Properties

Label 722.2.a.h.1.2
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [722,2,Mod(1,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

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: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.23607 q^{3} +1.00000 q^{4} -1.38197 q^{5} -3.23607 q^{6} +1.23607 q^{7} -1.00000 q^{8} +7.47214 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.23607 q^{3} +1.00000 q^{4} -1.38197 q^{5} -3.23607 q^{6} +1.23607 q^{7} -1.00000 q^{8} +7.47214 q^{9} +1.38197 q^{10} +1.23607 q^{11} +3.23607 q^{12} +3.61803 q^{13} -1.23607 q^{14} -4.47214 q^{15} +1.00000 q^{16} -5.61803 q^{17} -7.47214 q^{18} -1.38197 q^{20} +4.00000 q^{21} -1.23607 q^{22} +0.763932 q^{23} -3.23607 q^{24} -3.09017 q^{25} -3.61803 q^{26} +14.4721 q^{27} +1.23607 q^{28} +2.09017 q^{29} +4.47214 q^{30} +3.23607 q^{31} -1.00000 q^{32} +4.00000 q^{33} +5.61803 q^{34} -1.70820 q^{35} +7.47214 q^{36} +10.6180 q^{37} +11.7082 q^{39} +1.38197 q^{40} -5.85410 q^{41} -4.00000 q^{42} -4.76393 q^{43} +1.23607 q^{44} -10.3262 q^{45} -0.763932 q^{46} -4.47214 q^{47} +3.23607 q^{48} -5.47214 q^{49} +3.09017 q^{50} -18.1803 q^{51} +3.61803 q^{52} +5.09017 q^{53} -14.4721 q^{54} -1.70820 q^{55} -1.23607 q^{56} -2.09017 q^{58} +8.47214 q^{59} -4.47214 q^{60} +3.61803 q^{61} -3.23607 q^{62} +9.23607 q^{63} +1.00000 q^{64} -5.00000 q^{65} -4.00000 q^{66} -1.70820 q^{67} -5.61803 q^{68} +2.47214 q^{69} +1.70820 q^{70} -14.9443 q^{71} -7.47214 q^{72} -3.38197 q^{73} -10.6180 q^{74} -10.0000 q^{75} +1.52786 q^{77} -11.7082 q^{78} -7.23607 q^{79} -1.38197 q^{80} +24.4164 q^{81} +5.85410 q^{82} +8.47214 q^{83} +4.00000 q^{84} +7.76393 q^{85} +4.76393 q^{86} +6.76393 q^{87} -1.23607 q^{88} +2.14590 q^{89} +10.3262 q^{90} +4.47214 q^{91} +0.763932 q^{92} +10.4721 q^{93} +4.47214 q^{94} -3.23607 q^{96} +6.38197 q^{97} +5.47214 q^{98} +9.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 5 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 5 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 5 q^{10} - 2 q^{11} + 2 q^{12} + 5 q^{13} + 2 q^{14} + 2 q^{16} - 9 q^{17} - 6 q^{18} - 5 q^{20} + 8 q^{21} + 2 q^{22} + 6 q^{23} - 2 q^{24} + 5 q^{25} - 5 q^{26} + 20 q^{27} - 2 q^{28} - 7 q^{29} + 2 q^{31} - 2 q^{32} + 8 q^{33} + 9 q^{34} + 10 q^{35} + 6 q^{36} + 19 q^{37} + 10 q^{39} + 5 q^{40} - 5 q^{41} - 8 q^{42} - 14 q^{43} - 2 q^{44} - 5 q^{45} - 6 q^{46} + 2 q^{48} - 2 q^{49} - 5 q^{50} - 14 q^{51} + 5 q^{52} - q^{53} - 20 q^{54} + 10 q^{55} + 2 q^{56} + 7 q^{58} + 8 q^{59} + 5 q^{61} - 2 q^{62} + 14 q^{63} + 2 q^{64} - 10 q^{65} - 8 q^{66} + 10 q^{67} - 9 q^{68} - 4 q^{69} - 10 q^{70} - 12 q^{71} - 6 q^{72} - 9 q^{73} - 19 q^{74} - 20 q^{75} + 12 q^{77} - 10 q^{78} - 10 q^{79} - 5 q^{80} + 22 q^{81} + 5 q^{82} + 8 q^{83} + 8 q^{84} + 20 q^{85} + 14 q^{86} + 18 q^{87} + 2 q^{88} + 11 q^{89} + 5 q^{90} + 6 q^{92} + 12 q^{93} - 2 q^{96} + 15 q^{97} + 2 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.23607 1.86834 0.934172 0.356822i \(-0.116140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.38197 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(6\) −3.23607 −1.32112
\(7\) 1.23607 0.467190 0.233595 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) −1.00000 −0.353553
\(9\) 7.47214 2.49071
\(10\) 1.38197 0.437016
\(11\) 1.23607 0.372689 0.186344 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) 3.23607 0.934172
\(13\) 3.61803 1.00346 0.501731 0.865024i \(-0.332697\pi\)
0.501731 + 0.865024i \(0.332697\pi\)
\(14\) −1.23607 −0.330353
\(15\) −4.47214 −1.15470
\(16\) 1.00000 0.250000
\(17\) −5.61803 −1.36257 −0.681287 0.732017i \(-0.738579\pi\)
−0.681287 + 0.732017i \(0.738579\pi\)
\(18\) −7.47214 −1.76120
\(19\) 0 0
\(20\) −1.38197 −0.309017
\(21\) 4.00000 0.872872
\(22\) −1.23607 −0.263531
\(23\) 0.763932 0.159291 0.0796454 0.996823i \(-0.474621\pi\)
0.0796454 + 0.996823i \(0.474621\pi\)
\(24\) −3.23607 −0.660560
\(25\) −3.09017 −0.618034
\(26\) −3.61803 −0.709555
\(27\) 14.4721 2.78516
\(28\) 1.23607 0.233595
\(29\) 2.09017 0.388135 0.194067 0.980988i \(-0.437832\pi\)
0.194067 + 0.980988i \(0.437832\pi\)
\(30\) 4.47214 0.816497
\(31\) 3.23607 0.581215 0.290607 0.956842i \(-0.406143\pi\)
0.290607 + 0.956842i \(0.406143\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 5.61803 0.963485
\(35\) −1.70820 −0.288739
\(36\) 7.47214 1.24536
\(37\) 10.6180 1.74559 0.872797 0.488083i \(-0.162304\pi\)
0.872797 + 0.488083i \(0.162304\pi\)
\(38\) 0 0
\(39\) 11.7082 1.87481
\(40\) 1.38197 0.218508
\(41\) −5.85410 −0.914257 −0.457129 0.889401i \(-0.651122\pi\)
−0.457129 + 0.889401i \(0.651122\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.76393 −0.726493 −0.363246 0.931693i \(-0.618332\pi\)
−0.363246 + 0.931693i \(0.618332\pi\)
\(44\) 1.23607 0.186344
\(45\) −10.3262 −1.53934
\(46\) −0.763932 −0.112636
\(47\) −4.47214 −0.652328 −0.326164 0.945313i \(-0.605756\pi\)
−0.326164 + 0.945313i \(0.605756\pi\)
\(48\) 3.23607 0.467086
\(49\) −5.47214 −0.781734
\(50\) 3.09017 0.437016
\(51\) −18.1803 −2.54576
\(52\) 3.61803 0.501731
\(53\) 5.09017 0.699189 0.349594 0.936901i \(-0.386319\pi\)
0.349594 + 0.936901i \(0.386319\pi\)
\(54\) −14.4721 −1.96941
\(55\) −1.70820 −0.230334
\(56\) −1.23607 −0.165177
\(57\) 0 0
\(58\) −2.09017 −0.274453
\(59\) 8.47214 1.10298 0.551489 0.834182i \(-0.314060\pi\)
0.551489 + 0.834182i \(0.314060\pi\)
\(60\) −4.47214 −0.577350
\(61\) 3.61803 0.463242 0.231621 0.972806i \(-0.425597\pi\)
0.231621 + 0.972806i \(0.425597\pi\)
\(62\) −3.23607 −0.410981
\(63\) 9.23607 1.16364
\(64\) 1.00000 0.125000
\(65\) −5.00000 −0.620174
\(66\) −4.00000 −0.492366
\(67\) −1.70820 −0.208690 −0.104345 0.994541i \(-0.533275\pi\)
−0.104345 + 0.994541i \(0.533275\pi\)
\(68\) −5.61803 −0.681287
\(69\) 2.47214 0.297610
\(70\) 1.70820 0.204169
\(71\) −14.9443 −1.77356 −0.886779 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(72\) −7.47214 −0.880600
\(73\) −3.38197 −0.395829 −0.197915 0.980219i \(-0.563417\pi\)
−0.197915 + 0.980219i \(0.563417\pi\)
\(74\) −10.6180 −1.23432
\(75\) −10.0000 −1.15470
\(76\) 0 0
\(77\) 1.52786 0.174116
\(78\) −11.7082 −1.32569
\(79\) −7.23607 −0.814121 −0.407061 0.913401i \(-0.633446\pi\)
−0.407061 + 0.913401i \(0.633446\pi\)
\(80\) −1.38197 −0.154508
\(81\) 24.4164 2.71293
\(82\) 5.85410 0.646477
\(83\) 8.47214 0.929938 0.464969 0.885327i \(-0.346066\pi\)
0.464969 + 0.885327i \(0.346066\pi\)
\(84\) 4.00000 0.436436
\(85\) 7.76393 0.842117
\(86\) 4.76393 0.513708
\(87\) 6.76393 0.725170
\(88\) −1.23607 −0.131765
\(89\) 2.14590 0.227465 0.113732 0.993511i \(-0.463719\pi\)
0.113732 + 0.993511i \(0.463719\pi\)
\(90\) 10.3262 1.08848
\(91\) 4.47214 0.468807
\(92\) 0.763932 0.0796454
\(93\) 10.4721 1.08591
\(94\) 4.47214 0.461266
\(95\) 0 0
\(96\) −3.23607 −0.330280
\(97\) 6.38197 0.647990 0.323995 0.946059i \(-0.394974\pi\)
0.323995 + 0.946059i \(0.394974\pi\)
\(98\) 5.47214 0.552769
\(99\) 9.23607 0.928260
\(100\) −3.09017 −0.309017
\(101\) −13.3820 −1.33156 −0.665778 0.746150i \(-0.731900\pi\)
−0.665778 + 0.746150i \(0.731900\pi\)
\(102\) 18.1803 1.80012
\(103\) −16.6525 −1.64082 −0.820409 0.571778i \(-0.806254\pi\)
−0.820409 + 0.571778i \(0.806254\pi\)
\(104\) −3.61803 −0.354777
\(105\) −5.52786 −0.539464
\(106\) −5.09017 −0.494401
\(107\) −17.4164 −1.68371 −0.841854 0.539706i \(-0.818536\pi\)
−0.841854 + 0.539706i \(0.818536\pi\)
\(108\) 14.4721 1.39258
\(109\) 18.0344 1.72739 0.863693 0.504018i \(-0.168145\pi\)
0.863693 + 0.504018i \(0.168145\pi\)
\(110\) 1.70820 0.162871
\(111\) 34.3607 3.26137
\(112\) 1.23607 0.116797
\(113\) −3.14590 −0.295941 −0.147971 0.988992i \(-0.547274\pi\)
−0.147971 + 0.988992i \(0.547274\pi\)
\(114\) 0 0
\(115\) −1.05573 −0.0984472
\(116\) 2.09017 0.194067
\(117\) 27.0344 2.49934
\(118\) −8.47214 −0.779923
\(119\) −6.94427 −0.636580
\(120\) 4.47214 0.408248
\(121\) −9.47214 −0.861103
\(122\) −3.61803 −0.327561
\(123\) −18.9443 −1.70815
\(124\) 3.23607 0.290607
\(125\) 11.1803 1.00000
\(126\) −9.23607 −0.822814
\(127\) 15.4164 1.36798 0.683992 0.729489i \(-0.260242\pi\)
0.683992 + 0.729489i \(0.260242\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −15.4164 −1.35734
\(130\) 5.00000 0.438529
\(131\) −4.47214 −0.390732 −0.195366 0.980730i \(-0.562590\pi\)
−0.195366 + 0.980730i \(0.562590\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 1.70820 0.147566
\(135\) −20.0000 −1.72133
\(136\) 5.61803 0.481742
\(137\) 4.61803 0.394545 0.197273 0.980349i \(-0.436792\pi\)
0.197273 + 0.980349i \(0.436792\pi\)
\(138\) −2.47214 −0.210442
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) −1.70820 −0.144370
\(141\) −14.4721 −1.21877
\(142\) 14.9443 1.25410
\(143\) 4.47214 0.373979
\(144\) 7.47214 0.622678
\(145\) −2.88854 −0.239881
\(146\) 3.38197 0.279893
\(147\) −17.7082 −1.46055
\(148\) 10.6180 0.872797
\(149\) −1.90983 −0.156459 −0.0782297 0.996935i \(-0.524927\pi\)
−0.0782297 + 0.996935i \(0.524927\pi\)
\(150\) 10.0000 0.816497
\(151\) 5.52786 0.449851 0.224926 0.974376i \(-0.427786\pi\)
0.224926 + 0.974376i \(0.427786\pi\)
\(152\) 0 0
\(153\) −41.9787 −3.39378
\(154\) −1.52786 −0.123119
\(155\) −4.47214 −0.359211
\(156\) 11.7082 0.937407
\(157\) 11.6180 0.927220 0.463610 0.886039i \(-0.346554\pi\)
0.463610 + 0.886039i \(0.346554\pi\)
\(158\) 7.23607 0.575671
\(159\) 16.4721 1.30633
\(160\) 1.38197 0.109254
\(161\) 0.944272 0.0744191
\(162\) −24.4164 −1.91833
\(163\) −20.6525 −1.61763 −0.808813 0.588065i \(-0.799890\pi\)
−0.808813 + 0.588065i \(0.799890\pi\)
\(164\) −5.85410 −0.457129
\(165\) −5.52786 −0.430344
\(166\) −8.47214 −0.657565
\(167\) −3.52786 −0.272994 −0.136497 0.990640i \(-0.543584\pi\)
−0.136497 + 0.990640i \(0.543584\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0.0901699 0.00693615
\(170\) −7.76393 −0.595466
\(171\) 0 0
\(172\) −4.76393 −0.363246
\(173\) −14.3262 −1.08920 −0.544602 0.838695i \(-0.683319\pi\)
−0.544602 + 0.838695i \(0.683319\pi\)
\(174\) −6.76393 −0.512772
\(175\) −3.81966 −0.288739
\(176\) 1.23607 0.0931721
\(177\) 27.4164 2.06074
\(178\) −2.14590 −0.160842
\(179\) 3.41641 0.255354 0.127677 0.991816i \(-0.459248\pi\)
0.127677 + 0.991816i \(0.459248\pi\)
\(180\) −10.3262 −0.769672
\(181\) −21.4164 −1.59187 −0.795935 0.605383i \(-0.793020\pi\)
−0.795935 + 0.605383i \(0.793020\pi\)
\(182\) −4.47214 −0.331497
\(183\) 11.7082 0.865495
\(184\) −0.763932 −0.0563178
\(185\) −14.6738 −1.07884
\(186\) −10.4721 −0.767854
\(187\) −6.94427 −0.507815
\(188\) −4.47214 −0.326164
\(189\) 17.8885 1.30120
\(190\) 0 0
\(191\) 12.4721 0.902452 0.451226 0.892410i \(-0.350987\pi\)
0.451226 + 0.892410i \(0.350987\pi\)
\(192\) 3.23607 0.233543
\(193\) −20.4721 −1.47362 −0.736808 0.676102i \(-0.763668\pi\)
−0.736808 + 0.676102i \(0.763668\pi\)
\(194\) −6.38197 −0.458198
\(195\) −16.1803 −1.15870
\(196\) −5.47214 −0.390867
\(197\) −10.8541 −0.773323 −0.386661 0.922222i \(-0.626372\pi\)
−0.386661 + 0.922222i \(0.626372\pi\)
\(198\) −9.23607 −0.656379
\(199\) 1.52786 0.108307 0.0541537 0.998533i \(-0.482754\pi\)
0.0541537 + 0.998533i \(0.482754\pi\)
\(200\) 3.09017 0.218508
\(201\) −5.52786 −0.389905
\(202\) 13.3820 0.941552
\(203\) 2.58359 0.181333
\(204\) −18.1803 −1.27288
\(205\) 8.09017 0.565042
\(206\) 16.6525 1.16023
\(207\) 5.70820 0.396748
\(208\) 3.61803 0.250866
\(209\) 0 0
\(210\) 5.52786 0.381459
\(211\) −2.76393 −0.190277 −0.0951385 0.995464i \(-0.530329\pi\)
−0.0951385 + 0.995464i \(0.530329\pi\)
\(212\) 5.09017 0.349594
\(213\) −48.3607 −3.31362
\(214\) 17.4164 1.19056
\(215\) 6.58359 0.448997
\(216\) −14.4721 −0.984704
\(217\) 4.00000 0.271538
\(218\) −18.0344 −1.22145
\(219\) −10.9443 −0.739545
\(220\) −1.70820 −0.115167
\(221\) −20.3262 −1.36729
\(222\) −34.3607 −2.30614
\(223\) 14.2918 0.957049 0.478525 0.878074i \(-0.341172\pi\)
0.478525 + 0.878074i \(0.341172\pi\)
\(224\) −1.23607 −0.0825883
\(225\) −23.0902 −1.53934
\(226\) 3.14590 0.209262
\(227\) 8.94427 0.593652 0.296826 0.954932i \(-0.404072\pi\)
0.296826 + 0.954932i \(0.404072\pi\)
\(228\) 0 0
\(229\) 6.61803 0.437332 0.218666 0.975800i \(-0.429829\pi\)
0.218666 + 0.975800i \(0.429829\pi\)
\(230\) 1.05573 0.0696126
\(231\) 4.94427 0.325309
\(232\) −2.09017 −0.137226
\(233\) −18.5623 −1.21606 −0.608029 0.793915i \(-0.708039\pi\)
−0.608029 + 0.793915i \(0.708039\pi\)
\(234\) −27.0344 −1.76730
\(235\) 6.18034 0.403161
\(236\) 8.47214 0.551489
\(237\) −23.4164 −1.52106
\(238\) 6.94427 0.450130
\(239\) 7.81966 0.505812 0.252906 0.967491i \(-0.418614\pi\)
0.252906 + 0.967491i \(0.418614\pi\)
\(240\) −4.47214 −0.288675
\(241\) 1.41641 0.0912389 0.0456194 0.998959i \(-0.485474\pi\)
0.0456194 + 0.998959i \(0.485474\pi\)
\(242\) 9.47214 0.608892
\(243\) 35.5967 2.28353
\(244\) 3.61803 0.231621
\(245\) 7.56231 0.483138
\(246\) 18.9443 1.20784
\(247\) 0 0
\(248\) −3.23607 −0.205491
\(249\) 27.4164 1.73744
\(250\) −11.1803 −0.707107
\(251\) −15.7082 −0.991493 −0.495747 0.868467i \(-0.665105\pi\)
−0.495747 + 0.868467i \(0.665105\pi\)
\(252\) 9.23607 0.581818
\(253\) 0.944272 0.0593659
\(254\) −15.4164 −0.967311
\(255\) 25.1246 1.57336
\(256\) 1.00000 0.0625000
\(257\) 20.6180 1.28612 0.643059 0.765817i \(-0.277665\pi\)
0.643059 + 0.765817i \(0.277665\pi\)
\(258\) 15.4164 0.959784
\(259\) 13.1246 0.815524
\(260\) −5.00000 −0.310087
\(261\) 15.6180 0.966732
\(262\) 4.47214 0.276289
\(263\) −14.9443 −0.921503 −0.460752 0.887529i \(-0.652420\pi\)
−0.460752 + 0.887529i \(0.652420\pi\)
\(264\) −4.00000 −0.246183
\(265\) −7.03444 −0.432122
\(266\) 0 0
\(267\) 6.94427 0.424983
\(268\) −1.70820 −0.104345
\(269\) −1.32624 −0.0808622 −0.0404311 0.999182i \(-0.512873\pi\)
−0.0404311 + 0.999182i \(0.512873\pi\)
\(270\) 20.0000 1.21716
\(271\) 26.0000 1.57939 0.789694 0.613501i \(-0.210239\pi\)
0.789694 + 0.613501i \(0.210239\pi\)
\(272\) −5.61803 −0.340643
\(273\) 14.4721 0.875894
\(274\) −4.61803 −0.278986
\(275\) −3.81966 −0.230334
\(276\) 2.47214 0.148805
\(277\) 1.09017 0.0655020 0.0327510 0.999464i \(-0.489573\pi\)
0.0327510 + 0.999464i \(0.489573\pi\)
\(278\) 14.0000 0.839664
\(279\) 24.1803 1.44764
\(280\) 1.70820 0.102085
\(281\) −25.9787 −1.54976 −0.774880 0.632108i \(-0.782190\pi\)
−0.774880 + 0.632108i \(0.782190\pi\)
\(282\) 14.4721 0.861803
\(283\) 13.5279 0.804148 0.402074 0.915607i \(-0.368289\pi\)
0.402074 + 0.915607i \(0.368289\pi\)
\(284\) −14.9443 −0.886779
\(285\) 0 0
\(286\) −4.47214 −0.264443
\(287\) −7.23607 −0.427132
\(288\) −7.47214 −0.440300
\(289\) 14.5623 0.856606
\(290\) 2.88854 0.169621
\(291\) 20.6525 1.21067
\(292\) −3.38197 −0.197915
\(293\) 5.05573 0.295359 0.147679 0.989035i \(-0.452820\pi\)
0.147679 + 0.989035i \(0.452820\pi\)
\(294\) 17.7082 1.03276
\(295\) −11.7082 −0.681678
\(296\) −10.6180 −0.617161
\(297\) 17.8885 1.03800
\(298\) 1.90983 0.110633
\(299\) 2.76393 0.159842
\(300\) −10.0000 −0.577350
\(301\) −5.88854 −0.339410
\(302\) −5.52786 −0.318093
\(303\) −43.3050 −2.48780
\(304\) 0 0
\(305\) −5.00000 −0.286299
\(306\) 41.9787 2.39976
\(307\) −1.70820 −0.0974923 −0.0487462 0.998811i \(-0.515523\pi\)
−0.0487462 + 0.998811i \(0.515523\pi\)
\(308\) 1.52786 0.0870581
\(309\) −53.8885 −3.06561
\(310\) 4.47214 0.254000
\(311\) 26.7639 1.51764 0.758822 0.651298i \(-0.225775\pi\)
0.758822 + 0.651298i \(0.225775\pi\)
\(312\) −11.7082 −0.662847
\(313\) 22.5066 1.27215 0.636073 0.771628i \(-0.280558\pi\)
0.636073 + 0.771628i \(0.280558\pi\)
\(314\) −11.6180 −0.655644
\(315\) −12.7639 −0.719166
\(316\) −7.23607 −0.407061
\(317\) −14.3262 −0.804642 −0.402321 0.915499i \(-0.631796\pi\)
−0.402321 + 0.915499i \(0.631796\pi\)
\(318\) −16.4721 −0.923712
\(319\) 2.58359 0.144653
\(320\) −1.38197 −0.0772542
\(321\) −56.3607 −3.14575
\(322\) −0.944272 −0.0526222
\(323\) 0 0
\(324\) 24.4164 1.35647
\(325\) −11.1803 −0.620174
\(326\) 20.6525 1.14383
\(327\) 58.3607 3.22735
\(328\) 5.85410 0.323239
\(329\) −5.52786 −0.304761
\(330\) 5.52786 0.304299
\(331\) 12.2918 0.675618 0.337809 0.941215i \(-0.390314\pi\)
0.337809 + 0.941215i \(0.390314\pi\)
\(332\) 8.47214 0.464969
\(333\) 79.3394 4.34777
\(334\) 3.52786 0.193036
\(335\) 2.36068 0.128978
\(336\) 4.00000 0.218218
\(337\) 4.61803 0.251560 0.125780 0.992058i \(-0.459857\pi\)
0.125780 + 0.992058i \(0.459857\pi\)
\(338\) −0.0901699 −0.00490460
\(339\) −10.1803 −0.552920
\(340\) 7.76393 0.421058
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −15.4164 −0.832408
\(344\) 4.76393 0.256854
\(345\) −3.41641 −0.183933
\(346\) 14.3262 0.770183
\(347\) 28.6525 1.53815 0.769073 0.639161i \(-0.220718\pi\)
0.769073 + 0.639161i \(0.220718\pi\)
\(348\) 6.76393 0.362585
\(349\) −21.9098 −1.17281 −0.586403 0.810019i \(-0.699457\pi\)
−0.586403 + 0.810019i \(0.699457\pi\)
\(350\) 3.81966 0.204169
\(351\) 52.3607 2.79481
\(352\) −1.23607 −0.0658826
\(353\) 18.7426 0.997570 0.498785 0.866726i \(-0.333780\pi\)
0.498785 + 0.866726i \(0.333780\pi\)
\(354\) −27.4164 −1.45717
\(355\) 20.6525 1.09612
\(356\) 2.14590 0.113732
\(357\) −22.4721 −1.18935
\(358\) −3.41641 −0.180563
\(359\) 20.4721 1.08048 0.540239 0.841512i \(-0.318334\pi\)
0.540239 + 0.841512i \(0.318334\pi\)
\(360\) 10.3262 0.544241
\(361\) 0 0
\(362\) 21.4164 1.12562
\(363\) −30.6525 −1.60884
\(364\) 4.47214 0.234404
\(365\) 4.67376 0.244636
\(366\) −11.7082 −0.611998
\(367\) −34.8328 −1.81826 −0.909129 0.416514i \(-0.863252\pi\)
−0.909129 + 0.416514i \(0.863252\pi\)
\(368\) 0.763932 0.0398227
\(369\) −43.7426 −2.27715
\(370\) 14.6738 0.762853
\(371\) 6.29180 0.326654
\(372\) 10.4721 0.542955
\(373\) 2.67376 0.138442 0.0692211 0.997601i \(-0.477949\pi\)
0.0692211 + 0.997601i \(0.477949\pi\)
\(374\) 6.94427 0.359080
\(375\) 36.1803 1.86834
\(376\) 4.47214 0.230633
\(377\) 7.56231 0.389479
\(378\) −17.8885 −0.920087
\(379\) −8.58359 −0.440910 −0.220455 0.975397i \(-0.570754\pi\)
−0.220455 + 0.975397i \(0.570754\pi\)
\(380\) 0 0
\(381\) 49.8885 2.55587
\(382\) −12.4721 −0.638130
\(383\) 25.2361 1.28950 0.644751 0.764392i \(-0.276961\pi\)
0.644751 + 0.764392i \(0.276961\pi\)
\(384\) −3.23607 −0.165140
\(385\) −2.11146 −0.107610
\(386\) 20.4721 1.04200
\(387\) −35.5967 −1.80948
\(388\) 6.38197 0.323995
\(389\) −11.7984 −0.598201 −0.299101 0.954222i \(-0.596687\pi\)
−0.299101 + 0.954222i \(0.596687\pi\)
\(390\) 16.1803 0.819323
\(391\) −4.29180 −0.217045
\(392\) 5.47214 0.276385
\(393\) −14.4721 −0.730023
\(394\) 10.8541 0.546822
\(395\) 10.0000 0.503155
\(396\) 9.23607 0.464130
\(397\) 22.3607 1.12225 0.561125 0.827731i \(-0.310369\pi\)
0.561125 + 0.827731i \(0.310369\pi\)
\(398\) −1.52786 −0.0765849
\(399\) 0 0
\(400\) −3.09017 −0.154508
\(401\) 1.05573 0.0527205 0.0263603 0.999653i \(-0.491608\pi\)
0.0263603 + 0.999653i \(0.491608\pi\)
\(402\) 5.52786 0.275705
\(403\) 11.7082 0.583227
\(404\) −13.3820 −0.665778
\(405\) −33.7426 −1.67669
\(406\) −2.58359 −0.128222
\(407\) 13.1246 0.650563
\(408\) 18.1803 0.900061
\(409\) −17.2705 −0.853972 −0.426986 0.904258i \(-0.640425\pi\)
−0.426986 + 0.904258i \(0.640425\pi\)
\(410\) −8.09017 −0.399545
\(411\) 14.9443 0.737147
\(412\) −16.6525 −0.820409
\(413\) 10.4721 0.515300
\(414\) −5.70820 −0.280543
\(415\) −11.7082 −0.574733
\(416\) −3.61803 −0.177389
\(417\) −45.3050 −2.21859
\(418\) 0 0
\(419\) 2.47214 0.120772 0.0603859 0.998175i \(-0.480767\pi\)
0.0603859 + 0.998175i \(0.480767\pi\)
\(420\) −5.52786 −0.269732
\(421\) 25.0902 1.22282 0.611410 0.791314i \(-0.290603\pi\)
0.611410 + 0.791314i \(0.290603\pi\)
\(422\) 2.76393 0.134546
\(423\) −33.4164 −1.62476
\(424\) −5.09017 −0.247201
\(425\) 17.3607 0.842117
\(426\) 48.3607 2.34308
\(427\) 4.47214 0.216422
\(428\) −17.4164 −0.841854
\(429\) 14.4721 0.698721
\(430\) −6.58359 −0.317489
\(431\) −24.6525 −1.18747 −0.593734 0.804661i \(-0.702347\pi\)
−0.593734 + 0.804661i \(0.702347\pi\)
\(432\) 14.4721 0.696291
\(433\) 4.03444 0.193883 0.0969415 0.995290i \(-0.469094\pi\)
0.0969415 + 0.995290i \(0.469094\pi\)
\(434\) −4.00000 −0.192006
\(435\) −9.34752 −0.448179
\(436\) 18.0344 0.863693
\(437\) 0 0
\(438\) 10.9443 0.522938
\(439\) 24.7639 1.18192 0.590959 0.806702i \(-0.298749\pi\)
0.590959 + 0.806702i \(0.298749\pi\)
\(440\) 1.70820 0.0814354
\(441\) −40.8885 −1.94707
\(442\) 20.3262 0.966821
\(443\) 4.76393 0.226341 0.113171 0.993576i \(-0.463899\pi\)
0.113171 + 0.993576i \(0.463899\pi\)
\(444\) 34.3607 1.63069
\(445\) −2.96556 −0.140581
\(446\) −14.2918 −0.676736
\(447\) −6.18034 −0.292320
\(448\) 1.23607 0.0583987
\(449\) 1.79837 0.0848705 0.0424353 0.999099i \(-0.486488\pi\)
0.0424353 + 0.999099i \(0.486488\pi\)
\(450\) 23.0902 1.08848
\(451\) −7.23607 −0.340733
\(452\) −3.14590 −0.147971
\(453\) 17.8885 0.840477
\(454\) −8.94427 −0.419775
\(455\) −6.18034 −0.289739
\(456\) 0 0
\(457\) −26.2148 −1.22628 −0.613138 0.789976i \(-0.710093\pi\)
−0.613138 + 0.789976i \(0.710093\pi\)
\(458\) −6.61803 −0.309240
\(459\) −81.3050 −3.79499
\(460\) −1.05573 −0.0492236
\(461\) −5.41641 −0.252267 −0.126134 0.992013i \(-0.540257\pi\)
−0.126134 + 0.992013i \(0.540257\pi\)
\(462\) −4.94427 −0.230028
\(463\) −22.9443 −1.06631 −0.533155 0.846017i \(-0.678994\pi\)
−0.533155 + 0.846017i \(0.678994\pi\)
\(464\) 2.09017 0.0970337
\(465\) −14.4721 −0.671129
\(466\) 18.5623 0.859882
\(467\) 29.3050 1.35607 0.678036 0.735029i \(-0.262831\pi\)
0.678036 + 0.735029i \(0.262831\pi\)
\(468\) 27.0344 1.24967
\(469\) −2.11146 −0.0974980
\(470\) −6.18034 −0.285078
\(471\) 37.5967 1.73237
\(472\) −8.47214 −0.389962
\(473\) −5.88854 −0.270756
\(474\) 23.4164 1.07555
\(475\) 0 0
\(476\) −6.94427 −0.318290
\(477\) 38.0344 1.74148
\(478\) −7.81966 −0.357663
\(479\) −5.70820 −0.260814 −0.130407 0.991461i \(-0.541628\pi\)
−0.130407 + 0.991461i \(0.541628\pi\)
\(480\) 4.47214 0.204124
\(481\) 38.4164 1.75164
\(482\) −1.41641 −0.0645156
\(483\) 3.05573 0.139040
\(484\) −9.47214 −0.430552
\(485\) −8.81966 −0.400480
\(486\) −35.5967 −1.61470
\(487\) 14.7639 0.669018 0.334509 0.942393i \(-0.391430\pi\)
0.334509 + 0.942393i \(0.391430\pi\)
\(488\) −3.61803 −0.163781
\(489\) −66.8328 −3.02228
\(490\) −7.56231 −0.341630
\(491\) −10.4721 −0.472601 −0.236300 0.971680i \(-0.575935\pi\)
−0.236300 + 0.971680i \(0.575935\pi\)
\(492\) −18.9443 −0.854074
\(493\) −11.7426 −0.528862
\(494\) 0 0
\(495\) −12.7639 −0.573696
\(496\) 3.23607 0.145304
\(497\) −18.4721 −0.828589
\(498\) −27.4164 −1.22856
\(499\) 27.3050 1.22234 0.611169 0.791500i \(-0.290700\pi\)
0.611169 + 0.791500i \(0.290700\pi\)
\(500\) 11.1803 0.500000
\(501\) −11.4164 −0.510047
\(502\) 15.7082 0.701091
\(503\) 11.2361 0.500992 0.250496 0.968118i \(-0.419406\pi\)
0.250496 + 0.968118i \(0.419406\pi\)
\(504\) −9.23607 −0.411407
\(505\) 18.4934 0.822946
\(506\) −0.944272 −0.0419780
\(507\) 0.291796 0.0129591
\(508\) 15.4164 0.683992
\(509\) −17.6869 −0.783959 −0.391979 0.919974i \(-0.628210\pi\)
−0.391979 + 0.919974i \(0.628210\pi\)
\(510\) −25.1246 −1.11254
\(511\) −4.18034 −0.184927
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −20.6180 −0.909422
\(515\) 23.0132 1.01408
\(516\) −15.4164 −0.678670
\(517\) −5.52786 −0.243115
\(518\) −13.1246 −0.576662
\(519\) −46.3607 −2.03501
\(520\) 5.00000 0.219265
\(521\) 28.2705 1.23855 0.619277 0.785173i \(-0.287426\pi\)
0.619277 + 0.785173i \(0.287426\pi\)
\(522\) −15.6180 −0.683583
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −4.47214 −0.195366
\(525\) −12.3607 −0.539464
\(526\) 14.9443 0.651601
\(527\) −18.1803 −0.791948
\(528\) 4.00000 0.174078
\(529\) −22.4164 −0.974626
\(530\) 7.03444 0.305557
\(531\) 63.3050 2.74720
\(532\) 0 0
\(533\) −21.1803 −0.917422
\(534\) −6.94427 −0.300508
\(535\) 24.0689 1.04059
\(536\) 1.70820 0.0737832
\(537\) 11.0557 0.477090
\(538\) 1.32624 0.0571782
\(539\) −6.76393 −0.291343
\(540\) −20.0000 −0.860663
\(541\) −24.3820 −1.04826 −0.524131 0.851637i \(-0.675610\pi\)
−0.524131 + 0.851637i \(0.675610\pi\)
\(542\) −26.0000 −1.11680
\(543\) −69.3050 −2.97416
\(544\) 5.61803 0.240871
\(545\) −24.9230 −1.06758
\(546\) −14.4721 −0.619350
\(547\) 22.9443 0.981026 0.490513 0.871434i \(-0.336809\pi\)
0.490513 + 0.871434i \(0.336809\pi\)
\(548\) 4.61803 0.197273
\(549\) 27.0344 1.15380
\(550\) 3.81966 0.162871
\(551\) 0 0
\(552\) −2.47214 −0.105221
\(553\) −8.94427 −0.380349
\(554\) −1.09017 −0.0463169
\(555\) −47.4853 −2.01564
\(556\) −14.0000 −0.593732
\(557\) 22.9443 0.972180 0.486090 0.873909i \(-0.338423\pi\)
0.486090 + 0.873909i \(0.338423\pi\)
\(558\) −24.1803 −1.02364
\(559\) −17.2361 −0.729008
\(560\) −1.70820 −0.0721848
\(561\) −22.4721 −0.948774
\(562\) 25.9787 1.09585
\(563\) 39.7082 1.67350 0.836751 0.547584i \(-0.184452\pi\)
0.836751 + 0.547584i \(0.184452\pi\)
\(564\) −14.4721 −0.609387
\(565\) 4.34752 0.182902
\(566\) −13.5279 −0.568619
\(567\) 30.1803 1.26746
\(568\) 14.9443 0.627048
\(569\) 19.0902 0.800302 0.400151 0.916449i \(-0.368958\pi\)
0.400151 + 0.916449i \(0.368958\pi\)
\(570\) 0 0
\(571\) 34.5410 1.44550 0.722748 0.691111i \(-0.242879\pi\)
0.722748 + 0.691111i \(0.242879\pi\)
\(572\) 4.47214 0.186989
\(573\) 40.3607 1.68609
\(574\) 7.23607 0.302028
\(575\) −2.36068 −0.0984472
\(576\) 7.47214 0.311339
\(577\) −18.9230 −0.787774 −0.393887 0.919159i \(-0.628870\pi\)
−0.393887 + 0.919159i \(0.628870\pi\)
\(578\) −14.5623 −0.605712
\(579\) −66.2492 −2.75322
\(580\) −2.88854 −0.119940
\(581\) 10.4721 0.434457
\(582\) −20.6525 −0.856073
\(583\) 6.29180 0.260580
\(584\) 3.38197 0.139947
\(585\) −37.3607 −1.54467
\(586\) −5.05573 −0.208850
\(587\) 43.1246 1.77994 0.889972 0.456016i \(-0.150724\pi\)
0.889972 + 0.456016i \(0.150724\pi\)
\(588\) −17.7082 −0.730274
\(589\) 0 0
\(590\) 11.7082 0.482019
\(591\) −35.1246 −1.44483
\(592\) 10.6180 0.436399
\(593\) 29.9230 1.22879 0.614395 0.788999i \(-0.289400\pi\)
0.614395 + 0.788999i \(0.289400\pi\)
\(594\) −17.8885 −0.733976
\(595\) 9.59675 0.393428
\(596\) −1.90983 −0.0782297
\(597\) 4.94427 0.202356
\(598\) −2.76393 −0.113026
\(599\) 1.12461 0.0459504 0.0229752 0.999736i \(-0.492686\pi\)
0.0229752 + 0.999736i \(0.492686\pi\)
\(600\) 10.0000 0.408248
\(601\) 7.52786 0.307068 0.153534 0.988143i \(-0.450935\pi\)
0.153534 + 0.988143i \(0.450935\pi\)
\(602\) 5.88854 0.239999
\(603\) −12.7639 −0.519787
\(604\) 5.52786 0.224926
\(605\) 13.0902 0.532191
\(606\) 43.3050 1.75914
\(607\) 23.7771 0.965082 0.482541 0.875873i \(-0.339714\pi\)
0.482541 + 0.875873i \(0.339714\pi\)
\(608\) 0 0
\(609\) 8.36068 0.338792
\(610\) 5.00000 0.202444
\(611\) −16.1803 −0.654586
\(612\) −41.9787 −1.69689
\(613\) 10.6180 0.428858 0.214429 0.976740i \(-0.431211\pi\)
0.214429 + 0.976740i \(0.431211\pi\)
\(614\) 1.70820 0.0689375
\(615\) 26.1803 1.05569
\(616\) −1.52786 −0.0615594
\(617\) 25.0557 1.00871 0.504353 0.863498i \(-0.331731\pi\)
0.504353 + 0.863498i \(0.331731\pi\)
\(618\) 53.8885 2.16772
\(619\) −22.3607 −0.898752 −0.449376 0.893343i \(-0.648354\pi\)
−0.449376 + 0.893343i \(0.648354\pi\)
\(620\) −4.47214 −0.179605
\(621\) 11.0557 0.443651
\(622\) −26.7639 −1.07314
\(623\) 2.65248 0.106269
\(624\) 11.7082 0.468703
\(625\) 0 0
\(626\) −22.5066 −0.899544
\(627\) 0 0
\(628\) 11.6180 0.463610
\(629\) −59.6525 −2.37850
\(630\) 12.7639 0.508527
\(631\) 9.12461 0.363245 0.181623 0.983368i \(-0.441865\pi\)
0.181623 + 0.983368i \(0.441865\pi\)
\(632\) 7.23607 0.287835
\(633\) −8.94427 −0.355503
\(634\) 14.3262 0.568968
\(635\) −21.3050 −0.845461
\(636\) 16.4721 0.653163
\(637\) −19.7984 −0.784440
\(638\) −2.58359 −0.102285
\(639\) −111.666 −4.41742
\(640\) 1.38197 0.0546270
\(641\) 20.2705 0.800637 0.400319 0.916376i \(-0.368899\pi\)
0.400319 + 0.916376i \(0.368899\pi\)
\(642\) 56.3607 2.22438
\(643\) 14.8328 0.584949 0.292475 0.956273i \(-0.405521\pi\)
0.292475 + 0.956273i \(0.405521\pi\)
\(644\) 0.944272 0.0372095
\(645\) 21.3050 0.838882
\(646\) 0 0
\(647\) −32.0689 −1.26076 −0.630379 0.776288i \(-0.717100\pi\)
−0.630379 + 0.776288i \(0.717100\pi\)
\(648\) −24.4164 −0.959167
\(649\) 10.4721 0.411067
\(650\) 11.1803 0.438529
\(651\) 12.9443 0.507326
\(652\) −20.6525 −0.808813
\(653\) 38.3951 1.50252 0.751259 0.660008i \(-0.229447\pi\)
0.751259 + 0.660008i \(0.229447\pi\)
\(654\) −58.3607 −2.28208
\(655\) 6.18034 0.241486
\(656\) −5.85410 −0.228564
\(657\) −25.2705 −0.985896
\(658\) 5.52786 0.215499
\(659\) 18.5410 0.722256 0.361128 0.932516i \(-0.382392\pi\)
0.361128 + 0.932516i \(0.382392\pi\)
\(660\) −5.52786 −0.215172
\(661\) 5.41641 0.210674 0.105337 0.994437i \(-0.466408\pi\)
0.105337 + 0.994437i \(0.466408\pi\)
\(662\) −12.2918 −0.477734
\(663\) −65.7771 −2.55457
\(664\) −8.47214 −0.328783
\(665\) 0 0
\(666\) −79.3394 −3.07434
\(667\) 1.59675 0.0618263
\(668\) −3.52786 −0.136497
\(669\) 46.2492 1.78810
\(670\) −2.36068 −0.0912010
\(671\) 4.47214 0.172645
\(672\) −4.00000 −0.154303
\(673\) 32.6180 1.25733 0.628666 0.777675i \(-0.283601\pi\)
0.628666 + 0.777675i \(0.283601\pi\)
\(674\) −4.61803 −0.177880
\(675\) −44.7214 −1.72133
\(676\) 0.0901699 0.00346807
\(677\) −4.38197 −0.168413 −0.0842063 0.996448i \(-0.526835\pi\)
−0.0842063 + 0.996448i \(0.526835\pi\)
\(678\) 10.1803 0.390974
\(679\) 7.88854 0.302735
\(680\) −7.76393 −0.297733
\(681\) 28.9443 1.10915
\(682\) −4.00000 −0.153168
\(683\) 2.18034 0.0834284 0.0417142 0.999130i \(-0.486718\pi\)
0.0417142 + 0.999130i \(0.486718\pi\)
\(684\) 0 0
\(685\) −6.38197 −0.243842
\(686\) 15.4164 0.588601
\(687\) 21.4164 0.817087
\(688\) −4.76393 −0.181623
\(689\) 18.4164 0.701609
\(690\) 3.41641 0.130060
\(691\) 22.3607 0.850640 0.425320 0.905043i \(-0.360161\pi\)
0.425320 + 0.905043i \(0.360161\pi\)
\(692\) −14.3262 −0.544602
\(693\) 11.4164 0.433673
\(694\) −28.6525 −1.08763
\(695\) 19.3475 0.733893
\(696\) −6.76393 −0.256386
\(697\) 32.8885 1.24574
\(698\) 21.9098 0.829299
\(699\) −60.0689 −2.27201
\(700\) −3.81966 −0.144370
\(701\) −39.6312 −1.49685 −0.748425 0.663220i \(-0.769189\pi\)
−0.748425 + 0.663220i \(0.769189\pi\)
\(702\) −52.3607 −1.97623
\(703\) 0 0
\(704\) 1.23607 0.0465861
\(705\) 20.0000 0.753244
\(706\) −18.7426 −0.705389
\(707\) −16.5410 −0.622089
\(708\) 27.4164 1.03037
\(709\) 18.0902 0.679391 0.339695 0.940536i \(-0.389676\pi\)
0.339695 + 0.940536i \(0.389676\pi\)
\(710\) −20.6525 −0.775074
\(711\) −54.0689 −2.02774
\(712\) −2.14590 −0.0804209
\(713\) 2.47214 0.0925822
\(714\) 22.4721 0.840999
\(715\) −6.18034 −0.231132
\(716\) 3.41641 0.127677
\(717\) 25.3050 0.945031
\(718\) −20.4721 −0.764013
\(719\) 25.7082 0.958754 0.479377 0.877609i \(-0.340863\pi\)
0.479377 + 0.877609i \(0.340863\pi\)
\(720\) −10.3262 −0.384836
\(721\) −20.5836 −0.766573
\(722\) 0 0
\(723\) 4.58359 0.170466
\(724\) −21.4164 −0.795935
\(725\) −6.45898 −0.239881
\(726\) 30.6525 1.13762
\(727\) −13.0557 −0.484210 −0.242105 0.970250i \(-0.577838\pi\)
−0.242105 + 0.970250i \(0.577838\pi\)
\(728\) −4.47214 −0.165748
\(729\) 41.9443 1.55349
\(730\) −4.67376 −0.172984
\(731\) 26.7639 0.989900
\(732\) 11.7082 0.432748
\(733\) 10.0902 0.372689 0.186344 0.982484i \(-0.440336\pi\)
0.186344 + 0.982484i \(0.440336\pi\)
\(734\) 34.8328 1.28570
\(735\) 24.4721 0.902668
\(736\) −0.763932 −0.0281589
\(737\) −2.11146 −0.0777765
\(738\) 43.7426 1.61019
\(739\) −31.1246 −1.14494 −0.572469 0.819927i \(-0.694014\pi\)
−0.572469 + 0.819927i \(0.694014\pi\)
\(740\) −14.6738 −0.539418
\(741\) 0 0
\(742\) −6.29180 −0.230979
\(743\) 41.8885 1.53674 0.768371 0.640005i \(-0.221068\pi\)
0.768371 + 0.640005i \(0.221068\pi\)
\(744\) −10.4721 −0.383927
\(745\) 2.63932 0.0966972
\(746\) −2.67376 −0.0978934
\(747\) 63.3050 2.31621
\(748\) −6.94427 −0.253908
\(749\) −21.5279 −0.786611
\(750\) −36.1803 −1.32112
\(751\) 34.0689 1.24319 0.621596 0.783338i \(-0.286485\pi\)
0.621596 + 0.783338i \(0.286485\pi\)
\(752\) −4.47214 −0.163082
\(753\) −50.8328 −1.85245
\(754\) −7.56231 −0.275403
\(755\) −7.63932 −0.278023
\(756\) 17.8885 0.650600
\(757\) 24.5623 0.892732 0.446366 0.894850i \(-0.352718\pi\)
0.446366 + 0.894850i \(0.352718\pi\)
\(758\) 8.58359 0.311770
\(759\) 3.05573 0.110916
\(760\) 0 0
\(761\) −18.3607 −0.665574 −0.332787 0.943002i \(-0.607989\pi\)
−0.332787 + 0.943002i \(0.607989\pi\)
\(762\) −49.8885 −1.80727
\(763\) 22.2918 0.807017
\(764\) 12.4721 0.451226
\(765\) 58.0132 2.09747
\(766\) −25.2361 −0.911816
\(767\) 30.6525 1.10680
\(768\) 3.23607 0.116772
\(769\) −3.74265 −0.134963 −0.0674816 0.997721i \(-0.521496\pi\)
−0.0674816 + 0.997721i \(0.521496\pi\)
\(770\) 2.11146 0.0760916
\(771\) 66.7214 2.40291
\(772\) −20.4721 −0.736808
\(773\) −14.8541 −0.534265 −0.267132 0.963660i \(-0.586076\pi\)
−0.267132 + 0.963660i \(0.586076\pi\)
\(774\) 35.5967 1.27950
\(775\) −10.0000 −0.359211
\(776\) −6.38197 −0.229099
\(777\) 42.4721 1.52368
\(778\) 11.7984 0.422992
\(779\) 0 0
\(780\) −16.1803 −0.579349
\(781\) −18.4721 −0.660985
\(782\) 4.29180 0.153474
\(783\) 30.2492 1.08102
\(784\) −5.47214 −0.195433
\(785\) −16.0557 −0.573054
\(786\) 14.4721 0.516204
\(787\) 19.5279 0.696093 0.348047 0.937477i \(-0.386845\pi\)
0.348047 + 0.937477i \(0.386845\pi\)
\(788\) −10.8541 −0.386661
\(789\) −48.3607 −1.72169
\(790\) −10.0000 −0.355784
\(791\) −3.88854 −0.138261
\(792\) −9.23607 −0.328189
\(793\) 13.0902 0.464846
\(794\) −22.3607 −0.793551
\(795\) −22.7639 −0.807353
\(796\) 1.52786 0.0541537
\(797\) 48.5623 1.72017 0.860083 0.510155i \(-0.170412\pi\)
0.860083 + 0.510155i \(0.170412\pi\)
\(798\) 0 0
\(799\) 25.1246 0.888845
\(800\) 3.09017 0.109254
\(801\) 16.0344 0.566549
\(802\) −1.05573 −0.0372791
\(803\) −4.18034 −0.147521
\(804\) −5.52786 −0.194953
\(805\) −1.30495 −0.0459935
\(806\) −11.7082 −0.412404
\(807\) −4.29180 −0.151078
\(808\) 13.3820 0.470776
\(809\) −39.7426 −1.39728 −0.698639 0.715475i \(-0.746210\pi\)
−0.698639 + 0.715475i \(0.746210\pi\)
\(810\) 33.7426 1.18560
\(811\) −43.5967 −1.53089 −0.765444 0.643502i \(-0.777481\pi\)
−0.765444 + 0.643502i \(0.777481\pi\)
\(812\) 2.58359 0.0906663
\(813\) 84.1378 2.95084
\(814\) −13.1246 −0.460017
\(815\) 28.5410 0.999748
\(816\) −18.1803 −0.636439
\(817\) 0 0
\(818\) 17.2705 0.603849
\(819\) 33.4164 1.16766
\(820\) 8.09017 0.282521
\(821\) −14.2705 −0.498044 −0.249022 0.968498i \(-0.580109\pi\)
−0.249022 + 0.968498i \(0.580109\pi\)
\(822\) −14.9443 −0.521241
\(823\) 32.8328 1.14448 0.572240 0.820086i \(-0.306075\pi\)
0.572240 + 0.820086i \(0.306075\pi\)
\(824\) 16.6525 0.580116
\(825\) −12.3607 −0.430344
\(826\) −10.4721 −0.364372
\(827\) 36.5410 1.27066 0.635328 0.772243i \(-0.280865\pi\)
0.635328 + 0.772243i \(0.280865\pi\)
\(828\) 5.70820 0.198374
\(829\) −43.2705 −1.50285 −0.751423 0.659820i \(-0.770632\pi\)
−0.751423 + 0.659820i \(0.770632\pi\)
\(830\) 11.7082 0.406398
\(831\) 3.52786 0.122380
\(832\) 3.61803 0.125433
\(833\) 30.7426 1.06517
\(834\) 45.3050 1.56878
\(835\) 4.87539 0.168720
\(836\) 0 0
\(837\) 46.8328 1.61878
\(838\) −2.47214 −0.0853985
\(839\) 22.0000 0.759524 0.379762 0.925084i \(-0.376006\pi\)
0.379762 + 0.925084i \(0.376006\pi\)
\(840\) 5.52786 0.190729
\(841\) −24.6312 −0.849351
\(842\) −25.0902 −0.864664
\(843\) −84.0689 −2.89549
\(844\) −2.76393 −0.0951385
\(845\) −0.124612 −0.00428678
\(846\) 33.4164 1.14888
\(847\) −11.7082 −0.402299
\(848\) 5.09017 0.174797
\(849\) 43.7771 1.50243
\(850\) −17.3607 −0.595466
\(851\) 8.11146 0.278057
\(852\) −48.3607 −1.65681
\(853\) −17.9098 −0.613221 −0.306610 0.951835i \(-0.599195\pi\)
−0.306610 + 0.951835i \(0.599195\pi\)
\(854\) −4.47214 −0.153033
\(855\) 0 0
\(856\) 17.4164 0.595281
\(857\) −14.5623 −0.497439 −0.248719 0.968576i \(-0.580010\pi\)
−0.248719 + 0.968576i \(0.580010\pi\)
\(858\) −14.4721 −0.494071
\(859\) −30.0689 −1.02594 −0.512969 0.858407i \(-0.671454\pi\)
−0.512969 + 0.858407i \(0.671454\pi\)
\(860\) 6.58359 0.224499
\(861\) −23.4164 −0.798029
\(862\) 24.6525 0.839667
\(863\) 9.88854 0.336610 0.168305 0.985735i \(-0.446171\pi\)
0.168305 + 0.985735i \(0.446171\pi\)
\(864\) −14.4721 −0.492352
\(865\) 19.7984 0.673165
\(866\) −4.03444 −0.137096
\(867\) 47.1246 1.60044
\(868\) 4.00000 0.135769
\(869\) −8.94427 −0.303414
\(870\) 9.34752 0.316911
\(871\) −6.18034 −0.209413
\(872\) −18.0344 −0.610723
\(873\) 47.6869 1.61396
\(874\) 0 0
\(875\) 13.8197 0.467190
\(876\) −10.9443 −0.369773
\(877\) −40.7426 −1.37578 −0.687891 0.725814i \(-0.741463\pi\)
−0.687891 + 0.725814i \(0.741463\pi\)
\(878\) −24.7639 −0.835742
\(879\) 16.3607 0.551832
\(880\) −1.70820 −0.0575835
\(881\) −46.2148 −1.55702 −0.778508 0.627635i \(-0.784023\pi\)
−0.778508 + 0.627635i \(0.784023\pi\)
\(882\) 40.8885 1.37679
\(883\) −40.1803 −1.35218 −0.676088 0.736821i \(-0.736326\pi\)
−0.676088 + 0.736821i \(0.736326\pi\)
\(884\) −20.3262 −0.683645
\(885\) −37.8885 −1.27361
\(886\) −4.76393 −0.160047
\(887\) −10.9443 −0.367473 −0.183736 0.982976i \(-0.558819\pi\)
−0.183736 + 0.982976i \(0.558819\pi\)
\(888\) −34.3607 −1.15307
\(889\) 19.0557 0.639109
\(890\) 2.96556 0.0994057
\(891\) 30.1803 1.01108
\(892\) 14.2918 0.478525
\(893\) 0 0
\(894\) 6.18034 0.206701
\(895\) −4.72136 −0.157818
\(896\) −1.23607 −0.0412941
\(897\) 8.94427 0.298641
\(898\) −1.79837 −0.0600125
\(899\) 6.76393 0.225590
\(900\) −23.0902 −0.769672
\(901\) −28.5967 −0.952696
\(902\) 7.23607 0.240935
\(903\) −19.0557 −0.634135
\(904\) 3.14590 0.104631
\(905\) 29.5967 0.983829
\(906\) −17.8885 −0.594307
\(907\) −13.7082 −0.455173 −0.227587 0.973758i \(-0.573084\pi\)
−0.227587 + 0.973758i \(0.573084\pi\)
\(908\) 8.94427 0.296826
\(909\) −99.9919 −3.31652
\(910\) 6.18034 0.204876
\(911\) −46.0689 −1.52633 −0.763165 0.646204i \(-0.776356\pi\)
−0.763165 + 0.646204i \(0.776356\pi\)
\(912\) 0 0
\(913\) 10.4721 0.346577
\(914\) 26.2148 0.867108
\(915\) −16.1803 −0.534906
\(916\) 6.61803 0.218666
\(917\) −5.52786 −0.182546
\(918\) 81.3050 2.68346
\(919\) 10.1115 0.333546 0.166773 0.985995i \(-0.446665\pi\)
0.166773 + 0.985995i \(0.446665\pi\)
\(920\) 1.05573 0.0348063
\(921\) −5.52786 −0.182149
\(922\) 5.41641 0.178380
\(923\) −54.0689 −1.77970
\(924\) 4.94427 0.162655
\(925\) −32.8115 −1.07884
\(926\) 22.9443 0.753996
\(927\) −124.430 −4.08680
\(928\) −2.09017 −0.0686132
\(929\) 10.1591 0.333308 0.166654 0.986015i \(-0.446704\pi\)
0.166654 + 0.986015i \(0.446704\pi\)
\(930\) 14.4721 0.474560
\(931\) 0 0
\(932\) −18.5623 −0.608029
\(933\) 86.6099 2.83548
\(934\) −29.3050 −0.958887
\(935\) 9.59675 0.313847
\(936\) −27.0344 −0.883648
\(937\) −19.8885 −0.649730 −0.324865 0.945760i \(-0.605319\pi\)
−0.324865 + 0.945760i \(0.605319\pi\)
\(938\) 2.11146 0.0689415
\(939\) 72.8328 2.37681
\(940\) 6.18034 0.201580
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) −37.5967 −1.22497
\(943\) −4.47214 −0.145633
\(944\) 8.47214 0.275745
\(945\) −24.7214 −0.804186
\(946\) 5.88854 0.191453
\(947\) −14.1803 −0.460799 −0.230400 0.973096i \(-0.574003\pi\)
−0.230400 + 0.973096i \(0.574003\pi\)
\(948\) −23.4164 −0.760530
\(949\) −12.2361 −0.397200
\(950\) 0 0
\(951\) −46.3607 −1.50335
\(952\) 6.94427 0.225065
\(953\) −52.8115 −1.71073 −0.855367 0.518023i \(-0.826668\pi\)
−0.855367 + 0.518023i \(0.826668\pi\)
\(954\) −38.0344 −1.23141
\(955\) −17.2361 −0.557746
\(956\) 7.81966 0.252906
\(957\) 8.36068 0.270262
\(958\) 5.70820 0.184424
\(959\) 5.70820 0.184328
\(960\) −4.47214 −0.144338
\(961\) −20.5279 −0.662189
\(962\) −38.4164 −1.23859
\(963\) −130.138 −4.19363
\(964\) 1.41641 0.0456194
\(965\) 28.2918 0.910745
\(966\) −3.05573 −0.0983164
\(967\) −44.8328 −1.44173 −0.720863 0.693077i \(-0.756254\pi\)
−0.720863 + 0.693077i \(0.756254\pi\)
\(968\) 9.47214 0.304446
\(969\) 0 0
\(970\) 8.81966 0.283182
\(971\) −6.54102 −0.209911 −0.104956 0.994477i \(-0.533470\pi\)
−0.104956 + 0.994477i \(0.533470\pi\)
\(972\) 35.5967 1.14177
\(973\) −17.3050 −0.554771
\(974\) −14.7639 −0.473067
\(975\) −36.1803 −1.15870
\(976\) 3.61803 0.115810
\(977\) 20.0344 0.640959 0.320479 0.947256i \(-0.396156\pi\)
0.320479 + 0.947256i \(0.396156\pi\)
\(978\) 66.8328 2.13708
\(979\) 2.65248 0.0847735
\(980\) 7.56231 0.241569
\(981\) 134.756 4.30242
\(982\) 10.4721 0.334179
\(983\) −29.5967 −0.943990 −0.471995 0.881601i \(-0.656466\pi\)
−0.471995 + 0.881601i \(0.656466\pi\)
\(984\) 18.9443 0.603921
\(985\) 15.0000 0.477940
\(986\) 11.7426 0.373962
\(987\) −17.8885 −0.569399
\(988\) 0 0
\(989\) −3.63932 −0.115724
\(990\) 12.7639 0.405664
\(991\) 0.652476 0.0207266 0.0103633 0.999946i \(-0.496701\pi\)
0.0103633 + 0.999946i \(0.496701\pi\)
\(992\) −3.23607 −0.102745
\(993\) 39.7771 1.26229
\(994\) 18.4721 0.585901
\(995\) −2.11146 −0.0669377
\(996\) 27.4164 0.868722
\(997\) 24.5066 0.776131 0.388066 0.921632i \(-0.373143\pi\)
0.388066 + 0.921632i \(0.373143\pi\)
\(998\) −27.3050 −0.864323
\(999\) 153.666 4.86177
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 722.2.a.h.1.2 2
3.2 odd 2 6498.2.a.bk.1.1 2
4.3 odd 2 5776.2.a.t.1.1 2
19.2 odd 18 722.2.e.p.99.1 12
19.3 odd 18 722.2.e.p.389.1 12
19.4 even 9 722.2.e.q.415.1 12
19.5 even 9 722.2.e.q.595.1 12
19.6 even 9 722.2.e.q.245.2 12
19.7 even 3 722.2.c.i.429.1 4
19.8 odd 6 722.2.c.h.653.2 4
19.9 even 9 722.2.e.q.423.2 12
19.10 odd 18 722.2.e.p.423.1 12
19.11 even 3 722.2.c.i.653.1 4
19.12 odd 6 722.2.c.h.429.2 4
19.13 odd 18 722.2.e.p.245.1 12
19.14 odd 18 722.2.e.p.595.2 12
19.15 odd 18 722.2.e.p.415.2 12
19.16 even 9 722.2.e.q.389.2 12
19.17 even 9 722.2.e.q.99.2 12
19.18 odd 2 722.2.a.i.1.1 yes 2
57.56 even 2 6498.2.a.be.1.1 2
76.75 even 2 5776.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.h.1.2 2 1.1 even 1 trivial
722.2.a.i.1.1 yes 2 19.18 odd 2
722.2.c.h.429.2 4 19.12 odd 6
722.2.c.h.653.2 4 19.8 odd 6
722.2.c.i.429.1 4 19.7 even 3
722.2.c.i.653.1 4 19.11 even 3
722.2.e.p.99.1 12 19.2 odd 18
722.2.e.p.245.1 12 19.13 odd 18
722.2.e.p.389.1 12 19.3 odd 18
722.2.e.p.415.2 12 19.15 odd 18
722.2.e.p.423.1 12 19.10 odd 18
722.2.e.p.595.2 12 19.14 odd 18
722.2.e.q.99.2 12 19.17 even 9
722.2.e.q.245.2 12 19.6 even 9
722.2.e.q.389.2 12 19.16 even 9
722.2.e.q.415.1 12 19.4 even 9
722.2.e.q.423.2 12 19.9 even 9
722.2.e.q.595.1 12 19.5 even 9
5776.2.a.t.1.1 2 4.3 odd 2
5776.2.a.be.1.2 2 76.75 even 2
6498.2.a.be.1.1 2 57.56 even 2
6498.2.a.bk.1.1 2 3.2 odd 2