Properties

Label 76.3.j.a.13.1
Level $76$
Weight $3$
Character 76.13
Analytic conductor $2.071$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,3,Mod(13,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.j (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 13.1
Root \(3.09175i\) of defining polynomial
Character \(\chi\) \(=\) 76.13
Dual form 76.3.j.a.41.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.617749 + 1.69725i) q^{3} +(0.757040 + 4.29338i) q^{5} +(-2.41011 + 4.17443i) q^{7} +(4.39535 + 3.68814i) q^{9} +O(q^{10})\) \(q+(-0.617749 + 1.69725i) q^{3} +(0.757040 + 4.29338i) q^{5} +(-2.41011 + 4.17443i) q^{7} +(4.39535 + 3.68814i) q^{9} +(0.0945208 + 0.163715i) q^{11} +(1.39295 + 3.82710i) q^{13} +(-7.75461 - 1.36735i) q^{15} +(13.1145 - 11.0044i) q^{17} +(-8.88753 - 16.7932i) q^{19} +(-5.59622 - 6.66931i) q^{21} +(2.45173 - 13.9044i) q^{23} +(5.63227 - 2.04998i) q^{25} +(-23.0527 + 13.3095i) q^{27} +(18.3700 - 21.8925i) q^{29} +(-5.92457 - 3.42055i) q^{31} +(-0.336255 + 0.0592909i) q^{33} +(-19.7470 - 7.18732i) q^{35} +31.9736i q^{37} -7.35605 q^{39} +(19.4006 - 53.3028i) q^{41} +(7.16167 + 40.6158i) q^{43} +(-12.5071 + 21.6630i) q^{45} +(44.9103 + 37.6842i) q^{47} +(12.8827 + 22.3136i) q^{49} +(10.5757 + 29.0566i) q^{51} +(-5.32437 - 0.938830i) q^{53} +(-0.631335 + 0.529753i) q^{55} +(33.9926 - 4.71039i) q^{57} +(-36.3323 - 43.2991i) q^{59} +(-15.7238 + 89.1742i) q^{61} +(-25.9892 + 9.45928i) q^{63} +(-15.3767 + 8.87775i) q^{65} +(5.04936 - 6.01760i) q^{67} +(22.0848 + 12.7506i) q^{69} +(23.9368 - 4.22071i) q^{71} +(-111.851 - 40.7104i) q^{73} +10.8258i q^{75} -0.911222 q^{77} +(47.8267 - 131.403i) q^{79} +(0.618357 + 3.50688i) q^{81} +(52.0220 - 90.1047i) q^{83} +(57.1743 + 47.9749i) q^{85} +(25.8090 + 44.7025i) q^{87} +(-52.7400 - 144.902i) q^{89} +(-19.3332 - 3.40896i) q^{91} +(9.46544 - 7.94245i) q^{93} +(65.3715 - 50.8707i) q^{95} +(-52.4375 - 62.4925i) q^{97} +(-0.188351 + 1.06819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9} - 15 q^{11} + 51 q^{13} + 21 q^{15} - 45 q^{17} + 30 q^{19} - 63 q^{21} + 48 q^{23} - 54 q^{25} - 198 q^{27} - 39 q^{29} - 108 q^{31} - 105 q^{33} + 51 q^{35} + 48 q^{39} + 54 q^{41} + 75 q^{43} + 288 q^{45} + 339 q^{47} - 24 q^{49} + 360 q^{51} + 69 q^{53} - 51 q^{55} + 510 q^{57} - 483 q^{59} - 36 q^{61} - 267 q^{63} - 585 q^{65} - 87 q^{67} - 351 q^{69} - 234 q^{71} - 132 q^{73} + 108 q^{77} + 363 q^{79} + 258 q^{81} + 279 q^{83} + 666 q^{85} + 600 q^{89} + 270 q^{91} - 456 q^{93} - 39 q^{95} - 801 q^{97} - 267 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.617749 + 1.69725i −0.205916 + 0.565751i −0.999063 0.0432802i \(-0.986219\pi\)
0.793147 + 0.609031i \(0.208441\pi\)
\(4\) 0 0
\(5\) 0.757040 + 4.29338i 0.151408 + 0.858677i 0.961997 + 0.273061i \(0.0880361\pi\)
−0.810589 + 0.585616i \(0.800853\pi\)
\(6\) 0 0
\(7\) −2.41011 + 4.17443i −0.344301 + 0.596347i −0.985227 0.171256i \(-0.945218\pi\)
0.640925 + 0.767603i \(0.278551\pi\)
\(8\) 0 0
\(9\) 4.39535 + 3.68814i 0.488372 + 0.409793i
\(10\) 0 0
\(11\) 0.0945208 + 0.163715i 0.00859280 + 0.0148832i 0.870290 0.492540i \(-0.163931\pi\)
−0.861697 + 0.507423i \(0.830598\pi\)
\(12\) 0 0
\(13\) 1.39295 + 3.82710i 0.107150 + 0.294393i 0.981667 0.190602i \(-0.0610439\pi\)
−0.874517 + 0.484994i \(0.838822\pi\)
\(14\) 0 0
\(15\) −7.75461 1.36735i −0.516974 0.0911565i
\(16\) 0 0
\(17\) 13.1145 11.0044i 0.771443 0.647317i −0.169635 0.985507i \(-0.554259\pi\)
0.941078 + 0.338190i \(0.109814\pi\)
\(18\) 0 0
\(19\) −8.88753 16.7932i −0.467765 0.883853i
\(20\) 0 0
\(21\) −5.59622 6.66931i −0.266487 0.317586i
\(22\) 0 0
\(23\) 2.45173 13.9044i 0.106597 0.604541i −0.883974 0.467537i \(-0.845142\pi\)
0.990571 0.137004i \(-0.0437473\pi\)
\(24\) 0 0
\(25\) 5.63227 2.04998i 0.225291 0.0819992i
\(26\) 0 0
\(27\) −23.0527 + 13.3095i −0.853803 + 0.492943i
\(28\) 0 0
\(29\) 18.3700 21.8925i 0.633447 0.754913i −0.349873 0.936797i \(-0.613775\pi\)
0.983320 + 0.181884i \(0.0582196\pi\)
\(30\) 0 0
\(31\) −5.92457 3.42055i −0.191115 0.110340i 0.401389 0.915908i \(-0.368527\pi\)
−0.592505 + 0.805567i \(0.701861\pi\)
\(32\) 0 0
\(33\) −0.336255 + 0.0592909i −0.0101896 + 0.00179669i
\(34\) 0 0
\(35\) −19.7470 7.18732i −0.564200 0.205352i
\(36\) 0 0
\(37\) 31.9736i 0.864152i 0.901837 + 0.432076i \(0.142219\pi\)
−0.901837 + 0.432076i \(0.857781\pi\)
\(38\) 0 0
\(39\) −7.35605 −0.188617
\(40\) 0 0
\(41\) 19.4006 53.3028i 0.473186 1.30007i −0.441993 0.897019i \(-0.645728\pi\)
0.915178 0.403049i \(-0.132050\pi\)
\(42\) 0 0
\(43\) 7.16167 + 40.6158i 0.166550 + 0.944554i 0.947451 + 0.319899i \(0.103649\pi\)
−0.780901 + 0.624655i \(0.785240\pi\)
\(44\) 0 0
\(45\) −12.5071 + 21.6630i −0.277936 + 0.481400i
\(46\) 0 0
\(47\) 44.9103 + 37.6842i 0.955538 + 0.801791i 0.980221 0.197904i \(-0.0634134\pi\)
−0.0246837 + 0.999695i \(0.507858\pi\)
\(48\) 0 0
\(49\) 12.8827 + 22.3136i 0.262913 + 0.455379i
\(50\) 0 0
\(51\) 10.5757 + 29.0566i 0.207367 + 0.569737i
\(52\) 0 0
\(53\) −5.32437 0.938830i −0.100460 0.0177138i 0.123192 0.992383i \(-0.460687\pi\)
−0.223652 + 0.974669i \(0.571798\pi\)
\(54\) 0 0
\(55\) −0.631335 + 0.529753i −0.0114788 + 0.00963187i
\(56\) 0 0
\(57\) 33.9926 4.71039i 0.596361 0.0826385i
\(58\) 0 0
\(59\) −36.3323 43.2991i −0.615801 0.733883i 0.364541 0.931187i \(-0.381226\pi\)
−0.980342 + 0.197304i \(0.936781\pi\)
\(60\) 0 0
\(61\) −15.7238 + 89.1742i −0.257767 + 1.46187i 0.531100 + 0.847309i \(0.321779\pi\)
−0.788868 + 0.614563i \(0.789332\pi\)
\(62\) 0 0
\(63\) −25.9892 + 9.45928i −0.412526 + 0.150147i
\(64\) 0 0
\(65\) −15.3767 + 8.87775i −0.236565 + 0.136581i
\(66\) 0 0
\(67\) 5.04936 6.01760i 0.0753637 0.0898149i −0.727045 0.686590i \(-0.759107\pi\)
0.802409 + 0.596775i \(0.203551\pi\)
\(68\) 0 0
\(69\) 22.0848 + 12.7506i 0.320069 + 0.184792i
\(70\) 0 0
\(71\) 23.9368 4.22071i 0.337138 0.0594466i −0.00251622 0.999997i \(-0.500801\pi\)
0.339655 + 0.940550i \(0.389690\pi\)
\(72\) 0 0
\(73\) −111.851 40.7104i −1.53220 0.557677i −0.568044 0.822998i \(-0.692300\pi\)
−0.964160 + 0.265322i \(0.914522\pi\)
\(74\) 0 0
\(75\) 10.8258i 0.144343i
\(76\) 0 0
\(77\) −0.911222 −0.0118340
\(78\) 0 0
\(79\) 47.8267 131.403i 0.605401 1.66333i −0.134745 0.990880i \(-0.543022\pi\)
0.740146 0.672446i \(-0.234756\pi\)
\(80\) 0 0
\(81\) 0.618357 + 3.50688i 0.00763404 + 0.0432948i
\(82\) 0 0
\(83\) 52.0220 90.1047i 0.626770 1.08560i −0.361425 0.932401i \(-0.617710\pi\)
0.988196 0.153197i \(-0.0489570\pi\)
\(84\) 0 0
\(85\) 57.1743 + 47.9749i 0.672639 + 0.564411i
\(86\) 0 0
\(87\) 25.8090 + 44.7025i 0.296655 + 0.513822i
\(88\) 0 0
\(89\) −52.7400 144.902i −0.592584 1.62811i −0.765692 0.643208i \(-0.777603\pi\)
0.173108 0.984903i \(-0.444619\pi\)
\(90\) 0 0
\(91\) −19.3332 3.40896i −0.212452 0.0374611i
\(92\) 0 0
\(93\) 9.46544 7.94245i 0.101779 0.0854027i
\(94\) 0 0
\(95\) 65.3715 50.8707i 0.688121 0.535481i
\(96\) 0 0
\(97\) −52.4375 62.4925i −0.540592 0.644253i 0.424728 0.905321i \(-0.360370\pi\)
−0.965320 + 0.261068i \(0.915925\pi\)
\(98\) 0 0
\(99\) −0.188351 + 1.06819i −0.00190253 + 0.0107898i
\(100\) 0 0
\(101\) −110.337 + 40.1595i −1.09245 + 0.397619i −0.824528 0.565822i \(-0.808559\pi\)
−0.267922 + 0.963441i \(0.586337\pi\)
\(102\) 0 0
\(103\) −134.487 + 77.6462i −1.30570 + 0.753846i −0.981376 0.192099i \(-0.938470\pi\)
−0.324325 + 0.945946i \(0.605137\pi\)
\(104\) 0 0
\(105\) 24.3974 29.0756i 0.232356 0.276911i
\(106\) 0 0
\(107\) 138.583 + 80.0108i 1.29517 + 0.747765i 0.979565 0.201127i \(-0.0644604\pi\)
0.315601 + 0.948892i \(0.397794\pi\)
\(108\) 0 0
\(109\) −105.342 + 18.5747i −0.966443 + 0.170410i −0.634528 0.772900i \(-0.718806\pi\)
−0.331914 + 0.943310i \(0.607695\pi\)
\(110\) 0 0
\(111\) −54.2673 19.7517i −0.488894 0.177943i
\(112\) 0 0
\(113\) 54.9633i 0.486401i −0.969976 0.243201i \(-0.921803\pi\)
0.969976 0.243201i \(-0.0781973\pi\)
\(114\) 0 0
\(115\) 61.5531 0.535245
\(116\) 0 0
\(117\) −7.99237 + 21.9589i −0.0683109 + 0.187683i
\(118\) 0 0
\(119\) 14.3297 + 81.2675i 0.120417 + 0.682920i
\(120\) 0 0
\(121\) 60.4821 104.758i 0.499852 0.865770i
\(122\) 0 0
\(123\) 78.4835 + 65.8555i 0.638077 + 0.535410i
\(124\) 0 0
\(125\) 67.5604 + 117.018i 0.540483 + 0.936145i
\(126\) 0 0
\(127\) 57.6206 + 158.311i 0.453705 + 1.24655i 0.930098 + 0.367312i \(0.119722\pi\)
−0.476392 + 0.879233i \(0.658056\pi\)
\(128\) 0 0
\(129\) −73.3594 12.9352i −0.568677 0.100273i
\(130\) 0 0
\(131\) 26.6313 22.3463i 0.203292 0.170582i −0.535458 0.844562i \(-0.679861\pi\)
0.738750 + 0.673980i \(0.235416\pi\)
\(132\) 0 0
\(133\) 91.5220 + 3.37306i 0.688135 + 0.0253613i
\(134\) 0 0
\(135\) −74.5945 88.8982i −0.552552 0.658505i
\(136\) 0 0
\(137\) −28.3002 + 160.498i −0.206571 + 1.17152i 0.688378 + 0.725352i \(0.258323\pi\)
−0.894949 + 0.446169i \(0.852788\pi\)
\(138\) 0 0
\(139\) 24.9206 9.07036i 0.179285 0.0652544i −0.250818 0.968034i \(-0.580700\pi\)
0.430103 + 0.902780i \(0.358477\pi\)
\(140\) 0 0
\(141\) −91.7028 + 52.9447i −0.650375 + 0.375494i
\(142\) 0 0
\(143\) −0.494891 + 0.589788i −0.00346077 + 0.00412439i
\(144\) 0 0
\(145\) 107.900 + 62.2959i 0.744135 + 0.429627i
\(146\) 0 0
\(147\) −45.8301 + 8.08107i −0.311769 + 0.0549733i
\(148\) 0 0
\(149\) −147.677 53.7500i −0.991121 0.360739i −0.204967 0.978769i \(-0.565709\pi\)
−0.786154 + 0.618030i \(0.787931\pi\)
\(150\) 0 0
\(151\) 26.7672i 0.177267i −0.996064 0.0886333i \(-0.971750\pi\)
0.996064 0.0886333i \(-0.0282499\pi\)
\(152\) 0 0
\(153\) 98.2287 0.642017
\(154\) 0 0
\(155\) 10.2006 28.0260i 0.0658104 0.180813i
\(156\) 0 0
\(157\) 35.9951 + 204.138i 0.229268 + 1.30024i 0.854355 + 0.519689i \(0.173952\pi\)
−0.625087 + 0.780555i \(0.714937\pi\)
\(158\) 0 0
\(159\) 4.88255 8.45683i 0.0307079 0.0531876i
\(160\) 0 0
\(161\) 52.1342 + 43.7458i 0.323815 + 0.271713i
\(162\) 0 0
\(163\) −78.8734 136.613i −0.483886 0.838115i 0.515943 0.856623i \(-0.327442\pi\)
−0.999829 + 0.0185078i \(0.994108\pi\)
\(164\) 0 0
\(165\) −0.509117 1.39879i −0.00308556 0.00847750i
\(166\) 0 0
\(167\) 11.7322 + 2.06870i 0.0702526 + 0.0123874i 0.208664 0.977987i \(-0.433088\pi\)
−0.138411 + 0.990375i \(0.544200\pi\)
\(168\) 0 0
\(169\) 116.755 97.9692i 0.690859 0.579699i
\(170\) 0 0
\(171\) 22.8718 106.590i 0.133753 0.623336i
\(172\) 0 0
\(173\) −86.5340 103.127i −0.500197 0.596111i 0.455584 0.890193i \(-0.349431\pi\)
−0.955780 + 0.294082i \(0.904986\pi\)
\(174\) 0 0
\(175\) −5.01689 + 28.4522i −0.0286680 + 0.162584i
\(176\) 0 0
\(177\) 95.9337 34.9170i 0.541998 0.197271i
\(178\) 0 0
\(179\) −278.151 + 160.591i −1.55392 + 0.897154i −0.556099 + 0.831116i \(0.687702\pi\)
−0.997817 + 0.0660376i \(0.978964\pi\)
\(180\) 0 0
\(181\) 47.0632 56.0877i 0.260017 0.309877i −0.620203 0.784441i \(-0.712950\pi\)
0.880221 + 0.474564i \(0.157394\pi\)
\(182\) 0 0
\(183\) −141.638 81.7745i −0.773976 0.446855i
\(184\) 0 0
\(185\) −137.275 + 24.2053i −0.742027 + 0.130839i
\(186\) 0 0
\(187\) 3.04118 + 1.10690i 0.0162630 + 0.00591924i
\(188\) 0 0
\(189\) 128.309i 0.678884i
\(190\) 0 0
\(191\) 133.689 0.699940 0.349970 0.936761i \(-0.386192\pi\)
0.349970 + 0.936761i \(0.386192\pi\)
\(192\) 0 0
\(193\) 73.5478 202.071i 0.381077 1.04700i −0.589827 0.807529i \(-0.700804\pi\)
0.970904 0.239470i \(-0.0769736\pi\)
\(194\) 0 0
\(195\) −5.56882 31.5824i −0.0285581 0.161961i
\(196\) 0 0
\(197\) 62.1141 107.585i 0.315300 0.546115i −0.664201 0.747554i \(-0.731228\pi\)
0.979501 + 0.201438i \(0.0645616\pi\)
\(198\) 0 0
\(199\) −212.314 178.153i −1.06691 0.895240i −0.0721368 0.997395i \(-0.522982\pi\)
−0.994769 + 0.102155i \(0.967426\pi\)
\(200\) 0 0
\(201\) 7.09414 + 12.2874i 0.0352942 + 0.0611314i
\(202\) 0 0
\(203\) 47.1150 + 129.447i 0.232094 + 0.637672i
\(204\) 0 0
\(205\) 243.536 + 42.9420i 1.18798 + 0.209473i
\(206\) 0 0
\(207\) 62.0577 52.0726i 0.299795 0.251558i
\(208\) 0 0
\(209\) 1.90924 3.04233i 0.00913512 0.0145566i
\(210\) 0 0
\(211\) −160.725 191.544i −0.761729 0.907793i 0.236227 0.971698i \(-0.424089\pi\)
−0.997956 + 0.0639050i \(0.979645\pi\)
\(212\) 0 0
\(213\) −7.62335 + 43.2342i −0.0357904 + 0.202977i
\(214\) 0 0
\(215\) −168.958 + 61.4956i −0.785850 + 0.286026i
\(216\) 0 0
\(217\) 28.5577 16.4878i 0.131602 0.0759807i
\(218\) 0 0
\(219\) 138.192 164.690i 0.631012 0.752010i
\(220\) 0 0
\(221\) 60.3829 + 34.8621i 0.273226 + 0.157747i
\(222\) 0 0
\(223\) 109.691 19.3415i 0.491887 0.0867330i 0.0777978 0.996969i \(-0.475211\pi\)
0.414090 + 0.910236i \(0.364100\pi\)
\(224\) 0 0
\(225\) 32.3164 + 11.7622i 0.143629 + 0.0522765i
\(226\) 0 0
\(227\) 345.131i 1.52040i 0.649689 + 0.760200i \(0.274899\pi\)
−0.649689 + 0.760200i \(0.725101\pi\)
\(228\) 0 0
\(229\) −138.138 −0.603225 −0.301612 0.953431i \(-0.597525\pi\)
−0.301612 + 0.953431i \(0.597525\pi\)
\(230\) 0 0
\(231\) 0.562906 1.54657i 0.00243682 0.00669512i
\(232\) 0 0
\(233\) −16.3333 92.6309i −0.0701002 0.397558i −0.999588 0.0287048i \(-0.990862\pi\)
0.929488 0.368853i \(-0.120249\pi\)
\(234\) 0 0
\(235\) −127.794 + 221.345i −0.543804 + 0.941896i
\(236\) 0 0
\(237\) 193.479 + 162.348i 0.816366 + 0.685012i
\(238\) 0 0
\(239\) 106.213 + 183.966i 0.444404 + 0.769731i 0.998011 0.0630478i \(-0.0200821\pi\)
−0.553606 + 0.832779i \(0.686749\pi\)
\(240\) 0 0
\(241\) 10.3102 + 28.3272i 0.0427811 + 0.117540i 0.959244 0.282581i \(-0.0911904\pi\)
−0.916462 + 0.400121i \(0.868968\pi\)
\(242\) 0 0
\(243\) −242.265 42.7178i −0.996975 0.175794i
\(244\) 0 0
\(245\) −86.0480 + 72.2028i −0.351216 + 0.294705i
\(246\) 0 0
\(247\) 51.8894 57.4057i 0.210079 0.232412i
\(248\) 0 0
\(249\) 120.794 + 143.956i 0.485116 + 0.578138i
\(250\) 0 0
\(251\) 0.289957 1.64443i 0.00115521 0.00655150i −0.984225 0.176923i \(-0.943386\pi\)
0.985380 + 0.170371i \(0.0544968\pi\)
\(252\) 0 0
\(253\) 2.50810 0.912874i 0.00991344 0.00360820i
\(254\) 0 0
\(255\) −116.745 + 67.4027i −0.457823 + 0.264324i
\(256\) 0 0
\(257\) −231.634 + 276.051i −0.901299 + 1.07413i 0.0955985 + 0.995420i \(0.469523\pi\)
−0.996898 + 0.0787067i \(0.974921\pi\)
\(258\) 0 0
\(259\) −133.472 77.0599i −0.515335 0.297529i
\(260\) 0 0
\(261\) 161.485 28.4742i 0.618716 0.109096i
\(262\) 0 0
\(263\) 126.255 + 45.9530i 0.480056 + 0.174726i 0.570702 0.821157i \(-0.306671\pi\)
−0.0906463 + 0.995883i \(0.528893\pi\)
\(264\) 0 0
\(265\) 23.5703i 0.0889445i
\(266\) 0 0
\(267\) 278.515 1.04313
\(268\) 0 0
\(269\) −18.1149 + 49.7703i −0.0673417 + 0.185020i −0.968799 0.247849i \(-0.920276\pi\)
0.901457 + 0.432869i \(0.142499\pi\)
\(270\) 0 0
\(271\) 25.0084 + 141.830i 0.0922819 + 0.523357i 0.995546 + 0.0942727i \(0.0300525\pi\)
−0.903264 + 0.429084i \(0.858836\pi\)
\(272\) 0 0
\(273\) 17.7289 30.7073i 0.0649410 0.112481i
\(274\) 0 0
\(275\) 0.867979 + 0.728321i 0.00315629 + 0.00264844i
\(276\) 0 0
\(277\) −36.9790 64.0496i −0.133498 0.231226i 0.791524 0.611137i \(-0.209288\pi\)
−0.925023 + 0.379912i \(0.875954\pi\)
\(278\) 0 0
\(279\) −13.4251 36.8852i −0.0481187 0.132205i
\(280\) 0 0
\(281\) −62.2447 10.9754i −0.221511 0.0390584i 0.0617909 0.998089i \(-0.480319\pi\)
−0.283302 + 0.959031i \(0.591430\pi\)
\(282\) 0 0
\(283\) −74.5778 + 62.5782i −0.263526 + 0.221125i −0.764971 0.644065i \(-0.777247\pi\)
0.501445 + 0.865190i \(0.332802\pi\)
\(284\) 0 0
\(285\) 45.9572 + 142.377i 0.161253 + 0.499569i
\(286\) 0 0
\(287\) 175.751 + 209.452i 0.612373 + 0.729798i
\(288\) 0 0
\(289\) 0.709798 4.02546i 0.00245605 0.0139289i
\(290\) 0 0
\(291\) 138.459 50.3949i 0.475803 0.173178i
\(292\) 0 0
\(293\) 337.975 195.130i 1.15350 0.665972i 0.203761 0.979021i \(-0.434684\pi\)
0.949737 + 0.313048i \(0.101350\pi\)
\(294\) 0 0
\(295\) 158.395 188.767i 0.536931 0.639890i
\(296\) 0 0
\(297\) −4.35791 2.51604i −0.0146731 0.00847153i
\(298\) 0 0
\(299\) 56.6289 9.98520i 0.189394 0.0333953i
\(300\) 0 0
\(301\) −186.808 67.9927i −0.620626 0.225889i
\(302\) 0 0
\(303\) 212.079i 0.699930i
\(304\) 0 0
\(305\) −394.763 −1.29430
\(306\) 0 0
\(307\) −16.9626 + 46.6044i −0.0552528 + 0.151806i −0.964248 0.265000i \(-0.914628\pi\)
0.908996 + 0.416805i \(0.136850\pi\)
\(308\) 0 0
\(309\) −48.7058 276.224i −0.157624 0.893930i
\(310\) 0 0
\(311\) −106.770 + 184.932i −0.343313 + 0.594636i −0.985046 0.172292i \(-0.944883\pi\)
0.641732 + 0.766929i \(0.278216\pi\)
\(312\) 0 0
\(313\) 148.248 + 124.395i 0.473635 + 0.397427i 0.848118 0.529807i \(-0.177736\pi\)
−0.374484 + 0.927234i \(0.622180\pi\)
\(314\) 0 0
\(315\) −60.2871 104.420i −0.191388 0.331493i
\(316\) 0 0
\(317\) −204.114 560.798i −0.643892 1.76908i −0.639140 0.769090i \(-0.720710\pi\)
−0.00475166 0.999989i \(-0.501513\pi\)
\(318\) 0 0
\(319\) 5.32047 + 0.938142i 0.0166786 + 0.00294088i
\(320\) 0 0
\(321\) −221.408 + 185.783i −0.689744 + 0.578764i
\(322\) 0 0
\(323\) −301.355 122.433i −0.932987 0.379050i
\(324\) 0 0
\(325\) 15.6910 + 18.6998i 0.0482799 + 0.0575378i
\(326\) 0 0
\(327\) 33.5492 190.267i 0.102597 0.581856i
\(328\) 0 0
\(329\) −265.549 + 96.6518i −0.807139 + 0.293775i
\(330\) 0 0
\(331\) 479.438 276.804i 1.44845 0.836265i 0.450065 0.892996i \(-0.351401\pi\)
0.998390 + 0.0567306i \(0.0180676\pi\)
\(332\) 0 0
\(333\) −117.923 + 140.535i −0.354123 + 0.422028i
\(334\) 0 0
\(335\) 29.6584 + 17.1233i 0.0885326 + 0.0511143i
\(336\) 0 0
\(337\) 60.2142 10.6174i 0.178677 0.0315056i −0.0835936 0.996500i \(-0.526640\pi\)
0.262271 + 0.964994i \(0.415529\pi\)
\(338\) 0 0
\(339\) 93.2866 + 33.9535i 0.275182 + 0.100158i
\(340\) 0 0
\(341\) 1.29325i 0.00379253i
\(342\) 0 0
\(343\) −360.386 −1.05069
\(344\) 0 0
\(345\) −38.0244 + 104.471i −0.110216 + 0.302815i
\(346\) 0 0
\(347\) −34.8091 197.412i −0.100315 0.568912i −0.992989 0.118209i \(-0.962285\pi\)
0.892674 0.450703i \(-0.148826\pi\)
\(348\) 0 0
\(349\) 127.380 220.629i 0.364986 0.632173i −0.623788 0.781593i \(-0.714407\pi\)
0.988774 + 0.149420i \(0.0477406\pi\)
\(350\) 0 0
\(351\) −83.0480 69.6856i −0.236604 0.198534i
\(352\) 0 0
\(353\) 48.6888 + 84.3315i 0.137929 + 0.238899i 0.926712 0.375771i \(-0.122622\pi\)
−0.788784 + 0.614671i \(0.789289\pi\)
\(354\) 0 0
\(355\) 36.2422 + 99.5747i 0.102091 + 0.280492i
\(356\) 0 0
\(357\) −146.784 25.8819i −0.411158 0.0724983i
\(358\) 0 0
\(359\) 435.611 365.521i 1.21340 1.01817i 0.214259 0.976777i \(-0.431266\pi\)
0.999143 0.0413888i \(-0.0131782\pi\)
\(360\) 0 0
\(361\) −203.023 + 298.500i −0.562392 + 0.826871i
\(362\) 0 0
\(363\) 140.438 + 167.368i 0.386882 + 0.461068i
\(364\) 0 0
\(365\) 90.1098 511.038i 0.246876 1.40010i
\(366\) 0 0
\(367\) 403.610 146.902i 1.09975 0.400278i 0.272529 0.962148i \(-0.412140\pi\)
0.827226 + 0.561870i \(0.189918\pi\)
\(368\) 0 0
\(369\) 281.860 162.732i 0.763849 0.441009i
\(370\) 0 0
\(371\) 16.7514 19.9635i 0.0451520 0.0538101i
\(372\) 0 0
\(373\) 168.596 + 97.3387i 0.451999 + 0.260962i 0.708674 0.705536i \(-0.249294\pi\)
−0.256675 + 0.966498i \(0.582627\pi\)
\(374\) 0 0
\(375\) −240.344 + 42.3792i −0.640919 + 0.113011i
\(376\) 0 0
\(377\) 109.373 + 39.8086i 0.290115 + 0.105593i
\(378\) 0 0
\(379\) 296.888i 0.783345i 0.920105 + 0.391672i \(0.128103\pi\)
−0.920105 + 0.391672i \(0.871897\pi\)
\(380\) 0 0
\(381\) −304.289 −0.798659
\(382\) 0 0
\(383\) −86.3957 + 237.370i −0.225576 + 0.619766i −0.999915 0.0130049i \(-0.995860\pi\)
0.774339 + 0.632771i \(0.218083\pi\)
\(384\) 0 0
\(385\) −0.689831 3.91222i −0.00179177 0.0101616i
\(386\) 0 0
\(387\) −118.319 + 204.934i −0.305733 + 0.529545i
\(388\) 0 0
\(389\) −218.384 183.246i −0.561398 0.471069i 0.317381 0.948298i \(-0.397197\pi\)
−0.878779 + 0.477229i \(0.841641\pi\)
\(390\) 0 0
\(391\) −120.857 209.330i −0.309096 0.535371i
\(392\) 0 0
\(393\) 21.4758 + 59.0044i 0.0546459 + 0.150138i
\(394\) 0 0
\(395\) 600.369 + 105.861i 1.51992 + 0.268003i
\(396\) 0 0
\(397\) −515.532 + 432.583i −1.29857 + 1.08963i −0.308178 + 0.951329i \(0.599719\pi\)
−0.990390 + 0.138300i \(0.955836\pi\)
\(398\) 0 0
\(399\) −62.2626 + 153.252i −0.156047 + 0.384091i
\(400\) 0 0
\(401\) 349.553 + 416.581i 0.871703 + 1.03886i 0.998896 + 0.0469775i \(0.0149589\pi\)
−0.127193 + 0.991878i \(0.540597\pi\)
\(402\) 0 0
\(403\) 4.83817 27.4386i 0.0120054 0.0680859i
\(404\) 0 0
\(405\) −14.5883 + 5.30969i −0.0360204 + 0.0131103i
\(406\) 0 0
\(407\) −5.23455 + 3.02217i −0.0128613 + 0.00742548i
\(408\) 0 0
\(409\) −388.659 + 463.185i −0.950266 + 1.13248i 0.0408077 + 0.999167i \(0.487007\pi\)
−0.991074 + 0.133316i \(0.957438\pi\)
\(410\) 0 0
\(411\) −254.924 147.180i −0.620253 0.358103i
\(412\) 0 0
\(413\) 268.314 47.3110i 0.649670 0.114554i
\(414\) 0 0
\(415\) 426.237 + 155.137i 1.02708 + 0.373825i
\(416\) 0 0
\(417\) 47.8997i 0.114867i
\(418\) 0 0
\(419\) 66.8368 0.159515 0.0797575 0.996814i \(-0.474585\pi\)
0.0797575 + 0.996814i \(0.474585\pi\)
\(420\) 0 0
\(421\) −70.5506 + 193.836i −0.167579 + 0.460419i −0.994847 0.101389i \(-0.967671\pi\)
0.827268 + 0.561807i \(0.189894\pi\)
\(422\) 0 0
\(423\) 58.4119 + 331.271i 0.138090 + 0.783145i
\(424\) 0 0
\(425\) 51.3058 88.8643i 0.120720 0.209092i
\(426\) 0 0
\(427\) −334.355 280.558i −0.783034 0.657043i
\(428\) 0 0
\(429\) −0.695300 1.20429i −0.00162075 0.00280721i
\(430\) 0 0
\(431\) 89.0241 + 244.592i 0.206552 + 0.567498i 0.999105 0.0423058i \(-0.0134704\pi\)
−0.792552 + 0.609804i \(0.791248\pi\)
\(432\) 0 0
\(433\) −207.053 36.5090i −0.478182 0.0843164i −0.0706382 0.997502i \(-0.522504\pi\)
−0.407544 + 0.913186i \(0.633615\pi\)
\(434\) 0 0
\(435\) −172.387 + 144.650i −0.396291 + 0.332528i
\(436\) 0 0
\(437\) −255.290 + 82.4038i −0.584187 + 0.188567i
\(438\) 0 0
\(439\) 312.137 + 371.990i 0.711018 + 0.847359i 0.993725 0.111848i \(-0.0356768\pi\)
−0.282707 + 0.959206i \(0.591232\pi\)
\(440\) 0 0
\(441\) −25.6713 + 145.589i −0.0582116 + 0.330135i
\(442\) 0 0
\(443\) 381.133 138.721i 0.860346 0.313140i 0.126095 0.992018i \(-0.459756\pi\)
0.734251 + 0.678878i \(0.237533\pi\)
\(444\) 0 0
\(445\) 582.193 336.129i 1.30830 0.755347i
\(446\) 0 0
\(447\) 182.455 217.441i 0.408176 0.486445i
\(448\) 0 0
\(449\) 8.43341 + 4.86903i 0.0187826 + 0.0108442i 0.509362 0.860552i \(-0.329881\pi\)
−0.490579 + 0.871397i \(0.663215\pi\)
\(450\) 0 0
\(451\) 10.5602 1.86205i 0.0234151 0.00412872i
\(452\) 0 0
\(453\) 45.4308 + 16.5354i 0.100289 + 0.0365021i
\(454\) 0 0
\(455\) 85.5854i 0.188100i
\(456\) 0 0
\(457\) 233.468 0.510871 0.255435 0.966826i \(-0.417781\pi\)
0.255435 + 0.966826i \(0.417781\pi\)
\(458\) 0 0
\(459\) −155.862 + 428.228i −0.339569 + 0.932959i
\(460\) 0 0
\(461\) −121.198 687.346i −0.262902 1.49099i −0.774944 0.632030i \(-0.782222\pi\)
0.512043 0.858960i \(-0.328889\pi\)
\(462\) 0 0
\(463\) −380.012 + 658.199i −0.820759 + 1.42160i 0.0843581 + 0.996436i \(0.473116\pi\)
−0.905117 + 0.425162i \(0.860217\pi\)
\(464\) 0 0
\(465\) 41.2657 + 34.6260i 0.0887434 + 0.0744646i
\(466\) 0 0
\(467\) −197.668 342.370i −0.423271 0.733127i 0.572986 0.819565i \(-0.305785\pi\)
−0.996257 + 0.0864380i \(0.972452\pi\)
\(468\) 0 0
\(469\) 12.9505 + 35.5813i 0.0276131 + 0.0758663i
\(470\) 0 0
\(471\) −368.710 65.0135i −0.782824 0.138033i
\(472\) 0 0
\(473\) −5.97249 + 5.01151i −0.0126268 + 0.0105952i
\(474\) 0 0
\(475\) −84.4828 76.3647i −0.177858 0.160768i
\(476\) 0 0
\(477\) −19.9399 23.7635i −0.0418028 0.0498186i
\(478\) 0 0
\(479\) 69.4029 393.603i 0.144891 0.821719i −0.822563 0.568674i \(-0.807457\pi\)
0.967454 0.253045i \(-0.0814322\pi\)
\(480\) 0 0
\(481\) −122.366 + 44.5377i −0.254400 + 0.0925940i
\(482\) 0 0
\(483\) −106.453 + 61.4609i −0.220400 + 0.127248i
\(484\) 0 0
\(485\) 228.607 272.443i 0.471355 0.561739i
\(486\) 0 0
\(487\) −833.507 481.226i −1.71151 0.988143i −0.932525 0.361105i \(-0.882400\pi\)
−0.778989 0.627038i \(-0.784267\pi\)
\(488\) 0 0
\(489\) 280.590 49.4756i 0.573804 0.101177i
\(490\) 0 0
\(491\) 881.263 + 320.754i 1.79483 + 0.653266i 0.998850 + 0.0479482i \(0.0152682\pi\)
0.795984 + 0.605318i \(0.206954\pi\)
\(492\) 0 0
\(493\) 489.260i 0.992414i
\(494\) 0 0
\(495\) −4.72874 −0.00955301
\(496\) 0 0
\(497\) −40.0713 + 110.095i −0.0806264 + 0.221519i
\(498\) 0 0
\(499\) −53.2720 302.121i −0.106758 0.605452i −0.990504 0.137486i \(-0.956098\pi\)
0.883746 0.467966i \(-0.155013\pi\)
\(500\) 0 0
\(501\) −10.7587 + 18.6345i −0.0214744 + 0.0371947i
\(502\) 0 0
\(503\) −705.700 592.153i −1.40298 1.17724i −0.959757 0.280832i \(-0.909390\pi\)
−0.443226 0.896410i \(-0.646166\pi\)
\(504\) 0 0
\(505\) −255.950 443.318i −0.506832 0.877858i
\(506\) 0 0
\(507\) 94.1530 + 258.683i 0.185706 + 0.510223i
\(508\) 0 0
\(509\) 751.063 + 132.433i 1.47557 + 0.260182i 0.852805 0.522229i \(-0.174900\pi\)
0.622762 + 0.782412i \(0.286011\pi\)
\(510\) 0 0
\(511\) 439.516 368.797i 0.860109 0.721717i
\(512\) 0 0
\(513\) 428.390 + 268.840i 0.835069 + 0.524055i
\(514\) 0 0
\(515\) −435.177 518.624i −0.845004 1.00704i
\(516\) 0 0
\(517\) −1.92451 + 10.9144i −0.00372245 + 0.0211111i
\(518\) 0 0
\(519\) 228.489 83.1633i 0.440249 0.160237i
\(520\) 0 0
\(521\) −37.4959 + 21.6483i −0.0719691 + 0.0415514i −0.535553 0.844502i \(-0.679897\pi\)
0.463584 + 0.886053i \(0.346563\pi\)
\(522\) 0 0
\(523\) −511.250 + 609.284i −0.977533 + 1.16498i 0.00875793 + 0.999962i \(0.497212\pi\)
−0.986291 + 0.165017i \(0.947232\pi\)
\(524\) 0 0
\(525\) −45.1914 26.0913i −0.0860788 0.0496976i
\(526\) 0 0
\(527\) −115.339 + 20.3374i −0.218860 + 0.0385909i
\(528\) 0 0
\(529\) 309.775 + 112.749i 0.585586 + 0.213136i
\(530\) 0 0
\(531\) 324.313i 0.610759i
\(532\) 0 0
\(533\) 231.019 0.433432
\(534\) 0 0
\(535\) −238.605 + 655.561i −0.445990 + 1.22535i
\(536\) 0 0
\(537\) −100.735 571.297i −0.187589 1.06387i
\(538\) 0 0
\(539\) −2.43537 + 4.21819i −0.00451832 + 0.00782596i
\(540\) 0 0
\(541\) −407.967 342.325i −0.754097 0.632763i 0.182486 0.983209i \(-0.441586\pi\)
−0.936583 + 0.350446i \(0.886030\pi\)
\(542\) 0 0
\(543\) 66.1217 + 114.526i 0.121771 + 0.210914i
\(544\) 0 0
\(545\) −159.496 438.213i −0.292654 0.804061i
\(546\) 0 0
\(547\) 6.49765 + 1.14571i 0.0118787 + 0.00209454i 0.179584 0.983743i \(-0.442525\pi\)
−0.167706 + 0.985837i \(0.553636\pi\)
\(548\) 0 0
\(549\) −397.998 + 333.960i −0.724952 + 0.608307i
\(550\) 0 0
\(551\) −530.909 113.921i −0.963536 0.206752i
\(552\) 0 0
\(553\) 433.264 + 516.344i 0.783480 + 0.933715i
\(554\) 0 0
\(555\) 43.7190 247.943i 0.0787731 0.446744i
\(556\) 0 0
\(557\) −158.030 + 57.5181i −0.283716 + 0.103264i −0.479958 0.877291i \(-0.659348\pi\)
0.196242 + 0.980555i \(0.437126\pi\)
\(558\) 0 0
\(559\) −145.465 + 83.9844i −0.260224 + 0.150240i
\(560\) 0 0
\(561\) −3.75737 + 4.47786i −0.00669763 + 0.00798192i
\(562\) 0 0
\(563\) −692.075 399.570i −1.22926 0.709715i −0.262387 0.964963i \(-0.584510\pi\)
−0.966876 + 0.255248i \(0.917843\pi\)
\(564\) 0 0
\(565\) 235.979 41.6094i 0.417661 0.0736450i
\(566\) 0 0
\(567\) −16.1295 5.87067i −0.0284471 0.0103539i
\(568\) 0 0
\(569\) 799.192i 1.40456i 0.711903 + 0.702278i \(0.247834\pi\)
−0.711903 + 0.702278i \(0.752166\pi\)
\(570\) 0 0
\(571\) 761.851 1.33424 0.667120 0.744950i \(-0.267527\pi\)
0.667120 + 0.744950i \(0.267527\pi\)
\(572\) 0 0
\(573\) −82.5860 + 226.903i −0.144129 + 0.395991i
\(574\) 0 0
\(575\) −14.6950 83.3396i −0.0255565 0.144938i
\(576\) 0 0
\(577\) 88.2589 152.869i 0.152962 0.264937i −0.779353 0.626585i \(-0.784452\pi\)
0.932315 + 0.361647i \(0.117786\pi\)
\(578\) 0 0
\(579\) 297.531 + 249.658i 0.513871 + 0.431189i
\(580\) 0 0
\(581\) 250.757 + 434.324i 0.431596 + 0.747546i
\(582\) 0 0
\(583\) −0.349563 0.960417i −0.000599594 0.00164737i
\(584\) 0 0
\(585\) −100.328 17.6906i −0.171502 0.0302404i
\(586\) 0 0
\(587\) 260.508 218.592i 0.443796 0.372389i −0.393332 0.919397i \(-0.628678\pi\)
0.837128 + 0.547007i \(0.184233\pi\)
\(588\) 0 0
\(589\) −4.78722 + 129.893i −0.00812771 + 0.220531i
\(590\) 0 0
\(591\) 144.227 + 171.884i 0.244040 + 0.290835i
\(592\) 0 0
\(593\) 29.4003 166.737i 0.0495789 0.281176i −0.949932 0.312458i \(-0.898848\pi\)
0.999511 + 0.0312817i \(0.00995889\pi\)
\(594\) 0 0
\(595\) −338.064 + 123.045i −0.568176 + 0.206799i
\(596\) 0 0
\(597\) 433.527 250.297i 0.726176 0.419258i
\(598\) 0 0
\(599\) −27.9949 + 33.3630i −0.0467361 + 0.0556979i −0.788906 0.614514i \(-0.789352\pi\)
0.742170 + 0.670212i \(0.233797\pi\)
\(600\) 0 0
\(601\) 829.079 + 478.669i 1.37950 + 0.796454i 0.992099 0.125458i \(-0.0400400\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(602\) 0 0
\(603\) 44.3875 7.82671i 0.0736110 0.0129796i
\(604\) 0 0
\(605\) 495.554 + 180.367i 0.819098 + 0.298127i
\(606\) 0 0
\(607\) 383.700i 0.632124i −0.948738 0.316062i \(-0.897639\pi\)
0.948738 0.316062i \(-0.102361\pi\)
\(608\) 0 0
\(609\) −248.810 −0.408555
\(610\) 0 0
\(611\) −81.6635 + 224.369i −0.133655 + 0.367215i
\(612\) 0 0
\(613\) 92.6661 + 525.535i 0.151168 + 0.857317i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.811038 + 0.584994i \(0.801097\pi\)
\(614\) 0 0
\(615\) −223.328 + 386.815i −0.363135 + 0.628967i
\(616\) 0 0
\(617\) −522.562 438.481i −0.846940 0.710667i 0.112174 0.993689i \(-0.464219\pi\)
−0.959113 + 0.283022i \(0.908663\pi\)
\(618\) 0 0
\(619\) −90.2029 156.236i −0.145724 0.252401i 0.783919 0.620863i \(-0.213218\pi\)
−0.929643 + 0.368462i \(0.879884\pi\)
\(620\) 0 0
\(621\) 128.542 + 353.166i 0.206992 + 0.568705i
\(622\) 0 0
\(623\) 731.992 + 129.070i 1.17495 + 0.207175i
\(624\) 0 0
\(625\) −336.471 + 282.333i −0.538353 + 0.451732i
\(626\) 0 0
\(627\) 3.98416 + 5.11985i 0.00635433 + 0.00816564i
\(628\) 0 0
\(629\) 351.850 + 419.319i 0.559380 + 0.666644i
\(630\) 0 0
\(631\) −104.292 + 591.469i −0.165280 + 0.937351i 0.783495 + 0.621399i \(0.213435\pi\)
−0.948775 + 0.315953i \(0.897676\pi\)
\(632\) 0 0
\(633\) 424.386 154.464i 0.670437 0.244019i
\(634\) 0 0
\(635\) −636.070 + 367.235i −1.00169 + 0.578323i
\(636\) 0 0
\(637\) −67.4513 + 80.3854i −0.105889 + 0.126194i
\(638\) 0 0
\(639\) 120.777 + 69.7308i 0.189010 + 0.109125i
\(640\) 0 0
\(641\) −762.921 + 134.524i −1.19020 + 0.209865i −0.733460 0.679733i \(-0.762096\pi\)
−0.456744 + 0.889598i \(0.650985\pi\)
\(642\) 0 0
\(643\) 101.877 + 37.0801i 0.158440 + 0.0576674i 0.420023 0.907514i \(-0.362022\pi\)
−0.261583 + 0.965181i \(0.584244\pi\)
\(644\) 0 0
\(645\) 324.753i 0.503492i
\(646\) 0 0
\(647\) −230.970 −0.356986 −0.178493 0.983941i \(-0.557122\pi\)
−0.178493 + 0.983941i \(0.557122\pi\)
\(648\) 0 0
\(649\) 3.65455 10.0408i 0.00563105 0.0154712i
\(650\) 0 0
\(651\) 10.3425 + 58.6550i 0.0158870 + 0.0900998i
\(652\) 0 0
\(653\) 341.320 591.183i 0.522695 0.905334i −0.476956 0.878927i \(-0.658260\pi\)
0.999651 0.0264072i \(-0.00840665\pi\)
\(654\) 0 0
\(655\) 116.102 + 97.4213i 0.177255 + 0.148735i
\(656\) 0 0
\(657\) −341.478 591.458i −0.519754 0.900240i
\(658\) 0 0
\(659\) 212.474 + 583.768i 0.322419 + 0.885839i 0.989970 + 0.141275i \(0.0451203\pi\)
−0.667551 + 0.744564i \(0.732657\pi\)
\(660\) 0 0
\(661\) −1006.10 177.403i −1.52209 0.268386i −0.650838 0.759217i \(-0.725582\pi\)
−0.871255 + 0.490831i \(0.836693\pi\)
\(662\) 0 0
\(663\) −96.4712 + 80.9489i −0.145507 + 0.122095i
\(664\) 0 0
\(665\) 54.8039 + 395.493i 0.0824120 + 0.594726i
\(666\) 0 0
\(667\) −259.364 309.098i −0.388852 0.463416i
\(668\) 0 0
\(669\) −34.9341 + 198.121i −0.0522184 + 0.296145i
\(670\) 0 0
\(671\) −16.0854 + 5.85459i −0.0239722 + 0.00872518i
\(672\) 0 0
\(673\) 753.056 434.777i 1.11895 0.646028i 0.177820 0.984063i \(-0.443095\pi\)
0.941134 + 0.338035i \(0.109762\pi\)
\(674\) 0 0
\(675\) −102.555 + 122.220i −0.151933 + 0.181067i
\(676\) 0 0
\(677\) 523.041 + 301.978i 0.772587 + 0.446053i 0.833797 0.552072i \(-0.186163\pi\)
−0.0612099 + 0.998125i \(0.519496\pi\)
\(678\) 0 0
\(679\) 387.251 68.2828i 0.570325 0.100564i
\(680\) 0 0
\(681\) −585.774 213.204i −0.860167 0.313075i
\(682\) 0 0
\(683\) 192.200i 0.281406i 0.990052 + 0.140703i \(0.0449363\pi\)
−0.990052 + 0.140703i \(0.955064\pi\)
\(684\) 0 0
\(685\) −710.506 −1.03724
\(686\) 0 0
\(687\) 85.3349 234.456i 0.124214 0.341275i
\(688\) 0 0
\(689\) −3.82359 21.6847i −0.00554948 0.0314727i
\(690\) 0 0
\(691\) 119.347 206.715i 0.172716 0.299154i −0.766652 0.642063i \(-0.778079\pi\)
0.939369 + 0.342909i \(0.111412\pi\)
\(692\) 0 0
\(693\) −4.00514 3.36071i −0.00577942 0.00484951i
\(694\) 0 0
\(695\) 57.8084 + 100.127i 0.0831776 + 0.144068i
\(696\) 0 0
\(697\) −332.135 912.533i −0.476520 1.30923i
\(698\) 0 0
\(699\) 167.308 + 29.5009i 0.239353 + 0.0422044i
\(700\) 0 0
\(701\) 13.1679 11.0492i 0.0187845 0.0157621i −0.633347 0.773868i \(-0.718319\pi\)
0.652132 + 0.758106i \(0.273875\pi\)
\(702\) 0 0
\(703\) 536.939 284.167i 0.763783 0.404220i
\(704\) 0 0
\(705\) −296.734 353.634i −0.420900 0.501609i
\(706\) 0 0
\(707\) 98.2820 557.385i 0.139013 0.788380i
\(708\) 0 0
\(709\) 89.6670 32.6361i 0.126470 0.0460312i −0.278010 0.960578i \(-0.589675\pi\)
0.404480 + 0.914547i \(0.367453\pi\)
\(710\) 0 0
\(711\) 694.847 401.170i 0.977281 0.564233i
\(712\) 0 0
\(713\) −62.0863 + 73.9916i −0.0870776 + 0.103775i
\(714\) 0 0
\(715\) −2.90684 1.67826i −0.00406551 0.00234722i
\(716\) 0 0
\(717\) −377.849 + 66.6249i −0.526986 + 0.0929218i
\(718\) 0 0
\(719\) −458.305 166.809i −0.637420 0.232002i 0.00303783 0.999995i \(-0.499033\pi\)
−0.640458 + 0.767994i \(0.721255\pi\)
\(720\) 0 0
\(721\) 748.543i 1.03820i
\(722\) 0 0
\(723\) −54.4475 −0.0753077
\(724\) 0 0
\(725\) 58.5856 160.962i 0.0808077 0.222017i
\(726\) 0 0
\(727\) −1.77453 10.0639i −0.00244090 0.0138430i 0.983563 0.180565i \(-0.0577927\pi\)
−0.986004 + 0.166722i \(0.946682\pi\)
\(728\) 0 0
\(729\) 206.137 357.041i 0.282767 0.489768i
\(730\) 0 0
\(731\) 540.875 + 453.848i 0.739910 + 0.620859i
\(732\) 0 0
\(733\) 251.354 + 435.357i 0.342911 + 0.593939i 0.984972 0.172714i \(-0.0552538\pi\)
−0.642061 + 0.766653i \(0.721920\pi\)
\(734\) 0 0
\(735\) −69.3903 190.648i −0.0944086 0.259385i
\(736\) 0 0
\(737\) 1.46244 + 0.257868i 0.00198431 + 0.000349888i
\(738\) 0 0
\(739\) −172.968 + 145.138i −0.234057 + 0.196397i −0.752271 0.658854i \(-0.771042\pi\)
0.518214 + 0.855251i \(0.326597\pi\)
\(740\) 0 0
\(741\) 65.3772 + 123.532i 0.0882283 + 0.166709i
\(742\) 0 0
\(743\) 22.9359 + 27.3339i 0.0308693 + 0.0367886i 0.781258 0.624208i \(-0.214578\pi\)
−0.750389 + 0.660997i \(0.770134\pi\)
\(744\) 0 0
\(745\) 118.972 674.725i 0.159694 0.905671i
\(746\) 0 0
\(747\) 560.973 204.178i 0.750968 0.273330i
\(748\) 0 0
\(749\) −667.999 + 385.670i −0.891855 + 0.514913i
\(750\) 0 0
\(751\) 662.322 789.325i 0.881920 1.05103i −0.116406 0.993202i \(-0.537137\pi\)
0.998326 0.0578298i \(-0.0184181\pi\)
\(752\) 0 0
\(753\) 2.61189 + 1.50797i 0.00346864 + 0.00200262i
\(754\) 0 0
\(755\) 114.922 20.2639i 0.152215 0.0268396i
\(756\) 0 0
\(757\) −1300.99 473.521i −1.71861 0.625524i −0.720894 0.693045i \(-0.756269\pi\)
−0.997718 + 0.0675215i \(0.978491\pi\)
\(758\) 0 0
\(759\) 4.82081i 0.00635152i
\(760\) 0 0
\(761\) −723.684 −0.950965 −0.475483 0.879725i \(-0.657727\pi\)
−0.475483 + 0.879725i \(0.657727\pi\)
\(762\) 0 0
\(763\) 176.348 484.511i 0.231124 0.635008i
\(764\) 0 0
\(765\) 74.3630 + 421.733i 0.0972065 + 0.551286i
\(766\) 0 0
\(767\) 115.101 199.361i 0.150067 0.259923i
\(768\) 0 0
\(769\) 726.061 + 609.238i 0.944163 + 0.792247i 0.978305 0.207170i \(-0.0664254\pi\)
−0.0341420 + 0.999417i \(0.510870\pi\)
\(770\) 0 0
\(771\) −325.436 563.671i −0.422095 0.731091i
\(772\) 0 0
\(773\) −190.354 522.993i −0.246253 0.676575i −0.999816 0.0191990i \(-0.993888\pi\)
0.753562 0.657376i \(-0.228334\pi\)
\(774\) 0 0
\(775\) −40.3809 7.12024i −0.0521044 0.00918740i
\(776\) 0 0
\(777\) 213.242 178.931i 0.274443 0.230285i
\(778\) 0 0
\(779\) −1067.55 + 147.932i −1.37041 + 0.189899i
\(780\) 0 0
\(781\) 2.95352 + 3.51987i 0.00378172 + 0.00450687i
\(782\) 0 0
\(783\) −132.100 + 749.175i −0.168710 + 0.956801i
\(784\) 0 0
\(785\) −849.194 + 309.082i −1.08178 + 0.393734i
\(786\) 0 0
\(787\) −296.622 + 171.255i −0.376902 + 0.217604i −0.676469 0.736471i \(-0.736491\pi\)
0.299568 + 0.954075i \(0.403158\pi\)
\(788\) 0 0
\(789\) −155.987 + 185.899i −0.197703 + 0.235613i
\(790\) 0 0
\(791\) 229.441 + 132.468i 0.290064 + 0.167469i
\(792\) 0 0
\(793\) −363.182 + 64.0387i −0.457984 + 0.0807550i
\(794\) 0 0
\(795\) 40.0047 + 14.5605i 0.0503204 + 0.0183151i
\(796\) 0 0
\(797\) 1411.17i 1.77060i 0.465019 + 0.885301i \(0.346048\pi\)
−0.465019 + 0.885301i \(0.653952\pi\)
\(798\) 0 0
\(799\) 1003.67 1.25616
\(800\) 0 0
\(801\) 302.607 831.407i 0.377787 1.03796i
\(802\) 0 0
\(803\) −3.90734 22.1596i −0.00486593 0.0275960i
\(804\) 0 0
\(805\) −148.350 + 256.949i −0.184285 + 0.319192i
\(806\) 0 0
\(807\) −73.2823 61.4911i −0.0908083 0.0761972i
\(808\) 0 0
\(809\) 703.686 + 1218.82i 0.869822 + 1.50658i 0.862178 + 0.506605i \(0.169100\pi\)
0.00764386 + 0.999971i \(0.497567\pi\)
\(810\) 0 0
\(811\) 172.417 + 473.711i 0.212598 + 0.584108i 0.999454 0.0330281i \(-0.0105151\pi\)
−0.786857 + 0.617136i \(0.788293\pi\)
\(812\) 0 0
\(813\) −256.170 45.1696i −0.315092 0.0555592i
\(814\) 0 0
\(815\) 526.821 442.055i 0.646406 0.542399i
\(816\) 0 0
\(817\) 618.420 481.242i 0.756941 0.589035i
\(818\) 0 0
\(819\) −72.4033 86.2869i −0.0884045 0.105356i
\(820\) 0 0
\(821\) −210.354 + 1192.98i −0.256216 + 1.45308i 0.536714 + 0.843764i \(0.319665\pi\)
−0.792931 + 0.609312i \(0.791446\pi\)
\(822\) 0 0
\(823\) −385.490 + 140.307i −0.468396 + 0.170482i −0.565426 0.824799i \(-0.691288\pi\)
0.0970294 + 0.995282i \(0.469066\pi\)
\(824\) 0 0
\(825\) −1.77234 + 1.02326i −0.00214829 + 0.00124031i
\(826\) 0 0
\(827\) −283.443 + 337.795i −0.342737 + 0.408458i −0.909687 0.415294i \(-0.863679\pi\)
0.566950 + 0.823752i \(0.308123\pi\)
\(828\) 0 0
\(829\) 134.275 + 77.5237i 0.161972 + 0.0935147i 0.578795 0.815473i \(-0.303523\pi\)
−0.416823 + 0.908988i \(0.636856\pi\)
\(830\) 0 0
\(831\) 131.552 23.1962i 0.158306 0.0279136i
\(832\) 0 0
\(833\) 414.499 + 150.865i 0.497597 + 0.181111i
\(834\) 0 0
\(835\) 51.9369i 0.0621999i
\(836\) 0 0
\(837\) 182.103 0.217566
\(838\) 0 0
\(839\) 250.499 688.241i 0.298569 0.820311i −0.696171 0.717876i \(-0.745115\pi\)
0.994740 0.102435i \(-0.0326633\pi\)
\(840\) 0 0
\(841\) 4.21334 + 23.8950i 0.00500991 + 0.0284126i
\(842\) 0 0
\(843\) 57.0796 98.8648i 0.0677101 0.117277i
\(844\) 0 0
\(845\) 509.007 + 427.108i 0.602376 + 0.505453i
\(846\) 0 0
\(847\) 291.537 + 504.957i 0.344200 + 0.596171i
\(848\) 0 0
\(849\) −60.1406 165.235i −0.0708370 0.194623i
\(850\) 0 0
\(851\) 444.575 + 78.3906i 0.522415 + 0.0921158i
\(852\) 0 0
\(853\) −226.059 + 189.686i −0.265016 + 0.222375i −0.765606 0.643309i \(-0.777561\pi\)
0.500590 + 0.865685i \(0.333116\pi\)
\(854\) 0 0
\(855\) 474.949 + 17.5043i 0.555496 + 0.0204729i
\(856\) 0 0
\(857\) 24.3399 + 29.0071i 0.0284013 + 0.0338473i 0.780058 0.625707i \(-0.215189\pi\)
−0.751657 + 0.659554i \(0.770745\pi\)
\(858\) 0 0
\(859\) 29.0571 164.791i 0.0338266 0.191840i −0.963212 0.268743i \(-0.913392\pi\)
0.997039 + 0.0769024i \(0.0245030\pi\)
\(860\) 0 0
\(861\) −464.063 + 168.905i −0.538981 + 0.196173i
\(862\) 0 0
\(863\) −1171.31 + 676.254i −1.35725 + 0.783608i −0.989252 0.146219i \(-0.953290\pi\)
−0.367997 + 0.929827i \(0.619956\pi\)
\(864\) 0 0
\(865\) 377.255 449.595i 0.436133 0.519763i
\(866\) 0 0
\(867\) 6.39375 + 3.69143i 0.00737456 + 0.00425771i
\(868\) 0 0
\(869\) 26.0332 4.59035i 0.0299576 0.00528234i
\(870\) 0 0
\(871\) 30.0635 + 10.9422i 0.0345161 + 0.0125628i
\(872\) 0 0
\(873\) 468.073i 0.536166i
\(874\) 0 0
\(875\) −651.312 −0.744356
\(876\) 0 0
\(877\) −171.809 + 472.041i −0.195905 + 0.538245i −0.998283 0.0585713i \(-0.981346\pi\)
0.802378 + 0.596816i \(0.203568\pi\)
\(878\) 0 0
\(879\) 122.401 + 694.170i 0.139250 + 0.789727i
\(880\) 0 0
\(881\) −170.295 + 294.960i −0.193297 + 0.334801i −0.946341 0.323170i \(-0.895252\pi\)
0.753044 + 0.657971i \(0.228585\pi\)
\(882\) 0 0
\(883\) 1203.63 + 1009.96i 1.36311 + 1.14379i 0.975008 + 0.222169i \(0.0713137\pi\)
0.388102 + 0.921616i \(0.373131\pi\)
\(884\) 0 0
\(885\) 222.538 + 385.447i 0.251455 + 0.435533i
\(886\) 0 0
\(887\) −26.4753 72.7402i −0.0298481 0.0820070i 0.923874 0.382698i \(-0.125005\pi\)
−0.953722 + 0.300691i \(0.902783\pi\)
\(888\) 0 0
\(889\) −799.731 141.014i −0.899585 0.158621i
\(890\) 0 0
\(891\) −0.515680 + 0.432707i −0.000578766 + 0.000485642i
\(892\) 0 0
\(893\) 233.697 1089.11i 0.261699 1.21960i
\(894\) 0 0
\(895\) −900.048 1072.64i −1.00564 1.19848i
\(896\) 0 0
\(897\) −18.0350 + 102.282i −0.0201059 + 0.114027i
\(898\) 0 0
\(899\) −183.719 + 66.8681i −0.204359 + 0.0743805i
\(900\) 0 0
\(901\) −80.1578 + 46.2792i −0.0889654 + 0.0513642i
\(902\) 0 0
\(903\) 230.801 275.058i 0.255594 0.304605i
\(904\) 0 0
\(905\) 276.435 + 159.600i 0.305453 + 0.176353i
\(906\) 0 0
\(907\) 1554.83 274.159i 1.71426 0.302270i 0.771620 0.636083i \(-0.219447\pi\)
0.942639 + 0.333813i \(0.108335\pi\)
\(908\) 0 0
\(909\) −633.085 230.424i −0.696464 0.253492i
\(910\) 0 0
\(911\) 1736.18i 1.90580i 0.303290 + 0.952898i \(0.401915\pi\)
−0.303290 + 0.952898i \(0.598085\pi\)
\(912\) 0 0
\(913\) 19.6686 0.0215428
\(914\) 0 0
\(915\) 243.864 670.012i 0.266518 0.732253i
\(916\) 0 0
\(917\) 29.0988 + 165.028i 0.0317326 + 0.179965i
\(918\) 0 0
\(919\) −621.707 + 1076.83i −0.676503 + 1.17174i 0.299524 + 0.954089i \(0.403172\pi\)
−0.976027 + 0.217649i \(0.930161\pi\)
\(920\) 0 0
\(921\) −68.6207 57.5796i −0.0745067 0.0625186i
\(922\) 0 0
\(923\) 49.4959 + 85.7295i 0.0536251 + 0.0928814i
\(924\) 0 0
\(925\) 65.5453 + 180.084i 0.0708597 + 0.194686i
\(926\) 0 0
\(927\) −877.488 154.725i −0.946589 0.166909i
\(928\) 0 0
\(929\) 437.894 367.436i 0.471360 0.395518i −0.375930 0.926648i \(-0.622677\pi\)
0.847291 + 0.531130i \(0.178232\pi\)
\(930\) 0 0
\(931\) 260.221 414.655i 0.279507 0.445387i
\(932\) 0 0
\(933\) −247.919 295.458i −0.265722 0.316675i
\(934\) 0 0
\(935\) −2.45005 + 13.8949i −0.00262037 + 0.0148609i
\(936\) 0 0
\(937\) 76.0200 27.6690i 0.0811313 0.0295294i −0.301136 0.953581i \(-0.597366\pi\)
0.382267 + 0.924052i \(0.375143\pi\)
\(938\) 0 0
\(939\) −302.709 + 174.769i −0.322374 + 0.186122i
\(940\) 0 0
\(941\) 617.870 736.349i 0.656610 0.782518i −0.330285 0.943881i \(-0.607145\pi\)
0.986895 + 0.161364i \(0.0515892\pi\)
\(942\) 0 0
\(943\) −693.580 400.438i −0.735503 0.424643i
\(944\) 0 0
\(945\) 550.880 97.1351i 0.582942 0.102788i
\(946\) 0 0
\(947\) −1623.43 590.881i −1.71429 0.623951i −0.716970 0.697104i \(-0.754472\pi\)
−0.997321 + 0.0731531i \(0.976694\pi\)
\(948\) 0 0
\(949\) 484.773i 0.510825i
\(950\) 0 0
\(951\) 1077.91 1.13345
\(952\) 0 0
\(953\) 520.185 1429.20i 0.545840 1.49968i −0.293437 0.955978i \(-0.594799\pi\)
0.839277 0.543704i \(-0.182979\pi\)
\(954\) 0 0
\(955\) 101.208 + 573.976i 0.105976 + 0.601022i
\(956\) 0 0
\(957\) −4.87898 + 8.45063i −0.00509820 + 0.00883034i
\(958\) 0 0
\(959\) −601.783 504.956i −0.627511 0.526544i
\(960\) 0 0
\(961\) −457.100 791.720i −0.475650 0.823850i
\(962\) 0 0
\(963\) 314.029 + 862.788i 0.326095 + 0.895938i
\(964\) 0 0
\(965\) 923.247 + 162.793i 0.956732 + 0.168698i
\(966\) 0 0
\(967\) 347.163 291.304i 0.359010 0.301245i −0.445386 0.895339i \(-0.646934\pi\)
0.804396 + 0.594093i \(0.202489\pi\)
\(968\) 0 0
\(969\) 393.961 435.842i 0.406565 0.449786i
\(970\) 0 0
\(971\) −566.677 675.340i −0.583602 0.695510i 0.390761 0.920492i \(-0.372212\pi\)
−0.974363 + 0.224983i \(0.927767\pi\)
\(972\) 0 0
\(973\) −22.1978 + 125.890i −0.0228138 + 0.129383i
\(974\) 0 0
\(975\) −41.4313 + 15.0798i −0.0424937 + 0.0154664i
\(976\) 0 0
\(977\) −1269.34 + 732.855i −1.29922 + 0.750108i −0.980270 0.197662i \(-0.936665\pi\)
−0.318954 + 0.947770i \(0.603332\pi\)
\(978\) 0 0
\(979\) 18.7376 22.3305i 0.0191395 0.0228095i
\(980\) 0 0
\(981\) −531.522 306.874i −0.541817 0.312818i
\(982\) 0 0
\(983\) −1852.95 + 326.724i −1.88499 + 0.332375i −0.992849 0.119380i \(-0.961909\pi\)
−0.892142 + 0.451754i \(0.850798\pi\)
\(984\) 0 0
\(985\) 508.925 + 185.234i 0.516675 + 0.188054i
\(986\) 0 0
\(987\) 510.410i 0.517132i
\(988\) 0 0
\(989\) 582.299 0.588775
\(990\) 0 0
\(991\) 179.003 491.807i 0.180629 0.496274i −0.816024 0.578017i \(-0.803827\pi\)
0.996653 + 0.0817435i \(0.0260488\pi\)
\(992\) 0 0
\(993\) 173.633 + 984.723i 0.174857 + 0.991664i
\(994\) 0 0
\(995\) 604.148 1046.42i 0.607184 1.05167i
\(996\) 0 0
\(997\) 512.180 + 429.770i 0.513721 + 0.431063i 0.862436 0.506165i \(-0.168937\pi\)
−0.348715 + 0.937229i \(0.613382\pi\)
\(998\) 0 0
\(999\) −425.552 737.077i −0.425978 0.737815i
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 76.3.j.a.13.1 18
4.3 odd 2 304.3.z.b.241.3 18
19.3 odd 18 inner 76.3.j.a.41.1 yes 18
19.4 even 9 1444.3.c.c.721.8 18
19.15 odd 18 1444.3.c.c.721.11 18
76.3 even 18 304.3.z.b.193.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.j.a.13.1 18 1.1 even 1 trivial
76.3.j.a.41.1 yes 18 19.3 odd 18 inner
304.3.z.b.193.3 18 76.3 even 18
304.3.z.b.241.3 18 4.3 odd 2
1444.3.c.c.721.8 18 19.4 even 9
1444.3.c.c.721.11 18 19.15 odd 18