.835335\pi\)
0.869153 0.494543i \(-0.164665\pi\)
\(692\) −6.00000 −0.228086
\(693\) −12.0000 −0.455842
\(694\) − 30.0000i − 1.13878i
\(695\) 12.0000i 0.455186i
\(696\) − 3.00000i − 0.113715i
\(697\) 9.00000i 0.340899i
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) 8.00000i 0.302372i
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) − 6.00000i − 0.226134i
\(705\) 18.0000 0.677919
\(706\) −15.0000 −0.564532
\(707\) − 30.0000i − 1.12827i
\(708\) 0 0
\(709\) 5.00000i 0.187779i 0.995583 + 0.0938895i \(0.0299300\pi\)
−0.995583 + 0.0938895i \(0.970070\pi\)
\(710\) − 18.0000i − 0.675528i
\(711\) −4.00000 −0.150012
\(712\) 18.0000 0.674579
\(713\) 24.0000i 0.898807i
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) − 6.00000i − 0.224074i
\(718\) −6.00000 −0.223918
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) − 3.00000i − 0.111803i
\(721\) − 28.0000i − 1.04277i
\(722\) 15.0000i 0.558242i
\(723\) − 1.00000i − 0.0371904i
\(724\) −7.00000 −0.260153
\(725\) −12.0000 −0.445669
\(726\) − 25.0000i − 0.927837i
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 39.0000i − 1.44345i
\(731\) 30.0000 1.10959
\(732\) 7.00000 0.258727
\(733\) 31.0000i 1.14501i 0.819901 + 0.572506i \(0.194029\pi\)
−0.819901 + 0.572506i \(0.805971\pi\)
\(734\) 2.00000i 0.0738213i
\(735\) − 9.00000i − 0.331970i
\(736\) 6.00000i 0.221163i
\(737\) −60.0000 −2.21013
\(738\) −3.00000 −0.110432
\(739\) − 16.0000i − 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 21.0000 0.771975
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) 4.00000 0.146647
\(745\) 27.0000 0.989203
\(746\) 29.0000i 1.06177i
\(747\) 6.00000i 0.219529i
\(748\) − 18.0000i − 0.658145i
\(749\) − 12.0000i − 0.438470i
\(750\) 3.00000 0.109545
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) − 2.00000i − 0.0727393i
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 20.0000 0.726433
\(759\) 36.0000i 1.30672i
\(760\) 6.00000i 0.217643i
\(761\) − 30.0000i − 1.08750i −0.839248 0.543750i \(-0.817004\pi\)
0.839248 0.543750i \(-0.182996\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 28.0000 1.01367
\(764\) 12.0000 0.434145
\(765\) − 9.00000i − 0.325396i
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 14.0000i − 0.504853i −0.967616 0.252426i \(-0.918771\pi\)
0.967616 0.252426i \(-0.0812286\pi\)
\(770\) −36.0000 −1.29735
\(771\) 3.00000 0.108042
\(772\) − 23.0000i − 0.827788i
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 10.0000i 0.359443i
\(775\) − 16.0000i − 0.574737i
\(776\) −14.0000 −0.502571
\(777\) 14.0000 0.502247
\(778\) − 39.0000i − 1.39822i
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 18.0000i 0.643679i
\(783\) 3.00000 0.107211
\(784\) 3.00000 0.107143
\(785\) − 15.0000i − 0.535373i
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 6.00000i 0.213741i
\(789\) −6.00000 −0.213606
\(790\) −12.0000 −0.426941
\(791\) − 6.00000i − 0.213335i
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) − 9.00000i − 0.319197i
\(796\) −10.0000 −0.354441
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 18.0000i 0.636794i
\(800\) − 4.00000i − 0.141421i
\(801\) 18.0000i 0.635999i
\(802\) 3.00000 0.105934
\(803\) 78.0000 2.75256
\(804\) − 10.0000i − 0.352673i
\(805\) 36.0000 1.26883
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 15.0000i 0.527698i
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 3.00000 0.105409
\(811\) 4.00000i 0.140459i 0.997531 + 0.0702295i \(0.0223732\pi\)
−0.997531 + 0.0702295i \(0.977627\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) − 16.0000i − 0.561144i
\(814\) 42.0000i 1.47210i
\(815\) −12.0000 −0.420342
\(816\) 3.00000 0.105021
\(817\) − 20.0000i − 0.699711i
\(818\) −1.00000 −0.0349642
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) − 18.0000i − 0.628204i −0.949389 0.314102i \(-0.898297\pi\)
0.949389 0.314102i \(-0.101703\pi\)
\(822\) −9.00000 −0.313911
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 14.0000i 0.487713i
\(825\) − 24.0000i − 0.835573i
\(826\) 0 0
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) −6.00000 −0.208514
\(829\) −17.0000 −0.590434 −0.295217 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(830\) 18.0000i 0.624789i
\(831\) −17.0000 −0.589723
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) − 4.00000i − 0.138509i
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 4.00000i 0.138260i
\(838\) 24.0000i 0.829066i
\(839\) − 12.0000i − 0.414286i −0.978311 0.207143i \(-0.933583\pi\)
0.978311 0.207143i \(-0.0664165\pi\)
\(840\) − 6.00000i − 0.207020i
\(841\) −20.0000 −0.689655
\(842\) 29.0000 0.999406
\(843\) 9.00000i 0.309976i
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) − 50.0000i − 1.71802i
\(848\) 3.00000 0.103020
\(849\) −14.0000 −0.480479
\(850\) − 12.0000i − 0.411597i
\(851\) − 42.0000i − 1.43974i
\(852\) 6.00000i 0.205557i
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) 14.0000 0.479070
\(855\) −6.00000 −0.205196
\(856\) 6.00000i 0.205076i
\(857\) −21.0000 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 0 0
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) 30.0000i 1.02299i
\(861\) −6.00000 −0.204479
\(862\) −6.00000 −0.204361
\(863\) − 18.0000i − 0.612727i −0.951915 0.306364i \(-0.900888\pi\)
0.951915 0.306364i \(-0.0991123\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 18.0000i − 0.612018i
\(866\) 13.0000i 0.441758i
\(867\) −8.00000 −0.271694
\(868\) 8.00000 0.271538
\(869\) − 24.0000i − 0.814144i
\(870\) 9.00000 0.305129
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) − 14.0000i − 0.473828i
\(874\) 12.0000 0.405906
\(875\) 6.00000 0.202837
\(876\) 13.0000i 0.439229i
\(877\) − 41.0000i − 1.38447i −0.721671 0.692236i \(-0.756626\pi\)
0.721671 0.692236i \(-0.243374\pi\)
\(878\) − 14.0000i − 0.472477i
\(879\) − 21.0000i − 0.708312i
\(880\) 18.0000 0.606780
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 3.00000i 0.101015i
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 36.0000i − 1.20944i
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −7.00000 −0.234905
\(889\) 8.00000i 0.268311i
\(890\) 54.0000i 1.81008i
\(891\) 6.00000i 0.201008i
\(892\) 8.00000i 0.267860i
\(893\) 12.0000 0.401565
\(894\) −9.00000 −0.301005
\(895\) 18.0000i 0.601674i
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 12.0000i 0.400222i
\(900\) 4.00000 0.133333
\(901\) 9.00000 0.299833
\(902\) − 18.0000i − 0.599334i
\(903\) 20.0000i 0.665558i
\(904\) 3.00000i 0.0997785i
\(905\) − 21.0000i − 0.698064i
\(906\) 10.0000 0.332228
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 18.0000i 0.597351i
\(909\) −15.0000 −0.497519
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) − 2.00000i − 0.0662266i
\(913\) −36.0000 −1.19143
\(914\) 11.0000 0.363848
\(915\) 21.0000i 0.694239i
\(916\) 22.0000i 0.726900i
\(917\) 0 0
\(918\) 3.00000i 0.0990148i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −18.0000 −0.593442
\(921\) − 10.0000i − 0.329511i
\(922\) 15.0000 0.493999
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 28.0000i 0.920634i
\(926\) −38.0000 −1.24876
\(927\) −14.0000 −0.459820
\(928\) 3.00000i 0.0984798i
\(929\) − 33.0000i − 1.08269i −0.840799 0.541347i \(-0.817914\pi\)
0.840799 0.541347i \(-0.182086\pi\)
\(930\) 12.0000i 0.393496i
\(931\) − 6.00000i − 0.196642i
\(932\) −6.00000 −0.196537
\(933\) 30.0000 0.982156
\(934\) 18.0000i 0.588978i
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) − 20.0000i − 0.653023i
\(939\) −10.0000 −0.326338
\(940\) −18.0000 −0.587095
\(941\) − 42.0000i − 1.36916i −0.728937 0.684580i \(-0.759985\pi\)
0.728937 0.684580i \(-0.240015\pi\)
\(942\) 5.00000i 0.162909i
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) 6.00000 0.195180
\(946\) −60.0000 −1.95077
\(947\) 24.0000i 0.779895i 0.920837 + 0.389948i \(0.127507\pi\)
−0.920837 + 0.389948i \(0.872493\pi\)
\(948\) 4.00000 0.129914
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) − 3.00000i − 0.0972817i
\(952\) 6.00000 0.194461
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 3.00000i 0.0971286i
\(955\) 36.0000i 1.16493i
\(956\) 6.00000i 0.194054i
\(957\) 18.0000i 0.581857i
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 3.00000i 0.0968246i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 1.00000i 0.0322078i
\(965\) 69.0000 2.22119
\(966\) −12.0000 −0.386094
\(967\) 22.0000i 0.707472i 0.935345 + 0.353736i \(0.115089\pi\)
−0.935345 + 0.353736i \(0.884911\pi\)
\(968\) 25.0000i 0.803530i
\(969\) − 6.00000i − 0.192748i
\(970\) − 42.0000i − 1.34854i
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 8.00000i − 0.256468i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 3.00000i 0.0959785i 0.998848 + 0.0479893i \(0.0152813\pi\)
−0.998848 + 0.0479893i \(0.984719\pi\)
\(978\) 4.00000 0.127906
\(979\) −108.000 −3.45169
\(980\) 9.00000i 0.287494i
\(981\) − 14.0000i − 0.446986i
\(982\) 18.0000i 0.574403i
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 3.00000 0.0956365
\(985\) −18.0000 −0.573528
\(986\) 9.00000i 0.286618i
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 18.0000i 0.572078i
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) −4.00000 −0.127000
\(993\) 4.00000i 0.126936i
\(994\) 12.0000i 0.380617i
\(995\) − 30.0000i − 0.951064i
\(996\) − 6.00000i − 0.190117i
\(997\) 5.00000 0.158352 0.0791758 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) 32.0000 1.01294
\(999\) − 7.00000i − 0.221470i
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 1014.2.b.c.337.2 2
3.2 odd 2 3042.2.b.h.1351.1 2
13.2 odd 12 1014.2.e.a.529.1 2
13.3 even 3 1014.2.i.b.823.1 4
13.4 even 6 1014.2.i.b.361.1 4
13.5 odd 4 1014.2.a.f.1.1 1
13.6 odd 12 1014.2.e.a.991.1 2
13.7 odd 12 78.2.e.a.55.1 2
13.8 odd 4 1014.2.a.c.1.1 1
13.9 even 3 1014.2.i.b.361.2 4
13.10 even 6 1014.2.i.b.823.2 4
13.11 odd 12 78.2.e.a.61.1 yes 2
13.12 even 2 inner 1014.2.b.c.337.1 2
39.5 even 4 3042.2.a.h.1.1 1
39.8 even 4 3042.2.a.i.1.1 1
39.11 even 12 234.2.h.a.217.1 2
39.20 even 12 234.2.h.a.55.1 2
39.38 odd 2 3042.2.b.h.1351.2 2
52.7 even 12 624.2.q.g.289.1 2
52.11 even 12 624.2.q.g.529.1 2
52.31 even 4 8112.2.a.c.1.1 1
52.47 even 4 8112.2.a.m.1.1 1
65.7 even 12 1950.2.z.g.1849.1 4
65.24 odd 12 1950.2.i.m.451.1 2
65.33 even 12 1950.2.z.g.1849.2 4
65.37 even 12 1950.2.z.g.1699.2 4
65.59 odd 12 1950.2.i.m.601.1 2
65.63 even 12 1950.2.z.g.1699.1 4
156.11 odd 12 1872.2.t.c.1153.1 2
156.59 odd 12 1872.2.t.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 2 13.7 odd 12
78.2.e.a.61.1 yes 2 13.11 odd 12
234.2.h.a.55.1 2 39.20 even 12
234.2.h.a.217.1 2 39.11 even 12
624.2.q.g.289.1 2 52.7 even 12
624.2.q.g.529.1 2 52.11 even 12
1014.2.a.c.1.1 1 13.8 odd 4
1014.2.a.f.1.1 1 13.5 odd 4
1014.2.b.c.337.1 2 13.12 even 2 inner
1014.2.b.c.337.2 2 1.1 even 1 trivial
1014.2.e.a.529.1 2 13.2 odd 12
1014.2.e.a.991.1 2 13.6 odd 12
1014.2.i.b.361.1 4 13.4 even 6
1014.2.i.b.361.2 4 13.9 even 3
1014.2.i.b.823.1 4 13.3 even 3
1014.2.i.b.823.2 4 13.10 even 6
1872.2.t.c.289.1 2 156.59 odd 12
1872.2.t.c.1153.1 2 156.11 odd 12
1950.2.i.m.451.1 2 65.24 odd 12
1950.2.i.m.601.1 2 65.59 odd 12
1950.2.z.g.1699.1 4 65.63 even 12
1950.2.z.g.1699.2 4 65.37 even 12
1950.2.z.g.1849.1 4 65.7 even 12
1950.2.z.g.1849.2 4 65.33 even 12
3042.2.a.h.1.1 1 39.5 even 4
3042.2.a.i.1.1 1 39.8 even 4
3042.2.b.h.1351.1 2 3.2 odd 2
3042.2.b.h.1351.2 2 39.38 odd 2
8112.2.a.c.1.1 1 52.31 even 4
8112.2.a.m.1.1 1 52.47 even 4