Properties

Label 3174.2.a.ba.1.3
Level $3174$
Weight $2$
Character 3174.1
Self dual yes
Analytic conductor $25.345$
Analytic rank $0$
Dimension $5$
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] = [3174,2,Mod(1,3174)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3174, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3174.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.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: \(25.3445176016\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.68251\) of defining polynomial
Character \(\chi\) \(=\) 3174.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.194262 q^{5} -1.00000 q^{6} +1.95185 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.194262 q^{5} -1.00000 q^{6} +1.95185 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.194262 q^{10} +6.30195 q^{11} -1.00000 q^{12} +4.81288 q^{13} +1.95185 q^{14} +0.194262 q^{15} +1.00000 q^{16} -3.20786 q^{17} +1.00000 q^{18} +0.859449 q^{19} -0.194262 q^{20} -1.95185 q^{21} +6.30195 q^{22} -1.00000 q^{24} -4.96226 q^{25} +4.81288 q^{26} -1.00000 q^{27} +1.95185 q^{28} +5.41741 q^{29} +0.194262 q^{30} -1.01371 q^{31} +1.00000 q^{32} -6.30195 q^{33} -3.20786 q^{34} -0.379170 q^{35} +1.00000 q^{36} +6.89037 q^{37} +0.859449 q^{38} -4.81288 q^{39} -0.194262 q^{40} -0.972011 q^{41} -1.95185 q^{42} -5.14282 q^{43} +6.30195 q^{44} -0.194262 q^{45} -7.41216 q^{47} -1.00000 q^{48} -3.19028 q^{49} -4.96226 q^{50} +3.20786 q^{51} +4.81288 q^{52} +2.00797 q^{53} -1.00000 q^{54} -1.22423 q^{55} +1.95185 q^{56} -0.859449 q^{57} +5.41741 q^{58} -8.60221 q^{59} +0.194262 q^{60} +12.1201 q^{61} -1.01371 q^{62} +1.95185 q^{63} +1.00000 q^{64} -0.934959 q^{65} -6.30195 q^{66} +7.09327 q^{67} -3.20786 q^{68} -0.379170 q^{70} +1.10938 q^{71} +1.00000 q^{72} -13.1859 q^{73} +6.89037 q^{74} +4.96226 q^{75} +0.859449 q^{76} +12.3005 q^{77} -4.81288 q^{78} +1.12979 q^{79} -0.194262 q^{80} +1.00000 q^{81} -0.972011 q^{82} +4.00714 q^{83} -1.95185 q^{84} +0.623165 q^{85} -5.14282 q^{86} -5.41741 q^{87} +6.30195 q^{88} -9.55158 q^{89} -0.194262 q^{90} +9.39402 q^{91} +1.01371 q^{93} -7.41216 q^{94} -0.166958 q^{95} -1.00000 q^{96} -1.12094 q^{97} -3.19028 q^{98} +6.30195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9} - q^{10} + 11 q^{11} - 5 q^{12} + 12 q^{13} + 11 q^{14} + q^{15} + 5 q^{16} - q^{17} + 5 q^{18} + 15 q^{19} - q^{20} - 11 q^{21} + 11 q^{22} - 5 q^{24} + 6 q^{25} + 12 q^{26} - 5 q^{27} + 11 q^{28} + q^{29} + q^{30} - 18 q^{31} + 5 q^{32} - 11 q^{33} - q^{34} - 11 q^{35} + 5 q^{36} + 10 q^{37} + 15 q^{38} - 12 q^{39} - q^{40} - 16 q^{41} - 11 q^{42} + 18 q^{43} + 11 q^{44} - q^{45} - 4 q^{47} - 5 q^{48} + 20 q^{49} + 6 q^{50} + q^{51} + 12 q^{52} - q^{53} - 5 q^{54} - 22 q^{55} + 11 q^{56} - 15 q^{57} + q^{58} + 2 q^{59} + q^{60} - q^{61} - 18 q^{62} + 11 q^{63} + 5 q^{64} + 24 q^{65} - 11 q^{66} + 29 q^{67} - q^{68} - 11 q^{70} - 11 q^{71} + 5 q^{72} - 8 q^{73} + 10 q^{74} - 6 q^{75} + 15 q^{76} + 11 q^{77} - 12 q^{78} + 40 q^{79} - q^{80} + 5 q^{81} - 16 q^{82} + 8 q^{83} - 11 q^{84} - 13 q^{85} + 18 q^{86} - q^{87} + 11 q^{88} + 2 q^{89} - q^{90} + 33 q^{91} + 18 q^{93} - 4 q^{94} - 3 q^{95} - 5 q^{96} + 17 q^{97} + 20 q^{98} + 11 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.194262 −0.0868765 −0.0434383 0.999056i \(-0.513831\pi\)
−0.0434383 + 0.999056i \(0.513831\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.95185 0.737730 0.368865 0.929483i \(-0.379747\pi\)
0.368865 + 0.929483i \(0.379747\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.194262 −0.0614310
\(11\) 6.30195 1.90011 0.950055 0.312083i \(-0.101027\pi\)
0.950055 + 0.312083i \(0.101027\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.81288 1.33485 0.667426 0.744676i \(-0.267396\pi\)
0.667426 + 0.744676i \(0.267396\pi\)
\(14\) 1.95185 0.521654
\(15\) 0.194262 0.0501582
\(16\) 1.00000 0.250000
\(17\) −3.20786 −0.778020 −0.389010 0.921233i \(-0.627183\pi\)
−0.389010 + 0.921233i \(0.627183\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.859449 0.197171 0.0985855 0.995129i \(-0.468568\pi\)
0.0985855 + 0.995129i \(0.468568\pi\)
\(20\) −0.194262 −0.0434383
\(21\) −1.95185 −0.425928
\(22\) 6.30195 1.34358
\(23\) 0 0
\(24\) −1.00000 −0.204124
\(25\) −4.96226 −0.992452
\(26\) 4.81288 0.943883
\(27\) −1.00000 −0.192450
\(28\) 1.95185 0.368865
\(29\) 5.41741 1.00599 0.502994 0.864290i \(-0.332232\pi\)
0.502994 + 0.864290i \(0.332232\pi\)
\(30\) 0.194262 0.0354672
\(31\) −1.01371 −0.182067 −0.0910334 0.995848i \(-0.529017\pi\)
−0.0910334 + 0.995848i \(0.529017\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.30195 −1.09703
\(34\) −3.20786 −0.550144
\(35\) −0.379170 −0.0640914
\(36\) 1.00000 0.166667
\(37\) 6.89037 1.13277 0.566385 0.824141i \(-0.308342\pi\)
0.566385 + 0.824141i \(0.308342\pi\)
\(38\) 0.859449 0.139421
\(39\) −4.81288 −0.770678
\(40\) −0.194262 −0.0307155
\(41\) −0.972011 −0.151803 −0.0759013 0.997115i \(-0.524183\pi\)
−0.0759013 + 0.997115i \(0.524183\pi\)
\(42\) −1.95185 −0.301177
\(43\) −5.14282 −0.784273 −0.392136 0.919907i \(-0.628264\pi\)
−0.392136 + 0.919907i \(0.628264\pi\)
\(44\) 6.30195 0.950055
\(45\) −0.194262 −0.0289588
\(46\) 0 0
\(47\) −7.41216 −1.08117 −0.540587 0.841288i \(-0.681798\pi\)
−0.540587 + 0.841288i \(0.681798\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.19028 −0.455755
\(50\) −4.96226 −0.701770
\(51\) 3.20786 0.449190
\(52\) 4.81288 0.667426
\(53\) 2.00797 0.275815 0.137908 0.990445i \(-0.455962\pi\)
0.137908 + 0.990445i \(0.455962\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.22423 −0.165075
\(56\) 1.95185 0.260827
\(57\) −0.859449 −0.113837
\(58\) 5.41741 0.711341
\(59\) −8.60221 −1.11991 −0.559956 0.828522i \(-0.689182\pi\)
−0.559956 + 0.828522i \(0.689182\pi\)
\(60\) 0.194262 0.0250791
\(61\) 12.1201 1.55182 0.775912 0.630841i \(-0.217290\pi\)
0.775912 + 0.630841i \(0.217290\pi\)
\(62\) −1.01371 −0.128741
\(63\) 1.95185 0.245910
\(64\) 1.00000 0.125000
\(65\) −0.934959 −0.115967
\(66\) −6.30195 −0.775716
\(67\) 7.09327 0.866580 0.433290 0.901254i \(-0.357353\pi\)
0.433290 + 0.901254i \(0.357353\pi\)
\(68\) −3.20786 −0.389010
\(69\) 0 0
\(70\) −0.379170 −0.0453194
\(71\) 1.10938 0.131659 0.0658294 0.997831i \(-0.479031\pi\)
0.0658294 + 0.997831i \(0.479031\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.1859 −1.54330 −0.771648 0.636049i \(-0.780567\pi\)
−0.771648 + 0.636049i \(0.780567\pi\)
\(74\) 6.89037 0.800989
\(75\) 4.96226 0.572993
\(76\) 0.859449 0.0985855
\(77\) 12.3005 1.40177
\(78\) −4.81288 −0.544951
\(79\) 1.12979 0.127112 0.0635559 0.997978i \(-0.479756\pi\)
0.0635559 + 0.997978i \(0.479756\pi\)
\(80\) −0.194262 −0.0217191
\(81\) 1.00000 0.111111
\(82\) −0.972011 −0.107341
\(83\) 4.00714 0.439841 0.219920 0.975518i \(-0.429420\pi\)
0.219920 + 0.975518i \(0.429420\pi\)
\(84\) −1.95185 −0.212964
\(85\) 0.623165 0.0675917
\(86\) −5.14282 −0.554564
\(87\) −5.41741 −0.580807
\(88\) 6.30195 0.671790
\(89\) −9.55158 −1.01247 −0.506233 0.862397i \(-0.668962\pi\)
−0.506233 + 0.862397i \(0.668962\pi\)
\(90\) −0.194262 −0.0204770
\(91\) 9.39402 0.984760
\(92\) 0 0
\(93\) 1.01371 0.105116
\(94\) −7.41216 −0.764506
\(95\) −0.166958 −0.0171295
\(96\) −1.00000 −0.102062
\(97\) −1.12094 −0.113815 −0.0569073 0.998379i \(-0.518124\pi\)
−0.0569073 + 0.998379i \(0.518124\pi\)
\(98\) −3.19028 −0.322267
\(99\) 6.30195 0.633370
\(100\) −4.96226 −0.496226
\(101\) 12.5714 1.25090 0.625451 0.780263i \(-0.284915\pi\)
0.625451 + 0.780263i \(0.284915\pi\)
\(102\) 3.20786 0.317626
\(103\) 19.3963 1.91117 0.955587 0.294708i \(-0.0952223\pi\)
0.955587 + 0.294708i \(0.0952223\pi\)
\(104\) 4.81288 0.471942
\(105\) 0.379170 0.0370032
\(106\) 2.00797 0.195031
\(107\) 3.07594 0.297363 0.148681 0.988885i \(-0.452497\pi\)
0.148681 + 0.988885i \(0.452497\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.94929 0.186708 0.0933542 0.995633i \(-0.470241\pi\)
0.0933542 + 0.995633i \(0.470241\pi\)
\(110\) −1.22423 −0.116726
\(111\) −6.89037 −0.654005
\(112\) 1.95185 0.184432
\(113\) −17.1492 −1.61326 −0.806632 0.591054i \(-0.798712\pi\)
−0.806632 + 0.591054i \(0.798712\pi\)
\(114\) −0.859449 −0.0804947
\(115\) 0 0
\(116\) 5.41741 0.502994
\(117\) 4.81288 0.444951
\(118\) −8.60221 −0.791898
\(119\) −6.26126 −0.573969
\(120\) 0.194262 0.0177336
\(121\) 28.7146 2.61042
\(122\) 12.1201 1.09731
\(123\) 0.972011 0.0876433
\(124\) −1.01371 −0.0910334
\(125\) 1.93529 0.173097
\(126\) 1.95185 0.173885
\(127\) −21.6118 −1.91774 −0.958869 0.283850i \(-0.908388\pi\)
−0.958869 + 0.283850i \(0.908388\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.14282 0.452800
\(130\) −0.934959 −0.0820013
\(131\) −14.9571 −1.30681 −0.653404 0.757009i \(-0.726660\pi\)
−0.653404 + 0.757009i \(0.726660\pi\)
\(132\) −6.30195 −0.548514
\(133\) 1.67751 0.145459
\(134\) 7.09327 0.612765
\(135\) 0.194262 0.0167194
\(136\) −3.20786 −0.275072
\(137\) 18.1570 1.55126 0.775628 0.631190i \(-0.217433\pi\)
0.775628 + 0.631190i \(0.217433\pi\)
\(138\) 0 0
\(139\) −7.36718 −0.624876 −0.312438 0.949938i \(-0.601146\pi\)
−0.312438 + 0.949938i \(0.601146\pi\)
\(140\) −0.379170 −0.0320457
\(141\) 7.41216 0.624217
\(142\) 1.10938 0.0930969
\(143\) 30.3305 2.53637
\(144\) 1.00000 0.0833333
\(145\) −1.05240 −0.0873967
\(146\) −13.1859 −1.09128
\(147\) 3.19028 0.263130
\(148\) 6.89037 0.566385
\(149\) 15.6333 1.28073 0.640365 0.768070i \(-0.278783\pi\)
0.640365 + 0.768070i \(0.278783\pi\)
\(150\) 4.96226 0.405167
\(151\) 20.7042 1.68488 0.842440 0.538790i \(-0.181118\pi\)
0.842440 + 0.538790i \(0.181118\pi\)
\(152\) 0.859449 0.0697105
\(153\) −3.20786 −0.259340
\(154\) 12.3005 0.991199
\(155\) 0.196924 0.0158173
\(156\) −4.81288 −0.385339
\(157\) −3.97631 −0.317344 −0.158672 0.987331i \(-0.550721\pi\)
−0.158672 + 0.987331i \(0.550721\pi\)
\(158\) 1.12979 0.0898816
\(159\) −2.00797 −0.159242
\(160\) −0.194262 −0.0153577
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 24.5129 1.91999 0.959997 0.280009i \(-0.0903375\pi\)
0.959997 + 0.280009i \(0.0903375\pi\)
\(164\) −0.972011 −0.0759013
\(165\) 1.22423 0.0953060
\(166\) 4.00714 0.311015
\(167\) −1.95444 −0.151239 −0.0756194 0.997137i \(-0.524093\pi\)
−0.0756194 + 0.997137i \(0.524093\pi\)
\(168\) −1.95185 −0.150588
\(169\) 10.1638 0.781832
\(170\) 0.623165 0.0477945
\(171\) 0.859449 0.0657237
\(172\) −5.14282 −0.392136
\(173\) 6.14165 0.466941 0.233471 0.972364i \(-0.424992\pi\)
0.233471 + 0.972364i \(0.424992\pi\)
\(174\) −5.41741 −0.410693
\(175\) −9.68559 −0.732162
\(176\) 6.30195 0.475027
\(177\) 8.60221 0.646582
\(178\) −9.55158 −0.715921
\(179\) 6.47091 0.483658 0.241829 0.970319i \(-0.422253\pi\)
0.241829 + 0.970319i \(0.422253\pi\)
\(180\) −0.194262 −0.0144794
\(181\) 3.26146 0.242423 0.121211 0.992627i \(-0.461322\pi\)
0.121211 + 0.992627i \(0.461322\pi\)
\(182\) 9.39402 0.696331
\(183\) −12.1201 −0.895946
\(184\) 0 0
\(185\) −1.33853 −0.0984110
\(186\) 1.01371 0.0743285
\(187\) −20.2158 −1.47832
\(188\) −7.41216 −0.540587
\(189\) −1.95185 −0.141976
\(190\) −0.166958 −0.0121124
\(191\) −22.8536 −1.65363 −0.826815 0.562474i \(-0.809850\pi\)
−0.826815 + 0.562474i \(0.809850\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −16.9414 −1.21947 −0.609734 0.792606i \(-0.708724\pi\)
−0.609734 + 0.792606i \(0.708724\pi\)
\(194\) −1.12094 −0.0804790
\(195\) 0.934959 0.0669538
\(196\) −3.19028 −0.227877
\(197\) 1.25480 0.0894010 0.0447005 0.999000i \(-0.485767\pi\)
0.0447005 + 0.999000i \(0.485767\pi\)
\(198\) 6.30195 0.447860
\(199\) 24.8140 1.75902 0.879509 0.475883i \(-0.157871\pi\)
0.879509 + 0.475883i \(0.157871\pi\)
\(200\) −4.96226 −0.350885
\(201\) −7.09327 −0.500320
\(202\) 12.5714 0.884521
\(203\) 10.5740 0.742147
\(204\) 3.20786 0.224595
\(205\) 0.188825 0.0131881
\(206\) 19.3963 1.35140
\(207\) 0 0
\(208\) 4.81288 0.333713
\(209\) 5.41620 0.374646
\(210\) 0.379170 0.0261652
\(211\) 7.25827 0.499680 0.249840 0.968287i \(-0.419622\pi\)
0.249840 + 0.968287i \(0.419622\pi\)
\(212\) 2.00797 0.137908
\(213\) −1.10938 −0.0760133
\(214\) 3.07594 0.210267
\(215\) 0.999053 0.0681349
\(216\) −1.00000 −0.0680414
\(217\) −1.97860 −0.134316
\(218\) 1.94929 0.132023
\(219\) 13.1859 0.891023
\(220\) −1.22423 −0.0825374
\(221\) −15.4390 −1.03854
\(222\) −6.89037 −0.462451
\(223\) 20.7342 1.38846 0.694231 0.719752i \(-0.255745\pi\)
0.694231 + 0.719752i \(0.255745\pi\)
\(224\) 1.95185 0.130413
\(225\) −4.96226 −0.330817
\(226\) −17.1492 −1.14075
\(227\) −21.0504 −1.39716 −0.698581 0.715531i \(-0.746185\pi\)
−0.698581 + 0.715531i \(0.746185\pi\)
\(228\) −0.859449 −0.0569184
\(229\) −1.91595 −0.126609 −0.0633047 0.997994i \(-0.520164\pi\)
−0.0633047 + 0.997994i \(0.520164\pi\)
\(230\) 0 0
\(231\) −12.3005 −0.809311
\(232\) 5.41741 0.355670
\(233\) −14.4306 −0.945378 −0.472689 0.881229i \(-0.656717\pi\)
−0.472689 + 0.881229i \(0.656717\pi\)
\(234\) 4.81288 0.314628
\(235\) 1.43990 0.0939287
\(236\) −8.60221 −0.559956
\(237\) −1.12979 −0.0733880
\(238\) −6.26126 −0.405857
\(239\) −9.81701 −0.635010 −0.317505 0.948257i \(-0.602845\pi\)
−0.317505 + 0.948257i \(0.602845\pi\)
\(240\) 0.194262 0.0125395
\(241\) −14.9039 −0.960043 −0.480022 0.877257i \(-0.659371\pi\)
−0.480022 + 0.877257i \(0.659371\pi\)
\(242\) 28.7146 1.84584
\(243\) −1.00000 −0.0641500
\(244\) 12.1201 0.775912
\(245\) 0.619750 0.0395944
\(246\) 0.972011 0.0619732
\(247\) 4.13642 0.263194
\(248\) −1.01371 −0.0643703
\(249\) −4.00714 −0.253942
\(250\) 1.93529 0.122398
\(251\) −2.14471 −0.135373 −0.0676864 0.997707i \(-0.521562\pi\)
−0.0676864 + 0.997707i \(0.521562\pi\)
\(252\) 1.95185 0.122955
\(253\) 0 0
\(254\) −21.6118 −1.35604
\(255\) −0.623165 −0.0390241
\(256\) 1.00000 0.0625000
\(257\) −15.8488 −0.988620 −0.494310 0.869286i \(-0.664579\pi\)
−0.494310 + 0.869286i \(0.664579\pi\)
\(258\) 5.14282 0.320178
\(259\) 13.4490 0.835678
\(260\) −0.934959 −0.0579837
\(261\) 5.41741 0.335329
\(262\) −14.9571 −0.924053
\(263\) 0.130608 0.00805362 0.00402681 0.999992i \(-0.498718\pi\)
0.00402681 + 0.999992i \(0.498718\pi\)
\(264\) −6.30195 −0.387858
\(265\) −0.390071 −0.0239619
\(266\) 1.67751 0.102855
\(267\) 9.55158 0.584547
\(268\) 7.09327 0.433290
\(269\) −23.9688 −1.46140 −0.730702 0.682697i \(-0.760807\pi\)
−0.730702 + 0.682697i \(0.760807\pi\)
\(270\) 0.194262 0.0118224
\(271\) 3.22188 0.195715 0.0978575 0.995200i \(-0.468801\pi\)
0.0978575 + 0.995200i \(0.468801\pi\)
\(272\) −3.20786 −0.194505
\(273\) −9.39402 −0.568552
\(274\) 18.1570 1.09690
\(275\) −31.2719 −1.88577
\(276\) 0 0
\(277\) −13.0026 −0.781250 −0.390625 0.920550i \(-0.627741\pi\)
−0.390625 + 0.920550i \(0.627741\pi\)
\(278\) −7.36718 −0.441854
\(279\) −1.01371 −0.0606889
\(280\) −0.379170 −0.0226597
\(281\) 9.71850 0.579757 0.289878 0.957063i \(-0.406385\pi\)
0.289878 + 0.957063i \(0.406385\pi\)
\(282\) 7.41216 0.441388
\(283\) 19.9847 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(284\) 1.10938 0.0658294
\(285\) 0.166958 0.00988974
\(286\) 30.3305 1.79348
\(287\) −1.89722 −0.111989
\(288\) 1.00000 0.0589256
\(289\) −6.70963 −0.394684
\(290\) −1.05240 −0.0617988
\(291\) 1.12094 0.0657108
\(292\) −13.1859 −0.771648
\(293\) −17.8821 −1.04468 −0.522341 0.852737i \(-0.674941\pi\)
−0.522341 + 0.852737i \(0.674941\pi\)
\(294\) 3.19028 0.186061
\(295\) 1.67108 0.0972941
\(296\) 6.89037 0.400494
\(297\) −6.30195 −0.365676
\(298\) 15.6333 0.905613
\(299\) 0 0
\(300\) 4.96226 0.286496
\(301\) −10.0380 −0.578581
\(302\) 20.7042 1.19139
\(303\) −12.5714 −0.722209
\(304\) 0.859449 0.0492928
\(305\) −2.35448 −0.134817
\(306\) −3.20786 −0.183381
\(307\) −12.5139 −0.714205 −0.357102 0.934065i \(-0.616235\pi\)
−0.357102 + 0.934065i \(0.616235\pi\)
\(308\) 12.3005 0.700884
\(309\) −19.3963 −1.10342
\(310\) 0.196924 0.0111845
\(311\) 5.45773 0.309479 0.154740 0.987955i \(-0.450546\pi\)
0.154740 + 0.987955i \(0.450546\pi\)
\(312\) −4.81288 −0.272476
\(313\) −23.9234 −1.35223 −0.676114 0.736797i \(-0.736337\pi\)
−0.676114 + 0.736797i \(0.736337\pi\)
\(314\) −3.97631 −0.224396
\(315\) −0.379170 −0.0213638
\(316\) 1.12979 0.0635559
\(317\) −16.1225 −0.905532 −0.452766 0.891629i \(-0.649563\pi\)
−0.452766 + 0.891629i \(0.649563\pi\)
\(318\) −2.00797 −0.112601
\(319\) 34.1402 1.91149
\(320\) −0.194262 −0.0108596
\(321\) −3.07594 −0.171682
\(322\) 0 0
\(323\) −2.75699 −0.153403
\(324\) 1.00000 0.0555556
\(325\) −23.8828 −1.32478
\(326\) 24.5129 1.35764
\(327\) −1.94929 −0.107796
\(328\) −0.972011 −0.0536703
\(329\) −14.4674 −0.797615
\(330\) 1.22423 0.0673915
\(331\) −0.517893 −0.0284660 −0.0142330 0.999899i \(-0.504531\pi\)
−0.0142330 + 0.999899i \(0.504531\pi\)
\(332\) 4.00714 0.219920
\(333\) 6.89037 0.377590
\(334\) −1.95444 −0.106942
\(335\) −1.37795 −0.0752855
\(336\) −1.95185 −0.106482
\(337\) −0.530119 −0.0288774 −0.0144387 0.999896i \(-0.504596\pi\)
−0.0144387 + 0.999896i \(0.504596\pi\)
\(338\) 10.1638 0.552839
\(339\) 17.1492 0.931418
\(340\) 0.623165 0.0337958
\(341\) −6.38832 −0.345947
\(342\) 0.859449 0.0464737
\(343\) −19.8899 −1.07395
\(344\) −5.14282 −0.277282
\(345\) 0 0
\(346\) 6.14165 0.330177
\(347\) 26.8126 1.43937 0.719687 0.694298i \(-0.244285\pi\)
0.719687 + 0.694298i \(0.244285\pi\)
\(348\) −5.41741 −0.290404
\(349\) 1.45728 0.0780066 0.0390033 0.999239i \(-0.487582\pi\)
0.0390033 + 0.999239i \(0.487582\pi\)
\(350\) −9.68559 −0.517716
\(351\) −4.81288 −0.256893
\(352\) 6.30195 0.335895
\(353\) 22.4537 1.19509 0.597545 0.801835i \(-0.296143\pi\)
0.597545 + 0.801835i \(0.296143\pi\)
\(354\) 8.60221 0.457202
\(355\) −0.215510 −0.0114381
\(356\) −9.55158 −0.506233
\(357\) 6.26126 0.331381
\(358\) 6.47091 0.341998
\(359\) 0.168999 0.00891940 0.00445970 0.999990i \(-0.498580\pi\)
0.00445970 + 0.999990i \(0.498580\pi\)
\(360\) −0.194262 −0.0102385
\(361\) −18.2613 −0.961124
\(362\) 3.26146 0.171419
\(363\) −28.7146 −1.50712
\(364\) 9.39402 0.492380
\(365\) 2.56152 0.134076
\(366\) −12.1201 −0.633530
\(367\) 17.8312 0.930782 0.465391 0.885105i \(-0.345914\pi\)
0.465391 + 0.885105i \(0.345914\pi\)
\(368\) 0 0
\(369\) −0.972011 −0.0506009
\(370\) −1.33853 −0.0695871
\(371\) 3.91925 0.203477
\(372\) 1.01371 0.0525582
\(373\) −27.3884 −1.41812 −0.709058 0.705150i \(-0.750879\pi\)
−0.709058 + 0.705150i \(0.750879\pi\)
\(374\) −20.2158 −1.04533
\(375\) −1.93529 −0.0999378
\(376\) −7.41216 −0.382253
\(377\) 26.0733 1.34285
\(378\) −1.95185 −0.100392
\(379\) −2.73604 −0.140541 −0.0702704 0.997528i \(-0.522386\pi\)
−0.0702704 + 0.997528i \(0.522386\pi\)
\(380\) −0.166958 −0.00856476
\(381\) 21.6118 1.10721
\(382\) −22.8536 −1.16929
\(383\) −14.4109 −0.736362 −0.368181 0.929754i \(-0.620019\pi\)
−0.368181 + 0.929754i \(0.620019\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.38951 −0.121781
\(386\) −16.9414 −0.862294
\(387\) −5.14282 −0.261424
\(388\) −1.12094 −0.0569073
\(389\) 1.08404 0.0549630 0.0274815 0.999622i \(-0.491251\pi\)
0.0274815 + 0.999622i \(0.491251\pi\)
\(390\) 0.934959 0.0473435
\(391\) 0 0
\(392\) −3.19028 −0.161134
\(393\) 14.9571 0.754486
\(394\) 1.25480 0.0632161
\(395\) −0.219476 −0.0110430
\(396\) 6.30195 0.316685
\(397\) −27.0241 −1.35630 −0.678149 0.734924i \(-0.737218\pi\)
−0.678149 + 0.734924i \(0.737218\pi\)
\(398\) 24.8140 1.24381
\(399\) −1.67751 −0.0839807
\(400\) −4.96226 −0.248113
\(401\) 9.27409 0.463126 0.231563 0.972820i \(-0.425616\pi\)
0.231563 + 0.972820i \(0.425616\pi\)
\(402\) −7.09327 −0.353780
\(403\) −4.87884 −0.243032
\(404\) 12.5714 0.625451
\(405\) −0.194262 −0.00965294
\(406\) 10.5740 0.524777
\(407\) 43.4228 2.15239
\(408\) 3.20786 0.158813
\(409\) 36.7657 1.81795 0.908974 0.416852i \(-0.136867\pi\)
0.908974 + 0.416852i \(0.136867\pi\)
\(410\) 0.188825 0.00932538
\(411\) −18.1570 −0.895619
\(412\) 19.3963 0.955587
\(413\) −16.7902 −0.826193
\(414\) 0 0
\(415\) −0.778434 −0.0382118
\(416\) 4.81288 0.235971
\(417\) 7.36718 0.360772
\(418\) 5.41620 0.264915
\(419\) −26.4958 −1.29441 −0.647203 0.762318i \(-0.724061\pi\)
−0.647203 + 0.762318i \(0.724061\pi\)
\(420\) 0.379170 0.0185016
\(421\) 9.95095 0.484980 0.242490 0.970154i \(-0.422036\pi\)
0.242490 + 0.970154i \(0.422036\pi\)
\(422\) 7.25827 0.353327
\(423\) −7.41216 −0.360392
\(424\) 2.00797 0.0975155
\(425\) 15.9182 0.772148
\(426\) −1.10938 −0.0537495
\(427\) 23.6567 1.14483
\(428\) 3.07594 0.148681
\(429\) −30.3305 −1.46437
\(430\) 0.999053 0.0481786
\(431\) 8.77287 0.422575 0.211287 0.977424i \(-0.432234\pi\)
0.211287 + 0.977424i \(0.432234\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −7.99061 −0.384004 −0.192002 0.981395i \(-0.561498\pi\)
−0.192002 + 0.981395i \(0.561498\pi\)
\(434\) −1.97860 −0.0949758
\(435\) 1.05240 0.0504585
\(436\) 1.94929 0.0933542
\(437\) 0 0
\(438\) 13.1859 0.630048
\(439\) −1.41063 −0.0673257 −0.0336628 0.999433i \(-0.510717\pi\)
−0.0336628 + 0.999433i \(0.510717\pi\)
\(440\) −1.22423 −0.0583628
\(441\) −3.19028 −0.151918
\(442\) −15.4390 −0.734361
\(443\) 12.5935 0.598335 0.299167 0.954201i \(-0.403291\pi\)
0.299167 + 0.954201i \(0.403291\pi\)
\(444\) −6.89037 −0.327002
\(445\) 1.85551 0.0879594
\(446\) 20.7342 0.981791
\(447\) −15.6333 −0.739430
\(448\) 1.95185 0.0922162
\(449\) −38.3533 −1.81000 −0.905002 0.425407i \(-0.860131\pi\)
−0.905002 + 0.425407i \(0.860131\pi\)
\(450\) −4.96226 −0.233923
\(451\) −6.12557 −0.288442
\(452\) −17.1492 −0.806632
\(453\) −20.7042 −0.972766
\(454\) −21.0504 −0.987943
\(455\) −1.82490 −0.0855525
\(456\) −0.859449 −0.0402474
\(457\) 26.6357 1.24597 0.622983 0.782235i \(-0.285921\pi\)
0.622983 + 0.782235i \(0.285921\pi\)
\(458\) −1.91595 −0.0895264
\(459\) 3.20786 0.149730
\(460\) 0 0
\(461\) 7.65301 0.356436 0.178218 0.983991i \(-0.442967\pi\)
0.178218 + 0.983991i \(0.442967\pi\)
\(462\) −12.3005 −0.572269
\(463\) −4.15149 −0.192936 −0.0964681 0.995336i \(-0.530755\pi\)
−0.0964681 + 0.995336i \(0.530755\pi\)
\(464\) 5.41741 0.251497
\(465\) −0.196924 −0.00913214
\(466\) −14.4306 −0.668483
\(467\) −32.1155 −1.48613 −0.743063 0.669221i \(-0.766628\pi\)
−0.743063 + 0.669221i \(0.766628\pi\)
\(468\) 4.81288 0.222475
\(469\) 13.8450 0.639302
\(470\) 1.43990 0.0664176
\(471\) 3.97631 0.183219
\(472\) −8.60221 −0.395949
\(473\) −32.4098 −1.49020
\(474\) −1.12979 −0.0518932
\(475\) −4.26481 −0.195683
\(476\) −6.26126 −0.286984
\(477\) 2.00797 0.0919385
\(478\) −9.81701 −0.449020
\(479\) 1.81509 0.0829337 0.0414668 0.999140i \(-0.486797\pi\)
0.0414668 + 0.999140i \(0.486797\pi\)
\(480\) 0.194262 0.00886680
\(481\) 33.1625 1.51208
\(482\) −14.9039 −0.678853
\(483\) 0 0
\(484\) 28.7146 1.30521
\(485\) 0.217756 0.00988781
\(486\) −1.00000 −0.0453609
\(487\) −16.1873 −0.733515 −0.366758 0.930317i \(-0.619532\pi\)
−0.366758 + 0.930317i \(0.619532\pi\)
\(488\) 12.1201 0.548653
\(489\) −24.5129 −1.10851
\(490\) 0.619750 0.0279975
\(491\) −1.42155 −0.0641536 −0.0320768 0.999485i \(-0.510212\pi\)
−0.0320768 + 0.999485i \(0.510212\pi\)
\(492\) 0.972011 0.0438216
\(493\) −17.3783 −0.782679
\(494\) 4.13642 0.186106
\(495\) −1.22423 −0.0550250
\(496\) −1.01371 −0.0455167
\(497\) 2.16534 0.0971286
\(498\) −4.00714 −0.179564
\(499\) 20.5750 0.921061 0.460531 0.887644i \(-0.347659\pi\)
0.460531 + 0.887644i \(0.347659\pi\)
\(500\) 1.93529 0.0865487
\(501\) 1.95444 0.0873177
\(502\) −2.14471 −0.0957230
\(503\) 14.7875 0.659342 0.329671 0.944096i \(-0.393062\pi\)
0.329671 + 0.944096i \(0.393062\pi\)
\(504\) 1.95185 0.0869423
\(505\) −2.44214 −0.108674
\(506\) 0 0
\(507\) −10.1638 −0.451391
\(508\) −21.6118 −0.958869
\(509\) 33.1496 1.46933 0.734666 0.678429i \(-0.237339\pi\)
0.734666 + 0.678429i \(0.237339\pi\)
\(510\) −0.623165 −0.0275942
\(511\) −25.7370 −1.13854
\(512\) 1.00000 0.0441942
\(513\) −0.859449 −0.0379456
\(514\) −15.8488 −0.699060
\(515\) −3.76796 −0.166036
\(516\) 5.14282 0.226400
\(517\) −46.7111 −2.05435
\(518\) 13.4490 0.590913
\(519\) −6.14165 −0.269589
\(520\) −0.934959 −0.0410006
\(521\) −28.9660 −1.26902 −0.634511 0.772913i \(-0.718799\pi\)
−0.634511 + 0.772913i \(0.718799\pi\)
\(522\) 5.41741 0.237114
\(523\) 33.6692 1.47225 0.736126 0.676845i \(-0.236653\pi\)
0.736126 + 0.676845i \(0.236653\pi\)
\(524\) −14.9571 −0.653404
\(525\) 9.68559 0.422714
\(526\) 0.130608 0.00569477
\(527\) 3.25182 0.141652
\(528\) −6.30195 −0.274257
\(529\) 0 0
\(530\) −0.390071 −0.0169436
\(531\) −8.60221 −0.373304
\(532\) 1.67751 0.0727295
\(533\) −4.67817 −0.202634
\(534\) 9.55158 0.413337
\(535\) −0.597538 −0.0258338
\(536\) 7.09327 0.306382
\(537\) −6.47091 −0.279240
\(538\) −23.9688 −1.03337
\(539\) −20.1050 −0.865984
\(540\) 0.194262 0.00835970
\(541\) −40.1990 −1.72829 −0.864144 0.503245i \(-0.832139\pi\)
−0.864144 + 0.503245i \(0.832139\pi\)
\(542\) 3.22188 0.138391
\(543\) −3.26146 −0.139963
\(544\) −3.20786 −0.137536
\(545\) −0.378673 −0.0162206
\(546\) −9.39402 −0.402027
\(547\) 22.7001 0.970588 0.485294 0.874351i \(-0.338713\pi\)
0.485294 + 0.874351i \(0.338713\pi\)
\(548\) 18.1570 0.775628
\(549\) 12.1201 0.517275
\(550\) −31.2719 −1.33344
\(551\) 4.65598 0.198352
\(552\) 0 0
\(553\) 2.20519 0.0937741
\(554\) −13.0026 −0.552427
\(555\) 1.33853 0.0568176
\(556\) −7.36718 −0.312438
\(557\) 3.95623 0.167631 0.0838154 0.996481i \(-0.473289\pi\)
0.0838154 + 0.996481i \(0.473289\pi\)
\(558\) −1.01371 −0.0429136
\(559\) −24.7518 −1.04689
\(560\) −0.379170 −0.0160228
\(561\) 20.2158 0.853511
\(562\) 9.71850 0.409950
\(563\) −27.2143 −1.14695 −0.573473 0.819224i \(-0.694404\pi\)
−0.573473 + 0.819224i \(0.694404\pi\)
\(564\) 7.41216 0.312108
\(565\) 3.33144 0.140155
\(566\) 19.9847 0.840022
\(567\) 1.95185 0.0819700
\(568\) 1.10938 0.0465484
\(569\) −20.6417 −0.865344 −0.432672 0.901551i \(-0.642429\pi\)
−0.432672 + 0.901551i \(0.642429\pi\)
\(570\) 0.166958 0.00699310
\(571\) 30.3575 1.27042 0.635212 0.772338i \(-0.280913\pi\)
0.635212 + 0.772338i \(0.280913\pi\)
\(572\) 30.3305 1.26818
\(573\) 22.8536 0.954723
\(574\) −1.89722 −0.0791884
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −14.1163 −0.587667 −0.293834 0.955857i \(-0.594931\pi\)
−0.293834 + 0.955857i \(0.594931\pi\)
\(578\) −6.70963 −0.279084
\(579\) 16.9414 0.704060
\(580\) −1.05240 −0.0436983
\(581\) 7.82134 0.324484
\(582\) 1.12094 0.0464646
\(583\) 12.6541 0.524079
\(584\) −13.1859 −0.545638
\(585\) −0.934959 −0.0386558
\(586\) −17.8821 −0.738702
\(587\) 36.5538 1.50874 0.754369 0.656451i \(-0.227943\pi\)
0.754369 + 0.656451i \(0.227943\pi\)
\(588\) 3.19028 0.131565
\(589\) −0.871227 −0.0358983
\(590\) 1.67108 0.0687973
\(591\) −1.25480 −0.0516157
\(592\) 6.89037 0.283192
\(593\) 6.64351 0.272816 0.136408 0.990653i \(-0.456444\pi\)
0.136408 + 0.990653i \(0.456444\pi\)
\(594\) −6.30195 −0.258572
\(595\) 1.21632 0.0498644
\(596\) 15.6333 0.640365
\(597\) −24.8140 −1.01557
\(598\) 0 0
\(599\) 28.0099 1.14445 0.572226 0.820096i \(-0.306080\pi\)
0.572226 + 0.820096i \(0.306080\pi\)
\(600\) 4.96226 0.202584
\(601\) 17.5864 0.717363 0.358682 0.933460i \(-0.383226\pi\)
0.358682 + 0.933460i \(0.383226\pi\)
\(602\) −10.0380 −0.409119
\(603\) 7.09327 0.288860
\(604\) 20.7042 0.842440
\(605\) −5.57814 −0.226784
\(606\) −12.5714 −0.510679
\(607\) −23.5948 −0.957684 −0.478842 0.877901i \(-0.658943\pi\)
−0.478842 + 0.877901i \(0.658943\pi\)
\(608\) 0.859449 0.0348552
\(609\) −10.5740 −0.428479
\(610\) −2.35448 −0.0953301
\(611\) −35.6738 −1.44321
\(612\) −3.20786 −0.129670
\(613\) 32.9938 1.33261 0.666303 0.745681i \(-0.267876\pi\)
0.666303 + 0.745681i \(0.267876\pi\)
\(614\) −12.5139 −0.505019
\(615\) −0.188825 −0.00761414
\(616\) 12.3005 0.495599
\(617\) −38.0582 −1.53216 −0.766082 0.642743i \(-0.777796\pi\)
−0.766082 + 0.642743i \(0.777796\pi\)
\(618\) −19.3963 −0.780234
\(619\) −8.49110 −0.341286 −0.170643 0.985333i \(-0.554584\pi\)
−0.170643 + 0.985333i \(0.554584\pi\)
\(620\) 0.196924 0.00790866
\(621\) 0 0
\(622\) 5.45773 0.218835
\(623\) −18.6432 −0.746926
\(624\) −4.81288 −0.192669
\(625\) 24.4354 0.977414
\(626\) −23.9234 −0.956170
\(627\) −5.41620 −0.216302
\(628\) −3.97631 −0.158672
\(629\) −22.1033 −0.881318
\(630\) −0.379170 −0.0151065
\(631\) 11.1468 0.443747 0.221874 0.975075i \(-0.428783\pi\)
0.221874 + 0.975075i \(0.428783\pi\)
\(632\) 1.12979 0.0449408
\(633\) −7.25827 −0.288490
\(634\) −16.1225 −0.640308
\(635\) 4.19835 0.166606
\(636\) −2.00797 −0.0796210
\(637\) −15.3545 −0.608366
\(638\) 34.1402 1.35163
\(639\) 1.10938 0.0438863
\(640\) −0.194262 −0.00767887
\(641\) −37.8358 −1.49442 −0.747212 0.664586i \(-0.768608\pi\)
−0.747212 + 0.664586i \(0.768608\pi\)
\(642\) −3.07594 −0.121398
\(643\) −22.3733 −0.882317 −0.441158 0.897429i \(-0.645432\pi\)
−0.441158 + 0.897429i \(0.645432\pi\)
\(644\) 0 0
\(645\) −0.999053 −0.0393377
\(646\) −2.75699 −0.108472
\(647\) −10.9467 −0.430359 −0.215180 0.976575i \(-0.569034\pi\)
−0.215180 + 0.976575i \(0.569034\pi\)
\(648\) 1.00000 0.0392837
\(649\) −54.2107 −2.12796
\(650\) −23.8828 −0.936759
\(651\) 1.97860 0.0775474
\(652\) 24.5129 0.959997
\(653\) 5.78807 0.226505 0.113252 0.993566i \(-0.463873\pi\)
0.113252 + 0.993566i \(0.463873\pi\)
\(654\) −1.94929 −0.0762234
\(655\) 2.90559 0.113531
\(656\) −0.972011 −0.0379507
\(657\) −13.1859 −0.514432
\(658\) −14.4674 −0.563999
\(659\) −8.31443 −0.323884 −0.161942 0.986800i \(-0.551776\pi\)
−0.161942 + 0.986800i \(0.551776\pi\)
\(660\) 1.22423 0.0476530
\(661\) −12.1685 −0.473301 −0.236650 0.971595i \(-0.576050\pi\)
−0.236650 + 0.971595i \(0.576050\pi\)
\(662\) −0.517893 −0.0201285
\(663\) 15.4390 0.599603
\(664\) 4.00714 0.155507
\(665\) −0.325877 −0.0126370
\(666\) 6.89037 0.266996
\(667\) 0 0
\(668\) −1.95444 −0.0756194
\(669\) −20.7342 −0.801629
\(670\) −1.37795 −0.0532349
\(671\) 76.3805 2.94864
\(672\) −1.95185 −0.0752942
\(673\) 4.87132 0.187775 0.0938877 0.995583i \(-0.470071\pi\)
0.0938877 + 0.995583i \(0.470071\pi\)
\(674\) −0.530119 −0.0204194
\(675\) 4.96226 0.190998
\(676\) 10.1638 0.390916
\(677\) −20.0060 −0.768892 −0.384446 0.923147i \(-0.625608\pi\)
−0.384446 + 0.923147i \(0.625608\pi\)
\(678\) 17.1492 0.658612
\(679\) −2.18791 −0.0839643
\(680\) 0.623165 0.0238973
\(681\) 21.0504 0.806652
\(682\) −6.38832 −0.244621
\(683\) −23.9883 −0.917888 −0.458944 0.888465i \(-0.651772\pi\)
−0.458944 + 0.888465i \(0.651772\pi\)
\(684\) 0.859449 0.0328618
\(685\) −3.52721 −0.134768
\(686\) −19.8899 −0.759400
\(687\) 1.91595 0.0730980
\(688\) −5.14282 −0.196068
\(689\) 9.66410 0.368173
\(690\) 0 0
\(691\) 16.5438 0.629356 0.314678 0.949199i \(-0.398104\pi\)
0.314678 + 0.949199i \(0.398104\pi\)
\(692\) 6.14165 0.233471
\(693\) 12.3005 0.467256
\(694\) 26.8126 1.01779
\(695\) 1.43116 0.0542870
\(696\) −5.41741 −0.205346
\(697\) 3.11808 0.118106
\(698\) 1.45728 0.0551590
\(699\) 14.4306 0.545814
\(700\) −9.68559 −0.366081
\(701\) −15.9822 −0.603640 −0.301820 0.953365i \(-0.597594\pi\)
−0.301820 + 0.953365i \(0.597594\pi\)
\(702\) −4.81288 −0.181650
\(703\) 5.92192 0.223349
\(704\) 6.30195 0.237514
\(705\) −1.43990 −0.0542298
\(706\) 22.4537 0.845056
\(707\) 24.5375 0.922828
\(708\) 8.60221 0.323291
\(709\) 6.40478 0.240536 0.120268 0.992741i \(-0.461625\pi\)
0.120268 + 0.992741i \(0.461625\pi\)
\(710\) −0.215510 −0.00808793
\(711\) 1.12979 0.0423706
\(712\) −9.55158 −0.357960
\(713\) 0 0
\(714\) 6.26126 0.234322
\(715\) −5.89206 −0.220351
\(716\) 6.47091 0.241829
\(717\) 9.81701 0.366623
\(718\) 0.168999 0.00630697
\(719\) −41.6971 −1.55504 −0.777519 0.628859i \(-0.783522\pi\)
−0.777519 + 0.628859i \(0.783522\pi\)
\(720\) −0.194262 −0.00723971
\(721\) 37.8587 1.40993
\(722\) −18.2613 −0.679617
\(723\) 14.9039 0.554281
\(724\) 3.26146 0.121211
\(725\) −26.8826 −0.998395
\(726\) −28.7146 −1.06570
\(727\) −10.0060 −0.371101 −0.185551 0.982635i \(-0.559407\pi\)
−0.185551 + 0.982635i \(0.559407\pi\)
\(728\) 9.39402 0.348165
\(729\) 1.00000 0.0370370
\(730\) 2.56152 0.0948062
\(731\) 16.4974 0.610180
\(732\) −12.1201 −0.447973
\(733\) −23.3610 −0.862858 −0.431429 0.902147i \(-0.641991\pi\)
−0.431429 + 0.902147i \(0.641991\pi\)
\(734\) 17.8312 0.658163
\(735\) −0.619750 −0.0228598
\(736\) 0 0
\(737\) 44.7014 1.64660
\(738\) −0.972011 −0.0357802
\(739\) 26.7250 0.983094 0.491547 0.870851i \(-0.336432\pi\)
0.491547 + 0.870851i \(0.336432\pi\)
\(740\) −1.33853 −0.0492055
\(741\) −4.13642 −0.151955
\(742\) 3.91925 0.143880
\(743\) −15.1711 −0.556575 −0.278288 0.960498i \(-0.589767\pi\)
−0.278288 + 0.960498i \(0.589767\pi\)
\(744\) 1.01371 0.0371642
\(745\) −3.03695 −0.111265
\(746\) −27.3884 −1.00276
\(747\) 4.00714 0.146614
\(748\) −20.2158 −0.739162
\(749\) 6.00378 0.219373
\(750\) −1.93529 −0.0706667
\(751\) 21.6684 0.790691 0.395346 0.918532i \(-0.370625\pi\)
0.395346 + 0.918532i \(0.370625\pi\)
\(752\) −7.41216 −0.270294
\(753\) 2.14471 0.0781575
\(754\) 26.0733 0.949535
\(755\) −4.02203 −0.146377
\(756\) −1.95185 −0.0709881
\(757\) −4.05195 −0.147271 −0.0736354 0.997285i \(-0.523460\pi\)
−0.0736354 + 0.997285i \(0.523460\pi\)
\(758\) −2.73604 −0.0993774
\(759\) 0 0
\(760\) −0.166958 −0.00605620
\(761\) −44.2112 −1.60266 −0.801328 0.598225i \(-0.795873\pi\)
−0.801328 + 0.598225i \(0.795873\pi\)
\(762\) 21.6118 0.782913
\(763\) 3.80473 0.137740
\(764\) −22.8536 −0.826815
\(765\) 0.623165 0.0225306
\(766\) −14.4109 −0.520687
\(767\) −41.4014 −1.49492
\(768\) −1.00000 −0.0360844
\(769\) 8.81046 0.317713 0.158857 0.987302i \(-0.449219\pi\)
0.158857 + 0.987302i \(0.449219\pi\)
\(770\) −2.38951 −0.0861119
\(771\) 15.8488 0.570780
\(772\) −16.9414 −0.609734
\(773\) −0.634119 −0.0228077 −0.0114038 0.999935i \(-0.503630\pi\)
−0.0114038 + 0.999935i \(0.503630\pi\)
\(774\) −5.14282 −0.184855
\(775\) 5.03027 0.180693
\(776\) −1.12094 −0.0402395
\(777\) −13.4490 −0.482479
\(778\) 1.08404 0.0388647
\(779\) −0.835394 −0.0299311
\(780\) 0.934959 0.0334769
\(781\) 6.99124 0.250166
\(782\) 0 0
\(783\) −5.41741 −0.193602
\(784\) −3.19028 −0.113939
\(785\) 0.772445 0.0275698
\(786\) 14.9571 0.533502
\(787\) 15.9809 0.569659 0.284829 0.958578i \(-0.408063\pi\)
0.284829 + 0.958578i \(0.408063\pi\)
\(788\) 1.25480 0.0447005
\(789\) −0.130608 −0.00464976
\(790\) −0.219476 −0.00780860
\(791\) −33.4727 −1.19015
\(792\) 6.30195 0.223930
\(793\) 58.3328 2.07146
\(794\) −27.0241 −0.959048
\(795\) 0.390071 0.0138344
\(796\) 24.8140 0.879509
\(797\) −40.9716 −1.45129 −0.725644 0.688070i \(-0.758458\pi\)
−0.725644 + 0.688070i \(0.758458\pi\)
\(798\) −1.67751 −0.0593833
\(799\) 23.7772 0.841176
\(800\) −4.96226 −0.175442
\(801\) −9.55158 −0.337488
\(802\) 9.27409 0.327479
\(803\) −83.0971 −2.93243
\(804\) −7.09327 −0.250160
\(805\) 0 0
\(806\) −4.87884 −0.171850
\(807\) 23.9688 0.843742
\(808\) 12.5714 0.442261
\(809\) −28.2257 −0.992363 −0.496181 0.868219i \(-0.665265\pi\)
−0.496181 + 0.868219i \(0.665265\pi\)
\(810\) −0.194262 −0.00682566
\(811\) −3.61726 −0.127019 −0.0635096 0.997981i \(-0.520229\pi\)
−0.0635096 + 0.997981i \(0.520229\pi\)
\(812\) 10.5740 0.371073
\(813\) −3.22188 −0.112996
\(814\) 43.4228 1.52197
\(815\) −4.76191 −0.166802
\(816\) 3.20786 0.112298
\(817\) −4.41999 −0.154636
\(818\) 36.7657 1.28548
\(819\) 9.39402 0.328253
\(820\) 0.188825 0.00659404
\(821\) 30.8947 1.07823 0.539116 0.842231i \(-0.318758\pi\)
0.539116 + 0.842231i \(0.318758\pi\)
\(822\) −18.1570 −0.633298
\(823\) 4.00604 0.139642 0.0698209 0.997560i \(-0.477757\pi\)
0.0698209 + 0.997560i \(0.477757\pi\)
\(824\) 19.3963 0.675702
\(825\) 31.2719 1.08875
\(826\) −16.7902 −0.584206
\(827\) −26.0104 −0.904469 −0.452234 0.891899i \(-0.649373\pi\)
−0.452234 + 0.891899i \(0.649373\pi\)
\(828\) 0 0
\(829\) 39.4460 1.37002 0.685008 0.728535i \(-0.259798\pi\)
0.685008 + 0.728535i \(0.259798\pi\)
\(830\) −0.778434 −0.0270199
\(831\) 13.0026 0.451055
\(832\) 4.81288 0.166857
\(833\) 10.2340 0.354587
\(834\) 7.36718 0.255104
\(835\) 0.379672 0.0131391
\(836\) 5.41620 0.187323
\(837\) 1.01371 0.0350388
\(838\) −26.4958 −0.915283
\(839\) 3.26103 0.112583 0.0562916 0.998414i \(-0.482072\pi\)
0.0562916 + 0.998414i \(0.482072\pi\)
\(840\) 0.379170 0.0130826
\(841\) 0.348328 0.0120113
\(842\) 9.95095 0.342932
\(843\) −9.71850 −0.334723
\(844\) 7.25827 0.249840
\(845\) −1.97444 −0.0679228
\(846\) −7.41216 −0.254835
\(847\) 56.0465 1.92578
\(848\) 2.00797 0.0689539
\(849\) −19.9847 −0.685875
\(850\) 15.9182 0.545991
\(851\) 0 0
\(852\) −1.10938 −0.0380066
\(853\) −7.71650 −0.264208 −0.132104 0.991236i \(-0.542173\pi\)
−0.132104 + 0.991236i \(0.542173\pi\)
\(854\) 23.6567 0.809515
\(855\) −0.166958 −0.00570984
\(856\) 3.07594 0.105134
\(857\) 4.63538 0.158342 0.0791708 0.996861i \(-0.474773\pi\)
0.0791708 + 0.996861i \(0.474773\pi\)
\(858\) −30.3305 −1.03547
\(859\) −49.1319 −1.67636 −0.838179 0.545396i \(-0.816379\pi\)
−0.838179 + 0.545396i \(0.816379\pi\)
\(860\) 0.999053 0.0340674
\(861\) 1.89722 0.0646571
\(862\) 8.77287 0.298805
\(863\) −40.8058 −1.38905 −0.694523 0.719470i \(-0.744385\pi\)
−0.694523 + 0.719470i \(0.744385\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.19309 −0.0405662
\(866\) −7.99061 −0.271532
\(867\) 6.70963 0.227871
\(868\) −1.97860 −0.0671580
\(869\) 7.11991 0.241526
\(870\) 1.05240 0.0356796
\(871\) 34.1390 1.15676
\(872\) 1.94929 0.0660114
\(873\) −1.12094 −0.0379382
\(874\) 0 0
\(875\) 3.77739 0.127699
\(876\) 13.1859 0.445511
\(877\) 25.6472 0.866043 0.433021 0.901384i \(-0.357447\pi\)
0.433021 + 0.901384i \(0.357447\pi\)
\(878\) −1.41063 −0.0476064
\(879\) 17.8821 0.603147
\(880\) −1.22423 −0.0412687
\(881\) 43.0288 1.44968 0.724838 0.688919i \(-0.241915\pi\)
0.724838 + 0.688919i \(0.241915\pi\)
\(882\) −3.19028 −0.107422
\(883\) −10.0370 −0.337771 −0.168885 0.985636i \(-0.554017\pi\)
−0.168885 + 0.985636i \(0.554017\pi\)
\(884\) −15.4390 −0.519271
\(885\) −1.67108 −0.0561728
\(886\) 12.5935 0.423087
\(887\) −10.5222 −0.353300 −0.176650 0.984274i \(-0.556526\pi\)
−0.176650 + 0.984274i \(0.556526\pi\)
\(888\) −6.89037 −0.231226
\(889\) −42.1830 −1.41477
\(890\) 1.85551 0.0621967
\(891\) 6.30195 0.211123
\(892\) 20.7342 0.694231
\(893\) −6.37037 −0.213176
\(894\) −15.6333 −0.522856
\(895\) −1.25705 −0.0420185
\(896\) 1.95185 0.0652067
\(897\) 0 0
\(898\) −38.3533 −1.27987
\(899\) −5.49166 −0.183157
\(900\) −4.96226 −0.165409
\(901\) −6.44128 −0.214590
\(902\) −6.12557 −0.203959
\(903\) 10.0380 0.334044
\(904\) −17.1492 −0.570375
\(905\) −0.633578 −0.0210608
\(906\) −20.7042 −0.687850
\(907\) 39.0117 1.29536 0.647680 0.761912i \(-0.275739\pi\)
0.647680 + 0.761912i \(0.275739\pi\)
\(908\) −21.0504 −0.698581
\(909\) 12.5714 0.416967
\(910\) −1.82490 −0.0604948
\(911\) −22.7667 −0.754294 −0.377147 0.926153i \(-0.623095\pi\)
−0.377147 + 0.926153i \(0.623095\pi\)
\(912\) −0.859449 −0.0284592
\(913\) 25.2528 0.835746
\(914\) 26.6357 0.881031
\(915\) 2.35448 0.0778367
\(916\) −1.91595 −0.0633047
\(917\) −29.1940 −0.964071
\(918\) 3.20786 0.105875
\(919\) −21.7361 −0.717007 −0.358504 0.933528i \(-0.616713\pi\)
−0.358504 + 0.933528i \(0.616713\pi\)
\(920\) 0 0
\(921\) 12.5139 0.412346
\(922\) 7.65301 0.252039
\(923\) 5.33930 0.175745
\(924\) −12.3005 −0.404655
\(925\) −34.1918 −1.12422
\(926\) −4.15149 −0.136426
\(927\) 19.3963 0.637058
\(928\) 5.41741 0.177835
\(929\) −40.7826 −1.33803 −0.669016 0.743248i \(-0.733284\pi\)
−0.669016 + 0.743248i \(0.733284\pi\)
\(930\) −0.196924 −0.00645740
\(931\) −2.74189 −0.0898617
\(932\) −14.4306 −0.472689
\(933\) −5.45773 −0.178678
\(934\) −32.1155 −1.05085
\(935\) 3.92715 0.128432
\(936\) 4.81288 0.157314
\(937\) −7.22235 −0.235944 −0.117972 0.993017i \(-0.537639\pi\)
−0.117972 + 0.993017i \(0.537639\pi\)
\(938\) 13.8450 0.452055
\(939\) 23.9234 0.780710
\(940\) 1.43990 0.0469643
\(941\) 13.2043 0.430449 0.215225 0.976565i \(-0.430952\pi\)
0.215225 + 0.976565i \(0.430952\pi\)
\(942\) 3.97631 0.129555
\(943\) 0 0
\(944\) −8.60221 −0.279978
\(945\) 0.379170 0.0123344
\(946\) −32.4098 −1.05373
\(947\) 33.7912 1.09807 0.549034 0.835800i \(-0.314996\pi\)
0.549034 + 0.835800i \(0.314996\pi\)
\(948\) −1.12979 −0.0366940
\(949\) −63.4623 −2.06007
\(950\) −4.26481 −0.138369
\(951\) 16.1225 0.522809
\(952\) −6.26126 −0.202929
\(953\) −22.5608 −0.730815 −0.365407 0.930848i \(-0.619070\pi\)
−0.365407 + 0.930848i \(0.619070\pi\)
\(954\) 2.00797 0.0650103
\(955\) 4.43958 0.143662
\(956\) −9.81701 −0.317505
\(957\) −34.1402 −1.10360
\(958\) 1.81509 0.0586430
\(959\) 35.4397 1.14441
\(960\) 0.194262 0.00626977
\(961\) −29.9724 −0.966852
\(962\) 33.1625 1.06920
\(963\) 3.07594 0.0991209
\(964\) −14.9039 −0.480022
\(965\) 3.29107 0.105943
\(966\) 0 0
\(967\) −15.9309 −0.512303 −0.256152 0.966637i \(-0.582455\pi\)
−0.256152 + 0.966637i \(0.582455\pi\)
\(968\) 28.7146 0.922921
\(969\) 2.75699 0.0885673
\(970\) 0.217756 0.00699174
\(971\) 42.1496 1.35264 0.676322 0.736606i \(-0.263573\pi\)
0.676322 + 0.736606i \(0.263573\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −14.3796 −0.460989
\(974\) −16.1873 −0.518674
\(975\) 23.8828 0.764861
\(976\) 12.1201 0.387956
\(977\) −31.8104 −1.01770 −0.508852 0.860854i \(-0.669930\pi\)
−0.508852 + 0.860854i \(0.669930\pi\)
\(978\) −24.5129 −0.783835
\(979\) −60.1936 −1.92379
\(980\) 0.619750 0.0197972
\(981\) 1.94929 0.0622361
\(982\) −1.42155 −0.0453634
\(983\) 19.8293 0.632457 0.316229 0.948683i \(-0.397583\pi\)
0.316229 + 0.948683i \(0.397583\pi\)
\(984\) 0.972011 0.0309866
\(985\) −0.243760 −0.00776685
\(986\) −17.3783 −0.553438
\(987\) 14.4674 0.460503
\(988\) 4.13642 0.131597
\(989\) 0 0
\(990\) −1.22423 −0.0389085
\(991\) 6.30934 0.200423 0.100211 0.994966i \(-0.468048\pi\)
0.100211 + 0.994966i \(0.468048\pi\)
\(992\) −1.01371 −0.0321852
\(993\) 0.517893 0.0164348
\(994\) 2.16534 0.0686803
\(995\) −4.82041 −0.152817
\(996\) −4.00714 −0.126971
\(997\) −44.5390 −1.41056 −0.705282 0.708927i \(-0.749180\pi\)
−0.705282 + 0.708927i \(0.749180\pi\)
\(998\) 20.5750 0.651289
\(999\) −6.89037 −0.218002
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 3174.2.a.ba.1.3 5
3.2 odd 2 9522.2.a.bs.1.3 5
23.9 even 11 138.2.e.b.127.1 yes 10
23.18 even 11 138.2.e.b.25.1 10
23.22 odd 2 3174.2.a.bb.1.3 5
69.32 odd 22 414.2.i.e.127.1 10
69.41 odd 22 414.2.i.e.163.1 10
69.68 even 2 9522.2.a.br.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.b.25.1 10 23.18 even 11
138.2.e.b.127.1 yes 10 23.9 even 11
414.2.i.e.127.1 10 69.32 odd 22
414.2.i.e.163.1 10 69.41 odd 22
3174.2.a.ba.1.3 5 1.1 even 1 trivial
3174.2.a.bb.1.3 5 23.22 odd 2
9522.2.a.br.1.3 5 69.68 even 2
9522.2.a.bs.1.3 5 3.2 odd 2