Properties

Label 4004.2.m.b.2157.20
Level $4004$
Weight $2$
Character 4004.2157
Analytic conductor $31.972$
Analytic rank $0$
Dimension $30$
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] = [4004,2,Mod(2157,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4004.2157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.m (of order \(2\), degree \(1\), 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: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.20
Character \(\chi\) \(=\) 4004.2157
Dual form 4004.2.m.b.2157.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.726270 q^{3} +2.96350i q^{5} -1.00000i q^{7} -2.47253 q^{9} +O(q^{10})\) \(q+0.726270 q^{3} +2.96350i q^{5} -1.00000i q^{7} -2.47253 q^{9} -1.00000i q^{11} +(-0.483609 - 3.57297i) q^{13} +2.15230i q^{15} +4.91968 q^{17} +6.32548i q^{19} -0.726270i q^{21} +0.763996 q^{23} -3.78231 q^{25} -3.97454 q^{27} +6.71850 q^{29} -5.63535i q^{31} -0.726270i q^{33} +2.96350 q^{35} +7.09762i q^{37} +(-0.351231 - 2.59494i) q^{39} +7.75835i q^{41} +1.96634 q^{43} -7.32734i q^{45} +9.93798i q^{47} -1.00000 q^{49} +3.57302 q^{51} -6.03131 q^{53} +2.96350 q^{55} +4.59400i q^{57} +4.32831i q^{59} -7.15178 q^{61} +2.47253i q^{63} +(10.5885 - 1.43317i) q^{65} -8.48905i q^{67} +0.554867 q^{69} +8.48248i q^{71} -0.437137i q^{73} -2.74697 q^{75} -1.00000 q^{77} +2.91924 q^{79} +4.53101 q^{81} +16.2024i q^{83} +14.5795i q^{85} +4.87945 q^{87} +5.41370i q^{89} +(-3.57297 + 0.483609i) q^{91} -4.09279i q^{93} -18.7455 q^{95} -9.24011i q^{97} +2.47253i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 0.726270 0.419312 0.209656 0.977775i \(-0.432766\pi\)
0.209656 + 0.977775i \(0.432766\pi\)
\(4\) 0 0
\(5\) 2.96350i 1.32532i 0.748922 + 0.662658i \(0.230572\pi\)
−0.748922 + 0.662658i \(0.769428\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.47253 −0.824177
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) −0.483609 3.57297i −0.134129 0.990964i
\(14\) 0 0
\(15\) 2.15230i 0.555721i
\(16\) 0 0
\(17\) 4.91968 1.19320 0.596599 0.802539i \(-0.296518\pi\)
0.596599 + 0.802539i \(0.296518\pi\)
\(18\) 0 0
\(19\) 6.32548i 1.45116i 0.688136 + 0.725582i \(0.258429\pi\)
−0.688136 + 0.725582i \(0.741571\pi\)
\(20\) 0 0
\(21\) 0.726270i 0.158485i
\(22\) 0 0
\(23\) 0.763996 0.159304 0.0796521 0.996823i \(-0.474619\pi\)
0.0796521 + 0.996823i \(0.474619\pi\)
\(24\) 0 0
\(25\) −3.78231 −0.756461
\(26\) 0 0
\(27\) −3.97454 −0.764900
\(28\) 0 0
\(29\) 6.71850 1.24759 0.623797 0.781586i \(-0.285589\pi\)
0.623797 + 0.781586i \(0.285589\pi\)
\(30\) 0 0
\(31\) 5.63535i 1.01214i −0.862493 0.506070i \(-0.831098\pi\)
0.862493 0.506070i \(-0.168902\pi\)
\(32\) 0 0
\(33\) 0.726270i 0.126427i
\(34\) 0 0
\(35\) 2.96350 0.500922
\(36\) 0 0
\(37\) 7.09762i 1.16684i 0.812170 + 0.583421i \(0.198286\pi\)
−0.812170 + 0.583421i \(0.801714\pi\)
\(38\) 0 0
\(39\) −0.351231 2.59494i −0.0562419 0.415523i
\(40\) 0 0
\(41\) 7.75835i 1.21165i 0.795598 + 0.605825i \(0.207157\pi\)
−0.795598 + 0.605825i \(0.792843\pi\)
\(42\) 0 0
\(43\) 1.96634 0.299864 0.149932 0.988696i \(-0.452095\pi\)
0.149932 + 0.988696i \(0.452095\pi\)
\(44\) 0 0
\(45\) 7.32734i 1.09230i
\(46\) 0 0
\(47\) 9.93798i 1.44960i 0.688958 + 0.724802i \(0.258069\pi\)
−0.688958 + 0.724802i \(0.741931\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.57302 0.500323
\(52\) 0 0
\(53\) −6.03131 −0.828464 −0.414232 0.910171i \(-0.635950\pi\)
−0.414232 + 0.910171i \(0.635950\pi\)
\(54\) 0 0
\(55\) 2.96350 0.399598
\(56\) 0 0
\(57\) 4.59400i 0.608491i
\(58\) 0 0
\(59\) 4.32831i 0.563498i 0.959488 + 0.281749i \(0.0909145\pi\)
−0.959488 + 0.281749i \(0.909085\pi\)
\(60\) 0 0
\(61\) −7.15178 −0.915692 −0.457846 0.889032i \(-0.651379\pi\)
−0.457846 + 0.889032i \(0.651379\pi\)
\(62\) 0 0
\(63\) 2.47253i 0.311510i
\(64\) 0 0
\(65\) 10.5885 1.43317i 1.31334 0.177763i
\(66\) 0 0
\(67\) 8.48905i 1.03710i −0.855047 0.518551i \(-0.826472\pi\)
0.855047 0.518551i \(-0.173528\pi\)
\(68\) 0 0
\(69\) 0.554867 0.0667982
\(70\) 0 0
\(71\) 8.48248i 1.00668i 0.864087 + 0.503342i \(0.167897\pi\)
−0.864087 + 0.503342i \(0.832103\pi\)
\(72\) 0 0
\(73\) 0.437137i 0.0511630i −0.999673 0.0255815i \(-0.991856\pi\)
0.999673 0.0255815i \(-0.00814373\pi\)
\(74\) 0 0
\(75\) −2.74697 −0.317193
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 2.91924 0.328440 0.164220 0.986424i \(-0.447489\pi\)
0.164220 + 0.986424i \(0.447489\pi\)
\(80\) 0 0
\(81\) 4.53101 0.503446
\(82\) 0 0
\(83\) 16.2024i 1.77845i 0.457473 + 0.889224i \(0.348755\pi\)
−0.457473 + 0.889224i \(0.651245\pi\)
\(84\) 0 0
\(85\) 14.5795i 1.58136i
\(86\) 0 0
\(87\) 4.87945 0.523132
\(88\) 0 0
\(89\) 5.41370i 0.573851i 0.957953 + 0.286925i \(0.0926332\pi\)
−0.957953 + 0.286925i \(0.907367\pi\)
\(90\) 0 0
\(91\) −3.57297 + 0.483609i −0.374549 + 0.0506960i
\(92\) 0 0
\(93\) 4.09279i 0.424402i
\(94\) 0 0
\(95\) −18.7455 −1.92325
\(96\) 0 0
\(97\) 9.24011i 0.938191i −0.883147 0.469096i \(-0.844580\pi\)
0.883147 0.469096i \(-0.155420\pi\)
\(98\) 0 0
\(99\) 2.47253i 0.248499i
\(100\) 0 0
\(101\) −2.22666 −0.221561 −0.110781 0.993845i \(-0.535335\pi\)
−0.110781 + 0.993845i \(0.535335\pi\)
\(102\) 0 0
\(103\) 6.86947 0.676869 0.338434 0.940990i \(-0.390103\pi\)
0.338434 + 0.940990i \(0.390103\pi\)
\(104\) 0 0
\(105\) 2.15230 0.210043
\(106\) 0 0
\(107\) 14.5787 1.40938 0.704688 0.709518i \(-0.251087\pi\)
0.704688 + 0.709518i \(0.251087\pi\)
\(108\) 0 0
\(109\) 10.7570i 1.03033i 0.857090 + 0.515166i \(0.172270\pi\)
−0.857090 + 0.515166i \(0.827730\pi\)
\(110\) 0 0
\(111\) 5.15479i 0.489271i
\(112\) 0 0
\(113\) −9.08338 −0.854492 −0.427246 0.904135i \(-0.640516\pi\)
−0.427246 + 0.904135i \(0.640516\pi\)
\(114\) 0 0
\(115\) 2.26410i 0.211128i
\(116\) 0 0
\(117\) 1.19574 + 8.83429i 0.110546 + 0.816730i
\(118\) 0 0
\(119\) 4.91968i 0.450987i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 5.63465i 0.508060i
\(124\) 0 0
\(125\) 3.60863i 0.322766i
\(126\) 0 0
\(127\) 6.18388 0.548730 0.274365 0.961626i \(-0.411532\pi\)
0.274365 + 0.961626i \(0.411532\pi\)
\(128\) 0 0
\(129\) 1.42809 0.125737
\(130\) 0 0
\(131\) −7.46908 −0.652577 −0.326288 0.945270i \(-0.605798\pi\)
−0.326288 + 0.945270i \(0.605798\pi\)
\(132\) 0 0
\(133\) 6.32548 0.548488
\(134\) 0 0
\(135\) 11.7785i 1.01373i
\(136\) 0 0
\(137\) 7.21732i 0.616617i 0.951286 + 0.308308i \(0.0997629\pi\)
−0.951286 + 0.308308i \(0.900237\pi\)
\(138\) 0 0
\(139\) −2.64119 −0.224023 −0.112011 0.993707i \(-0.535729\pi\)
−0.112011 + 0.993707i \(0.535729\pi\)
\(140\) 0 0
\(141\) 7.21766i 0.607836i
\(142\) 0 0
\(143\) −3.57297 + 0.483609i −0.298787 + 0.0404414i
\(144\) 0 0
\(145\) 19.9103i 1.65346i
\(146\) 0 0
\(147\) −0.726270 −0.0599017
\(148\) 0 0
\(149\) 10.1639i 0.832659i −0.909214 0.416329i \(-0.863316\pi\)
0.909214 0.416329i \(-0.136684\pi\)
\(150\) 0 0
\(151\) 10.2313i 0.832611i 0.909225 + 0.416305i \(0.136675\pi\)
−0.909225 + 0.416305i \(0.863325\pi\)
\(152\) 0 0
\(153\) −12.1641 −0.983407
\(154\) 0 0
\(155\) 16.7003 1.34140
\(156\) 0 0
\(157\) −9.66954 −0.771714 −0.385857 0.922559i \(-0.626094\pi\)
−0.385857 + 0.922559i \(0.626094\pi\)
\(158\) 0 0
\(159\) −4.38036 −0.347385
\(160\) 0 0
\(161\) 0.763996i 0.0602113i
\(162\) 0 0
\(163\) 2.94966i 0.231035i 0.993305 + 0.115518i \(0.0368527\pi\)
−0.993305 + 0.115518i \(0.963147\pi\)
\(164\) 0 0
\(165\) 2.15230 0.167556
\(166\) 0 0
\(167\) 16.6935i 1.29178i 0.763430 + 0.645891i \(0.223514\pi\)
−0.763430 + 0.645891i \(0.776486\pi\)
\(168\) 0 0
\(169\) −12.5322 + 3.45584i −0.964019 + 0.265834i
\(170\) 0 0
\(171\) 15.6399i 1.19602i
\(172\) 0 0
\(173\) −15.2932 −1.16272 −0.581361 0.813646i \(-0.697480\pi\)
−0.581361 + 0.813646i \(0.697480\pi\)
\(174\) 0 0
\(175\) 3.78231i 0.285915i
\(176\) 0 0
\(177\) 3.14352i 0.236282i
\(178\) 0 0
\(179\) −1.49826 −0.111985 −0.0559927 0.998431i \(-0.517832\pi\)
−0.0559927 + 0.998431i \(0.517832\pi\)
\(180\) 0 0
\(181\) −21.0631 −1.56561 −0.782805 0.622267i \(-0.786212\pi\)
−0.782805 + 0.622267i \(0.786212\pi\)
\(182\) 0 0
\(183\) −5.19412 −0.383961
\(184\) 0 0
\(185\) −21.0338 −1.54643
\(186\) 0 0
\(187\) 4.91968i 0.359763i
\(188\) 0 0
\(189\) 3.97454i 0.289105i
\(190\) 0 0
\(191\) 6.42418 0.464837 0.232419 0.972616i \(-0.425336\pi\)
0.232419 + 0.972616i \(0.425336\pi\)
\(192\) 0 0
\(193\) 3.03861i 0.218724i −0.994002 0.109362i \(-0.965119\pi\)
0.994002 0.109362i \(-0.0348807\pi\)
\(194\) 0 0
\(195\) 7.69010 1.04087i 0.550699 0.0745383i
\(196\) 0 0
\(197\) 16.9844i 1.21009i −0.796191 0.605045i \(-0.793155\pi\)
0.796191 0.605045i \(-0.206845\pi\)
\(198\) 0 0
\(199\) −0.342520 −0.0242806 −0.0121403 0.999926i \(-0.503864\pi\)
−0.0121403 + 0.999926i \(0.503864\pi\)
\(200\) 0 0
\(201\) 6.16534i 0.434870i
\(202\) 0 0
\(203\) 6.71850i 0.471547i
\(204\) 0 0
\(205\) −22.9918 −1.60582
\(206\) 0 0
\(207\) −1.88900 −0.131295
\(208\) 0 0
\(209\) 6.32548 0.437542
\(210\) 0 0
\(211\) −7.75600 −0.533945 −0.266973 0.963704i \(-0.586023\pi\)
−0.266973 + 0.963704i \(0.586023\pi\)
\(212\) 0 0
\(213\) 6.16057i 0.422115i
\(214\) 0 0
\(215\) 5.82724i 0.397414i
\(216\) 0 0
\(217\) −5.63535 −0.382553
\(218\) 0 0
\(219\) 0.317479i 0.0214533i
\(220\) 0 0
\(221\) −2.37921 17.5779i −0.160043 1.18242i
\(222\) 0 0
\(223\) 3.96125i 0.265265i 0.991165 + 0.132632i \(0.0423430\pi\)
−0.991165 + 0.132632i \(0.957657\pi\)
\(224\) 0 0
\(225\) 9.35187 0.623458
\(226\) 0 0
\(227\) 4.40000i 0.292038i 0.989282 + 0.146019i \(0.0466461\pi\)
−0.989282 + 0.146019i \(0.953354\pi\)
\(228\) 0 0
\(229\) 3.91264i 0.258555i 0.991608 + 0.129277i \(0.0412658\pi\)
−0.991608 + 0.129277i \(0.958734\pi\)
\(230\) 0 0
\(231\) −0.726270 −0.0477851
\(232\) 0 0
\(233\) −12.8867 −0.844234 −0.422117 0.906541i \(-0.638713\pi\)
−0.422117 + 0.906541i \(0.638713\pi\)
\(234\) 0 0
\(235\) −29.4512 −1.92118
\(236\) 0 0
\(237\) 2.12016 0.137719
\(238\) 0 0
\(239\) 26.1370i 1.69066i −0.534242 0.845332i \(-0.679403\pi\)
0.534242 0.845332i \(-0.320597\pi\)
\(240\) 0 0
\(241\) 20.5354i 1.32280i −0.750032 0.661401i \(-0.769962\pi\)
0.750032 0.661401i \(-0.230038\pi\)
\(242\) 0 0
\(243\) 15.2143 0.976001
\(244\) 0 0
\(245\) 2.96350i 0.189331i
\(246\) 0 0
\(247\) 22.6007 3.05906i 1.43805 0.194643i
\(248\) 0 0
\(249\) 11.7673i 0.745724i
\(250\) 0 0
\(251\) 27.2009 1.71691 0.858453 0.512893i \(-0.171426\pi\)
0.858453 + 0.512893i \(0.171426\pi\)
\(252\) 0 0
\(253\) 0.763996i 0.0480320i
\(254\) 0 0
\(255\) 10.5886i 0.663085i
\(256\) 0 0
\(257\) 10.9853 0.685242 0.342621 0.939474i \(-0.388685\pi\)
0.342621 + 0.939474i \(0.388685\pi\)
\(258\) 0 0
\(259\) 7.09762 0.441025
\(260\) 0 0
\(261\) −16.6117 −1.02824
\(262\) 0 0
\(263\) 6.16621 0.380225 0.190112 0.981762i \(-0.439115\pi\)
0.190112 + 0.981762i \(0.439115\pi\)
\(264\) 0 0
\(265\) 17.8738i 1.09798i
\(266\) 0 0
\(267\) 3.93181i 0.240623i
\(268\) 0 0
\(269\) 14.0961 0.859455 0.429728 0.902959i \(-0.358610\pi\)
0.429728 + 0.902959i \(0.358610\pi\)
\(270\) 0 0
\(271\) 28.2242i 1.71450i 0.514901 + 0.857249i \(0.327829\pi\)
−0.514901 + 0.857249i \(0.672171\pi\)
\(272\) 0 0
\(273\) −2.59494 + 0.351231i −0.157053 + 0.0212575i
\(274\) 0 0
\(275\) 3.78231i 0.228082i
\(276\) 0 0
\(277\) 6.01644 0.361493 0.180746 0.983530i \(-0.442149\pi\)
0.180746 + 0.983530i \(0.442149\pi\)
\(278\) 0 0
\(279\) 13.9336i 0.834182i
\(280\) 0 0
\(281\) 19.4529i 1.16046i 0.814451 + 0.580232i \(0.197038\pi\)
−0.814451 + 0.580232i \(0.802962\pi\)
\(282\) 0 0
\(283\) −23.4758 −1.39549 −0.697747 0.716344i \(-0.745814\pi\)
−0.697747 + 0.716344i \(0.745814\pi\)
\(284\) 0 0
\(285\) −13.6143 −0.806442
\(286\) 0 0
\(287\) 7.75835 0.457961
\(288\) 0 0
\(289\) 7.20330 0.423723
\(290\) 0 0
\(291\) 6.71081i 0.393395i
\(292\) 0 0
\(293\) 1.05645i 0.0617185i −0.999524 0.0308592i \(-0.990176\pi\)
0.999524 0.0308592i \(-0.00982436\pi\)
\(294\) 0 0
\(295\) −12.8269 −0.746813
\(296\) 0 0
\(297\) 3.97454i 0.230626i
\(298\) 0 0
\(299\) −0.369475 2.72974i −0.0213673 0.157865i
\(300\) 0 0
\(301\) 1.96634i 0.113338i
\(302\) 0 0
\(303\) −1.61716 −0.0929033
\(304\) 0 0
\(305\) 21.1943i 1.21358i
\(306\) 0 0
\(307\) 0.356927i 0.0203709i −0.999948 0.0101854i \(-0.996758\pi\)
0.999948 0.0101854i \(-0.00324218\pi\)
\(308\) 0 0
\(309\) 4.98909 0.283819
\(310\) 0 0
\(311\) 33.0350 1.87324 0.936621 0.350343i \(-0.113935\pi\)
0.936621 + 0.350343i \(0.113935\pi\)
\(312\) 0 0
\(313\) 6.66203 0.376560 0.188280 0.982115i \(-0.439709\pi\)
0.188280 + 0.982115i \(0.439709\pi\)
\(314\) 0 0
\(315\) −7.32734 −0.412849
\(316\) 0 0
\(317\) 29.1498i 1.63722i −0.574351 0.818609i \(-0.694745\pi\)
0.574351 0.818609i \(-0.305255\pi\)
\(318\) 0 0
\(319\) 6.71850i 0.376164i
\(320\) 0 0
\(321\) 10.5881 0.590968
\(322\) 0 0
\(323\) 31.1194i 1.73153i
\(324\) 0 0
\(325\) 1.82916 + 13.5141i 0.101463 + 0.749626i
\(326\) 0 0
\(327\) 7.81248i 0.432031i
\(328\) 0 0
\(329\) 9.93798 0.547899
\(330\) 0 0
\(331\) 19.2592i 1.05858i 0.848440 + 0.529292i \(0.177542\pi\)
−0.848440 + 0.529292i \(0.822458\pi\)
\(332\) 0 0
\(333\) 17.5491i 0.961685i
\(334\) 0 0
\(335\) 25.1573 1.37449
\(336\) 0 0
\(337\) 32.1061 1.74893 0.874465 0.485089i \(-0.161213\pi\)
0.874465 + 0.485089i \(0.161213\pi\)
\(338\) 0 0
\(339\) −6.59698 −0.358299
\(340\) 0 0
\(341\) −5.63535 −0.305171
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 1.64435i 0.0885286i
\(346\) 0 0
\(347\) 13.0697 0.701621 0.350810 0.936447i \(-0.385906\pi\)
0.350810 + 0.936447i \(0.385906\pi\)
\(348\) 0 0
\(349\) 31.7152i 1.69768i 0.528652 + 0.848839i \(0.322698\pi\)
−0.528652 + 0.848839i \(0.677302\pi\)
\(350\) 0 0
\(351\) 1.92212 + 14.2009i 0.102595 + 0.757988i
\(352\) 0 0
\(353\) 18.5166i 0.985537i 0.870160 + 0.492769i \(0.164015\pi\)
−0.870160 + 0.492769i \(0.835985\pi\)
\(354\) 0 0
\(355\) −25.1378 −1.33418
\(356\) 0 0
\(357\) 3.57302i 0.189104i
\(358\) 0 0
\(359\) 24.8442i 1.31123i 0.755096 + 0.655614i \(0.227590\pi\)
−0.755096 + 0.655614i \(0.772410\pi\)
\(360\) 0 0
\(361\) −21.0117 −1.10588
\(362\) 0 0
\(363\) −0.726270 −0.0381193
\(364\) 0 0
\(365\) 1.29545 0.0678071
\(366\) 0 0
\(367\) 16.1723 0.844187 0.422093 0.906552i \(-0.361295\pi\)
0.422093 + 0.906552i \(0.361295\pi\)
\(368\) 0 0
\(369\) 19.1828i 0.998615i
\(370\) 0 0
\(371\) 6.03131i 0.313130i
\(372\) 0 0
\(373\) 29.3452 1.51944 0.759718 0.650253i \(-0.225337\pi\)
0.759718 + 0.650253i \(0.225337\pi\)
\(374\) 0 0
\(375\) 2.62084i 0.135340i
\(376\) 0 0
\(377\) −3.24913 24.0050i −0.167339 1.23632i
\(378\) 0 0
\(379\) 13.9422i 0.716165i 0.933690 + 0.358083i \(0.116569\pi\)
−0.933690 + 0.358083i \(0.883431\pi\)
\(380\) 0 0
\(381\) 4.49116 0.230089
\(382\) 0 0
\(383\) 9.38159i 0.479377i −0.970850 0.239688i \(-0.922955\pi\)
0.970850 0.239688i \(-0.0770453\pi\)
\(384\) 0 0
\(385\) 2.96350i 0.151034i
\(386\) 0 0
\(387\) −4.86184 −0.247141
\(388\) 0 0
\(389\) 30.5581 1.54936 0.774679 0.632355i \(-0.217911\pi\)
0.774679 + 0.632355i \(0.217911\pi\)
\(390\) 0 0
\(391\) 3.75862 0.190082
\(392\) 0 0
\(393\) −5.42457 −0.273633
\(394\) 0 0
\(395\) 8.65116i 0.435287i
\(396\) 0 0
\(397\) 3.84234i 0.192841i −0.995341 0.0964207i \(-0.969261\pi\)
0.995341 0.0964207i \(-0.0307394\pi\)
\(398\) 0 0
\(399\) 4.59400 0.229988
\(400\) 0 0
\(401\) 5.75461i 0.287371i 0.989623 + 0.143686i \(0.0458954\pi\)
−0.989623 + 0.143686i \(0.954105\pi\)
\(402\) 0 0
\(403\) −20.1350 + 2.72531i −1.00299 + 0.135757i
\(404\) 0 0
\(405\) 13.4276i 0.667224i
\(406\) 0 0
\(407\) 7.09762 0.351816
\(408\) 0 0
\(409\) 30.5803i 1.51210i 0.654515 + 0.756049i \(0.272873\pi\)
−0.654515 + 0.756049i \(0.727127\pi\)
\(410\) 0 0
\(411\) 5.24172i 0.258555i
\(412\) 0 0
\(413\) 4.32831 0.212982
\(414\) 0 0
\(415\) −48.0158 −2.35700
\(416\) 0 0
\(417\) −1.91822 −0.0939355
\(418\) 0 0
\(419\) −39.9466 −1.95152 −0.975759 0.218847i \(-0.929770\pi\)
−0.975759 + 0.218847i \(0.929770\pi\)
\(420\) 0 0
\(421\) 30.3486i 1.47910i −0.673101 0.739551i \(-0.735038\pi\)
0.673101 0.739551i \(-0.264962\pi\)
\(422\) 0 0
\(423\) 24.5720i 1.19473i
\(424\) 0 0
\(425\) −18.6078 −0.902609
\(426\) 0 0
\(427\) 7.15178i 0.346099i
\(428\) 0 0
\(429\) −2.59494 + 0.351231i −0.125285 + 0.0169576i
\(430\) 0 0
\(431\) 35.6955i 1.71939i −0.510806 0.859696i \(-0.670653\pi\)
0.510806 0.859696i \(-0.329347\pi\)
\(432\) 0 0
\(433\) 36.3314 1.74598 0.872989 0.487740i \(-0.162179\pi\)
0.872989 + 0.487740i \(0.162179\pi\)
\(434\) 0 0
\(435\) 14.4602i 0.693314i
\(436\) 0 0
\(437\) 4.83264i 0.231176i
\(438\) 0 0
\(439\) −19.8340 −0.946626 −0.473313 0.880894i \(-0.656942\pi\)
−0.473313 + 0.880894i \(0.656942\pi\)
\(440\) 0 0
\(441\) 2.47253 0.117740
\(442\) 0 0
\(443\) 9.64529 0.458262 0.229131 0.973396i \(-0.426412\pi\)
0.229131 + 0.973396i \(0.426412\pi\)
\(444\) 0 0
\(445\) −16.0435 −0.760534
\(446\) 0 0
\(447\) 7.38173i 0.349144i
\(448\) 0 0
\(449\) 26.7295i 1.26144i −0.776009 0.630722i \(-0.782759\pi\)
0.776009 0.630722i \(-0.217241\pi\)
\(450\) 0 0
\(451\) 7.75835 0.365326
\(452\) 0 0
\(453\) 7.43068i 0.349124i
\(454\) 0 0
\(455\) −1.43317 10.5885i −0.0671882 0.496396i
\(456\) 0 0
\(457\) 41.1316i 1.92406i 0.272950 + 0.962028i \(0.412001\pi\)
−0.272950 + 0.962028i \(0.587999\pi\)
\(458\) 0 0
\(459\) −19.5535 −0.912677
\(460\) 0 0
\(461\) 12.7142i 0.592158i −0.955163 0.296079i \(-0.904321\pi\)
0.955163 0.296079i \(-0.0956792\pi\)
\(462\) 0 0
\(463\) 9.55527i 0.444071i 0.975039 + 0.222036i \(0.0712701\pi\)
−0.975039 + 0.222036i \(0.928730\pi\)
\(464\) 0 0
\(465\) 12.1290 0.562467
\(466\) 0 0
\(467\) 13.9493 0.645495 0.322748 0.946485i \(-0.395393\pi\)
0.322748 + 0.946485i \(0.395393\pi\)
\(468\) 0 0
\(469\) −8.48905 −0.391988
\(470\) 0 0
\(471\) −7.02270 −0.323589
\(472\) 0 0
\(473\) 1.96634i 0.0904124i
\(474\) 0 0
\(475\) 23.9249i 1.09775i
\(476\) 0 0
\(477\) 14.9126 0.682802
\(478\) 0 0
\(479\) 18.8755i 0.862446i −0.902245 0.431223i \(-0.858082\pi\)
0.902245 0.431223i \(-0.141918\pi\)
\(480\) 0 0
\(481\) 25.3596 3.43248i 1.15630 0.156507i
\(482\) 0 0
\(483\) 0.554867i 0.0252473i
\(484\) 0 0
\(485\) 27.3830 1.24340
\(486\) 0 0
\(487\) 28.8145i 1.30571i −0.757483 0.652855i \(-0.773571\pi\)
0.757483 0.652855i \(-0.226429\pi\)
\(488\) 0 0
\(489\) 2.14225i 0.0968758i
\(490\) 0 0
\(491\) −21.5351 −0.971865 −0.485932 0.873996i \(-0.661520\pi\)
−0.485932 + 0.873996i \(0.661520\pi\)
\(492\) 0 0
\(493\) 33.0529 1.48863
\(494\) 0 0
\(495\) −7.32734 −0.329339
\(496\) 0 0
\(497\) 8.48248 0.380491
\(498\) 0 0
\(499\) 25.9872i 1.16335i 0.813423 + 0.581673i \(0.197602\pi\)
−0.813423 + 0.581673i \(0.802398\pi\)
\(500\) 0 0
\(501\) 12.1240i 0.541659i
\(502\) 0 0
\(503\) 39.5711 1.76439 0.882193 0.470887i \(-0.156066\pi\)
0.882193 + 0.470887i \(0.156066\pi\)
\(504\) 0 0
\(505\) 6.59871i 0.293639i
\(506\) 0 0
\(507\) −9.10179 + 2.50988i −0.404225 + 0.111467i
\(508\) 0 0
\(509\) 16.2151i 0.718721i −0.933199 0.359360i \(-0.882995\pi\)
0.933199 0.359360i \(-0.117005\pi\)
\(510\) 0 0
\(511\) −0.437137 −0.0193378
\(512\) 0 0
\(513\) 25.1408i 1.10999i
\(514\) 0 0
\(515\) 20.3576i 0.897064i
\(516\) 0 0
\(517\) 9.93798 0.437072
\(518\) 0 0
\(519\) −11.1070 −0.487544
\(520\) 0 0
\(521\) 26.8071 1.17444 0.587220 0.809427i \(-0.300222\pi\)
0.587220 + 0.809427i \(0.300222\pi\)
\(522\) 0 0
\(523\) −0.569536 −0.0249041 −0.0124520 0.999922i \(-0.503964\pi\)
−0.0124520 + 0.999922i \(0.503964\pi\)
\(524\) 0 0
\(525\) 2.74697i 0.119888i
\(526\) 0 0
\(527\) 27.7242i 1.20768i
\(528\) 0 0
\(529\) −22.4163 −0.974622
\(530\) 0 0
\(531\) 10.7019i 0.464422i
\(532\) 0 0
\(533\) 27.7204 3.75201i 1.20070 0.162518i
\(534\) 0 0
\(535\) 43.2039i 1.86787i
\(536\) 0 0
\(537\) −1.08814 −0.0469568
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 24.1535i 1.03844i −0.854641 0.519220i \(-0.826223\pi\)
0.854641 0.519220i \(-0.173777\pi\)
\(542\) 0 0
\(543\) −15.2975 −0.656479
\(544\) 0 0
\(545\) −31.8783 −1.36552
\(546\) 0 0
\(547\) −7.19909 −0.307811 −0.153905 0.988086i \(-0.549185\pi\)
−0.153905 + 0.988086i \(0.549185\pi\)
\(548\) 0 0
\(549\) 17.6830 0.754692
\(550\) 0 0
\(551\) 42.4977i 1.81046i
\(552\) 0 0
\(553\) 2.91924i 0.124139i
\(554\) 0 0
\(555\) −15.2762 −0.648439
\(556\) 0 0
\(557\) 17.8131i 0.754766i −0.926057 0.377383i \(-0.876824\pi\)
0.926057 0.377383i \(-0.123176\pi\)
\(558\) 0 0
\(559\) −0.950940 7.02567i −0.0402205 0.297154i
\(560\) 0 0
\(561\) 3.57302i 0.150853i
\(562\) 0 0
\(563\) −11.6963 −0.492941 −0.246471 0.969150i \(-0.579271\pi\)
−0.246471 + 0.969150i \(0.579271\pi\)
\(564\) 0 0
\(565\) 26.9185i 1.13247i
\(566\) 0 0
\(567\) 4.53101i 0.190285i
\(568\) 0 0
\(569\) −35.4110 −1.48451 −0.742253 0.670120i \(-0.766243\pi\)
−0.742253 + 0.670120i \(0.766243\pi\)
\(570\) 0 0
\(571\) −11.2151 −0.469336 −0.234668 0.972076i \(-0.575400\pi\)
−0.234668 + 0.972076i \(0.575400\pi\)
\(572\) 0 0
\(573\) 4.66569 0.194912
\(574\) 0 0
\(575\) −2.88967 −0.120507
\(576\) 0 0
\(577\) 21.0166i 0.874932i −0.899235 0.437466i \(-0.855876\pi\)
0.899235 0.437466i \(-0.144124\pi\)
\(578\) 0 0
\(579\) 2.20685i 0.0917135i
\(580\) 0 0
\(581\) 16.2024 0.672190
\(582\) 0 0
\(583\) 6.03131i 0.249791i
\(584\) 0 0
\(585\) −26.1804 + 3.54357i −1.08242 + 0.146509i
\(586\) 0 0
\(587\) 22.3533i 0.922619i −0.887239 0.461309i \(-0.847380\pi\)
0.887239 0.461309i \(-0.152620\pi\)
\(588\) 0 0
\(589\) 35.6463 1.46878
\(590\) 0 0
\(591\) 12.3353i 0.507405i
\(592\) 0 0
\(593\) 13.4125i 0.550786i −0.961332 0.275393i \(-0.911192\pi\)
0.961332 0.275393i \(-0.0888080\pi\)
\(594\) 0 0
\(595\) 14.5795 0.597700
\(596\) 0 0
\(597\) −0.248762 −0.0101812
\(598\) 0 0
\(599\) −18.4283 −0.752962 −0.376481 0.926424i \(-0.622866\pi\)
−0.376481 + 0.926424i \(0.622866\pi\)
\(600\) 0 0
\(601\) 18.4793 0.753786 0.376893 0.926257i \(-0.376992\pi\)
0.376893 + 0.926257i \(0.376992\pi\)
\(602\) 0 0
\(603\) 20.9894i 0.854756i
\(604\) 0 0
\(605\) 2.96350i 0.120483i
\(606\) 0 0
\(607\) −8.05677 −0.327014 −0.163507 0.986542i \(-0.552281\pi\)
−0.163507 + 0.986542i \(0.552281\pi\)
\(608\) 0 0
\(609\) 4.87945i 0.197725i
\(610\) 0 0
\(611\) 35.5081 4.80610i 1.43650 0.194434i
\(612\) 0 0
\(613\) 17.0743i 0.689626i 0.938671 + 0.344813i \(0.112058\pi\)
−0.938671 + 0.344813i \(0.887942\pi\)
\(614\) 0 0
\(615\) −16.6983 −0.673339
\(616\) 0 0
\(617\) 36.2886i 1.46092i −0.682953 0.730462i \(-0.739305\pi\)
0.682953 0.730462i \(-0.260695\pi\)
\(618\) 0 0
\(619\) 25.1271i 1.00994i 0.863136 + 0.504972i \(0.168497\pi\)
−0.863136 + 0.504972i \(0.831503\pi\)
\(620\) 0 0
\(621\) −3.03653 −0.121852
\(622\) 0 0
\(623\) 5.41370 0.216895
\(624\) 0 0
\(625\) −29.6057 −1.18423
\(626\) 0 0
\(627\) 4.59400 0.183467
\(628\) 0 0
\(629\) 34.9181i 1.39227i
\(630\) 0 0
\(631\) 2.66774i 0.106201i −0.998589 0.0531005i \(-0.983090\pi\)
0.998589 0.0531005i \(-0.0169104\pi\)
\(632\) 0 0
\(633\) −5.63295 −0.223890
\(634\) 0 0
\(635\) 18.3259i 0.727241i
\(636\) 0 0
\(637\) 0.483609 + 3.57297i 0.0191613 + 0.141566i
\(638\) 0 0
\(639\) 20.9732i 0.829687i
\(640\) 0 0
\(641\) −10.7926 −0.426281 −0.213141 0.977022i \(-0.568369\pi\)
−0.213141 + 0.977022i \(0.568369\pi\)
\(642\) 0 0
\(643\) 20.9566i 0.826449i 0.910629 + 0.413224i \(0.135598\pi\)
−0.910629 + 0.413224i \(0.864402\pi\)
\(644\) 0 0
\(645\) 4.23215i 0.166641i
\(646\) 0 0
\(647\) 21.2358 0.834866 0.417433 0.908708i \(-0.362930\pi\)
0.417433 + 0.908708i \(0.362930\pi\)
\(648\) 0 0
\(649\) 4.32831 0.169901
\(650\) 0 0
\(651\) −4.09279 −0.160409
\(652\) 0 0
\(653\) −16.8540 −0.659546 −0.329773 0.944060i \(-0.606972\pi\)
−0.329773 + 0.944060i \(0.606972\pi\)
\(654\) 0 0
\(655\) 22.1346i 0.864870i
\(656\) 0 0
\(657\) 1.08084i 0.0421674i
\(658\) 0 0
\(659\) 0.138324 0.00538832 0.00269416 0.999996i \(-0.499142\pi\)
0.00269416 + 0.999996i \(0.499142\pi\)
\(660\) 0 0
\(661\) 24.1473i 0.939221i −0.882874 0.469610i \(-0.844394\pi\)
0.882874 0.469610i \(-0.155606\pi\)
\(662\) 0 0
\(663\) −1.72794 12.7663i −0.0671078 0.495802i
\(664\) 0 0
\(665\) 18.7455i 0.726920i
\(666\) 0 0
\(667\) 5.13291 0.198747
\(668\) 0 0
\(669\) 2.87694i 0.111229i
\(670\) 0 0
\(671\) 7.15178i 0.276091i
\(672\) 0 0
\(673\) 42.2043 1.62685 0.813427 0.581667i \(-0.197599\pi\)
0.813427 + 0.581667i \(0.197599\pi\)
\(674\) 0 0
\(675\) 15.0329 0.578617
\(676\) 0 0
\(677\) 16.5801 0.637223 0.318612 0.947885i \(-0.396783\pi\)
0.318612 + 0.947885i \(0.396783\pi\)
\(678\) 0 0
\(679\) −9.24011 −0.354603
\(680\) 0 0
\(681\) 3.19559i 0.122455i
\(682\) 0 0
\(683\) 1.96449i 0.0751693i 0.999293 + 0.0375846i \(0.0119664\pi\)
−0.999293 + 0.0375846i \(0.988034\pi\)
\(684\) 0 0
\(685\) −21.3885 −0.817212
\(686\) 0 0
\(687\) 2.84163i 0.108415i
\(688\) 0 0
\(689\) 2.91680 + 21.5497i 0.111121 + 0.820978i
\(690\) 0 0
\(691\) 41.2966i 1.57100i 0.618865 + 0.785498i \(0.287593\pi\)
−0.618865 + 0.785498i \(0.712407\pi\)
\(692\) 0 0
\(693\) 2.47253 0.0939237
\(694\) 0 0
\(695\) 7.82716i 0.296901i
\(696\) 0 0
\(697\) 38.1686i 1.44574i
\(698\) 0 0
\(699\) −9.35920 −0.353998
\(700\) 0 0
\(701\) −17.2844 −0.652821 −0.326411 0.945228i \(-0.605839\pi\)
−0.326411 + 0.945228i \(0.605839\pi\)
\(702\) 0 0
\(703\) −44.8959 −1.69328
\(704\) 0 0
\(705\) −21.3895 −0.805575
\(706\) 0 0
\(707\) 2.22666i 0.0837423i
\(708\) 0 0
\(709\) 45.9143i 1.72435i −0.506613 0.862173i \(-0.669103\pi\)
0.506613 0.862173i \(-0.330897\pi\)
\(710\) 0 0
\(711\) −7.21792 −0.270693
\(712\) 0 0
\(713\) 4.30539i 0.161238i
\(714\) 0 0
\(715\) −1.43317 10.5885i −0.0535977 0.395987i
\(716\) 0 0
\(717\) 18.9825i 0.708916i
\(718\) 0 0
\(719\) 38.5039 1.43595 0.717976 0.696068i \(-0.245069\pi\)
0.717976 + 0.696068i \(0.245069\pi\)
\(720\) 0 0
\(721\) 6.86947i 0.255832i
\(722\) 0 0
\(723\) 14.9142i 0.554667i
\(724\) 0 0
\(725\) −25.4114 −0.943757
\(726\) 0 0
\(727\) 15.2100 0.564108 0.282054 0.959399i \(-0.408984\pi\)
0.282054 + 0.959399i \(0.408984\pi\)
\(728\) 0 0
\(729\) −2.54331 −0.0941968
\(730\) 0 0
\(731\) 9.67377 0.357797
\(732\) 0 0
\(733\) 43.5219i 1.60752i −0.594955 0.803759i \(-0.702830\pi\)
0.594955 0.803759i \(-0.297170\pi\)
\(734\) 0 0
\(735\) 2.15230i 0.0793887i
\(736\) 0 0
\(737\) −8.48905 −0.312698
\(738\) 0 0
\(739\) 18.2929i 0.672916i −0.941698 0.336458i \(-0.890771\pi\)
0.941698 0.336458i \(-0.109229\pi\)
\(740\) 0 0
\(741\) 16.4142 2.22170i 0.602992 0.0816163i
\(742\) 0 0
\(743\) 6.14492i 0.225435i 0.993627 + 0.112718i \(0.0359556\pi\)
−0.993627 + 0.112718i \(0.964044\pi\)
\(744\) 0 0
\(745\) 30.1207 1.10354
\(746\) 0 0
\(747\) 40.0610i 1.46576i
\(748\) 0 0
\(749\) 14.5787i 0.532694i
\(750\) 0 0
\(751\) 20.0322 0.730986 0.365493 0.930814i \(-0.380900\pi\)
0.365493 + 0.930814i \(0.380900\pi\)
\(752\) 0 0
\(753\) 19.7552 0.719919
\(754\) 0 0
\(755\) −30.3204 −1.10347
\(756\) 0 0
\(757\) −36.4448 −1.32461 −0.662305 0.749234i \(-0.730422\pi\)
−0.662305 + 0.749234i \(0.730422\pi\)
\(758\) 0 0
\(759\) 0.554867i 0.0201404i
\(760\) 0 0
\(761\) 17.5763i 0.637142i −0.947899 0.318571i \(-0.896797\pi\)
0.947899 0.318571i \(-0.103203\pi\)
\(762\) 0 0
\(763\) 10.7570 0.389429
\(764\) 0 0
\(765\) 36.0482i 1.30333i
\(766\) 0 0
\(767\) 15.4649 2.09321i 0.558406 0.0755815i
\(768\) 0 0
\(769\) 18.7179i 0.674984i −0.941328 0.337492i \(-0.890421\pi\)
0.941328 0.337492i \(-0.109579\pi\)
\(770\) 0 0
\(771\) 7.97827 0.287330
\(772\) 0 0
\(773\) 8.15430i 0.293290i 0.989189 + 0.146645i \(0.0468475\pi\)
−0.989189 + 0.146645i \(0.953153\pi\)
\(774\) 0 0
\(775\) 21.3146i 0.765644i
\(776\) 0 0
\(777\) 5.15479 0.184927
\(778\) 0 0
\(779\) −49.0752 −1.75830
\(780\) 0 0
\(781\) 8.48248 0.303527
\(782\) 0 0
\(783\) −26.7029 −0.954285
\(784\) 0 0
\(785\) 28.6556i 1.02276i
\(786\) 0 0
\(787\) 45.0189i 1.60475i −0.596820 0.802375i \(-0.703569\pi\)
0.596820 0.802375i \(-0.296431\pi\)
\(788\) 0 0
\(789\) 4.47833 0.159433
\(790\) 0 0
\(791\) 9.08338i 0.322968i
\(792\) 0 0
\(793\) 3.45867 + 25.5531i 0.122821 + 0.907418i
\(794\) 0 0
\(795\) 12.9812i 0.460395i
\(796\) 0 0
\(797\) −21.6870 −0.768194 −0.384097 0.923293i \(-0.625487\pi\)
−0.384097 + 0.923293i \(0.625487\pi\)
\(798\) 0 0
\(799\) 48.8917i 1.72966i
\(800\) 0 0
\(801\) 13.3855i 0.472955i
\(802\) 0 0
\(803\) −0.437137 −0.0154262
\(804\) 0 0
\(805\) 2.26410 0.0797990
\(806\) 0 0
\(807\) 10.2376 0.360380
\(808\) 0 0
\(809\) −42.2729 −1.48623 −0.743117 0.669161i \(-0.766654\pi\)
−0.743117 + 0.669161i \(0.766654\pi\)
\(810\) 0 0
\(811\) 17.4347i 0.612215i −0.951997 0.306107i \(-0.900973\pi\)
0.951997 0.306107i \(-0.0990266\pi\)
\(812\) 0 0
\(813\) 20.4984i 0.718910i
\(814\) 0 0
\(815\) −8.74130 −0.306194
\(816\) 0 0
\(817\) 12.4380i 0.435152i
\(818\) 0 0
\(819\) 8.83429 1.19574i 0.308695 0.0417825i
\(820\) 0 0
\(821\) 0.272128i 0.00949734i −0.999989 0.00474867i \(-0.998488\pi\)
0.999989 0.00474867i \(-0.00151155\pi\)
\(822\) 0 0
\(823\) 40.1727 1.40033 0.700167 0.713979i \(-0.253109\pi\)
0.700167 + 0.713979i \(0.253109\pi\)
\(824\) 0 0
\(825\) 2.74697i 0.0956374i
\(826\) 0 0
\(827\) 4.21908i 0.146712i 0.997306 + 0.0733559i \(0.0233709\pi\)
−0.997306 + 0.0733559i \(0.976629\pi\)
\(828\) 0 0
\(829\) 26.2263 0.910878 0.455439 0.890267i \(-0.349482\pi\)
0.455439 + 0.890267i \(0.349482\pi\)
\(830\) 0 0
\(831\) 4.36956 0.151578
\(832\) 0 0
\(833\) −4.91968 −0.170457
\(834\) 0 0
\(835\) −49.4711 −1.71202
\(836\) 0 0
\(837\) 22.3979i 0.774185i
\(838\) 0 0
\(839\) 23.1046i 0.797661i −0.917025 0.398830i \(-0.869416\pi\)
0.917025 0.398830i \(-0.130584\pi\)
\(840\) 0 0
\(841\) 16.1383 0.556493
\(842\) 0 0
\(843\) 14.1281i 0.486597i
\(844\) 0 0
\(845\) −10.2414 37.1392i −0.352314 1.27763i
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) 0 0
\(849\) −17.0498 −0.585147
\(850\) 0 0
\(851\) 5.42256i 0.185883i
\(852\) 0 0
\(853\) 10.1300i 0.346845i 0.984848 + 0.173423i \(0.0554826\pi\)
−0.984848 + 0.173423i \(0.944517\pi\)
\(854\) 0 0
\(855\) 46.3489 1.58510
\(856\) 0 0
\(857\) 25.6379 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(858\) 0 0
\(859\) −37.0070 −1.26266 −0.631331 0.775514i \(-0.717491\pi\)
−0.631331 + 0.775514i \(0.717491\pi\)
\(860\) 0 0
\(861\) 5.63465 0.192029
\(862\) 0 0
\(863\) 26.3294i 0.896262i 0.893968 + 0.448131i \(0.147910\pi\)
−0.893968 + 0.448131i \(0.852090\pi\)
\(864\) 0 0
\(865\) 45.3214i 1.54097i
\(866\) 0 0
\(867\) 5.23154 0.177672
\(868\) 0 0
\(869\) 2.91924i 0.0990285i
\(870\) 0 0
\(871\) −30.3311 + 4.10538i −1.02773 + 0.139106i
\(872\) 0 0
\(873\) 22.8465i 0.773236i
\(874\) 0 0
\(875\) 3.60863 0.121994
\(876\) 0 0
\(877\) 1.30338i 0.0440121i −0.999758 0.0220061i \(-0.992995\pi\)
0.999758 0.0220061i \(-0.00700531\pi\)
\(878\) 0 0
\(879\) 0.767268i 0.0258793i
\(880\) 0 0
\(881\) −19.9213 −0.671167 −0.335583 0.942010i \(-0.608933\pi\)
−0.335583 + 0.942010i \(0.608933\pi\)
\(882\) 0 0
\(883\) 14.7982 0.497999 0.248999 0.968504i \(-0.419898\pi\)
0.248999 + 0.968504i \(0.419898\pi\)
\(884\) 0 0
\(885\) −9.31581 −0.313148
\(886\) 0 0
\(887\) −43.8158 −1.47119 −0.735595 0.677421i \(-0.763097\pi\)
−0.735595 + 0.677421i \(0.763097\pi\)
\(888\) 0 0
\(889\) 6.18388i 0.207401i
\(890\) 0 0
\(891\) 4.53101i 0.151795i
\(892\) 0 0
\(893\) −62.8625 −2.10361
\(894\) 0 0
\(895\) 4.44009i 0.148416i
\(896\) 0 0
\(897\) −0.268339 1.98252i −0.00895958 0.0661946i
\(898\) 0 0
\(899\) 37.8611i 1.26274i
\(900\) 0 0
\(901\) −29.6721 −0.988523
\(902\) 0 0
\(903\) 1.42809i 0.0475239i
\(904\) 0 0
\(905\) 62.4205i 2.07493i
\(906\) 0 0
\(907\) −47.2132 −1.56769 −0.783844 0.620958i \(-0.786744\pi\)
−0.783844 + 0.620958i \(0.786744\pi\)
\(908\) 0 0
\(909\) 5.50550 0.182606
\(910\) 0 0
\(911\) −12.6688 −0.419735 −0.209867 0.977730i \(-0.567303\pi\)
−0.209867 + 0.977730i \(0.567303\pi\)
\(912\) 0 0
\(913\) 16.2024 0.536222
\(914\) 0 0
\(915\) 15.3928i 0.508869i
\(916\) 0 0
\(917\) 7.46908i 0.246651i
\(918\) 0 0
\(919\) 53.6094 1.76841 0.884206 0.467097i \(-0.154700\pi\)
0.884206 + 0.467097i \(0.154700\pi\)
\(920\) 0 0
\(921\) 0.259225i 0.00854176i
\(922\) 0 0
\(923\) 30.3076 4.10221i 0.997588 0.135026i
\(924\) 0 0
\(925\) 26.8454i 0.882671i
\(926\) 0 0
\(927\) −16.9850 −0.557860
\(928\) 0 0
\(929\) 52.0062i 1.70627i 0.521692 + 0.853134i \(0.325301\pi\)
−0.521692 + 0.853134i \(0.674699\pi\)
\(930\) 0 0
\(931\) 6.32548i 0.207309i
\(932\) 0 0
\(933\) 23.9923 0.785473
\(934\) 0 0
\(935\) 14.5795 0.476799
\(936\) 0 0
\(937\) −26.8918 −0.878517 −0.439259 0.898361i \(-0.644759\pi\)
−0.439259 + 0.898361i \(0.644759\pi\)
\(938\) 0 0
\(939\) 4.83843 0.157896
\(940\) 0 0
\(941\) 15.8480i 0.516630i −0.966061 0.258315i \(-0.916833\pi\)
0.966061 0.258315i \(-0.0831672\pi\)
\(942\) 0 0
\(943\) 5.92735i 0.193021i
\(944\) 0 0
\(945\) −11.7785 −0.383155
\(946\) 0 0
\(947\) 2.83310i 0.0920634i −0.998940 0.0460317i \(-0.985342\pi\)
0.998940 0.0460317i \(-0.0146575\pi\)
\(948\) 0 0
\(949\) −1.56188 + 0.211403i −0.0507007 + 0.00686245i
\(950\) 0 0
\(951\) 21.1707i 0.686506i
\(952\) 0 0
\(953\) 7.26334 0.235283 0.117641 0.993056i \(-0.462467\pi\)
0.117641 + 0.993056i \(0.462467\pi\)
\(954\) 0 0
\(955\) 19.0380i 0.616056i
\(956\) 0 0
\(957\) 4.87945i 0.157730i
\(958\) 0 0
\(959\) 7.21732 0.233059
\(960\) 0 0
\(961\) −0.757201 −0.0244258
\(962\) 0 0
\(963\) −36.0463 −1.16158
\(964\) 0 0
\(965\) 9.00490 0.289878
\(966\) 0 0
\(967\) 35.4836i 1.14108i −0.821271 0.570539i \(-0.806734\pi\)
0.821271 0.570539i \(-0.193266\pi\)
\(968\) 0 0
\(969\) 22.6010i 0.726050i
\(970\) 0 0
\(971\) −48.5557 −1.55823 −0.779113 0.626883i \(-0.784330\pi\)
−0.779113 + 0.626883i \(0.784330\pi\)
\(972\) 0 0
\(973\) 2.64119i 0.0846727i
\(974\) 0 0
\(975\) 1.32846 + 9.81486i 0.0425448 + 0.314327i
\(976\) 0 0
\(977\) 20.9442i 0.670066i −0.942206 0.335033i \(-0.891253\pi\)
0.942206 0.335033i \(-0.108747\pi\)
\(978\) 0 0
\(979\) 5.41370 0.173023
\(980\) 0 0
\(981\) 26.5970i 0.849177i
\(982\) 0 0
\(983\) 9.13085i 0.291229i 0.989341 + 0.145615i \(0.0465159\pi\)
−0.989341 + 0.145615i \(0.953484\pi\)
\(984\) 0 0
\(985\) 50.3332 1.60375
\(986\) 0 0
\(987\) 7.21766 0.229741
\(988\) 0 0
\(989\) 1.50227 0.0477696
\(990\) 0 0
\(991\) −22.9738 −0.729788 −0.364894 0.931049i \(-0.618895\pi\)
−0.364894 + 0.931049i \(0.618895\pi\)
\(992\) 0 0
\(993\) 13.9874i 0.443877i
\(994\) 0 0
\(995\) 1.01506i 0.0321795i
\(996\) 0 0
\(997\) −59.3596 −1.87994 −0.939969 0.341259i \(-0.889147\pi\)
−0.939969 + 0.341259i \(0.889147\pi\)
\(998\) 0 0
\(999\) 28.2098i 0.892517i
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 4004.2.m.b.2157.20 yes 30
13.12 even 2 inner 4004.2.m.b.2157.19 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.m.b.2157.19 30 13.12 even 2 inner
4004.2.m.b.2157.20 yes 30 1.1 even 1 trivial