Properties

Label 1155.2.a.o.1.2
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
Analytic rank $1$
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] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1155.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155 q^{2} -1.00000 q^{3} +0.438447 q^{4} +1.00000 q^{5} -1.56155 q^{6} -1.00000 q^{7} -2.43845 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.56155 q^{2} -1.00000 q^{3} +0.438447 q^{4} +1.00000 q^{5} -1.56155 q^{6} -1.00000 q^{7} -2.43845 q^{8} +1.00000 q^{9} +1.56155 q^{10} +1.00000 q^{11} -0.438447 q^{12} -7.12311 q^{13} -1.56155 q^{14} -1.00000 q^{15} -4.68466 q^{16} +0.561553 q^{17} +1.56155 q^{18} -2.56155 q^{19} +0.438447 q^{20} +1.00000 q^{21} +1.56155 q^{22} -1.43845 q^{23} +2.43845 q^{24} +1.00000 q^{25} -11.1231 q^{26} -1.00000 q^{27} -0.438447 q^{28} +1.68466 q^{29} -1.56155 q^{30} -5.12311 q^{31} -2.43845 q^{32} -1.00000 q^{33} +0.876894 q^{34} -1.00000 q^{35} +0.438447 q^{36} -7.12311 q^{37} -4.00000 q^{38} +7.12311 q^{39} -2.43845 q^{40} +2.00000 q^{41} +1.56155 q^{42} +2.56155 q^{43} +0.438447 q^{44} +1.00000 q^{45} -2.24621 q^{46} -5.12311 q^{47} +4.68466 q^{48} +1.00000 q^{49} +1.56155 q^{50} -0.561553 q^{51} -3.12311 q^{52} -13.6847 q^{53} -1.56155 q^{54} +1.00000 q^{55} +2.43845 q^{56} +2.56155 q^{57} +2.63068 q^{58} +10.5616 q^{59} -0.438447 q^{60} -3.43845 q^{61} -8.00000 q^{62} -1.00000 q^{63} +5.56155 q^{64} -7.12311 q^{65} -1.56155 q^{66} +6.87689 q^{67} +0.246211 q^{68} +1.43845 q^{69} -1.56155 q^{70} +5.12311 q^{71} -2.43845 q^{72} -11.1231 q^{73} -11.1231 q^{74} -1.00000 q^{75} -1.12311 q^{76} -1.00000 q^{77} +11.1231 q^{78} -10.2462 q^{79} -4.68466 q^{80} +1.00000 q^{81} +3.12311 q^{82} -5.43845 q^{83} +0.438447 q^{84} +0.561553 q^{85} +4.00000 q^{86} -1.68466 q^{87} -2.43845 q^{88} -2.31534 q^{89} +1.56155 q^{90} +7.12311 q^{91} -0.630683 q^{92} +5.12311 q^{93} -8.00000 q^{94} -2.56155 q^{95} +2.43845 q^{96} +2.80776 q^{97} +1.56155 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} - 2 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} - 2 q^{7} - 9 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} - 5 q^{12} - 6 q^{13} + q^{14} - 2 q^{15} + 3 q^{16} - 3 q^{17} - q^{18} - q^{19} + 5 q^{20} + 2 q^{21} - q^{22} - 7 q^{23} + 9 q^{24} + 2 q^{25} - 14 q^{26} - 2 q^{27} - 5 q^{28} - 9 q^{29} + q^{30} - 2 q^{31} - 9 q^{32} - 2 q^{33} + 10 q^{34} - 2 q^{35} + 5 q^{36} - 6 q^{37} - 8 q^{38} + 6 q^{39} - 9 q^{40} + 4 q^{41} - q^{42} + q^{43} + 5 q^{44} + 2 q^{45} + 12 q^{46} - 2 q^{47} - 3 q^{48} + 2 q^{49} - q^{50} + 3 q^{51} + 2 q^{52} - 15 q^{53} + q^{54} + 2 q^{55} + 9 q^{56} + q^{57} + 30 q^{58} + 17 q^{59} - 5 q^{60} - 11 q^{61} - 16 q^{62} - 2 q^{63} + 7 q^{64} - 6 q^{65} + q^{66} + 22 q^{67} - 16 q^{68} + 7 q^{69} + q^{70} + 2 q^{71} - 9 q^{72} - 14 q^{73} - 14 q^{74} - 2 q^{75} + 6 q^{76} - 2 q^{77} + 14 q^{78} - 4 q^{79} + 3 q^{80} + 2 q^{81} - 2 q^{82} - 15 q^{83} + 5 q^{84} - 3 q^{85} + 8 q^{86} + 9 q^{87} - 9 q^{88} - 17 q^{89} - q^{90} + 6 q^{91} - 26 q^{92} + 2 q^{93} - 16 q^{94} - q^{95} + 9 q^{96} - 15 q^{97} - q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 1.10418 0.552092 0.833783i \(-0.313830\pi\)
0.552092 + 0.833783i \(0.313830\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.438447 0.219224
\(5\) 1.00000 0.447214
\(6\) −1.56155 −0.637501
\(7\) −1.00000 −0.377964
\(8\) −2.43845 −0.862121
\(9\) 1.00000 0.333333
\(10\) 1.56155 0.493806
\(11\) 1.00000 0.301511
\(12\) −0.438447 −0.126569
\(13\) −7.12311 −1.97559 −0.987797 0.155747i \(-0.950222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) −1.56155 −0.417343
\(15\) −1.00000 −0.258199
\(16\) −4.68466 −1.17116
\(17\) 0.561553 0.136197 0.0680983 0.997679i \(-0.478307\pi\)
0.0680983 + 0.997679i \(0.478307\pi\)
\(18\) 1.56155 0.368062
\(19\) −2.56155 −0.587661 −0.293830 0.955858i \(-0.594930\pi\)
−0.293830 + 0.955858i \(0.594930\pi\)
\(20\) 0.438447 0.0980398
\(21\) 1.00000 0.218218
\(22\) 1.56155 0.332924
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) 2.43845 0.497746
\(25\) 1.00000 0.200000
\(26\) −11.1231 −2.18142
\(27\) −1.00000 −0.192450
\(28\) −0.438447 −0.0828587
\(29\) 1.68466 0.312833 0.156417 0.987691i \(-0.450006\pi\)
0.156417 + 0.987691i \(0.450006\pi\)
\(30\) −1.56155 −0.285099
\(31\) −5.12311 −0.920137 −0.460068 0.887883i \(-0.652175\pi\)
−0.460068 + 0.887883i \(0.652175\pi\)
\(32\) −2.43845 −0.431061
\(33\) −1.00000 −0.174078
\(34\) 0.876894 0.150386
\(35\) −1.00000 −0.169031
\(36\) 0.438447 0.0730745
\(37\) −7.12311 −1.17103 −0.585516 0.810661i \(-0.699108\pi\)
−0.585516 + 0.810661i \(0.699108\pi\)
\(38\) −4.00000 −0.648886
\(39\) 7.12311 1.14061
\(40\) −2.43845 −0.385552
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.56155 0.240953
\(43\) 2.56155 0.390633 0.195317 0.980740i \(-0.437427\pi\)
0.195317 + 0.980740i \(0.437427\pi\)
\(44\) 0.438447 0.0660984
\(45\) 1.00000 0.149071
\(46\) −2.24621 −0.331186
\(47\) −5.12311 −0.747282 −0.373641 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(48\) 4.68466 0.676172
\(49\) 1.00000 0.142857
\(50\) 1.56155 0.220837
\(51\) −0.561553 −0.0786331
\(52\) −3.12311 −0.433097
\(53\) −13.6847 −1.87973 −0.939866 0.341543i \(-0.889051\pi\)
−0.939866 + 0.341543i \(0.889051\pi\)
\(54\) −1.56155 −0.212500
\(55\) 1.00000 0.134840
\(56\) 2.43845 0.325851
\(57\) 2.56155 0.339286
\(58\) 2.63068 0.345426
\(59\) 10.5616 1.37500 0.687499 0.726186i \(-0.258709\pi\)
0.687499 + 0.726186i \(0.258709\pi\)
\(60\) −0.438447 −0.0566033
\(61\) −3.43845 −0.440248 −0.220124 0.975472i \(-0.570646\pi\)
−0.220124 + 0.975472i \(0.570646\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 5.56155 0.695194
\(65\) −7.12311 −0.883513
\(66\) −1.56155 −0.192214
\(67\) 6.87689 0.840146 0.420073 0.907490i \(-0.362004\pi\)
0.420073 + 0.907490i \(0.362004\pi\)
\(68\) 0.246211 0.0298575
\(69\) 1.43845 0.173169
\(70\) −1.56155 −0.186641
\(71\) 5.12311 0.608001 0.304000 0.952672i \(-0.401678\pi\)
0.304000 + 0.952672i \(0.401678\pi\)
\(72\) −2.43845 −0.287374
\(73\) −11.1231 −1.30186 −0.650931 0.759137i \(-0.725621\pi\)
−0.650931 + 0.759137i \(0.725621\pi\)
\(74\) −11.1231 −1.29303
\(75\) −1.00000 −0.115470
\(76\) −1.12311 −0.128829
\(77\) −1.00000 −0.113961
\(78\) 11.1231 1.25944
\(79\) −10.2462 −1.15279 −0.576394 0.817172i \(-0.695541\pi\)
−0.576394 + 0.817172i \(0.695541\pi\)
\(80\) −4.68466 −0.523761
\(81\) 1.00000 0.111111
\(82\) 3.12311 0.344889
\(83\) −5.43845 −0.596947 −0.298474 0.954418i \(-0.596477\pi\)
−0.298474 + 0.954418i \(0.596477\pi\)
\(84\) 0.438447 0.0478385
\(85\) 0.561553 0.0609090
\(86\) 4.00000 0.431331
\(87\) −1.68466 −0.180614
\(88\) −2.43845 −0.259939
\(89\) −2.31534 −0.245426 −0.122713 0.992442i \(-0.539159\pi\)
−0.122713 + 0.992442i \(0.539159\pi\)
\(90\) 1.56155 0.164602
\(91\) 7.12311 0.746704
\(92\) −0.630683 −0.0657533
\(93\) 5.12311 0.531241
\(94\) −8.00000 −0.825137
\(95\) −2.56155 −0.262810
\(96\) 2.43845 0.248873
\(97\) 2.80776 0.285085 0.142543 0.989789i \(-0.454472\pi\)
0.142543 + 0.989789i \(0.454472\pi\)
\(98\) 1.56155 0.157741
\(99\) 1.00000 0.100504
\(100\) 0.438447 0.0438447
\(101\) −12.2462 −1.21854 −0.609272 0.792961i \(-0.708538\pi\)
−0.609272 + 0.792961i \(0.708538\pi\)
\(102\) −0.876894 −0.0868255
\(103\) 3.68466 0.363060 0.181530 0.983385i \(-0.441895\pi\)
0.181530 + 0.983385i \(0.441895\pi\)
\(104\) 17.3693 1.70320
\(105\) 1.00000 0.0975900
\(106\) −21.3693 −2.07557
\(107\) 19.3693 1.87250 0.936251 0.351331i \(-0.114271\pi\)
0.936251 + 0.351331i \(0.114271\pi\)
\(108\) −0.438447 −0.0421896
\(109\) 18.4924 1.77125 0.885626 0.464398i \(-0.153729\pi\)
0.885626 + 0.464398i \(0.153729\pi\)
\(110\) 1.56155 0.148888
\(111\) 7.12311 0.676095
\(112\) 4.68466 0.442659
\(113\) 8.56155 0.805403 0.402702 0.915331i \(-0.368071\pi\)
0.402702 + 0.915331i \(0.368071\pi\)
\(114\) 4.00000 0.374634
\(115\) −1.43845 −0.134136
\(116\) 0.738634 0.0685804
\(117\) −7.12311 −0.658531
\(118\) 16.4924 1.51825
\(119\) −0.561553 −0.0514775
\(120\) 2.43845 0.222599
\(121\) 1.00000 0.0909091
\(122\) −5.36932 −0.486115
\(123\) −2.00000 −0.180334
\(124\) −2.24621 −0.201716
\(125\) 1.00000 0.0894427
\(126\) −1.56155 −0.139114
\(127\) 0.807764 0.0716775 0.0358387 0.999358i \(-0.488590\pi\)
0.0358387 + 0.999358i \(0.488590\pi\)
\(128\) 13.5616 1.19868
\(129\) −2.56155 −0.225532
\(130\) −11.1231 −0.975561
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −0.438447 −0.0381619
\(133\) 2.56155 0.222115
\(134\) 10.7386 0.927677
\(135\) −1.00000 −0.0860663
\(136\) −1.36932 −0.117418
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 2.24621 0.191210
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −0.438447 −0.0370556
\(141\) 5.12311 0.431443
\(142\) 8.00000 0.671345
\(143\) −7.12311 −0.595664
\(144\) −4.68466 −0.390388
\(145\) 1.68466 0.139903
\(146\) −17.3693 −1.43749
\(147\) −1.00000 −0.0824786
\(148\) −3.12311 −0.256718
\(149\) 0.246211 0.0201704 0.0100852 0.999949i \(-0.496790\pi\)
0.0100852 + 0.999949i \(0.496790\pi\)
\(150\) −1.56155 −0.127500
\(151\) 5.12311 0.416912 0.208456 0.978032i \(-0.433156\pi\)
0.208456 + 0.978032i \(0.433156\pi\)
\(152\) 6.24621 0.506635
\(153\) 0.561553 0.0453989
\(154\) −1.56155 −0.125834
\(155\) −5.12311 −0.411498
\(156\) 3.12311 0.250049
\(157\) −11.4384 −0.912887 −0.456444 0.889752i \(-0.650877\pi\)
−0.456444 + 0.889752i \(0.650877\pi\)
\(158\) −16.0000 −1.27289
\(159\) 13.6847 1.08526
\(160\) −2.43845 −0.192776
\(161\) 1.43845 0.113366
\(162\) 1.56155 0.122687
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0.876894 0.0684739
\(165\) −1.00000 −0.0778499
\(166\) −8.49242 −0.659140
\(167\) 2.24621 0.173817 0.0869085 0.996216i \(-0.472301\pi\)
0.0869085 + 0.996216i \(0.472301\pi\)
\(168\) −2.43845 −0.188130
\(169\) 37.7386 2.90297
\(170\) 0.876894 0.0672547
\(171\) −2.56155 −0.195887
\(172\) 1.12311 0.0856360
\(173\) −4.24621 −0.322833 −0.161417 0.986886i \(-0.551606\pi\)
−0.161417 + 0.986886i \(0.551606\pi\)
\(174\) −2.63068 −0.199432
\(175\) −1.00000 −0.0755929
\(176\) −4.68466 −0.353119
\(177\) −10.5616 −0.793855
\(178\) −3.61553 −0.270995
\(179\) −19.3693 −1.44773 −0.723865 0.689941i \(-0.757636\pi\)
−0.723865 + 0.689941i \(0.757636\pi\)
\(180\) 0.438447 0.0326799
\(181\) −20.2462 −1.50489 −0.752445 0.658656i \(-0.771125\pi\)
−0.752445 + 0.658656i \(0.771125\pi\)
\(182\) 11.1231 0.824499
\(183\) 3.43845 0.254177
\(184\) 3.50758 0.258582
\(185\) −7.12311 −0.523701
\(186\) 8.00000 0.586588
\(187\) 0.561553 0.0410648
\(188\) −2.24621 −0.163822
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −5.56155 −0.401371
\(193\) 22.4924 1.61904 0.809520 0.587092i \(-0.199727\pi\)
0.809520 + 0.587092i \(0.199727\pi\)
\(194\) 4.38447 0.314787
\(195\) 7.12311 0.510096
\(196\) 0.438447 0.0313177
\(197\) −23.1231 −1.64745 −0.823727 0.566987i \(-0.808109\pi\)
−0.823727 + 0.566987i \(0.808109\pi\)
\(198\) 1.56155 0.110975
\(199\) 20.4924 1.45267 0.726335 0.687341i \(-0.241222\pi\)
0.726335 + 0.687341i \(0.241222\pi\)
\(200\) −2.43845 −0.172424
\(201\) −6.87689 −0.485059
\(202\) −19.1231 −1.34550
\(203\) −1.68466 −0.118240
\(204\) −0.246211 −0.0172382
\(205\) 2.00000 0.139686
\(206\) 5.75379 0.400885
\(207\) −1.43845 −0.0999790
\(208\) 33.3693 2.31375
\(209\) −2.56155 −0.177186
\(210\) 1.56155 0.107757
\(211\) −3.36932 −0.231953 −0.115977 0.993252i \(-0.537000\pi\)
−0.115977 + 0.993252i \(0.537000\pi\)
\(212\) −6.00000 −0.412082
\(213\) −5.12311 −0.351029
\(214\) 30.2462 2.06759
\(215\) 2.56155 0.174696
\(216\) 2.43845 0.165915
\(217\) 5.12311 0.347779
\(218\) 28.8769 1.95579
\(219\) 11.1231 0.751630
\(220\) 0.438447 0.0295601
\(221\) −4.00000 −0.269069
\(222\) 11.1231 0.746534
\(223\) 14.5616 0.975114 0.487557 0.873091i \(-0.337888\pi\)
0.487557 + 0.873091i \(0.337888\pi\)
\(224\) 2.43845 0.162926
\(225\) 1.00000 0.0666667
\(226\) 13.3693 0.889314
\(227\) −25.9309 −1.72109 −0.860546 0.509373i \(-0.829878\pi\)
−0.860546 + 0.509373i \(0.829878\pi\)
\(228\) 1.12311 0.0743795
\(229\) 10.4924 0.693359 0.346679 0.937984i \(-0.387309\pi\)
0.346679 + 0.937984i \(0.387309\pi\)
\(230\) −2.24621 −0.148111
\(231\) 1.00000 0.0657952
\(232\) −4.10795 −0.269700
\(233\) −23.6155 −1.54710 −0.773552 0.633732i \(-0.781522\pi\)
−0.773552 + 0.633732i \(0.781522\pi\)
\(234\) −11.1231 −0.727140
\(235\) −5.12311 −0.334195
\(236\) 4.63068 0.301432
\(237\) 10.2462 0.665563
\(238\) −0.876894 −0.0568406
\(239\) 19.6847 1.27329 0.636647 0.771155i \(-0.280321\pi\)
0.636647 + 0.771155i \(0.280321\pi\)
\(240\) 4.68466 0.302393
\(241\) −0.246211 −0.0158599 −0.00792993 0.999969i \(-0.502524\pi\)
−0.00792993 + 0.999969i \(0.502524\pi\)
\(242\) 1.56155 0.100380
\(243\) −1.00000 −0.0641500
\(244\) −1.50758 −0.0965128
\(245\) 1.00000 0.0638877
\(246\) −3.12311 −0.199122
\(247\) 18.2462 1.16098
\(248\) 12.4924 0.793270
\(249\) 5.43845 0.344648
\(250\) 1.56155 0.0987613
\(251\) 24.4924 1.54595 0.772974 0.634438i \(-0.218768\pi\)
0.772974 + 0.634438i \(0.218768\pi\)
\(252\) −0.438447 −0.0276196
\(253\) −1.43845 −0.0904344
\(254\) 1.26137 0.0791452
\(255\) −0.561553 −0.0351658
\(256\) 10.0540 0.628373
\(257\) 9.36932 0.584442 0.292221 0.956351i \(-0.405606\pi\)
0.292221 + 0.956351i \(0.405606\pi\)
\(258\) −4.00000 −0.249029
\(259\) 7.12311 0.442608
\(260\) −3.12311 −0.193687
\(261\) 1.68466 0.104278
\(262\) 6.24621 0.385892
\(263\) −18.2462 −1.12511 −0.562555 0.826760i \(-0.690181\pi\)
−0.562555 + 0.826760i \(0.690181\pi\)
\(264\) 2.43845 0.150076
\(265\) −13.6847 −0.840642
\(266\) 4.00000 0.245256
\(267\) 2.31534 0.141697
\(268\) 3.01515 0.184180
\(269\) 1.68466 0.102715 0.0513577 0.998680i \(-0.483645\pi\)
0.0513577 + 0.998680i \(0.483645\pi\)
\(270\) −1.56155 −0.0950331
\(271\) −13.9309 −0.846240 −0.423120 0.906074i \(-0.639065\pi\)
−0.423120 + 0.906074i \(0.639065\pi\)
\(272\) −2.63068 −0.159509
\(273\) −7.12311 −0.431110
\(274\) 15.6155 0.943369
\(275\) 1.00000 0.0603023
\(276\) 0.630683 0.0379627
\(277\) −24.7386 −1.48640 −0.743200 0.669069i \(-0.766693\pi\)
−0.743200 + 0.669069i \(0.766693\pi\)
\(278\) −31.2311 −1.87311
\(279\) −5.12311 −0.306712
\(280\) 2.43845 0.145725
\(281\) 14.4924 0.864545 0.432273 0.901743i \(-0.357712\pi\)
0.432273 + 0.901743i \(0.357712\pi\)
\(282\) 8.00000 0.476393
\(283\) 6.24621 0.371299 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(284\) 2.24621 0.133288
\(285\) 2.56155 0.151733
\(286\) −11.1231 −0.657723
\(287\) −2.00000 −0.118056
\(288\) −2.43845 −0.143687
\(289\) −16.6847 −0.981450
\(290\) 2.63068 0.154479
\(291\) −2.80776 −0.164594
\(292\) −4.87689 −0.285399
\(293\) −26.1771 −1.52928 −0.764641 0.644457i \(-0.777084\pi\)
−0.764641 + 0.644457i \(0.777084\pi\)
\(294\) −1.56155 −0.0910716
\(295\) 10.5616 0.614917
\(296\) 17.3693 1.00957
\(297\) −1.00000 −0.0580259
\(298\) 0.384472 0.0222719
\(299\) 10.2462 0.592554
\(300\) −0.438447 −0.0253138
\(301\) −2.56155 −0.147645
\(302\) 8.00000 0.460348
\(303\) 12.2462 0.703526
\(304\) 12.0000 0.688247
\(305\) −3.43845 −0.196885
\(306\) 0.876894 0.0501287
\(307\) 11.3693 0.648881 0.324441 0.945906i \(-0.394824\pi\)
0.324441 + 0.945906i \(0.394824\pi\)
\(308\) −0.438447 −0.0249828
\(309\) −3.68466 −0.209613
\(310\) −8.00000 −0.454369
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −17.3693 −0.983344
\(313\) −2.31534 −0.130871 −0.0654354 0.997857i \(-0.520844\pi\)
−0.0654354 + 0.997857i \(0.520844\pi\)
\(314\) −17.8617 −1.00800
\(315\) −1.00000 −0.0563436
\(316\) −4.49242 −0.252719
\(317\) −22.4924 −1.26330 −0.631650 0.775254i \(-0.717622\pi\)
−0.631650 + 0.775254i \(0.717622\pi\)
\(318\) 21.3693 1.19833
\(319\) 1.68466 0.0943228
\(320\) 5.56155 0.310900
\(321\) −19.3693 −1.08109
\(322\) 2.24621 0.125176
\(323\) −1.43845 −0.0800373
\(324\) 0.438447 0.0243582
\(325\) −7.12311 −0.395119
\(326\) 31.2311 1.72973
\(327\) −18.4924 −1.02263
\(328\) −4.87689 −0.269281
\(329\) 5.12311 0.282446
\(330\) −1.56155 −0.0859607
\(331\) −23.6847 −1.30183 −0.650913 0.759152i \(-0.725614\pi\)
−0.650913 + 0.759152i \(0.725614\pi\)
\(332\) −2.38447 −0.130865
\(333\) −7.12311 −0.390344
\(334\) 3.50758 0.191926
\(335\) 6.87689 0.375725
\(336\) −4.68466 −0.255569
\(337\) −9.05398 −0.493201 −0.246601 0.969117i \(-0.579314\pi\)
−0.246601 + 0.969117i \(0.579314\pi\)
\(338\) 58.9309 3.20542
\(339\) −8.56155 −0.465000
\(340\) 0.246211 0.0133527
\(341\) −5.12311 −0.277432
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) −6.24621 −0.336773
\(345\) 1.43845 0.0774434
\(346\) −6.63068 −0.356468
\(347\) −14.2462 −0.764777 −0.382388 0.924002i \(-0.624898\pi\)
−0.382388 + 0.924002i \(0.624898\pi\)
\(348\) −0.738634 −0.0395949
\(349\) −9.19224 −0.492049 −0.246025 0.969264i \(-0.579124\pi\)
−0.246025 + 0.969264i \(0.579124\pi\)
\(350\) −1.56155 −0.0834685
\(351\) 7.12311 0.380203
\(352\) −2.43845 −0.129970
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −16.4924 −0.876562
\(355\) 5.12311 0.271906
\(356\) −1.01515 −0.0538031
\(357\) 0.561553 0.0297205
\(358\) −30.2462 −1.59856
\(359\) −4.31534 −0.227755 −0.113878 0.993495i \(-0.536327\pi\)
−0.113878 + 0.993495i \(0.536327\pi\)
\(360\) −2.43845 −0.128517
\(361\) −12.4384 −0.654655
\(362\) −31.6155 −1.66168
\(363\) −1.00000 −0.0524864
\(364\) 3.12311 0.163695
\(365\) −11.1231 −0.582210
\(366\) 5.36932 0.280659
\(367\) 17.4384 0.910280 0.455140 0.890420i \(-0.349589\pi\)
0.455140 + 0.890420i \(0.349589\pi\)
\(368\) 6.73863 0.351276
\(369\) 2.00000 0.104116
\(370\) −11.1231 −0.578263
\(371\) 13.6847 0.710472
\(372\) 2.24621 0.116461
\(373\) −0.561553 −0.0290761 −0.0145381 0.999894i \(-0.504628\pi\)
−0.0145381 + 0.999894i \(0.504628\pi\)
\(374\) 0.876894 0.0453431
\(375\) −1.00000 −0.0516398
\(376\) 12.4924 0.644247
\(377\) −12.0000 −0.618031
\(378\) 1.56155 0.0803176
\(379\) 23.0540 1.18420 0.592102 0.805863i \(-0.298298\pi\)
0.592102 + 0.805863i \(0.298298\pi\)
\(380\) −1.12311 −0.0576141
\(381\) −0.807764 −0.0413830
\(382\) 0 0
\(383\) −12.4924 −0.638333 −0.319166 0.947699i \(-0.603403\pi\)
−0.319166 + 0.947699i \(0.603403\pi\)
\(384\) −13.5616 −0.692060
\(385\) −1.00000 −0.0509647
\(386\) 35.1231 1.78772
\(387\) 2.56155 0.130211
\(388\) 1.23106 0.0624974
\(389\) −21.8617 −1.10843 −0.554217 0.832372i \(-0.686982\pi\)
−0.554217 + 0.832372i \(0.686982\pi\)
\(390\) 11.1231 0.563240
\(391\) −0.807764 −0.0408504
\(392\) −2.43845 −0.123160
\(393\) −4.00000 −0.201773
\(394\) −36.1080 −1.81909
\(395\) −10.2462 −0.515543
\(396\) 0.438447 0.0220328
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 32.0000 1.60402
\(399\) −2.56155 −0.128238
\(400\) −4.68466 −0.234233
\(401\) −34.4924 −1.72247 −0.861235 0.508207i \(-0.830308\pi\)
−0.861235 + 0.508207i \(0.830308\pi\)
\(402\) −10.7386 −0.535594
\(403\) 36.4924 1.81782
\(404\) −5.36932 −0.267133
\(405\) 1.00000 0.0496904
\(406\) −2.63068 −0.130559
\(407\) −7.12311 −0.353079
\(408\) 1.36932 0.0677913
\(409\) −8.24621 −0.407749 −0.203874 0.978997i \(-0.565353\pi\)
−0.203874 + 0.978997i \(0.565353\pi\)
\(410\) 3.12311 0.154239
\(411\) −10.0000 −0.493264
\(412\) 1.61553 0.0795914
\(413\) −10.5616 −0.519700
\(414\) −2.24621 −0.110395
\(415\) −5.43845 −0.266963
\(416\) 17.3693 0.851601
\(417\) 20.0000 0.979404
\(418\) −4.00000 −0.195646
\(419\) 11.1922 0.546777 0.273388 0.961904i \(-0.411856\pi\)
0.273388 + 0.961904i \(0.411856\pi\)
\(420\) 0.438447 0.0213940
\(421\) 30.1771 1.47074 0.735370 0.677665i \(-0.237008\pi\)
0.735370 + 0.677665i \(0.237008\pi\)
\(422\) −5.26137 −0.256119
\(423\) −5.12311 −0.249094
\(424\) 33.3693 1.62056
\(425\) 0.561553 0.0272393
\(426\) −8.00000 −0.387601
\(427\) 3.43845 0.166398
\(428\) 8.49242 0.410497
\(429\) 7.12311 0.343907
\(430\) 4.00000 0.192897
\(431\) −14.7386 −0.709935 −0.354968 0.934879i \(-0.615508\pi\)
−0.354968 + 0.934879i \(0.615508\pi\)
\(432\) 4.68466 0.225391
\(433\) −12.7386 −0.612180 −0.306090 0.952003i \(-0.599021\pi\)
−0.306090 + 0.952003i \(0.599021\pi\)
\(434\) 8.00000 0.384012
\(435\) −1.68466 −0.0807732
\(436\) 8.10795 0.388300
\(437\) 3.68466 0.176261
\(438\) 17.3693 0.829938
\(439\) 21.9309 1.04670 0.523352 0.852117i \(-0.324681\pi\)
0.523352 + 0.852117i \(0.324681\pi\)
\(440\) −2.43845 −0.116248
\(441\) 1.00000 0.0476190
\(442\) −6.24621 −0.297102
\(443\) −18.7386 −0.890299 −0.445150 0.895456i \(-0.646850\pi\)
−0.445150 + 0.895456i \(0.646850\pi\)
\(444\) 3.12311 0.148216
\(445\) −2.31534 −0.109758
\(446\) 22.7386 1.07671
\(447\) −0.246211 −0.0116454
\(448\) −5.56155 −0.262759
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.56155 0.0736123
\(451\) 2.00000 0.0941763
\(452\) 3.75379 0.176563
\(453\) −5.12311 −0.240704
\(454\) −40.4924 −1.90040
\(455\) 7.12311 0.333936
\(456\) −6.24621 −0.292506
\(457\) −18.3153 −0.856756 −0.428378 0.903600i \(-0.640915\pi\)
−0.428378 + 0.903600i \(0.640915\pi\)
\(458\) 16.3845 0.765596
\(459\) −0.561553 −0.0262110
\(460\) −0.630683 −0.0294058
\(461\) 32.2462 1.50186 0.750928 0.660384i \(-0.229607\pi\)
0.750928 + 0.660384i \(0.229607\pi\)
\(462\) 1.56155 0.0726500
\(463\) 27.8617 1.29484 0.647422 0.762131i \(-0.275847\pi\)
0.647422 + 0.762131i \(0.275847\pi\)
\(464\) −7.89205 −0.366379
\(465\) 5.12311 0.237578
\(466\) −36.8769 −1.70829
\(467\) 16.4924 0.763178 0.381589 0.924332i \(-0.375377\pi\)
0.381589 + 0.924332i \(0.375377\pi\)
\(468\) −3.12311 −0.144366
\(469\) −6.87689 −0.317545
\(470\) −8.00000 −0.369012
\(471\) 11.4384 0.527056
\(472\) −25.7538 −1.18541
\(473\) 2.56155 0.117780
\(474\) 16.0000 0.734904
\(475\) −2.56155 −0.117532
\(476\) −0.246211 −0.0112851
\(477\) −13.6847 −0.626577
\(478\) 30.7386 1.40595
\(479\) −36.4924 −1.66738 −0.833691 0.552232i \(-0.813776\pi\)
−0.833691 + 0.552232i \(0.813776\pi\)
\(480\) 2.43845 0.111299
\(481\) 50.7386 2.31348
\(482\) −0.384472 −0.0175122
\(483\) −1.43845 −0.0654516
\(484\) 0.438447 0.0199294
\(485\) 2.80776 0.127494
\(486\) −1.56155 −0.0708335
\(487\) −9.61553 −0.435721 −0.217861 0.975980i \(-0.569908\pi\)
−0.217861 + 0.975980i \(0.569908\pi\)
\(488\) 8.38447 0.379547
\(489\) −20.0000 −0.904431
\(490\) 1.56155 0.0705438
\(491\) 33.3002 1.50282 0.751408 0.659838i \(-0.229375\pi\)
0.751408 + 0.659838i \(0.229375\pi\)
\(492\) −0.876894 −0.0395335
\(493\) 0.946025 0.0426068
\(494\) 28.4924 1.28193
\(495\) 1.00000 0.0449467
\(496\) 24.0000 1.07763
\(497\) −5.12311 −0.229803
\(498\) 8.49242 0.380555
\(499\) −7.05398 −0.315779 −0.157890 0.987457i \(-0.550469\pi\)
−0.157890 + 0.987457i \(0.550469\pi\)
\(500\) 0.438447 0.0196080
\(501\) −2.24621 −0.100353
\(502\) 38.2462 1.70701
\(503\) −40.8078 −1.81953 −0.909764 0.415126i \(-0.863738\pi\)
−0.909764 + 0.415126i \(0.863738\pi\)
\(504\) 2.43845 0.108617
\(505\) −12.2462 −0.544949
\(506\) −2.24621 −0.0998563
\(507\) −37.7386 −1.67603
\(508\) 0.354162 0.0157134
\(509\) −1.19224 −0.0528449 −0.0264225 0.999651i \(-0.508412\pi\)
−0.0264225 + 0.999651i \(0.508412\pi\)
\(510\) −0.876894 −0.0388295
\(511\) 11.1231 0.492057
\(512\) −11.4233 −0.504843
\(513\) 2.56155 0.113095
\(514\) 14.6307 0.645332
\(515\) 3.68466 0.162365
\(516\) −1.12311 −0.0494420
\(517\) −5.12311 −0.225314
\(518\) 11.1231 0.488721
\(519\) 4.24621 0.186388
\(520\) 17.3693 0.761695
\(521\) −3.93087 −0.172215 −0.0861073 0.996286i \(-0.527443\pi\)
−0.0861073 + 0.996286i \(0.527443\pi\)
\(522\) 2.63068 0.115142
\(523\) −30.2462 −1.32257 −0.661287 0.750133i \(-0.729990\pi\)
−0.661287 + 0.750133i \(0.729990\pi\)
\(524\) 1.75379 0.0766146
\(525\) 1.00000 0.0436436
\(526\) −28.4924 −1.24233
\(527\) −2.87689 −0.125319
\(528\) 4.68466 0.203874
\(529\) −20.9309 −0.910038
\(530\) −21.3693 −0.928224
\(531\) 10.5616 0.458332
\(532\) 1.12311 0.0486928
\(533\) −14.2462 −0.617072
\(534\) 3.61553 0.156459
\(535\) 19.3693 0.837409
\(536\) −16.7689 −0.724308
\(537\) 19.3693 0.835848
\(538\) 2.63068 0.113417
\(539\) 1.00000 0.0430730
\(540\) −0.438447 −0.0188678
\(541\) −22.4924 −0.967025 −0.483512 0.875338i \(-0.660639\pi\)
−0.483512 + 0.875338i \(0.660639\pi\)
\(542\) −21.7538 −0.934405
\(543\) 20.2462 0.868848
\(544\) −1.36932 −0.0587090
\(545\) 18.4924 0.792128
\(546\) −11.1231 −0.476025
\(547\) −23.0540 −0.985717 −0.492858 0.870110i \(-0.664048\pi\)
−0.492858 + 0.870110i \(0.664048\pi\)
\(548\) 4.38447 0.187295
\(549\) −3.43845 −0.146749
\(550\) 1.56155 0.0665848
\(551\) −4.31534 −0.183840
\(552\) −3.50758 −0.149292
\(553\) 10.2462 0.435713
\(554\) −38.6307 −1.64126
\(555\) 7.12311 0.302359
\(556\) −8.76894 −0.371886
\(557\) 18.4924 0.783549 0.391775 0.920061i \(-0.371861\pi\)
0.391775 + 0.920061i \(0.371861\pi\)
\(558\) −8.00000 −0.338667
\(559\) −18.2462 −0.771733
\(560\) 4.68466 0.197963
\(561\) −0.561553 −0.0237088
\(562\) 22.6307 0.954618
\(563\) 24.4924 1.03223 0.516116 0.856519i \(-0.327377\pi\)
0.516116 + 0.856519i \(0.327377\pi\)
\(564\) 2.24621 0.0945826
\(565\) 8.56155 0.360187
\(566\) 9.75379 0.409982
\(567\) −1.00000 −0.0419961
\(568\) −12.4924 −0.524170
\(569\) −22.8078 −0.956151 −0.478076 0.878319i \(-0.658666\pi\)
−0.478076 + 0.878319i \(0.658666\pi\)
\(570\) 4.00000 0.167542
\(571\) −14.2462 −0.596185 −0.298093 0.954537i \(-0.596350\pi\)
−0.298093 + 0.954537i \(0.596350\pi\)
\(572\) −3.12311 −0.130584
\(573\) 0 0
\(574\) −3.12311 −0.130356
\(575\) −1.43845 −0.0599874
\(576\) 5.56155 0.231731
\(577\) 26.9848 1.12339 0.561697 0.827343i \(-0.310149\pi\)
0.561697 + 0.827343i \(0.310149\pi\)
\(578\) −26.0540 −1.08370
\(579\) −22.4924 −0.934753
\(580\) 0.738634 0.0306701
\(581\) 5.43845 0.225625
\(582\) −4.38447 −0.181742
\(583\) −13.6847 −0.566761
\(584\) 27.1231 1.12236
\(585\) −7.12311 −0.294504
\(586\) −40.8769 −1.68861
\(587\) 28.9848 1.19633 0.598166 0.801372i \(-0.295896\pi\)
0.598166 + 0.801372i \(0.295896\pi\)
\(588\) −0.438447 −0.0180813
\(589\) 13.1231 0.540728
\(590\) 16.4924 0.678982
\(591\) 23.1231 0.951157
\(592\) 33.3693 1.37147
\(593\) −0.246211 −0.0101107 −0.00505534 0.999987i \(-0.501609\pi\)
−0.00505534 + 0.999987i \(0.501609\pi\)
\(594\) −1.56155 −0.0640713
\(595\) −0.561553 −0.0230214
\(596\) 0.107951 0.00442183
\(597\) −20.4924 −0.838699
\(598\) 16.0000 0.654289
\(599\) 0.630683 0.0257690 0.0128845 0.999917i \(-0.495899\pi\)
0.0128845 + 0.999917i \(0.495899\pi\)
\(600\) 2.43845 0.0995492
\(601\) −9.05398 −0.369319 −0.184660 0.982803i \(-0.559118\pi\)
−0.184660 + 0.982803i \(0.559118\pi\)
\(602\) −4.00000 −0.163028
\(603\) 6.87689 0.280049
\(604\) 2.24621 0.0913970
\(605\) 1.00000 0.0406558
\(606\) 19.1231 0.776823
\(607\) 23.3693 0.948531 0.474266 0.880382i \(-0.342714\pi\)
0.474266 + 0.880382i \(0.342714\pi\)
\(608\) 6.24621 0.253317
\(609\) 1.68466 0.0682658
\(610\) −5.36932 −0.217397
\(611\) 36.4924 1.47633
\(612\) 0.246211 0.00995250
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 17.7538 0.716485
\(615\) −2.00000 −0.0806478
\(616\) 2.43845 0.0982478
\(617\) 4.24621 0.170946 0.0854730 0.996340i \(-0.472760\pi\)
0.0854730 + 0.996340i \(0.472760\pi\)
\(618\) −5.75379 −0.231451
\(619\) 1.12311 0.0451414 0.0225707 0.999745i \(-0.492815\pi\)
0.0225707 + 0.999745i \(0.492815\pi\)
\(620\) −2.24621 −0.0902100
\(621\) 1.43845 0.0577229
\(622\) −24.9848 −1.00180
\(623\) 2.31534 0.0927622
\(624\) −33.3693 −1.33584
\(625\) 1.00000 0.0400000
\(626\) −3.61553 −0.144506
\(627\) 2.56155 0.102299
\(628\) −5.01515 −0.200126
\(629\) −4.00000 −0.159490
\(630\) −1.56155 −0.0622138
\(631\) 29.3002 1.16642 0.583211 0.812321i \(-0.301796\pi\)
0.583211 + 0.812321i \(0.301796\pi\)
\(632\) 24.9848 0.993844
\(633\) 3.36932 0.133918
\(634\) −35.1231 −1.39492
\(635\) 0.807764 0.0320551
\(636\) 6.00000 0.237915
\(637\) −7.12311 −0.282228
\(638\) 2.63068 0.104150
\(639\) 5.12311 0.202667
\(640\) 13.5616 0.536067
\(641\) −12.7386 −0.503146 −0.251573 0.967838i \(-0.580948\pi\)
−0.251573 + 0.967838i \(0.580948\pi\)
\(642\) −30.2462 −1.19372
\(643\) −25.3002 −0.997742 −0.498871 0.866676i \(-0.666252\pi\)
−0.498871 + 0.866676i \(0.666252\pi\)
\(644\) 0.630683 0.0248524
\(645\) −2.56155 −0.100861
\(646\) −2.24621 −0.0883760
\(647\) −10.2462 −0.402820 −0.201410 0.979507i \(-0.564552\pi\)
−0.201410 + 0.979507i \(0.564552\pi\)
\(648\) −2.43845 −0.0957913
\(649\) 10.5616 0.414577
\(650\) −11.1231 −0.436284
\(651\) −5.12311 −0.200790
\(652\) 8.76894 0.343418
\(653\) 6.17708 0.241728 0.120864 0.992669i \(-0.461434\pi\)
0.120864 + 0.992669i \(0.461434\pi\)
\(654\) −28.8769 −1.12918
\(655\) 4.00000 0.156293
\(656\) −9.36932 −0.365810
\(657\) −11.1231 −0.433954
\(658\) 8.00000 0.311872
\(659\) 0.315342 0.0122840 0.00614198 0.999981i \(-0.498045\pi\)
0.00614198 + 0.999981i \(0.498045\pi\)
\(660\) −0.438447 −0.0170665
\(661\) 10.4924 0.408108 0.204054 0.978960i \(-0.434588\pi\)
0.204054 + 0.978960i \(0.434588\pi\)
\(662\) −36.9848 −1.43746
\(663\) 4.00000 0.155347
\(664\) 13.2614 0.514641
\(665\) 2.56155 0.0993328
\(666\) −11.1231 −0.431012
\(667\) −2.42329 −0.0938302
\(668\) 0.984845 0.0381048
\(669\) −14.5616 −0.562982
\(670\) 10.7386 0.414870
\(671\) −3.43845 −0.132740
\(672\) −2.43845 −0.0940651
\(673\) 8.56155 0.330024 0.165012 0.986292i \(-0.447234\pi\)
0.165012 + 0.986292i \(0.447234\pi\)
\(674\) −14.1383 −0.544585
\(675\) −1.00000 −0.0384900
\(676\) 16.5464 0.636400
\(677\) 16.0691 0.617587 0.308793 0.951129i \(-0.400075\pi\)
0.308793 + 0.951129i \(0.400075\pi\)
\(678\) −13.3693 −0.513446
\(679\) −2.80776 −0.107752
\(680\) −1.36932 −0.0525109
\(681\) 25.9309 0.993673
\(682\) −8.00000 −0.306336
\(683\) −40.4924 −1.54940 −0.774700 0.632329i \(-0.782099\pi\)
−0.774700 + 0.632329i \(0.782099\pi\)
\(684\) −1.12311 −0.0429430
\(685\) 10.0000 0.382080
\(686\) −1.56155 −0.0596204
\(687\) −10.4924 −0.400311
\(688\) −12.0000 −0.457496
\(689\) 97.4773 3.71359
\(690\) 2.24621 0.0855118
\(691\) 30.8769 1.17461 0.587306 0.809365i \(-0.300188\pi\)
0.587306 + 0.809365i \(0.300188\pi\)
\(692\) −1.86174 −0.0707727
\(693\) −1.00000 −0.0379869
\(694\) −22.2462 −0.844455
\(695\) −20.0000 −0.758643
\(696\) 4.10795 0.155711
\(697\) 1.12311 0.0425407
\(698\) −14.3542 −0.543313
\(699\) 23.6155 0.893221
\(700\) −0.438447 −0.0165717
\(701\) 26.3153 0.993917 0.496958 0.867774i \(-0.334450\pi\)
0.496958 + 0.867774i \(0.334450\pi\)
\(702\) 11.1231 0.419815
\(703\) 18.2462 0.688169
\(704\) 5.56155 0.209609
\(705\) 5.12311 0.192947
\(706\) −46.8466 −1.76309
\(707\) 12.2462 0.460566
\(708\) −4.63068 −0.174032
\(709\) −42.8078 −1.60768 −0.803840 0.594846i \(-0.797213\pi\)
−0.803840 + 0.594846i \(0.797213\pi\)
\(710\) 8.00000 0.300235
\(711\) −10.2462 −0.384263
\(712\) 5.64584 0.211587
\(713\) 7.36932 0.275983
\(714\) 0.876894 0.0328169
\(715\) −7.12311 −0.266389
\(716\) −8.49242 −0.317377
\(717\) −19.6847 −0.735137
\(718\) −6.73863 −0.251484
\(719\) 23.5464 0.878132 0.439066 0.898455i \(-0.355309\pi\)
0.439066 + 0.898455i \(0.355309\pi\)
\(720\) −4.68466 −0.174587
\(721\) −3.68466 −0.137224
\(722\) −19.4233 −0.722860
\(723\) 0.246211 0.00915669
\(724\) −8.87689 −0.329907
\(725\) 1.68466 0.0625666
\(726\) −1.56155 −0.0579547
\(727\) 22.5616 0.836762 0.418381 0.908272i \(-0.362598\pi\)
0.418381 + 0.908272i \(0.362598\pi\)
\(728\) −17.3693 −0.643750
\(729\) 1.00000 0.0370370
\(730\) −17.3693 −0.642867
\(731\) 1.43845 0.0532029
\(732\) 1.50758 0.0557217
\(733\) −52.2462 −1.92976 −0.964879 0.262695i \(-0.915389\pi\)
−0.964879 + 0.262695i \(0.915389\pi\)
\(734\) 27.2311 1.00512
\(735\) −1.00000 −0.0368856
\(736\) 3.50758 0.129291
\(737\) 6.87689 0.253314
\(738\) 3.12311 0.114963
\(739\) −4.63068 −0.170342 −0.0851712 0.996366i \(-0.527144\pi\)
−0.0851712 + 0.996366i \(0.527144\pi\)
\(740\) −3.12311 −0.114808
\(741\) −18.2462 −0.670291
\(742\) 21.3693 0.784492
\(743\) 6.38447 0.234224 0.117112 0.993119i \(-0.462636\pi\)
0.117112 + 0.993119i \(0.462636\pi\)
\(744\) −12.4924 −0.457994
\(745\) 0.246211 0.00902048
\(746\) −0.876894 −0.0321054
\(747\) −5.43845 −0.198982
\(748\) 0.246211 0.00900237
\(749\) −19.3693 −0.707739
\(750\) −1.56155 −0.0570198
\(751\) −3.68466 −0.134455 −0.0672275 0.997738i \(-0.521415\pi\)
−0.0672275 + 0.997738i \(0.521415\pi\)
\(752\) 24.0000 0.875190
\(753\) −24.4924 −0.892553
\(754\) −18.7386 −0.682421
\(755\) 5.12311 0.186449
\(756\) 0.438447 0.0159462
\(757\) 1.50758 0.0547938 0.0273969 0.999625i \(-0.491278\pi\)
0.0273969 + 0.999625i \(0.491278\pi\)
\(758\) 36.0000 1.30758
\(759\) 1.43845 0.0522123
\(760\) 6.24621 0.226574
\(761\) −3.75379 −0.136075 −0.0680374 0.997683i \(-0.521674\pi\)
−0.0680374 + 0.997683i \(0.521674\pi\)
\(762\) −1.26137 −0.0456945
\(763\) −18.4924 −0.669471
\(764\) 0 0
\(765\) 0.561553 0.0203030
\(766\) −19.5076 −0.704837
\(767\) −75.2311 −2.71644
\(768\) −10.0540 −0.362792
\(769\) 1.82292 0.0657361 0.0328681 0.999460i \(-0.489536\pi\)
0.0328681 + 0.999460i \(0.489536\pi\)
\(770\) −1.56155 −0.0562745
\(771\) −9.36932 −0.337428
\(772\) 9.86174 0.354932
\(773\) −24.7386 −0.889787 −0.444893 0.895584i \(-0.646758\pi\)
−0.444893 + 0.895584i \(0.646758\pi\)
\(774\) 4.00000 0.143777
\(775\) −5.12311 −0.184027
\(776\) −6.84658 −0.245778
\(777\) −7.12311 −0.255540
\(778\) −34.1383 −1.22392
\(779\) −5.12311 −0.183554
\(780\) 3.12311 0.111825
\(781\) 5.12311 0.183319
\(782\) −1.26137 −0.0451064
\(783\) −1.68466 −0.0602048
\(784\) −4.68466 −0.167309
\(785\) −11.4384 −0.408256
\(786\) −6.24621 −0.222795
\(787\) −32.4924 −1.15823 −0.579115 0.815246i \(-0.696602\pi\)
−0.579115 + 0.815246i \(0.696602\pi\)
\(788\) −10.1383 −0.361161
\(789\) 18.2462 0.649582
\(790\) −16.0000 −0.569254
\(791\) −8.56155 −0.304414
\(792\) −2.43845 −0.0866464
\(793\) 24.4924 0.869751
\(794\) 21.8617 0.775844
\(795\) 13.6847 0.485345
\(796\) 8.98485 0.318459
\(797\) 12.7386 0.451226 0.225613 0.974217i \(-0.427562\pi\)
0.225613 + 0.974217i \(0.427562\pi\)
\(798\) −4.00000 −0.141598
\(799\) −2.87689 −0.101777
\(800\) −2.43845 −0.0862121
\(801\) −2.31534 −0.0818086
\(802\) −53.8617 −1.90192
\(803\) −11.1231 −0.392526
\(804\) −3.01515 −0.106336
\(805\) 1.43845 0.0506986
\(806\) 56.9848 2.00721
\(807\) −1.68466 −0.0593028
\(808\) 29.8617 1.05053
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.56155 0.0548674
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −0.738634 −0.0259210
\(813\) 13.9309 0.488577
\(814\) −11.1231 −0.389865
\(815\) 20.0000 0.700569
\(816\) 2.63068 0.0920923
\(817\) −6.56155 −0.229560
\(818\) −12.8769 −0.450230
\(819\) 7.12311 0.248901
\(820\) 0.876894 0.0306225
\(821\) 37.5464 1.31038 0.655189 0.755465i \(-0.272589\pi\)
0.655189 + 0.755465i \(0.272589\pi\)
\(822\) −15.6155 −0.544654
\(823\) −22.7386 −0.792619 −0.396309 0.918117i \(-0.629709\pi\)
−0.396309 + 0.918117i \(0.629709\pi\)
\(824\) −8.98485 −0.313002
\(825\) −1.00000 −0.0348155
\(826\) −16.4924 −0.573845
\(827\) −2.73863 −0.0952316 −0.0476158 0.998866i \(-0.515162\pi\)
−0.0476158 + 0.998866i \(0.515162\pi\)
\(828\) −0.630683 −0.0219178
\(829\) 33.2311 1.15416 0.577081 0.816687i \(-0.304192\pi\)
0.577081 + 0.816687i \(0.304192\pi\)
\(830\) −8.49242 −0.294776
\(831\) 24.7386 0.858174
\(832\) −39.6155 −1.37342
\(833\) 0.561553 0.0194567
\(834\) 31.2311 1.08144
\(835\) 2.24621 0.0777333
\(836\) −1.12311 −0.0388434
\(837\) 5.12311 0.177080
\(838\) 17.4773 0.603742
\(839\) −34.4233 −1.18842 −0.594212 0.804308i \(-0.702536\pi\)
−0.594212 + 0.804308i \(0.702536\pi\)
\(840\) −2.43845 −0.0841344
\(841\) −26.1619 −0.902135
\(842\) 47.1231 1.62397
\(843\) −14.4924 −0.499146
\(844\) −1.47727 −0.0508496
\(845\) 37.7386 1.29825
\(846\) −8.00000 −0.275046
\(847\) −1.00000 −0.0343604
\(848\) 64.1080 2.20148
\(849\) −6.24621 −0.214369
\(850\) 0.876894 0.0300772
\(851\) 10.2462 0.351236
\(852\) −2.24621 −0.0769539
\(853\) 2.49242 0.0853389 0.0426695 0.999089i \(-0.486414\pi\)
0.0426695 + 0.999089i \(0.486414\pi\)
\(854\) 5.36932 0.183734
\(855\) −2.56155 −0.0876033
\(856\) −47.2311 −1.61432
\(857\) 13.5076 0.461410 0.230705 0.973024i \(-0.425897\pi\)
0.230705 + 0.973024i \(0.425897\pi\)
\(858\) 11.1231 0.379737
\(859\) −41.1231 −1.40310 −0.701551 0.712619i \(-0.747509\pi\)
−0.701551 + 0.712619i \(0.747509\pi\)
\(860\) 1.12311 0.0382976
\(861\) 2.00000 0.0681598
\(862\) −23.0152 −0.783899
\(863\) 31.5464 1.07385 0.536926 0.843629i \(-0.319585\pi\)
0.536926 + 0.843629i \(0.319585\pi\)
\(864\) 2.43845 0.0829577
\(865\) −4.24621 −0.144376
\(866\) −19.8920 −0.675959
\(867\) 16.6847 0.566641
\(868\) 2.24621 0.0762414
\(869\) −10.2462 −0.347579
\(870\) −2.63068 −0.0891885
\(871\) −48.9848 −1.65979
\(872\) −45.0928 −1.52703
\(873\) 2.80776 0.0950284
\(874\) 5.75379 0.194625
\(875\) −1.00000 −0.0338062
\(876\) 4.87689 0.164775
\(877\) −46.6695 −1.57592 −0.787959 0.615728i \(-0.788862\pi\)
−0.787959 + 0.615728i \(0.788862\pi\)
\(878\) 34.2462 1.15575
\(879\) 26.1771 0.882931
\(880\) −4.68466 −0.157920
\(881\) 31.9309 1.07578 0.537889 0.843016i \(-0.319222\pi\)
0.537889 + 0.843016i \(0.319222\pi\)
\(882\) 1.56155 0.0525802
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −1.75379 −0.0589863
\(885\) −10.5616 −0.355023
\(886\) −29.2614 −0.983055
\(887\) −33.4384 −1.12275 −0.561377 0.827560i \(-0.689728\pi\)
−0.561377 + 0.827560i \(0.689728\pi\)
\(888\) −17.3693 −0.582876
\(889\) −0.807764 −0.0270915
\(890\) −3.61553 −0.121193
\(891\) 1.00000 0.0335013
\(892\) 6.38447 0.213768
\(893\) 13.1231 0.439148
\(894\) −0.384472 −0.0128587
\(895\) −19.3693 −0.647445
\(896\) −13.5616 −0.453060
\(897\) −10.2462 −0.342111
\(898\) 3.12311 0.104219
\(899\) −8.63068 −0.287849
\(900\) 0.438447 0.0146149
\(901\) −7.68466 −0.256013
\(902\) 3.12311 0.103988
\(903\) 2.56155 0.0852431
\(904\) −20.8769 −0.694355
\(905\) −20.2462 −0.673007
\(906\) −8.00000 −0.265782
\(907\) 57.4773 1.90850 0.954251 0.299008i \(-0.0966557\pi\)
0.954251 + 0.299008i \(0.0966557\pi\)
\(908\) −11.3693 −0.377304
\(909\) −12.2462 −0.406181
\(910\) 11.1231 0.368727
\(911\) 20.4924 0.678944 0.339472 0.940616i \(-0.389752\pi\)
0.339472 + 0.940616i \(0.389752\pi\)
\(912\) −12.0000 −0.397360
\(913\) −5.43845 −0.179986
\(914\) −28.6004 −0.946016
\(915\) 3.43845 0.113672
\(916\) 4.60037 0.152001
\(917\) −4.00000 −0.132092
\(918\) −0.876894 −0.0289418
\(919\) −19.8617 −0.655178 −0.327589 0.944820i \(-0.606236\pi\)
−0.327589 + 0.944820i \(0.606236\pi\)
\(920\) 3.50758 0.115641
\(921\) −11.3693 −0.374632
\(922\) 50.3542 1.65833
\(923\) −36.4924 −1.20116
\(924\) 0.438447 0.0144239
\(925\) −7.12311 −0.234206
\(926\) 43.5076 1.42975
\(927\) 3.68466 0.121020
\(928\) −4.10795 −0.134850
\(929\) −22.9848 −0.754108 −0.377054 0.926191i \(-0.623063\pi\)
−0.377054 + 0.926191i \(0.623063\pi\)
\(930\) 8.00000 0.262330
\(931\) −2.56155 −0.0839515
\(932\) −10.3542 −0.339162
\(933\) 16.0000 0.523816
\(934\) 25.7538 0.842690
\(935\) 0.561553 0.0183647
\(936\) 17.3693 0.567734
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −10.7386 −0.350629
\(939\) 2.31534 0.0755583
\(940\) −2.24621 −0.0732633
\(941\) 7.61553 0.248259 0.124130 0.992266i \(-0.460386\pi\)
0.124130 + 0.992266i \(0.460386\pi\)
\(942\) 17.8617 0.581967
\(943\) −2.87689 −0.0936846
\(944\) −49.4773 −1.61035
\(945\) 1.00000 0.0325300
\(946\) 4.00000 0.130051
\(947\) 31.0540 1.00912 0.504559 0.863377i \(-0.331655\pi\)
0.504559 + 0.863377i \(0.331655\pi\)
\(948\) 4.49242 0.145907
\(949\) 79.2311 2.57195
\(950\) −4.00000 −0.129777
\(951\) 22.4924 0.729367
\(952\) 1.36932 0.0443798
\(953\) 15.7538 0.510315 0.255158 0.966899i \(-0.417873\pi\)
0.255158 + 0.966899i \(0.417873\pi\)
\(954\) −21.3693 −0.691857
\(955\) 0 0
\(956\) 8.63068 0.279136
\(957\) −1.68466 −0.0544573
\(958\) −56.9848 −1.84110
\(959\) −10.0000 −0.322917
\(960\) −5.56155 −0.179498
\(961\) −4.75379 −0.153348
\(962\) 79.2311 2.55451
\(963\) 19.3693 0.624168
\(964\) −0.107951 −0.00347686
\(965\) 22.4924 0.724057
\(966\) −2.24621 −0.0722707
\(967\) 30.5616 0.982793 0.491397 0.870936i \(-0.336487\pi\)
0.491397 + 0.870936i \(0.336487\pi\)
\(968\) −2.43845 −0.0783747
\(969\) 1.43845 0.0462096
\(970\) 4.38447 0.140777
\(971\) 4.80776 0.154288 0.0771442 0.997020i \(-0.475420\pi\)
0.0771442 + 0.997020i \(0.475420\pi\)
\(972\) −0.438447 −0.0140632
\(973\) 20.0000 0.641171
\(974\) −15.0152 −0.481117
\(975\) 7.12311 0.228122
\(976\) 16.1080 0.515603
\(977\) −6.17708 −0.197622 −0.0988112 0.995106i \(-0.531504\pi\)
−0.0988112 + 0.995106i \(0.531504\pi\)
\(978\) −31.2311 −0.998659
\(979\) −2.31534 −0.0739986
\(980\) 0.438447 0.0140057
\(981\) 18.4924 0.590418
\(982\) 52.0000 1.65939
\(983\) 49.6155 1.58249 0.791245 0.611500i \(-0.209433\pi\)
0.791245 + 0.611500i \(0.209433\pi\)
\(984\) 4.87689 0.155470
\(985\) −23.1231 −0.736763
\(986\) 1.47727 0.0470458
\(987\) −5.12311 −0.163070
\(988\) 8.00000 0.254514
\(989\) −3.68466 −0.117165
\(990\) 1.56155 0.0496294
\(991\) 57.4384 1.82459 0.912296 0.409531i \(-0.134308\pi\)
0.912296 + 0.409531i \(0.134308\pi\)
\(992\) 12.4924 0.396635
\(993\) 23.6847 0.751610
\(994\) −8.00000 −0.253745
\(995\) 20.4924 0.649653
\(996\) 2.38447 0.0755549
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −11.0152 −0.348679
\(999\) 7.12311 0.225365
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 1155.2.a.o.1.2 2
3.2 odd 2 3465.2.a.z.1.1 2
5.4 even 2 5775.2.a.bm.1.1 2
7.6 odd 2 8085.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.o.1.2 2 1.1 even 1 trivial
3465.2.a.z.1.1 2 3.2 odd 2
5775.2.a.bm.1.1 2 5.4 even 2
8085.2.a.bb.1.2 2 7.6 odd 2