Properties

Label 1682.2.b.e.1681.2
Level $1682$
Weight $2$
Character 1682.1681
Analytic conductor $13.431$
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] = [1682,2,Mod(1681,1682)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1682.1681");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.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: \(13.4308376200\)
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 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1681.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1682.1681
Dual form 1682.2.b.e.1681.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{6} -2.00000 q^{7} -1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{6} -2.00000 q^{7} -1.00000i q^{8} -6.00000 q^{9} +3.00000i q^{10} +1.00000i q^{11} -3.00000i q^{12} -3.00000 q^{13} -2.00000i q^{14} +9.00000i q^{15} +1.00000 q^{16} +4.00000i q^{17} -6.00000i q^{18} +8.00000i q^{19} -3.00000 q^{20} -6.00000i q^{21} -1.00000 q^{22} +3.00000 q^{24} +4.00000 q^{25} -3.00000i q^{26} -9.00000i q^{27} +2.00000 q^{28} -9.00000 q^{30} -3.00000i q^{31} +1.00000i q^{32} -3.00000 q^{33} -4.00000 q^{34} -6.00000 q^{35} +6.00000 q^{36} -8.00000i q^{37} -8.00000 q^{38} -9.00000i q^{39} -3.00000i q^{40} -2.00000i q^{41} +6.00000 q^{42} -7.00000i q^{43} -1.00000i q^{44} -18.0000 q^{45} +11.0000i q^{47} +3.00000i q^{48} -3.00000 q^{49} +4.00000i q^{50} -12.0000 q^{51} +3.00000 q^{52} +1.00000 q^{53} +9.00000 q^{54} +3.00000i q^{55} +2.00000i q^{56} -24.0000 q^{57} -4.00000 q^{59} -9.00000i q^{60} -4.00000i q^{61} +3.00000 q^{62} +12.0000 q^{63} -1.00000 q^{64} -9.00000 q^{65} -3.00000i q^{66} +4.00000 q^{67} -4.00000i q^{68} -6.00000i q^{70} +2.00000 q^{71} +6.00000i q^{72} -12.0000i q^{73} +8.00000 q^{74} +12.0000i q^{75} -8.00000i q^{76} -2.00000i q^{77} +9.00000 q^{78} +7.00000i q^{79} +3.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +6.00000i q^{84} +12.0000i q^{85} +7.00000 q^{86} +1.00000 q^{88} +6.00000i q^{89} -18.0000i q^{90} +6.00000 q^{91} +9.00000 q^{93} -11.0000 q^{94} +24.0000i q^{95} -3.00000 q^{96} -6.00000i q^{97} -3.00000i q^{98} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 6 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 6 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{9} - 6 q^{13} + 2 q^{16} - 6 q^{20} - 2 q^{22} + 6 q^{24} + 8 q^{25} + 4 q^{28} - 18 q^{30} - 6 q^{33} - 8 q^{34} - 12 q^{35} + 12 q^{36} - 16 q^{38} + 12 q^{42} - 36 q^{45} - 6 q^{49} - 24 q^{51} + 6 q^{52} + 2 q^{53} + 18 q^{54} - 48 q^{57} - 8 q^{59} + 6 q^{62} + 24 q^{63} - 2 q^{64} - 18 q^{65} + 8 q^{67} + 4 q^{71} + 16 q^{74} + 18 q^{78} + 6 q^{80} + 18 q^{81} + 4 q^{82} + 14 q^{86} + 2 q^{88} + 12 q^{91} + 18 q^{93} - 22 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −3.00000 −1.22474
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −6.00000 −2.00000
\(10\) 3.00000i 0.948683i
\(11\) 1.00000i 0.301511i 0.988571 + 0.150756i \(0.0481707\pi\)
−0.988571 + 0.150756i \(0.951829\pi\)
\(12\) − 3.00000i − 0.866025i
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) 9.00000i 2.32379i
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) −3.00000 −0.670820
\(21\) − 6.00000i − 1.30931i
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 4.00000 0.800000
\(26\) − 3.00000i − 0.588348i
\(27\) − 9.00000i − 1.73205i
\(28\) 2.00000 0.377964
\(29\) 0 0
\(30\) −9.00000 −1.64317
\(31\) − 3.00000i − 0.538816i −0.963026 0.269408i \(-0.913172\pi\)
0.963026 0.269408i \(-0.0868280\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −3.00000 −0.522233
\(34\) −4.00000 −0.685994
\(35\) −6.00000 −1.01419
\(36\) 6.00000 1.00000
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) −8.00000 −1.29777
\(39\) − 9.00000i − 1.44115i
\(40\) − 3.00000i − 0.474342i
\(41\) − 2.00000i − 0.312348i −0.987730 0.156174i \(-0.950084\pi\)
0.987730 0.156174i \(-0.0499160\pi\)
\(42\) 6.00000 0.925820
\(43\) − 7.00000i − 1.06749i −0.845645 0.533745i \(-0.820784\pi\)
0.845645 0.533745i \(-0.179216\pi\)
\(44\) − 1.00000i − 0.150756i
\(45\) −18.0000 −2.68328
\(46\) 0 0
\(47\) 11.0000i 1.60451i 0.596978 + 0.802257i \(0.296368\pi\)
−0.596978 + 0.802257i \(0.703632\pi\)
\(48\) 3.00000i 0.433013i
\(49\) −3.00000 −0.428571
\(50\) 4.00000i 0.565685i
\(51\) −12.0000 −1.68034
\(52\) 3.00000 0.416025
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 9.00000 1.22474
\(55\) 3.00000i 0.404520i
\(56\) 2.00000i 0.267261i
\(57\) −24.0000 −3.17888
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) − 9.00000i − 1.16190i
\(61\) − 4.00000i − 0.512148i −0.966657 0.256074i \(-0.917571\pi\)
0.966657 0.256074i \(-0.0824290\pi\)
\(62\) 3.00000 0.381000
\(63\) 12.0000 1.51186
\(64\) −1.00000 −0.125000
\(65\) −9.00000 −1.11631
\(66\) − 3.00000i − 0.369274i
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) − 6.00000i − 0.717137i
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 6.00000i 0.707107i
\(73\) − 12.0000i − 1.40449i −0.711934 0.702247i \(-0.752180\pi\)
0.711934 0.702247i \(-0.247820\pi\)
\(74\) 8.00000 0.929981
\(75\) 12.0000i 1.38564i
\(76\) − 8.00000i − 0.917663i
\(77\) − 2.00000i − 0.227921i
\(78\) 9.00000 1.01905
\(79\) 7.00000i 0.787562i 0.919204 + 0.393781i \(0.128833\pi\)
−0.919204 + 0.393781i \(0.871167\pi\)
\(80\) 3.00000 0.335410
\(81\) 9.00000 1.00000
\(82\) 2.00000 0.220863
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 6.00000i 0.654654i
\(85\) 12.0000i 1.30158i
\(86\) 7.00000 0.754829
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) − 18.0000i − 1.89737i
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 9.00000 0.933257
\(94\) −11.0000 −1.13456
\(95\) 24.0000i 2.46235i
\(96\) −3.00000 −0.306186
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) − 6.00000i − 0.603023i
\(100\) −4.00000 −0.400000
\(101\) − 8.00000i − 0.796030i −0.917379 0.398015i \(-0.869699\pi\)
0.917379 0.398015i \(-0.130301\pi\)
\(102\) − 12.0000i − 1.18818i
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 3.00000i 0.294174i
\(105\) − 18.0000i − 1.75662i
\(106\) 1.00000i 0.0971286i
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 9.00000i 0.866025i
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) −3.00000 −0.286039
\(111\) 24.0000 2.27798
\(112\) −2.00000 −0.188982
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) − 24.0000i − 2.24781i
\(115\) 0 0
\(116\) 0 0
\(117\) 18.0000 1.66410
\(118\) − 4.00000i − 0.368230i
\(119\) − 8.00000i − 0.733359i
\(120\) 9.00000 0.821584
\(121\) 10.0000 0.909091
\(122\) 4.00000 0.362143
\(123\) 6.00000 0.541002
\(124\) 3.00000i 0.269408i
\(125\) −3.00000 −0.268328
\(126\) 12.0000i 1.06904i
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 21.0000 1.84895
\(130\) − 9.00000i − 0.789352i
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 3.00000 0.261116
\(133\) − 16.0000i − 1.38738i
\(134\) 4.00000i 0.345547i
\(135\) − 27.0000i − 2.32379i
\(136\) 4.00000 0.342997
\(137\) 20.0000i 1.70872i 0.519685 + 0.854358i \(0.326049\pi\)
−0.519685 + 0.854358i \(0.673951\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 6.00000 0.507093
\(141\) −33.0000 −2.77910
\(142\) 2.00000i 0.167836i
\(143\) − 3.00000i − 0.250873i
\(144\) −6.00000 −0.500000
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) − 9.00000i − 0.742307i
\(148\) 8.00000i 0.657596i
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) −12.0000 −0.979796
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 8.00000 0.648886
\(153\) − 24.0000i − 1.94029i
\(154\) 2.00000 0.161165
\(155\) − 9.00000i − 0.722897i
\(156\) 9.00000i 0.720577i
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) −7.00000 −0.556890
\(159\) 3.00000i 0.237915i
\(160\) 3.00000i 0.237171i
\(161\) 0 0
\(162\) 9.00000i 0.707107i
\(163\) 19.0000i 1.48819i 0.668071 + 0.744097i \(0.267120\pi\)
−0.668071 + 0.744097i \(0.732880\pi\)
\(164\) 2.00000i 0.156174i
\(165\) −9.00000 −0.700649
\(166\) 0 0
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) −6.00000 −0.462910
\(169\) −4.00000 −0.307692
\(170\) −12.0000 −0.920358
\(171\) − 48.0000i − 3.67065i
\(172\) 7.00000i 0.533745i
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 1.00000i 0.0753778i
\(177\) − 12.0000i − 0.901975i
\(178\) −6.00000 −0.449719
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 18.0000 1.34164
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) − 24.0000i − 1.76452i
\(186\) 9.00000i 0.659912i
\(187\) −4.00000 −0.292509
\(188\) − 11.0000i − 0.802257i
\(189\) 18.0000i 1.30931i
\(190\) −24.0000 −1.74114
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) − 3.00000i − 0.216506i
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 6.00000 0.430775
\(195\) − 27.0000i − 1.93351i
\(196\) 3.00000 0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 6.00000 0.426401
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) − 4.00000i − 0.282843i
\(201\) 12.0000i 0.846415i
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) − 6.00000i − 0.419058i
\(206\) − 6.00000i − 0.418040i
\(207\) 0 0
\(208\) −3.00000 −0.208013
\(209\) −8.00000 −0.553372
\(210\) 18.0000 1.24212
\(211\) − 25.0000i − 1.72107i −0.509390 0.860535i \(-0.670129\pi\)
0.509390 0.860535i \(-0.329871\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 6.00000i 0.411113i
\(214\) − 2.00000i − 0.136717i
\(215\) − 21.0000i − 1.43219i
\(216\) −9.00000 −0.612372
\(217\) 6.00000i 0.407307i
\(218\) − 1.00000i − 0.0677285i
\(219\) 36.0000 2.43265
\(220\) − 3.00000i − 0.202260i
\(221\) − 12.0000i − 0.807207i
\(222\) 24.0000i 1.61077i
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) − 2.00000i − 0.133631i
\(225\) −24.0000 −1.60000
\(226\) −18.0000 −1.19734
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 24.0000 1.58944
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) −25.0000 −1.63780 −0.818902 0.573933i \(-0.805417\pi\)
−0.818902 + 0.573933i \(0.805417\pi\)
\(234\) 18.0000i 1.17670i
\(235\) 33.0000i 2.15268i
\(236\) 4.00000 0.260378
\(237\) −21.0000 −1.36410
\(238\) 8.00000 0.518563
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 9.00000i 0.580948i
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 10.0000i 0.642824i
\(243\) 0 0
\(244\) 4.00000i 0.256074i
\(245\) −9.00000 −0.574989
\(246\) 6.00000i 0.382546i
\(247\) − 24.0000i − 1.52708i
\(248\) −3.00000 −0.190500
\(249\) 0 0
\(250\) − 3.00000i − 0.189737i
\(251\) 7.00000i 0.441836i 0.975292 + 0.220918i \(0.0709053\pi\)
−0.975292 + 0.220918i \(0.929095\pi\)
\(252\) −12.0000 −0.755929
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −36.0000 −2.25441
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) 21.0000i 1.30740i
\(259\) 16.0000i 0.994192i
\(260\) 9.00000 0.558156
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 17.0000i 1.04826i 0.851637 + 0.524132i \(0.175610\pi\)
−0.851637 + 0.524132i \(0.824390\pi\)
\(264\) 3.00000i 0.184637i
\(265\) 3.00000 0.184289
\(266\) 16.0000 0.981023
\(267\) −18.0000 −1.10158
\(268\) −4.00000 −0.244339
\(269\) 20.0000i 1.21942i 0.792624 + 0.609711i \(0.208714\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 27.0000 1.64317
\(271\) 13.0000i 0.789694i 0.918747 + 0.394847i \(0.129202\pi\)
−0.918747 + 0.394847i \(0.870798\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 18.0000i 1.08941i
\(274\) −20.0000 −1.20824
\(275\) 4.00000i 0.241209i
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) 18.0000i 1.07763i
\(280\) 6.00000i 0.358569i
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) − 33.0000i − 1.96512i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −2.00000 −0.118678
\(285\) −72.0000 −4.26491
\(286\) 3.00000 0.177394
\(287\) 4.00000i 0.236113i
\(288\) − 6.00000i − 0.353553i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 12.0000i 0.702247i
\(293\) 34.0000i 1.98630i 0.116841 + 0.993151i \(0.462723\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 9.00000 0.524891
\(295\) −12.0000 −0.698667
\(296\) −8.00000 −0.464991
\(297\) 9.00000 0.522233
\(298\) − 3.00000i − 0.173785i
\(299\) 0 0
\(300\) − 12.0000i − 0.692820i
\(301\) 14.0000i 0.806947i
\(302\) − 10.0000i − 0.575435i
\(303\) 24.0000 1.37876
\(304\) 8.00000i 0.458831i
\(305\) − 12.0000i − 0.687118i
\(306\) 24.0000 1.37199
\(307\) 29.0000i 1.65512i 0.561379 + 0.827559i \(0.310271\pi\)
−0.561379 + 0.827559i \(0.689729\pi\)
\(308\) 2.00000i 0.113961i
\(309\) − 18.0000i − 1.02398i
\(310\) 9.00000 0.511166
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) −9.00000 −0.509525
\(313\) 25.0000 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(314\) −22.0000 −1.24153
\(315\) 36.0000 2.02837
\(316\) − 7.00000i − 0.393781i
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) −3.00000 −0.168232
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) − 6.00000i − 0.334887i
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) −9.00000 −0.500000
\(325\) −12.0000 −0.665640
\(326\) −19.0000 −1.05231
\(327\) − 3.00000i − 0.165900i
\(328\) −2.00000 −0.110432
\(329\) − 22.0000i − 1.21290i
\(330\) − 9.00000i − 0.495434i
\(331\) 3.00000i 0.164895i 0.996595 + 0.0824475i \(0.0262737\pi\)
−0.996595 + 0.0824475i \(0.973726\pi\)
\(332\) 0 0
\(333\) 48.0000i 2.63038i
\(334\) 22.0000i 1.20379i
\(335\) 12.0000 0.655630
\(336\) − 6.00000i − 0.327327i
\(337\) 4.00000i 0.217894i 0.994048 + 0.108947i \(0.0347479\pi\)
−0.994048 + 0.108947i \(0.965252\pi\)
\(338\) − 4.00000i − 0.217571i
\(339\) −54.0000 −2.93288
\(340\) − 12.0000i − 0.650791i
\(341\) 3.00000 0.162459
\(342\) 48.0000 2.59554
\(343\) 20.0000 1.07990
\(344\) −7.00000 −0.377415
\(345\) 0 0
\(346\) 14.0000i 0.752645i
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) − 8.00000i − 0.427618i
\(351\) 27.0000i 1.44115i
\(352\) −1.00000 −0.0533002
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 12.0000 0.637793
\(355\) 6.00000 0.318447
\(356\) − 6.00000i − 0.317999i
\(357\) 24.0000 1.27021
\(358\) 14.0000i 0.739923i
\(359\) − 9.00000i − 0.475002i −0.971387 0.237501i \(-0.923672\pi\)
0.971387 0.237501i \(-0.0763283\pi\)
\(360\) 18.0000i 0.948683i
\(361\) −45.0000 −2.36842
\(362\) − 13.0000i − 0.683265i
\(363\) 30.0000i 1.57459i
\(364\) −6.00000 −0.314485
\(365\) − 36.0000i − 1.88433i
\(366\) 12.0000i 0.627250i
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) 12.0000i 0.624695i
\(370\) 24.0000 1.24770
\(371\) −2.00000 −0.103835
\(372\) −9.00000 −0.466628
\(373\) −1.00000 −0.0517780 −0.0258890 0.999665i \(-0.508242\pi\)
−0.0258890 + 0.999665i \(0.508242\pi\)
\(374\) − 4.00000i − 0.206835i
\(375\) − 9.00000i − 0.464758i
\(376\) 11.0000 0.567282
\(377\) 0 0
\(378\) −18.0000 −0.925820
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) − 24.0000i − 1.23117i
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) 34.0000 1.73732 0.868659 0.495410i \(-0.164982\pi\)
0.868659 + 0.495410i \(0.164982\pi\)
\(384\) 3.00000 0.153093
\(385\) − 6.00000i − 0.305788i
\(386\) 10.0000 0.508987
\(387\) 42.0000i 2.13498i
\(388\) 6.00000i 0.304604i
\(389\) 16.0000i 0.811232i 0.914044 + 0.405616i \(0.132943\pi\)
−0.914044 + 0.405616i \(0.867057\pi\)
\(390\) 27.0000 1.36720
\(391\) 0 0
\(392\) 3.00000i 0.151523i
\(393\) −36.0000 −1.81596
\(394\) 2.00000i 0.100759i
\(395\) 21.0000i 1.05662i
\(396\) 6.00000i 0.301511i
\(397\) 19.0000 0.953583 0.476791 0.879017i \(-0.341800\pi\)
0.476791 + 0.879017i \(0.341800\pi\)
\(398\) − 2.00000i − 0.100251i
\(399\) 48.0000 2.40301
\(400\) 4.00000 0.200000
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) −12.0000 −0.598506
\(403\) 9.00000i 0.448322i
\(404\) 8.00000i 0.398015i
\(405\) 27.0000 1.34164
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 12.0000i 0.594089i
\(409\) − 22.0000i − 1.08783i −0.839140 0.543915i \(-0.816941\pi\)
0.839140 0.543915i \(-0.183059\pi\)
\(410\) 6.00000 0.296319
\(411\) −60.0000 −2.95958
\(412\) 6.00000 0.295599
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) − 3.00000i − 0.147087i
\(417\) 0 0
\(418\) − 8.00000i − 0.391293i
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 18.0000i 0.878310i
\(421\) − 28.0000i − 1.36464i −0.731055 0.682318i \(-0.760972\pi\)
0.731055 0.682318i \(-0.239028\pi\)
\(422\) 25.0000 1.21698
\(423\) − 66.0000i − 3.20903i
\(424\) − 1.00000i − 0.0485643i
\(425\) 16.0000i 0.776114i
\(426\) −6.00000 −0.290701
\(427\) 8.00000i 0.387147i
\(428\) 2.00000 0.0966736
\(429\) 9.00000 0.434524
\(430\) 21.0000 1.01271
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) − 9.00000i − 0.433013i
\(433\) − 16.0000i − 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 0 0
\(438\) 36.0000i 1.72015i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 3.00000 0.143019
\(441\) 18.0000 0.857143
\(442\) 12.0000 0.570782
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) −24.0000 −1.13899
\(445\) 18.0000i 0.853282i
\(446\) 26.0000i 1.23114i
\(447\) − 9.00000i − 0.425685i
\(448\) 2.00000 0.0944911
\(449\) − 22.0000i − 1.03824i −0.854700 0.519122i \(-0.826259\pi\)
0.854700 0.519122i \(-0.173741\pi\)
\(450\) − 24.0000i − 1.13137i
\(451\) 2.00000 0.0941763
\(452\) − 18.0000i − 0.846649i
\(453\) − 30.0000i − 1.40952i
\(454\) − 18.0000i − 0.844782i
\(455\) 18.0000 0.843853
\(456\) 24.0000i 1.12390i
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −14.0000 −0.654177
\(459\) 36.0000 1.68034
\(460\) 0 0
\(461\) − 30.0000i − 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 6.00000i 0.279145i
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 27.0000 1.25210
\(466\) − 25.0000i − 1.15810i
\(467\) − 23.0000i − 1.06431i −0.846646 0.532157i \(-0.821382\pi\)
0.846646 0.532157i \(-0.178618\pi\)
\(468\) −18.0000 −0.832050
\(469\) −8.00000 −0.369406
\(470\) −33.0000 −1.52218
\(471\) −66.0000 −3.04112
\(472\) 4.00000i 0.184115i
\(473\) 7.00000 0.321860
\(474\) − 21.0000i − 0.964562i
\(475\) 32.0000i 1.46826i
\(476\) 8.00000i 0.366679i
\(477\) −6.00000 −0.274721
\(478\) 20.0000i 0.914779i
\(479\) − 11.0000i − 0.502603i −0.967909 0.251301i \(-0.919141\pi\)
0.967909 0.251301i \(-0.0808585\pi\)
\(480\) −9.00000 −0.410792
\(481\) 24.0000i 1.09431i
\(482\) − 17.0000i − 0.774329i
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) − 18.0000i − 0.817338i
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −4.00000 −0.181071
\(489\) −57.0000 −2.57763
\(490\) − 9.00000i − 0.406579i
\(491\) 5.00000i 0.225647i 0.993615 + 0.112823i \(0.0359894\pi\)
−0.993615 + 0.112823i \(0.964011\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 24.0000 1.07981
\(495\) − 18.0000i − 0.809040i
\(496\) − 3.00000i − 0.134704i
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 3.00000 0.134164
\(501\) 66.0000i 2.94866i
\(502\) −7.00000 −0.312425
\(503\) − 19.0000i − 0.847168i −0.905857 0.423584i \(-0.860772\pi\)
0.905857 0.423584i \(-0.139228\pi\)
\(504\) − 12.0000i − 0.534522i
\(505\) − 24.0000i − 1.06799i
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) − 8.00000i − 0.354943i
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) − 36.0000i − 1.59411i
\(511\) 24.0000i 1.06170i
\(512\) 1.00000i 0.0441942i
\(513\) 72.0000 3.17888
\(514\) 21.0000i 0.926270i
\(515\) −18.0000 −0.793175
\(516\) −21.0000 −0.924473
\(517\) −11.0000 −0.483779
\(518\) −16.0000 −0.703000
\(519\) 42.0000i 1.84360i
\(520\) 9.00000i 0.394676i
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) − 12.0000i − 0.524222i
\(525\) − 24.0000i − 1.04745i
\(526\) −17.0000 −0.741235
\(527\) 12.0000 0.522728
\(528\) −3.00000 −0.130558
\(529\) −23.0000 −1.00000
\(530\) 3.00000i 0.130312i
\(531\) 24.0000 1.04151
\(532\) 16.0000i 0.693688i
\(533\) 6.00000i 0.259889i
\(534\) − 18.0000i − 0.778936i
\(535\) −6.00000 −0.259403
\(536\) − 4.00000i − 0.172774i
\(537\) 42.0000i 1.81243i
\(538\) −20.0000 −0.862261
\(539\) − 3.00000i − 0.129219i
\(540\) 27.0000i 1.16190i
\(541\) 8.00000i 0.343947i 0.985102 + 0.171973i \(0.0550143\pi\)
−0.985102 + 0.171973i \(0.944986\pi\)
\(542\) −13.0000 −0.558398
\(543\) − 39.0000i − 1.67365i
\(544\) −4.00000 −0.171499
\(545\) −3.00000 −0.128506
\(546\) −18.0000 −0.770329
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) − 20.0000i − 0.854358i
\(549\) 24.0000i 1.02430i
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) − 14.0000i − 0.595341i
\(554\) 14.0000i 0.594803i
\(555\) 72.0000 3.05623
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −18.0000 −0.762001
\(559\) 21.0000i 0.888205i
\(560\) −6.00000 −0.253546
\(561\) − 12.0000i − 0.506640i
\(562\) − 13.0000i − 0.548372i
\(563\) − 9.00000i − 0.379305i −0.981851 0.189652i \(-0.939264\pi\)
0.981851 0.189652i \(-0.0607361\pi\)
\(564\) 33.0000 1.38955
\(565\) 54.0000i 2.27180i
\(566\) 0 0
\(567\) −18.0000 −0.755929
\(568\) − 2.00000i − 0.0839181i
\(569\) − 30.0000i − 1.25767i −0.777541 0.628833i \(-0.783533\pi\)
0.777541 0.628833i \(-0.216467\pi\)
\(570\) − 72.0000i − 3.01575i
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 3.00000i 0.125436i
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) − 44.0000i − 1.83174i −0.401470 0.915872i \(-0.631501\pi\)
0.401470 0.915872i \(-0.368499\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 30.0000 1.24676
\(580\) 0 0
\(581\) 0 0
\(582\) 18.0000i 0.746124i
\(583\) 1.00000i 0.0414158i
\(584\) −12.0000 −0.496564
\(585\) 54.0000 2.23263
\(586\) −34.0000 −1.40453
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 9.00000i 0.371154i
\(589\) 24.0000 0.988903
\(590\) − 12.0000i − 0.494032i
\(591\) 6.00000i 0.246807i
\(592\) − 8.00000i − 0.328798i
\(593\) 17.0000 0.698106 0.349053 0.937103i \(-0.386503\pi\)
0.349053 + 0.937103i \(0.386503\pi\)
\(594\) 9.00000i 0.369274i
\(595\) − 24.0000i − 0.983904i
\(596\) 3.00000 0.122885
\(597\) − 6.00000i − 0.245564i
\(598\) 0 0
\(599\) − 13.0000i − 0.531166i −0.964088 0.265583i \(-0.914436\pi\)
0.964088 0.265583i \(-0.0855644\pi\)
\(600\) 12.0000 0.489898
\(601\) 10.0000i 0.407909i 0.978980 + 0.203954i \(0.0653794\pi\)
−0.978980 + 0.203954i \(0.934621\pi\)
\(602\) −14.0000 −0.570597
\(603\) −24.0000 −0.977356
\(604\) 10.0000 0.406894
\(605\) 30.0000 1.21967
\(606\) 24.0000i 0.974933i
\(607\) 29.0000i 1.17707i 0.808470 + 0.588537i \(0.200296\pi\)
−0.808470 + 0.588537i \(0.799704\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) − 33.0000i − 1.33504i
\(612\) 24.0000i 0.970143i
\(613\) 27.0000 1.09052 0.545260 0.838267i \(-0.316431\pi\)
0.545260 + 0.838267i \(0.316431\pi\)
\(614\) −29.0000 −1.17034
\(615\) 18.0000 0.725830
\(616\) −2.00000 −0.0805823
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 18.0000 0.724066
\(619\) 31.0000i 1.24600i 0.782224 + 0.622998i \(0.214085\pi\)
−0.782224 + 0.622998i \(0.785915\pi\)
\(620\) 9.00000i 0.361449i
\(621\) 0 0
\(622\) 0 0
\(623\) − 12.0000i − 0.480770i
\(624\) − 9.00000i − 0.360288i
\(625\) −29.0000 −1.16000
\(626\) 25.0000i 0.999201i
\(627\) − 24.0000i − 0.958468i
\(628\) − 22.0000i − 0.877896i
\(629\) 32.0000 1.27592
\(630\) 36.0000i 1.43427i
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 7.00000 0.278445
\(633\) 75.0000 2.98098
\(634\) 0 0
\(635\) 24.0000i 0.952411i
\(636\) − 3.00000i − 0.118958i
\(637\) 9.00000 0.356593
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) − 3.00000i − 0.118585i
\(641\) − 36.0000i − 1.42191i −0.703235 0.710957i \(-0.748262\pi\)
0.703235 0.710957i \(-0.251738\pi\)
\(642\) 6.00000 0.236801
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 63.0000 2.48062
\(646\) − 32.0000i − 1.25902i
\(647\) 20.0000 0.786281 0.393141 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) − 9.00000i − 0.353553i
\(649\) − 4.00000i − 0.157014i
\(650\) − 12.0000i − 0.470679i
\(651\) −18.0000 −0.705476
\(652\) − 19.0000i − 0.744097i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 3.00000 0.117309
\(655\) 36.0000i 1.40664i
\(656\) − 2.00000i − 0.0780869i
\(657\) 72.0000i 2.80899i
\(658\) 22.0000 0.857649
\(659\) − 21.0000i − 0.818044i −0.912525 0.409022i \(-0.865870\pi\)
0.912525 0.409022i \(-0.134130\pi\)
\(660\) 9.00000 0.350325
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −3.00000 −0.116598
\(663\) 36.0000 1.39812
\(664\) 0 0
\(665\) − 48.0000i − 1.86136i
\(666\) −48.0000 −1.85996
\(667\) 0 0
\(668\) −22.0000 −0.851206
\(669\) 78.0000i 3.01565i
\(670\) 12.0000i 0.463600i
\(671\) 4.00000 0.154418
\(672\) 6.00000 0.231455
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −4.00000 −0.154074
\(675\) − 36.0000i − 1.38564i
\(676\) 4.00000 0.153846
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) − 54.0000i − 2.07386i
\(679\) 12.0000i 0.460518i
\(680\) 12.0000 0.460179
\(681\) − 54.0000i − 2.06928i
\(682\) 3.00000i 0.114876i
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 48.0000i 1.83533i
\(685\) 60.0000i 2.29248i
\(686\) 20.0000i 0.763604i
\(687\) −42.0000 −1.60240
\(688\) − 7.00000i − 0.266872i
\(689\) −3.00000 −0.114291
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −14.0000 −0.532200
\(693\) 12.0000i 0.455842i
\(694\) − 18.0000i − 0.683271i
\(695\) 0 0
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) − 19.0000i − 0.719161i
\(699\) − 75.0000i − 2.83676i
\(700\) 8.00000 0.302372
\(701\) −31.0000 −1.17085 −0.585427 0.810725i \(-0.699073\pi\)
−0.585427 + 0.810725i \(0.699073\pi\)
\(702\) −27.0000 −1.01905
\(703\) 64.0000 2.41381
\(704\) − 1.00000i − 0.0376889i
\(705\) −99.0000 −3.72856
\(706\) 2.00000i 0.0752710i
\(707\) 16.0000i 0.601742i
\(708\) 12.0000i 0.450988i
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 6.00000i 0.225176i
\(711\) − 42.0000i − 1.57512i
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 24.0000i 0.898177i
\(715\) − 9.00000i − 0.336581i
\(716\) −14.0000 −0.523205
\(717\) 60.0000i 2.24074i
\(718\) 9.00000 0.335877
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) −18.0000 −0.670820
\(721\) 12.0000 0.446903
\(722\) − 45.0000i − 1.67473i
\(723\) − 51.0000i − 1.89671i
\(724\) 13.0000 0.483141
\(725\) 0 0
\(726\) −30.0000 −1.11340
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) − 6.00000i − 0.222375i
\(729\) 27.0000 1.00000
\(730\) 36.0000 1.33242
\(731\) 28.0000 1.03562
\(732\) −12.0000 −0.443533
\(733\) − 48.0000i − 1.77292i −0.462805 0.886460i \(-0.653157\pi\)
0.462805 0.886460i \(-0.346843\pi\)
\(734\) −16.0000 −0.590571
\(735\) − 27.0000i − 0.995910i
\(736\) 0 0
\(737\) 4.00000i 0.147342i
\(738\) −12.0000 −0.441726
\(739\) − 1.00000i − 0.0367856i −0.999831 0.0183928i \(-0.994145\pi\)
0.999831 0.0183928i \(-0.00585494\pi\)
\(740\) 24.0000i 0.882258i
\(741\) 72.0000 2.64499
\(742\) − 2.00000i − 0.0734223i
\(743\) − 28.0000i − 1.02722i −0.858024 0.513610i \(-0.828308\pi\)
0.858024 0.513610i \(-0.171692\pi\)
\(744\) − 9.00000i − 0.329956i
\(745\) −9.00000 −0.329734
\(746\) − 1.00000i − 0.0366126i
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 4.00000 0.146157
\(750\) 9.00000 0.328634
\(751\) 32.0000i 1.16770i 0.811863 + 0.583848i \(0.198454\pi\)
−0.811863 + 0.583848i \(0.801546\pi\)
\(752\) 11.0000i 0.401129i
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) − 18.0000i − 0.654654i
\(757\) 28.0000i 1.01768i 0.860862 + 0.508839i \(0.169925\pi\)
−0.860862 + 0.508839i \(0.830075\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) − 24.0000i − 0.869428i
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) − 72.0000i − 2.60317i
\(766\) 34.0000i 1.22847i
\(767\) 12.0000 0.433295
\(768\) 3.00000i 0.108253i
\(769\) 48.0000i 1.73092i 0.500974 + 0.865462i \(0.332975\pi\)
−0.500974 + 0.865462i \(0.667025\pi\)
\(770\) 6.00000 0.216225
\(771\) 63.0000i 2.26889i
\(772\) 10.0000i 0.359908i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) −42.0000 −1.50966
\(775\) − 12.0000i − 0.431053i
\(776\) −6.00000 −0.215387
\(777\) −48.0000 −1.72199
\(778\) −16.0000 −0.573628
\(779\) 16.0000 0.573259
\(780\) 27.0000i 0.966755i
\(781\) 2.00000i 0.0715656i
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 66.0000i 2.35564i
\(786\) − 36.0000i − 1.28408i
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −51.0000 −1.81565
\(790\) −21.0000 −0.747146
\(791\) − 36.0000i − 1.28001i
\(792\) −6.00000 −0.213201
\(793\) 12.0000i 0.426132i
\(794\) 19.0000i 0.674285i
\(795\) 9.00000i 0.319197i
\(796\) 2.00000 0.0708881
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 48.0000i 1.69918i
\(799\) −44.0000 −1.55661
\(800\) 4.00000i 0.141421i
\(801\) − 36.0000i − 1.27200i
\(802\) − 5.00000i − 0.176556i
\(803\) 12.0000 0.423471
\(804\) − 12.0000i − 0.423207i
\(805\) 0 0
\(806\) −9.00000 −0.317011
\(807\) −60.0000 −2.11210
\(808\) −8.00000 −0.281439
\(809\) 24.0000i 0.843795i 0.906644 + 0.421898i \(0.138636\pi\)
−0.906644 + 0.421898i \(0.861364\pi\)
\(810\) 27.0000i 0.948683i
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) 0 0
\(813\) −39.0000 −1.36779
\(814\) 8.00000i 0.280400i
\(815\) 57.0000i 1.99662i
\(816\) −12.0000 −0.420084
\(817\) 56.0000 1.95919
\(818\) 22.0000 0.769212
\(819\) −36.0000 −1.25794
\(820\) 6.00000i 0.209529i
\(821\) 37.0000 1.29131 0.645654 0.763630i \(-0.276585\pi\)
0.645654 + 0.763630i \(0.276585\pi\)
\(822\) − 60.0000i − 2.09274i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) 6.00000i 0.209020i
\(825\) −12.0000 −0.417786
\(826\) 8.00000i 0.278356i
\(827\) 23.0000i 0.799788i 0.916561 + 0.399894i \(0.130953\pi\)
−0.916561 + 0.399894i \(0.869047\pi\)
\(828\) 0 0
\(829\) 28.0000i 0.972480i 0.873825 + 0.486240i \(0.161632\pi\)
−0.873825 + 0.486240i \(0.838368\pi\)
\(830\) 0 0
\(831\) 42.0000i 1.45696i
\(832\) 3.00000 0.104006
\(833\) − 12.0000i − 0.415775i
\(834\) 0 0
\(835\) 66.0000 2.28402
\(836\) 8.00000 0.276686
\(837\) −27.0000 −0.933257
\(838\) 14.0000i 0.483622i
\(839\) − 45.0000i − 1.55357i −0.629764 0.776786i \(-0.716849\pi\)
0.629764 0.776786i \(-0.283151\pi\)
\(840\) −18.0000 −0.621059
\(841\) 0 0
\(842\) 28.0000 0.964944
\(843\) − 39.0000i − 1.34323i
\(844\) 25.0000i 0.860535i
\(845\) −12.0000 −0.412813
\(846\) 66.0000 2.26913
\(847\) −20.0000 −0.687208
\(848\) 1.00000 0.0343401
\(849\) 0 0
\(850\) −16.0000 −0.548795
\(851\) 0 0
\(852\) − 6.00000i − 0.205557i
\(853\) 30.0000i 1.02718i 0.858036 + 0.513590i \(0.171685\pi\)
−0.858036 + 0.513590i \(0.828315\pi\)
\(854\) −8.00000 −0.273754
\(855\) − 144.000i − 4.92470i
\(856\) 2.00000i 0.0683586i
\(857\) −11.0000 −0.375753 −0.187876 0.982193i \(-0.560160\pi\)
−0.187876 + 0.982193i \(0.560160\pi\)
\(858\) 9.00000i 0.307255i
\(859\) − 51.0000i − 1.74010i −0.492966 0.870049i \(-0.664087\pi\)
0.492966 0.870049i \(-0.335913\pi\)
\(860\) 21.0000i 0.716094i
\(861\) −12.0000 −0.408959
\(862\) − 20.0000i − 0.681203i
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 9.00000 0.306186
\(865\) 42.0000 1.42804
\(866\) 16.0000 0.543702
\(867\) 3.00000i 0.101885i
\(868\) − 6.00000i − 0.203653i
\(869\) −7.00000 −0.237459
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 1.00000i 0.0338643i
\(873\) 36.0000i 1.21842i
\(874\) 0 0
\(875\) 6.00000 0.202837
\(876\) −36.0000 −1.21633
\(877\) −23.0000 −0.776655 −0.388327 0.921521i \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) 8.00000i 0.269987i
\(879\) −102.000 −3.44037
\(880\) 3.00000i 0.101130i
\(881\) − 42.0000i − 1.41502i −0.706705 0.707508i \(-0.749819\pi\)
0.706705 0.707508i \(-0.250181\pi\)
\(882\) 18.0000i 0.606092i
\(883\) 42.0000 1.41341 0.706706 0.707507i \(-0.250180\pi\)
0.706706 + 0.707507i \(0.250180\pi\)
\(884\) 12.0000i 0.403604i
\(885\) − 36.0000i − 1.21013i
\(886\) −12.0000 −0.403148
\(887\) 1.00000i 0.0335767i 0.999859 + 0.0167884i \(0.00534415\pi\)
−0.999859 + 0.0167884i \(0.994656\pi\)
\(888\) − 24.0000i − 0.805387i
\(889\) − 16.0000i − 0.536623i
\(890\) −18.0000 −0.603361
\(891\) 9.00000i 0.301511i
\(892\) −26.0000 −0.870544
\(893\) −88.0000 −2.94481
\(894\) 9.00000 0.301005
\(895\) 42.0000 1.40391
\(896\) 2.00000i 0.0668153i
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 0 0
\(900\) 24.0000 0.800000
\(901\) 4.00000i 0.133259i
\(902\) 2.00000i 0.0665927i
\(903\) −42.0000 −1.39767
\(904\) 18.0000 0.598671
\(905\) −39.0000 −1.29640
\(906\) 30.0000 0.996683
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 18.0000 0.597351
\(909\) 48.0000i 1.59206i
\(910\) 18.0000i 0.596694i
\(911\) − 3.00000i − 0.0993944i −0.998764 0.0496972i \(-0.984174\pi\)
0.998764 0.0496972i \(-0.0158256\pi\)
\(912\) −24.0000 −0.794719
\(913\) 0 0
\(914\) 10.0000i 0.330771i
\(915\) 36.0000 1.19012
\(916\) − 14.0000i − 0.462573i
\(917\) − 24.0000i − 0.792550i
\(918\) 36.0000i 1.18818i
\(919\) −6.00000 −0.197922 −0.0989609 0.995091i \(-0.531552\pi\)
−0.0989609 + 0.995091i \(0.531552\pi\)
\(920\) 0 0
\(921\) −87.0000 −2.86675
\(922\) 30.0000 0.987997
\(923\) −6.00000 −0.197492
\(924\) −6.00000 −0.197386
\(925\) − 32.0000i − 1.05215i
\(926\) 16.0000i 0.525793i
\(927\) 36.0000 1.18240
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 27.0000i 0.885365i
\(931\) − 24.0000i − 0.786568i
\(932\) 25.0000 0.818902
\(933\) 0 0
\(934\) 23.0000 0.752583
\(935\) −12.0000 −0.392442
\(936\) − 18.0000i − 0.588348i
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 75.0000i 2.44753i
\(940\) − 33.0000i − 1.07634i
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) − 66.0000i − 2.15040i
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 54.0000i 1.75662i
\(946\) 7.00000i 0.227590i
\(947\) − 51.0000i − 1.65728i −0.559784 0.828639i \(-0.689116\pi\)
0.559784 0.828639i \(-0.310884\pi\)
\(948\) 21.0000 0.682048
\(949\) 36.0000i 1.16861i
\(950\) −32.0000 −1.03822
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) 15.0000 0.485898 0.242949 0.970039i \(-0.421885\pi\)
0.242949 + 0.970039i \(0.421885\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 0 0
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 11.0000 0.355394
\(959\) − 40.0000i − 1.29167i
\(960\) − 9.00000i − 0.290474i
\(961\) 22.0000 0.709677
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) 17.0000 0.547533
\(965\) − 30.0000i − 0.965734i
\(966\) 0 0
\(967\) 59.0000i 1.89731i 0.316310 + 0.948656i \(0.397556\pi\)
−0.316310 + 0.948656i \(0.602444\pi\)
\(968\) − 10.0000i − 0.321412i
\(969\) − 96.0000i − 3.08396i
\(970\) 18.0000 0.577945
\(971\) 52.0000i 1.66876i 0.551190 + 0.834380i \(0.314174\pi\)
−0.551190 + 0.834380i \(0.685826\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2.00000i 0.0640841i
\(975\) − 36.0000i − 1.15292i
\(976\) − 4.00000i − 0.128037i
\(977\) 37.0000 1.18373 0.591867 0.806035i \(-0.298391\pi\)
0.591867 + 0.806035i \(0.298391\pi\)
\(978\) − 57.0000i − 1.82266i
\(979\) −6.00000 −0.191761
\(980\) 9.00000 0.287494
\(981\) 6.00000 0.191565
\(982\) −5.00000 −0.159556
\(983\) 39.0000i 1.24391i 0.783054 + 0.621953i \(0.213661\pi\)
−0.783054 + 0.621953i \(0.786339\pi\)
\(984\) − 6.00000i − 0.191273i
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 66.0000 2.10080
\(988\) 24.0000i 0.763542i
\(989\) 0 0
\(990\) 18.0000 0.572078
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 3.00000 0.0952501
\(993\) −9.00000 −0.285606
\(994\) − 4.00000i − 0.126872i
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) − 16.0000i − 0.506725i −0.967371 0.253363i \(-0.918463\pi\)
0.967371 0.253363i \(-0.0815366\pi\)
\(998\) 8.00000i 0.253236i
\(999\) −72.0000 −2.27798
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 1682.2.b.e.1681.2 2
29.12 odd 4 58.2.a.a.1.1 1
29.17 odd 4 1682.2.a.j.1.1 1
29.28 even 2 inner 1682.2.b.e.1681.1 2
87.41 even 4 522.2.a.k.1.1 1
116.99 even 4 464.2.a.f.1.1 1
145.12 even 4 1450.2.b.f.349.1 2
145.99 odd 4 1450.2.a.i.1.1 1
145.128 even 4 1450.2.b.f.349.2 2
203.41 even 4 2842.2.a.d.1.1 1
232.99 even 4 1856.2.a.b.1.1 1
232.157 odd 4 1856.2.a.p.1.1 1
319.186 even 4 7018.2.a.c.1.1 1
348.215 odd 4 4176.2.a.bh.1.1 1
377.12 odd 4 9802.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.a.1.1 1 29.12 odd 4
464.2.a.f.1.1 1 116.99 even 4
522.2.a.k.1.1 1 87.41 even 4
1450.2.a.i.1.1 1 145.99 odd 4
1450.2.b.f.349.1 2 145.12 even 4
1450.2.b.f.349.2 2 145.128 even 4
1682.2.a.j.1.1 1 29.17 odd 4
1682.2.b.e.1681.1 2 29.28 even 2 inner
1682.2.b.e.1681.2 2 1.1 even 1 trivial
1856.2.a.b.1.1 1 232.99 even 4
1856.2.a.p.1.1 1 232.157 odd 4
2842.2.a.d.1.1 1 203.41 even 4
4176.2.a.bh.1.1 1 348.215 odd 4
7018.2.a.c.1.1 1 319.186 even 4
9802.2.a.d.1.1 1 377.12 odd 4