Properties

Label 390.2.b.a.181.1
Level $390$
Weight $2$
Character 390.181
Analytic conductor $3.114$
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] = [390,2,Mod(181,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, 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("390.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.b (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: \(3.11416567883\)
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 181.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 390.181
Dual form 390.2.b.a.181.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} +(-3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000i q^{18} +4.00000i q^{19} -1.00000i q^{20} +4.00000 q^{22} +6.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} +(2.00000 + 3.00000i) q^{26} -1.00000 q^{27} +4.00000 q^{29} -1.00000 q^{30} +10.0000i q^{31} -1.00000i q^{32} -4.00000i q^{33} -4.00000i q^{34} -1.00000 q^{36} -4.00000i q^{37} +4.00000 q^{38} +(3.00000 - 2.00000i) q^{39} -1.00000 q^{40} -2.00000i q^{41} -12.0000 q^{43} -4.00000i q^{44} +1.00000i q^{45} -6.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} +1.00000i q^{50} -4.00000 q^{51} +(3.00000 - 2.00000i) q^{52} -2.00000 q^{53} +1.00000i q^{54} -4.00000 q^{55} -4.00000i q^{57} -4.00000i q^{58} +4.00000i q^{59} +1.00000i q^{60} -10.0000 q^{61} +10.0000 q^{62} -1.00000 q^{64} +(-2.00000 - 3.00000i) q^{65} -4.00000 q^{66} -10.0000i q^{67} -4.00000 q^{68} -6.00000 q^{69} +12.0000i q^{71} +1.00000i q^{72} +6.00000i q^{73} -4.00000 q^{74} +1.00000 q^{75} -4.00000i q^{76} +(-2.00000 - 3.00000i) q^{78} -8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000i q^{85} +12.0000i q^{86} -4.00000 q^{87} -4.00000 q^{88} -10.0000i q^{89} +1.00000 q^{90} -6.00000 q^{92} -10.0000i q^{93} -4.00000 q^{95} +1.00000i q^{96} -14.0000i q^{97} -7.00000i q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 6 q^{13} + 2 q^{16} + 8 q^{17} + 8 q^{22} + 12 q^{23} - 2 q^{25} + 4 q^{26} - 2 q^{27} + 8 q^{29} - 2 q^{30} - 2 q^{36} + 8 q^{38} + 6 q^{39} - 2 q^{40} - 24 q^{43} - 2 q^{48} + 14 q^{49} - 8 q^{51} + 6 q^{52} - 4 q^{53} - 8 q^{55} - 20 q^{61} + 20 q^{62} - 2 q^{64} - 4 q^{65} - 8 q^{66} - 8 q^{68} - 12 q^{69} - 8 q^{74} + 2 q^{75} - 4 q^{78} - 16 q^{79} + 2 q^{81} - 4 q^{82} - 8 q^{87} - 8 q^{88} + 2 q^{90} - 12 q^{92} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.00000 + 2.00000i −0.832050 + 0.554700i
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 2.00000 + 3.00000i 0.392232 + 0.588348i
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) −1.00000 −0.182574
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) 4.00000i 0.685994i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 4.00000i 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 4.00000 0.648886
\(39\) 3.00000 2.00000i 0.480384 0.320256i
\(40\) −1.00000 −0.158114
\(41\) 2.00000i 0.312348i −0.987730 0.156174i \(-0.950084\pi\)
0.987730 0.156174i \(-0.0499160\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 4.00000i 0.603023i
\(45\) 1.00000i 0.149071i
\(46\) 6.00000i 0.884652i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −0.144338
\(49\) 7.00000 1.00000
\(50\) 1.00000i 0.141421i
\(51\) −4.00000 −0.560112
\(52\) 3.00000 2.00000i 0.416025 0.277350i
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 4.00000i 0.525226i
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −2.00000 3.00000i −0.248069 0.372104i
\(66\) −4.00000 −0.492366
\(67\) 10.0000i 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) −4.00000 −0.485071
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.00000 0.115470
\(76\) 4.00000i 0.458831i
\(77\) 0 0
\(78\) −2.00000 3.00000i −0.226455 0.339683i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 12.0000i 1.29399i
\(87\) −4.00000 −0.428845
\(88\) −4.00000 −0.426401
\(89\) 10.0000i 1.06000i −0.847998 0.529999i \(-0.822192\pi\)
0.847998 0.529999i \(-0.177808\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 10.0000i 1.03695i
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 1.00000i 0.102062i
\(97\) 14.0000i 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 4.00000i 0.402015i
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 4.00000i 0.396059i
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −2.00000 3.00000i −0.196116 0.294174i
\(105\) 0 0
\(106\) 2.00000i 0.194257i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 4.00000i 0.381385i
\(111\) 4.00000i 0.379663i
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −4.00000 −0.374634
\(115\) 6.00000i 0.559503i
\(116\) −4.00000 −0.371391
\(117\) −3.00000 + 2.00000i −0.277350 + 0.184900i
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −5.00000 −0.454545
\(122\) 10.0000i 0.905357i
\(123\) 2.00000i 0.180334i
\(124\) 10.0000i 0.898027i
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 12.0000 1.05654
\(130\) −3.00000 + 2.00000i −0.263117 + 0.175412i
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) 1.00000i 0.0860663i
\(136\) 4.00000i 0.342997i
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 6.00000i 0.510754i
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −8.00000 12.0000i −0.668994 1.00349i
\(144\) 1.00000 0.0833333
\(145\) 4.00000i 0.332182i
\(146\) 6.00000 0.496564
\(147\) −7.00000 −0.577350
\(148\) 4.00000i 0.328798i
\(149\) 2.00000i 0.163846i −0.996639 0.0819232i \(-0.973894\pi\)
0.996639 0.0819232i \(-0.0261062\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 22.0000i 1.79033i −0.445730 0.895167i \(-0.647056\pi\)
0.445730 0.895167i \(-0.352944\pi\)
\(152\) −4.00000 −0.324443
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) −3.00000 + 2.00000i −0.240192 + 0.160128i
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 2.00000 0.158610
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 2.00000i 0.156652i −0.996928 0.0783260i \(-0.975042\pi\)
0.996928 0.0783260i \(-0.0249575\pi\)
\(164\) 2.00000i 0.156174i
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 5.00000 12.0000i 0.384615 0.923077i
\(170\) 4.00000 0.306786
\(171\) 4.00000i 0.305888i
\(172\) 12.0000 0.914991
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 4.00000i 0.303239i
\(175\) 0 0
\(176\) 4.00000i 0.301511i
\(177\) 4.00000i 0.300658i
\(178\) −10.0000 −0.749532
\(179\) 22.0000 1.64436 0.822179 0.569230i \(-0.192758\pi\)
0.822179 + 0.569230i \(0.192758\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 6.00000i 0.442326i
\(185\) 4.00000 0.294086
\(186\) −10.0000 −0.733236
\(187\) 16.0000i 1.17004i
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000i 0.290191i
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −14.0000 −1.00514
\(195\) 2.00000 + 3.00000i 0.143223 + 0.214834i
\(196\) −7.00000 −0.500000
\(197\) 26.0000i 1.85242i −0.377004 0.926212i \(-0.623046\pi\)
0.377004 0.926212i \(-0.376954\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 10.0000i 0.705346i
\(202\) 8.00000i 0.562878i
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 2.00000 0.139686
\(206\) 4.00000i 0.278693i
\(207\) 6.00000 0.417029
\(208\) −3.00000 + 2.00000i −0.208013 + 0.138675i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 2.00000 0.137361
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) 12.0000i 0.818393i
\(216\) 1.00000i 0.0680414i
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 6.00000i 0.405442i
\(220\) 4.00000 0.269680
\(221\) −12.0000 + 8.00000i −0.807207 + 0.538138i
\(222\) 4.00000 0.268462
\(223\) 20.0000i 1.33930i 0.742677 + 0.669650i \(0.233556\pi\)
−0.742677 + 0.669650i \(0.766444\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 12.0000i 0.798228i
\(227\) 16.0000i 1.06196i 0.847385 + 0.530979i \(0.178176\pi\)
−0.847385 + 0.530979i \(0.821824\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 2.00000i 0.132164i 0.997814 + 0.0660819i \(0.0210498\pi\)
−0.997814 + 0.0660819i \(0.978950\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 4.00000i 0.262613i
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 2.00000 + 3.00000i 0.130744 + 0.196116i
\(235\) 0 0
\(236\) 4.00000i 0.260378i
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 4.00000i 0.257663i 0.991667 + 0.128831i \(0.0411226\pi\)
−0.991667 + 0.128831i \(0.958877\pi\)
\(242\) 5.00000i 0.321412i
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 7.00000i 0.447214i
\(246\) 2.00000 0.127515
\(247\) −8.00000 12.0000i −0.509028 0.763542i
\(248\) −10.0000 −0.635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 24.0000i 1.50887i
\(254\) 8.00000i 0.501965i
\(255\) 4.00000i 0.250490i
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 0 0
\(260\) 2.00000 + 3.00000i 0.124035 + 0.186052i
\(261\) 4.00000 0.247594
\(262\) 18.0000i 1.11204i
\(263\) −14.0000 −0.863277 −0.431638 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 4.00000 0.246183
\(265\) 2.00000i 0.122859i
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 10.0000i 0.610847i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 2.00000i 0.121491i −0.998153 0.0607457i \(-0.980652\pi\)
0.998153 0.0607457i \(-0.0193479\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000i 0.241209i
\(276\) 6.00000 0.361158
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 16.0000i 0.959616i
\(279\) 10.0000i 0.598684i
\(280\) 0 0
\(281\) 30.0000i 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000i 0.712069i
\(285\) 4.00000 0.236940
\(286\) −12.0000 + 8.00000i −0.709575 + 0.473050i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 4.00000 0.234888
\(291\) 14.0000i 0.820695i
\(292\) 6.00000i 0.351123i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 7.00000i 0.408248i
\(295\) −4.00000 −0.232889
\(296\) 4.00000 0.232495
\(297\) 4.00000i 0.232104i
\(298\) −2.00000 −0.115857
\(299\) −18.0000 + 12.0000i −1.04097 + 0.693978i
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −22.0000 −1.26596
\(303\) −8.00000 −0.459588
\(304\) 4.00000i 0.229416i
\(305\) 10.0000i 0.572598i
\(306\) 4.00000i 0.228665i
\(307\) 2.00000i 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 10.0000i 0.567962i
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.00000 + 3.00000i 0.113228 + 0.169842i
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 14.0000i 0.790066i
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 16.0000i 0.895828i
\(320\) 1.00000i 0.0559017i
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) −1.00000 −0.0555556
\(325\) 3.00000 2.00000i 0.166410 0.110940i
\(326\) −2.00000 −0.110770
\(327\) 6.00000i 0.331801i
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 4.00000i 0.220193i
\(331\) 32.0000i 1.75888i −0.476011 0.879440i \(-0.657918\pi\)
0.476011 0.879440i \(-0.342082\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −12.0000 5.00000i −0.652714 0.271964i
\(339\) −12.0000 −0.651751
\(340\) 4.00000i 0.216930i
\(341\) −40.0000 −2.16612
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 12.0000i 0.646997i
\(345\) 6.00000i 0.323029i
\(346\) 22.0000i 1.18273i
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 4.00000 0.214423
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 0 0
\(351\) 3.00000 2.00000i 0.160128 0.106752i
\(352\) 4.00000 0.213201
\(353\) 6.00000i 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) −4.00000 −0.212598
\(355\) −12.0000 −0.636894
\(356\) 10.0000i 0.529999i
\(357\) 0 0
\(358\) 22.0000i 1.16274i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 3.00000 0.157895
\(362\) 22.0000i 1.15629i
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 10.0000i 0.522708i
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 6.00000 0.312772
\(369\) 2.00000i 0.104116i
\(370\) 4.00000i 0.207950i
\(371\) 0 0
\(372\) 10.0000i 0.518476i
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 16.0000 0.827340
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) −12.0000 + 8.00000i −0.618031 + 0.412021i
\(378\) 0 0
\(379\) 16.0000i 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 4.00000 0.205196
\(381\) −8.00000 −0.409852
\(382\) 12.0000i 0.613973i
\(383\) 16.0000i 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −12.0000 −0.609994
\(388\) 14.0000i 0.710742i
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 3.00000 2.00000i 0.151911 0.101274i
\(391\) 24.0000 1.21373
\(392\) 7.00000i 0.353553i
\(393\) 18.0000 0.907980
\(394\) −26.0000 −1.30986
\(395\) 8.00000i 0.402524i
\(396\) 4.00000i 0.201008i
\(397\) 28.0000i 1.40528i 0.711546 + 0.702640i \(0.247995\pi\)
−0.711546 + 0.702640i \(0.752005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 26.0000i 1.29838i −0.760627 0.649189i \(-0.775108\pi\)
0.760627 0.649189i \(-0.224892\pi\)
\(402\) 10.0000 0.498755
\(403\) −20.0000 30.0000i −0.996271 1.49441i
\(404\) −8.00000 −0.398015
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) 4.00000i 0.198030i
\(409\) 8.00000i 0.395575i −0.980245 0.197787i \(-0.936624\pi\)
0.980245 0.197787i \(-0.0633755\pi\)
\(410\) 2.00000i 0.0987730i
\(411\) 2.00000i 0.0986527i
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 6.00000i 0.294884i
\(415\) 0 0
\(416\) 2.00000 + 3.00000i 0.0980581 + 0.147087i
\(417\) −16.0000 −0.783523
\(418\) 16.0000i 0.782586i
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 14.0000i 0.682318i −0.940006 0.341159i \(-0.889181\pi\)
0.940006 0.341159i \(-0.110819\pi\)
\(422\) 24.0000i 1.16830i
\(423\) 0 0
\(424\) 2.00000i 0.0971286i
\(425\) −4.00000 −0.194029
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 0 0
\(429\) 8.00000 + 12.0000i 0.386244 + 0.579365i
\(430\) −12.0000 −0.578691
\(431\) 12.0000i 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 4.00000i 0.191785i
\(436\) 6.00000i 0.287348i
\(437\) 24.0000i 1.14808i
\(438\) −6.00000 −0.286691
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 4.00000i 0.190693i
\(441\) 7.00000 0.333333
\(442\) 8.00000 + 12.0000i 0.380521 + 0.570782i
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 4.00000i 0.189832i
\(445\) 10.0000 0.474045
\(446\) 20.0000 0.947027
\(447\) 2.00000i 0.0945968i
\(448\) 0 0
\(449\) 30.0000i 1.41579i 0.706319 + 0.707894i \(0.250354\pi\)
−0.706319 + 0.707894i \(0.749646\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 8.00000 0.376705
\(452\) −12.0000 −0.564433
\(453\) 22.0000i 1.03365i
\(454\) 16.0000 0.750917
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 14.0000i 0.654892i −0.944870 0.327446i \(-0.893812\pi\)
0.944870 0.327446i \(-0.106188\pi\)
\(458\) 2.00000 0.0934539
\(459\) −4.00000 −0.186704
\(460\) 6.00000i 0.279751i
\(461\) 26.0000i 1.21094i 0.795868 + 0.605470i \(0.207015\pi\)
−0.795868 + 0.605470i \(0.792985\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 4.00000 0.185695
\(465\) 10.0000 0.463739
\(466\) 24.0000i 1.11178i
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 3.00000 2.00000i 0.138675 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −4.00000 −0.184115
\(473\) 48.0000i 2.20704i
\(474\) 8.00000i 0.367452i
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 16.0000i 0.731059i 0.930800 + 0.365529i \(0.119112\pi\)
−0.930800 + 0.365529i \(0.880888\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 8.00000 + 12.0000i 0.364769 + 0.547153i
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 14.0000 0.635707
\(486\) 1.00000i 0.0453609i
\(487\) 24.0000i 1.08754i −0.839233 0.543772i \(-0.816996\pi\)
0.839233 0.543772i \(-0.183004\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 2.00000i 0.0904431i
\(490\) 7.00000 0.316228
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 16.0000 0.720604
\(494\) −12.0000 + 8.00000i −0.539906 + 0.359937i
\(495\) −4.00000 −0.179787
\(496\) 10.0000i 0.449013i
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000i 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 0 0
\(502\) 2.00000i 0.0892644i
\(503\) −22.0000 −0.980932 −0.490466 0.871460i \(-0.663173\pi\)
−0.490466 + 0.871460i \(0.663173\pi\)
\(504\) 0 0
\(505\) 8.00000i 0.355995i
\(506\) 24.0000 1.06693
\(507\) −5.00000 + 12.0000i −0.222058 + 0.532939i
\(508\) −8.00000 −0.354943
\(509\) 6.00000i 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 12.0000i 0.529297i
\(515\) 4.00000i 0.176261i
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 3.00000 2.00000i 0.131559 0.0877058i
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 14.0000i 0.610429i
\(527\) 40.0000i 1.74243i
\(528\) 4.00000i 0.174078i
\(529\) 13.0000 0.565217
\(530\) −2.00000 −0.0868744
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) 4.00000 + 6.00000i 0.173259 + 0.259889i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) −22.0000 −0.949370
\(538\) 0 0
\(539\) 28.0000i 1.20605i
\(540\) 1.00000i 0.0430331i
\(541\) 18.0000i 0.773880i −0.922105 0.386940i \(-0.873532\pi\)
0.922105 0.386940i \(-0.126468\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −22.0000 −0.944110
\(544\) 4.00000i 0.171499i
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) 16.0000i 0.681623i
\(552\) 6.00000i 0.255377i
\(553\) 0 0
\(554\) 22.0000i 0.934690i
\(555\) −4.00000 −0.169791
\(556\) −16.0000 −0.678551
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 10.0000 0.423334
\(559\) 36.0000 24.0000i 1.52264 1.01509i
\(560\) 0 0
\(561\) 16.0000i 0.675521i
\(562\) −30.0000 −1.26547
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) 12.0000i 0.504844i
\(566\) 4.00000i 0.168133i
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 46.0000 1.92842 0.964210 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 4.00000i 0.167542i
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 8.00000 + 12.0000i 0.334497 + 0.501745i
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) −1.00000 −0.0416667
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 14.0000i 0.581820i
\(580\) 4.00000i 0.166091i
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) 8.00000i 0.331326i
\(584\) −6.00000 −0.248282
\(585\) −2.00000 3.00000i −0.0826898 0.124035i
\(586\) 14.0000 0.578335
\(587\) 44.0000i 1.81607i 0.418890 + 0.908037i \(0.362419\pi\)
−0.418890 + 0.908037i \(0.637581\pi\)
\(588\) 7.00000 0.288675
\(589\) −40.0000 −1.64817
\(590\) 4.00000i 0.164677i
\(591\) 26.0000i 1.06950i
\(592\) 4.00000i 0.164399i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 2.00000i 0.0819232i
\(597\) 0 0
\(598\) 12.0000 + 18.0000i 0.490716 + 0.736075i
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 10.0000i 0.407231i
\(604\) 22.0000i 0.895167i
\(605\) 5.00000i 0.203279i
\(606\) 8.00000i 0.324978i
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 36.0000i 1.45403i 0.686624 + 0.727013i \(0.259092\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −2.00000 −0.0807134
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 26.0000i 1.04672i −0.852111 0.523360i \(-0.824678\pi\)
0.852111 0.523360i \(-0.175322\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 40.0000i 1.60774i −0.594808 0.803868i \(-0.702772\pi\)
0.594808 0.803868i \(-0.297228\pi\)
\(620\) 10.0000 0.401610
\(621\) −6.00000 −0.240772
\(622\) 12.0000i 0.481156i
\(623\) 0 0
\(624\) 3.00000 2.00000i 0.120096 0.0800641i
\(625\) 1.00000 0.0400000
\(626\) 10.0000i 0.399680i
\(627\) 16.0000 0.638978
\(628\) 14.0000 0.558661
\(629\) 16.0000i 0.637962i
\(630\) 0 0
\(631\) 22.0000i 0.875806i 0.899022 + 0.437903i \(0.144279\pi\)
−0.899022 + 0.437903i \(0.855721\pi\)
\(632\) 8.00000i 0.318223i
\(633\) −24.0000 −0.953914
\(634\) 6.00000 0.238290
\(635\) 8.00000i 0.317470i
\(636\) −2.00000 −0.0793052
\(637\) −21.0000 + 14.0000i −0.832050 + 0.554700i
\(638\) 16.0000 0.633446
\(639\) 12.0000i 0.474713i
\(640\) −1.00000 −0.0395285
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) 50.0000i 1.97181i −0.167313 0.985904i \(-0.553509\pi\)
0.167313 0.985904i \(-0.446491\pi\)
\(644\) 0 0
\(645\) 12.0000i 0.472500i
\(646\) 16.0000 0.629512
\(647\) 38.0000 1.49393 0.746967 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) −2.00000 3.00000i −0.0784465 0.117670i
\(651\) 0 0
\(652\) 2.00000i 0.0783260i
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −6.00000 −0.234619
\(655\) 18.0000i 0.703318i
\(656\) 2.00000i 0.0780869i
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 42.0000 1.63609 0.818044 0.575156i \(-0.195059\pi\)
0.818044 + 0.575156i \(0.195059\pi\)
\(660\) −4.00000 −0.155700
\(661\) 22.0000i 0.855701i −0.903850 0.427850i \(-0.859271\pi\)
0.903850 0.427850i \(-0.140729\pi\)
\(662\) −32.0000 −1.24372
\(663\) 12.0000 8.00000i 0.466041 0.310694i
\(664\) 0 0
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 20.0000i 0.773245i
\(670\) 10.0000i 0.386334i
\(671\) 40.0000i 1.54418i
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 2.00000i 0.0770371i
\(675\) 1.00000 0.0384900
\(676\) −5.00000 + 12.0000i −0.192308 + 0.461538i
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 12.0000i 0.460857i
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) 16.0000i 0.613121i
\(682\) 40.0000i 1.53168i
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 4.00000i 0.152944i
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) −12.0000 −0.457496
\(689\) 6.00000 4.00000i 0.228582 0.152388i
\(690\) −6.00000 −0.228416
\(691\) 8.00000i 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000i 0.606915i
\(696\) 4.00000i 0.151620i
\(697\) 8.00000i 0.303022i
\(698\) 14.0000 0.529908
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) −2.00000 3.00000i −0.0754851 0.113228i
\(703\) 16.0000 0.603451
\(704\) 4.00000i 0.150756i
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 4.00000i 0.150329i
\(709\) 50.0000i 1.87779i 0.344204 + 0.938895i \(0.388149\pi\)
−0.344204 + 0.938895i \(0.611851\pi\)
\(710\) 12.0000i 0.450352i
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) 60.0000i 2.24702i
\(714\) 0 0
\(715\) 12.0000 8.00000i 0.448775 0.299183i
\(716\) −22.0000 −0.822179
\(717\) 0 0
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) 4.00000i 0.148762i
\(724\) −22.0000 −0.817624
\(725\) −4.00000 −0.148556
\(726\) 5.00000i 0.185567i
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000i 0.222070i
\(731\) −48.0000 −1.77534
\(732\) −10.0000 −0.369611
\(733\) 40.0000i 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) 16.0000i 0.590571i
\(735\) 7.00000i 0.258199i
\(736\) 6.00000i 0.221163i
\(737\) 40.0000 1.47342
\(738\) −2.00000 −0.0736210
\(739\) 8.00000i 0.294285i −0.989115 0.147142i \(-0.952992\pi\)
0.989115 0.147142i \(-0.0470076\pi\)
\(740\) −4.00000 −0.147043
\(741\) 8.00000 + 12.0000i 0.293887 + 0.440831i
\(742\) 0 0
\(743\) 32.0000i 1.17397i −0.809599 0.586983i \(-0.800316\pi\)
0.809599 0.586983i \(-0.199684\pi\)
\(744\) 10.0000 0.366618
\(745\) 2.00000 0.0732743
\(746\) 26.0000i 0.951928i
\(747\) 0 0
\(748\) 16.0000i 0.585018i
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) −2.00000 −0.0728841
\(754\) 8.00000 + 12.0000i 0.291343 + 0.437014i
\(755\) 22.0000 0.800662
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −16.0000 −0.581146
\(759\) 24.0000i 0.871145i
\(760\) 4.00000i 0.145095i
\(761\) 18.0000i 0.652499i 0.945284 + 0.326250i \(0.105785\pi\)
−0.945284 + 0.326250i \(0.894215\pi\)
\(762\) 8.00000i 0.289809i
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 4.00000i 0.144620i
\(766\) −16.0000 −0.578103
\(767\) −8.00000 12.0000i −0.288863 0.433295i
\(768\) −1.00000 −0.0360844
\(769\) 24.0000i 0.865462i −0.901523 0.432731i \(-0.857550\pi\)
0.901523 0.432731i \(-0.142450\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 14.0000i 0.503871i
\(773\) 26.0000i 0.935155i 0.883952 + 0.467578i \(0.154873\pi\)
−0.883952 + 0.467578i \(0.845127\pi\)
\(774\) 12.0000i 0.431331i
\(775\) 10.0000i 0.359211i
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 20.0000i 0.717035i
\(779\) 8.00000 0.286630
\(780\) −2.00000 3.00000i −0.0716115 0.107417i
\(781\) −48.0000 −1.71758
\(782\) 24.0000i 0.858238i
\(783\) −4.00000 −0.142948
\(784\) 7.00000 0.250000
\(785\) 14.0000i 0.499681i
\(786\) 18.0000i 0.642039i
\(787\) 38.0000i 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 26.0000i 0.926212i
\(789\) 14.0000 0.498413
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 30.0000 20.0000i 1.06533 0.710221i
\(794\) 28.0000 0.993683
\(795\) 2.00000i 0.0709327i
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 10.0000i 0.353333i
\(802\) −26.0000 −0.918092
\(803\) −24.0000 −0.846942
\(804\) 10.0000i 0.352673i
\(805\) 0 0
\(806\) −30.0000 + 20.0000i −1.05670 + 0.704470i
\(807\) 0 0
\(808\) 8.00000i 0.281439i
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 1.00000 0.0351364
\(811\) 52.0000i 1.82597i 0.407997 + 0.912983i \(0.366228\pi\)
−0.407997 + 0.912983i \(0.633772\pi\)
\(812\) 0 0
\(813\) 2.00000i 0.0701431i
\(814\) 16.0000i 0.560800i
\(815\) 2.00000 0.0700569
\(816\) −4.00000 −0.140028
\(817\) 48.0000i 1.67931i
\(818\) −8.00000 −0.279713
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 34.0000i 1.18661i −0.804978 0.593304i \(-0.797823\pi\)
0.804978 0.593304i \(-0.202177\pi\)
\(822\) 2.00000 0.0697580
\(823\) −12.0000 −0.418294 −0.209147 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 4.00000i 0.139347i
\(825\) 4.00000i 0.139262i
\(826\) 0 0
\(827\) 52.0000i 1.80822i −0.427303 0.904109i \(-0.640536\pi\)
0.427303 0.904109i \(-0.359464\pi\)
\(828\) −6.00000 −0.208514
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 3.00000 2.00000i 0.104006 0.0693375i
\(833\) 28.0000 0.970143
\(834\) 16.0000i 0.554035i
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 10.0000i 0.345651i
\(838\) 14.0000i 0.483622i
\(839\) 48.0000i 1.65714i −0.559883 0.828572i \(-0.689154\pi\)
0.559883 0.828572i \(-0.310846\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −14.0000 −0.482472
\(843\) 30.0000i 1.03325i
\(844\) −24.0000 −0.826114
\(845\) 12.0000 + 5.00000i 0.412813 + 0.172005i
\(846\) 0 0
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) 4.00000i 0.137199i
\(851\) 24.0000i 0.822709i
\(852\) 12.0000i 0.411113i
\(853\) 44.0000i 1.50653i 0.657716 + 0.753266i \(0.271523\pi\)
−0.657716 + 0.753266i \(0.728477\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −52.0000 −1.77629 −0.888143 0.459567i \(-0.848005\pi\)
−0.888143 + 0.459567i \(0.848005\pi\)
\(858\) 12.0000 8.00000i 0.409673 0.273115i
\(859\) −16.0000 −0.545913 −0.272956 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) 12.0000i 0.409197i
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 8.00000i 0.272323i −0.990687 0.136162i \(-0.956523\pi\)
0.990687 0.136162i \(-0.0434766\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 22.0000i 0.748022i
\(866\) 18.0000i 0.611665i
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 32.0000i 1.08553i
\(870\) −4.00000 −0.135613
\(871\) 20.0000 + 30.0000i 0.677674 + 1.01651i
\(872\) −6.00000 −0.203186
\(873\) 14.0000i 0.473828i
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 6.00000i 0.202721i
\(877\) 48.0000i 1.62084i 0.585846 + 0.810422i \(0.300762\pi\)
−0.585846 + 0.810422i \(0.699238\pi\)
\(878\) 32.0000i 1.07995i
\(879\) 14.0000i 0.472208i
\(880\) −4.00000 −0.134840
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 12.0000 8.00000i 0.403604 0.269069i
\(885\) 4.00000 0.134459
\(886\) 36.0000i 1.20944i
\(887\) −30.0000 −1.00730 −0.503651 0.863907i \(-0.668010\pi\)
−0.503651 + 0.863907i \(0.668010\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) 10.0000i 0.335201i
\(891\) 4.00000i 0.134005i
\(892\) 20.0000i 0.669650i
\(893\) 0 0
\(894\) 2.00000 0.0668900
\(895\) 22.0000i 0.735379i
\(896\) 0 0
\(897\) 18.0000 12.0000i 0.601003 0.400668i
\(898\) 30.0000 1.00111
\(899\) 40.0000i 1.33407i
\(900\) 1.00000 0.0333333
\(901\) −8.00000 −0.266519
\(902\) 8.00000i 0.266371i
\(903\) 0 0
\(904\) 12.0000i 0.399114i
\(905\) 22.0000i 0.731305i
\(906\) 22.0000 0.730901
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 16.0000i 0.530979i
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 0 0
\(914\) −14.0000 −0.463079
\(915\) 10.0000i 0.330590i
\(916\) 2.00000i 0.0660819i
\(917\) 0 0
\(918\) 4.00000i 0.132020i
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) −6.00000 −0.197814
\(921\) 2.00000i 0.0659022i
\(922\) 26.0000 0.856264
\(923\) −24.0000 36.0000i −0.789970 1.18495i
\(924\) 0 0
\(925\) 4.00000i 0.131519i
\(926\) 20.0000 0.657241
\(927\) 4.00000 0.131377
\(928\) 4.00000i 0.131306i
\(929\) 10.0000i 0.328089i −0.986453 0.164045i \(-0.947546\pi\)
0.986453 0.164045i \(-0.0524541\pi\)
\(930\) 10.0000i 0.327913i
\(931\) 28.0000i 0.917663i
\(932\) −24.0000 −0.786146
\(933\) −12.0000 −0.392862
\(934\) 4.00000i 0.130884i
\(935\) −16.0000 −0.523256
\(936\) −2.00000 3.00000i −0.0653720 0.0980581i
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 18.0000i 0.586783i 0.955992 + 0.293392i \(0.0947840\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(942\) 14.0000i 0.456145i
\(943\) 12.0000i 0.390774i
\(944\) 4.00000i 0.130189i
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 48.0000i 1.55979i −0.625910 0.779895i \(-0.715272\pi\)
0.625910 0.779895i \(-0.284728\pi\)
\(948\) −8.00000 −0.259828
\(949\) −12.0000 18.0000i −0.389536 0.584305i
\(950\) −4.00000 −0.129777
\(951\) 6.00000i 0.194563i
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 2.00000i 0.0647524i
\(955\) 12.0000i 0.388311i
\(956\) 0 0
\(957\) 16.0000i 0.517207i
\(958\) 16.0000 0.516937
\(959\) 0 0
\(960\) 1.00000i 0.0322749i
\(961\) −69.0000 −2.22581
\(962\) 12.0000 8.00000i 0.386896 0.257930i
\(963\) 0 0
\(964\) 4.00000i 0.128831i
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 16.0000i 0.513994i
\(970\) 14.0000i 0.449513i
\(971\) −34.0000 −1.09111 −0.545556 0.838074i \(-0.683681\pi\)
−0.545556 + 0.838074i \(0.683681\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −24.0000 −0.769010
\(975\) −3.00000 + 2.00000i −0.0960769 + 0.0640513i
\(976\) −10.0000 −0.320092
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) 2.00000 0.0639529
\(979\) 40.0000 1.27841
\(980\) 7.00000i 0.223607i
\(981\) 6.00000i 0.191565i
\(982\) 22.0000i 0.702048i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 26.0000 0.828429
\(986\) 16.0000i 0.509544i
\(987\) 0 0
\(988\) 8.00000 + 12.0000i 0.254514 + 0.381771i
\(989\) −72.0000 −2.28947
\(990\) 4.00000i 0.127128i
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 10.0000 0.317500
\(993\) 32.0000i 1.01549i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −4.00000 −0.126618
\(999\) 4.00000i 0.126554i
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 390.2.b.a.181.1 2
3.2 odd 2 1170.2.b.b.181.2 2
4.3 odd 2 3120.2.g.f.961.2 2
5.2 odd 4 1950.2.f.i.649.2 2
5.3 odd 4 1950.2.f.b.649.1 2
5.4 even 2 1950.2.b.g.1351.2 2
13.5 odd 4 5070.2.a.g.1.1 1
13.8 odd 4 5070.2.a.m.1.1 1
13.12 even 2 inner 390.2.b.a.181.2 yes 2
39.38 odd 2 1170.2.b.b.181.1 2
52.51 odd 2 3120.2.g.f.961.1 2
65.12 odd 4 1950.2.f.b.649.2 2
65.38 odd 4 1950.2.f.i.649.1 2
65.64 even 2 1950.2.b.g.1351.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.a.181.1 2 1.1 even 1 trivial
390.2.b.a.181.2 yes 2 13.12 even 2 inner
1170.2.b.b.181.1 2 39.38 odd 2
1170.2.b.b.181.2 2 3.2 odd 2
1950.2.b.g.1351.1 2 65.64 even 2
1950.2.b.g.1351.2 2 5.4 even 2
1950.2.f.b.649.1 2 5.3 odd 4
1950.2.f.b.649.2 2 65.12 odd 4
1950.2.f.i.649.1 2 65.38 odd 4
1950.2.f.i.649.2 2 5.2 odd 4
3120.2.g.f.961.1 2 52.51 odd 2
3120.2.g.f.961.2 2 4.3 odd 2
5070.2.a.g.1.1 1 13.5 odd 4
5070.2.a.m.1.1 1 13.8 odd 4