Properties

Label 189.2.c.b.188.2
Level $189$
Weight $2$
Character 189.188
Analytic conductor $1.509$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,2,Mod(188,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.c (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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 188.2
Root \(-1.93649 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 189.188
Dual form 189.2.c.b.188.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{2} -3.00000 q^{4} +3.87298 q^{5} +(-2.00000 - 1.73205i) q^{7} +2.23607i q^{8} +O(q^{10})\) \(q-2.23607i q^{2} -3.00000 q^{4} +3.87298 q^{5} +(-2.00000 - 1.73205i) q^{7} +2.23607i q^{8} -8.66025i q^{10} -2.23607i q^{11} +3.46410i q^{13} +(-3.87298 + 4.47214i) q^{14} -1.00000 q^{16} +5.19615i q^{19} -11.6190 q^{20} -5.00000 q^{22} -2.23607i q^{23} +10.0000 q^{25} +7.74597 q^{26} +(6.00000 + 5.19615i) q^{28} +4.47214i q^{29} -1.73205i q^{31} +6.70820i q^{32} +(-7.74597 - 6.70820i) q^{35} -1.00000 q^{37} +11.6190 q^{38} +8.66025i q^{40} +3.87298 q^{41} +2.00000 q^{43} +6.70820i q^{44} -5.00000 q^{46} +7.74597 q^{47} +(1.00000 + 6.92820i) q^{49} -22.3607i q^{50} -10.3923i q^{52} -8.94427i q^{53} -8.66025i q^{55} +(3.87298 - 4.47214i) q^{56} +10.0000 q^{58} -7.74597 q^{59} +6.92820i q^{61} -3.87298 q^{62} +13.0000 q^{64} +13.4164i q^{65} -10.0000 q^{67} +(-15.0000 + 17.3205i) q^{70} +11.1803i q^{71} +10.3923i q^{73} +2.23607i q^{74} -15.5885i q^{76} +(-3.87298 + 4.47214i) q^{77} +2.00000 q^{79} -3.87298 q^{80} -8.66025i q^{82} +7.74597 q^{83} -4.47214i q^{86} +5.00000 q^{88} -11.6190 q^{89} +(6.00000 - 6.92820i) q^{91} +6.70820i q^{92} -17.3205i q^{94} +20.1246i q^{95} -13.8564i q^{97} +(15.4919 - 2.23607i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 8 q^{7} - 4 q^{16} - 20 q^{22} + 40 q^{25} + 24 q^{28} - 4 q^{37} + 8 q^{43} - 20 q^{46} + 4 q^{49} + 40 q^{58} + 52 q^{64} - 40 q^{67} - 60 q^{70} + 8 q^{79} + 20 q^{88} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i −0.612372 0.790569i \(-0.709785\pi\)
0.612372 0.790569i \(-0.290215\pi\)
\(3\) 0 0
\(4\) −3.00000 −1.50000
\(5\) 3.87298 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 2.23607i 0.790569i
\(9\) 0 0
\(10\) 8.66025i 2.73861i
\(11\) 2.23607i 0.674200i −0.941469 0.337100i \(-0.890554\pi\)
0.941469 0.337100i \(-0.109446\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) −3.87298 + 4.47214i −1.03510 + 1.19523i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i 0.802955 + 0.596040i \(0.203260\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −11.6190 −2.59808
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 2.23607i 0.466252i −0.972446 0.233126i \(-0.925104\pi\)
0.972446 0.233126i \(-0.0748955\pi\)
\(24\) 0 0
\(25\) 10.0000 2.00000
\(26\) 7.74597 1.51911
\(27\) 0 0
\(28\) 6.00000 + 5.19615i 1.13389 + 0.981981i
\(29\) 4.47214i 0.830455i 0.909718 + 0.415227i \(0.136298\pi\)
−0.909718 + 0.415227i \(0.863702\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i −0.987829 0.155543i \(-0.950287\pi\)
0.987829 0.155543i \(-0.0497126\pi\)
\(32\) 6.70820i 1.18585i
\(33\) 0 0
\(34\) 0 0
\(35\) −7.74597 6.70820i −1.30931 1.13389i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 11.6190 1.88484
\(39\) 0 0
\(40\) 8.66025i 1.36931i
\(41\) 3.87298 0.604858 0.302429 0.953172i \(-0.402202\pi\)
0.302429 + 0.953172i \(0.402202\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 6.70820i 1.01130i
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 7.74597 1.12987 0.564933 0.825137i \(-0.308902\pi\)
0.564933 + 0.825137i \(0.308902\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 22.3607i 3.16228i
\(51\) 0 0
\(52\) 10.3923i 1.44115i
\(53\) 8.94427i 1.22859i −0.789076 0.614295i \(-0.789440\pi\)
0.789076 0.614295i \(-0.210560\pi\)
\(54\) 0 0
\(55\) 8.66025i 1.16775i
\(56\) 3.87298 4.47214i 0.517549 0.597614i
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −7.74597 −1.00844 −0.504219 0.863576i \(-0.668220\pi\)
−0.504219 + 0.863576i \(0.668220\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −3.87298 −0.491869
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 13.4164i 1.66410i
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −15.0000 + 17.3205i −1.79284 + 2.07020i
\(71\) 11.1803i 1.32686i 0.748237 + 0.663431i \(0.230900\pi\)
−0.748237 + 0.663431i \(0.769100\pi\)
\(72\) 0 0
\(73\) 10.3923i 1.21633i 0.793812 + 0.608164i \(0.208094\pi\)
−0.793812 + 0.608164i \(0.791906\pi\)
\(74\) 2.23607i 0.259938i
\(75\) 0 0
\(76\) 15.5885i 1.78812i
\(77\) −3.87298 + 4.47214i −0.441367 + 0.509647i
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) −3.87298 −0.433013
\(81\) 0 0
\(82\) 8.66025i 0.956365i
\(83\) 7.74597 0.850230 0.425115 0.905139i \(-0.360234\pi\)
0.425115 + 0.905139i \(0.360234\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.47214i 0.482243i
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) −11.6190 −1.23161 −0.615803 0.787900i \(-0.711168\pi\)
−0.615803 + 0.787900i \(0.711168\pi\)
\(90\) 0 0
\(91\) 6.00000 6.92820i 0.628971 0.726273i
\(92\) 6.70820i 0.699379i
\(93\) 0 0
\(94\) 17.3205i 1.78647i
\(95\) 20.1246i 2.06474i
\(96\) 0 0
\(97\) 13.8564i 1.40690i −0.710742 0.703452i \(-0.751641\pi\)
0.710742 0.703452i \(-0.248359\pi\)
\(98\) 15.4919 2.23607i 1.56492 0.225877i
\(99\) 0 0
\(100\) −30.0000 −3.00000
\(101\) −15.4919 −1.54150 −0.770752 0.637135i \(-0.780120\pi\)
−0.770752 + 0.637135i \(0.780120\pi\)
\(102\) 0 0
\(103\) 1.73205i 0.170664i −0.996353 0.0853320i \(-0.972805\pi\)
0.996353 0.0853320i \(-0.0271951\pi\)
\(104\) −7.74597 −0.759555
\(105\) 0 0
\(106\) −20.0000 −1.94257
\(107\) 8.94427i 0.864675i −0.901712 0.432338i \(-0.857689\pi\)
0.901712 0.432338i \(-0.142311\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −19.3649 −1.84637
\(111\) 0 0
\(112\) 2.00000 + 1.73205i 0.188982 + 0.163663i
\(113\) 8.94427i 0.841406i −0.907198 0.420703i \(-0.861783\pi\)
0.907198 0.420703i \(-0.138217\pi\)
\(114\) 0 0
\(115\) 8.66025i 0.807573i
\(116\) 13.4164i 1.24568i
\(117\) 0 0
\(118\) 17.3205i 1.59448i
\(119\) 0 0
\(120\) 0 0
\(121\) 6.00000 0.545455
\(122\) 15.4919 1.40257
\(123\) 0 0
\(124\) 5.19615i 0.466628i
\(125\) 19.3649 1.73205
\(126\) 0 0
\(127\) −10.0000 −0.887357 −0.443678 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) 15.6525i 1.38350i
\(129\) 0 0
\(130\) 30.0000 2.63117
\(131\) −7.74597 −0.676768 −0.338384 0.941008i \(-0.609880\pi\)
−0.338384 + 0.941008i \(0.609880\pi\)
\(132\) 0 0
\(133\) 9.00000 10.3923i 0.780399 0.901127i
\(134\) 22.3607i 1.93167i
\(135\) 0 0
\(136\) 0 0
\(137\) 22.3607i 1.91040i −0.295958 0.955201i \(-0.595639\pi\)
0.295958 0.955201i \(-0.404361\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i 0.989150 + 0.146911i \(0.0469330\pi\)
−0.989150 + 0.146911i \(0.953067\pi\)
\(140\) 23.2379 + 20.1246i 1.96396 + 1.70084i
\(141\) 0 0
\(142\) 25.0000 2.09795
\(143\) 7.74597 0.647750
\(144\) 0 0
\(145\) 17.3205i 1.43839i
\(146\) 23.2379 1.92318
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) 8.94427i 0.732743i −0.930469 0.366372i \(-0.880600\pi\)
0.930469 0.366372i \(-0.119400\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −11.6190 −0.942421
\(153\) 0 0
\(154\) 10.0000 + 8.66025i 0.805823 + 0.697863i
\(155\) 6.70820i 0.538816i
\(156\) 0 0
\(157\) 3.46410i 0.276465i 0.990400 + 0.138233i \(0.0441422\pi\)
−0.990400 + 0.138233i \(0.955858\pi\)
\(158\) 4.47214i 0.355784i
\(159\) 0 0
\(160\) 25.9808i 2.05396i
\(161\) −3.87298 + 4.47214i −0.305234 + 0.352454i
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −11.6190 −0.907288
\(165\) 0 0
\(166\) 17.3205i 1.34433i
\(167\) 15.4919 1.19880 0.599401 0.800449i \(-0.295406\pi\)
0.599401 + 0.800449i \(0.295406\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) −3.87298 −0.294457 −0.147229 0.989102i \(-0.547035\pi\)
−0.147229 + 0.989102i \(0.547035\pi\)
\(174\) 0 0
\(175\) −20.0000 17.3205i −1.51186 1.30931i
\(176\) 2.23607i 0.168550i
\(177\) 0 0
\(178\) 25.9808i 1.94734i
\(179\) 17.8885i 1.33705i 0.743689 + 0.668526i \(0.233075\pi\)
−0.743689 + 0.668526i \(0.766925\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) −15.4919 13.4164i −1.14834 0.994490i
\(183\) 0 0
\(184\) 5.00000 0.368605
\(185\) −3.87298 −0.284747
\(186\) 0 0
\(187\) 0 0
\(188\) −23.2379 −1.69480
\(189\) 0 0
\(190\) 45.0000 3.26464
\(191\) 2.23607i 0.161796i −0.996722 0.0808981i \(-0.974221\pi\)
0.996722 0.0808981i \(-0.0257788\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −30.9839 −2.22451
\(195\) 0 0
\(196\) −3.00000 20.7846i −0.214286 1.48461i
\(197\) 8.94427i 0.637253i −0.947880 0.318626i \(-0.896778\pi\)
0.947880 0.318626i \(-0.103222\pi\)
\(198\) 0 0
\(199\) 25.9808i 1.84173i −0.389885 0.920864i \(-0.627485\pi\)
0.389885 0.920864i \(-0.372515\pi\)
\(200\) 22.3607i 1.58114i
\(201\) 0 0
\(202\) 34.6410i 2.43733i
\(203\) 7.74597 8.94427i 0.543660 0.627765i
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) −3.87298 −0.269844
\(207\) 0 0
\(208\) 3.46410i 0.240192i
\(209\) 11.6190 0.803700
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 26.8328i 1.84289i
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) 7.74597 0.528271
\(216\) 0 0
\(217\) −3.00000 + 3.46410i −0.203653 + 0.235159i
\(218\) 15.6525i 1.06012i
\(219\) 0 0
\(220\) 25.9808i 1.75162i
\(221\) 0 0
\(222\) 0 0
\(223\) 1.73205i 0.115987i 0.998317 + 0.0579934i \(0.0184702\pi\)
−0.998317 + 0.0579934i \(0.981530\pi\)
\(224\) 11.6190 13.4164i 0.776324 0.896421i
\(225\) 0 0
\(226\) −20.0000 −1.33038
\(227\) −15.4919 −1.02824 −0.514118 0.857720i \(-0.671881\pi\)
−0.514118 + 0.857720i \(0.671881\pi\)
\(228\) 0 0
\(229\) 13.8564i 0.915657i 0.889041 + 0.457829i \(0.151373\pi\)
−0.889041 + 0.457829i \(0.848627\pi\)
\(230\) −19.3649 −1.27688
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) 17.8885i 1.17192i 0.810341 + 0.585959i \(0.199282\pi\)
−0.810341 + 0.585959i \(0.800718\pi\)
\(234\) 0 0
\(235\) 30.0000 1.95698
\(236\) 23.2379 1.51266
\(237\) 0 0
\(238\) 0 0
\(239\) 8.94427i 0.578557i −0.957245 0.289278i \(-0.906585\pi\)
0.957245 0.289278i \(-0.0934153\pi\)
\(240\) 0 0
\(241\) 17.3205i 1.11571i 0.829938 + 0.557856i \(0.188376\pi\)
−0.829938 + 0.557856i \(0.811624\pi\)
\(242\) 13.4164i 0.862439i
\(243\) 0 0
\(244\) 20.7846i 1.33060i
\(245\) 3.87298 + 26.8328i 0.247436 + 1.71429i
\(246\) 0 0
\(247\) −18.0000 −1.14531
\(248\) 3.87298 0.245935
\(249\) 0 0
\(250\) 43.3013i 2.73861i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 22.3607i 1.40303i
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) 27.1109 1.69113 0.845565 0.533872i \(-0.179264\pi\)
0.845565 + 0.533872i \(0.179264\pi\)
\(258\) 0 0
\(259\) 2.00000 + 1.73205i 0.124274 + 0.107624i
\(260\) 40.2492i 2.49615i
\(261\) 0 0
\(262\) 17.3205i 1.07006i
\(263\) 2.23607i 0.137882i −0.997621 0.0689409i \(-0.978038\pi\)
0.997621 0.0689409i \(-0.0219620\pi\)
\(264\) 0 0
\(265\) 34.6410i 2.12798i
\(266\) −23.2379 20.1246i −1.42481 1.23392i
\(267\) 0 0
\(268\) 30.0000 1.83254
\(269\) 11.6190 0.708420 0.354210 0.935166i \(-0.384750\pi\)
0.354210 + 0.935166i \(0.384750\pi\)
\(270\) 0 0
\(271\) 10.3923i 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −50.0000 −3.02061
\(275\) 22.3607i 1.34840i
\(276\) 0 0
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 7.74597 0.464572
\(279\) 0 0
\(280\) 15.0000 17.3205i 0.896421 1.03510i
\(281\) 22.3607i 1.33393i −0.745091 0.666963i \(-0.767594\pi\)
0.745091 0.666963i \(-0.232406\pi\)
\(282\) 0 0
\(283\) 24.2487i 1.44144i 0.693228 + 0.720718i \(0.256188\pi\)
−0.693228 + 0.720718i \(0.743812\pi\)
\(284\) 33.5410i 1.99029i
\(285\) 0 0
\(286\) 17.3205i 1.02418i
\(287\) −7.74597 6.70820i −0.457230 0.395973i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 38.7298 2.27429
\(291\) 0 0
\(292\) 31.1769i 1.82449i
\(293\) −30.9839 −1.81010 −0.905048 0.425309i \(-0.860166\pi\)
−0.905048 + 0.425309i \(0.860166\pi\)
\(294\) 0 0
\(295\) −30.0000 −1.74667
\(296\) 2.23607i 0.129969i
\(297\) 0 0
\(298\) −20.0000 −1.15857
\(299\) 7.74597 0.447961
\(300\) 0 0
\(301\) −4.00000 3.46410i −0.230556 0.199667i
\(302\) 35.7771i 2.05874i
\(303\) 0 0
\(304\) 5.19615i 0.298020i
\(305\) 26.8328i 1.53644i
\(306\) 0 0
\(307\) 5.19615i 0.296560i 0.988945 + 0.148280i \(0.0473737\pi\)
−0.988945 + 0.148280i \(0.952626\pi\)
\(308\) 11.6190 13.4164i 0.662051 0.764471i
\(309\) 0 0
\(310\) −15.0000 −0.851943
\(311\) −7.74597 −0.439233 −0.219617 0.975586i \(-0.570481\pi\)
−0.219617 + 0.975586i \(0.570481\pi\)
\(312\) 0 0
\(313\) 3.46410i 0.195803i −0.995196 0.0979013i \(-0.968787\pi\)
0.995196 0.0979013i \(-0.0312129\pi\)
\(314\) 7.74597 0.437130
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 4.47214i 0.251180i 0.992082 + 0.125590i \(0.0400824\pi\)
−0.992082 + 0.125590i \(0.959918\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 50.3488 2.81458
\(321\) 0 0
\(322\) 10.0000 + 8.66025i 0.557278 + 0.482617i
\(323\) 0 0
\(324\) 0 0
\(325\) 34.6410i 1.92154i
\(326\) 22.3607i 1.23844i
\(327\) 0 0
\(328\) 8.66025i 0.478183i
\(329\) −15.4919 13.4164i −0.854098 0.739671i
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −23.2379 −1.27535
\(333\) 0 0
\(334\) 34.6410i 1.89547i
\(335\) −38.7298 −2.11604
\(336\) 0 0
\(337\) 35.0000 1.90657 0.953286 0.302070i \(-0.0976776\pi\)
0.953286 + 0.302070i \(0.0976776\pi\)
\(338\) 2.23607i 0.121626i
\(339\) 0 0
\(340\) 0 0
\(341\) −3.87298 −0.209734
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 4.47214i 0.241121i
\(345\) 0 0
\(346\) 8.66025i 0.465578i
\(347\) 11.1803i 0.600192i 0.953909 + 0.300096i \(0.0970187\pi\)
−0.953909 + 0.300096i \(0.902981\pi\)
\(348\) 0 0
\(349\) 13.8564i 0.741716i −0.928689 0.370858i \(-0.879064\pi\)
0.928689 0.370858i \(-0.120936\pi\)
\(350\) −38.7298 + 44.7214i −2.07020 + 2.39046i
\(351\) 0 0
\(352\) 15.0000 0.799503
\(353\) 19.3649 1.03069 0.515345 0.856983i \(-0.327664\pi\)
0.515345 + 0.856983i \(0.327664\pi\)
\(354\) 0 0
\(355\) 43.3013i 2.29819i
\(356\) 34.8569 1.84741
\(357\) 0 0
\(358\) 40.0000 2.11407
\(359\) 8.94427i 0.472061i −0.971746 0.236030i \(-0.924154\pi\)
0.971746 0.236030i \(-0.0758465\pi\)
\(360\) 0 0
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) 0 0
\(364\) −18.0000 + 20.7846i −0.943456 + 1.08941i
\(365\) 40.2492i 2.10674i
\(366\) 0 0
\(367\) 1.73205i 0.0904123i 0.998978 + 0.0452062i \(0.0143945\pi\)
−0.998978 + 0.0452062i \(0.985606\pi\)
\(368\) 2.23607i 0.116563i
\(369\) 0 0
\(370\) 8.66025i 0.450225i
\(371\) −15.4919 + 17.8885i −0.804301 + 0.928727i
\(372\) 0 0
\(373\) 17.0000 0.880227 0.440113 0.897942i \(-0.354938\pi\)
0.440113 + 0.897942i \(0.354938\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 17.3205i 0.893237i
\(377\) −15.4919 −0.797875
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 60.3738i 3.09711i
\(381\) 0 0
\(382\) −5.00000 −0.255822
\(383\) −30.9839 −1.58320 −0.791601 0.611039i \(-0.790752\pi\)
−0.791601 + 0.611039i \(0.790752\pi\)
\(384\) 0 0
\(385\) −15.0000 + 17.3205i −0.764471 + 0.882735i
\(386\) 4.47214i 0.227626i
\(387\) 0 0
\(388\) 41.5692i 2.11036i
\(389\) 35.7771i 1.81397i −0.421163 0.906985i \(-0.638378\pi\)
0.421163 0.906985i \(-0.361622\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −15.4919 + 2.23607i −0.782461 + 0.112938i
\(393\) 0 0
\(394\) −20.0000 −1.00759
\(395\) 7.74597 0.389742
\(396\) 0 0
\(397\) 20.7846i 1.04315i −0.853206 0.521575i \(-0.825345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) −58.0948 −2.91203
\(399\) 0 0
\(400\) −10.0000 −0.500000
\(401\) 31.3050i 1.56329i 0.623721 + 0.781647i \(0.285620\pi\)
−0.623721 + 0.781647i \(0.714380\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 46.4758 2.31226
\(405\) 0 0
\(406\) −20.0000 17.3205i −0.992583 0.859602i
\(407\) 2.23607i 0.110838i
\(408\) 0 0
\(409\) 17.3205i 0.856444i −0.903674 0.428222i \(-0.859140\pi\)
0.903674 0.428222i \(-0.140860\pi\)
\(410\) 33.5410i 1.65647i
\(411\) 0 0
\(412\) 5.19615i 0.255996i
\(413\) 15.4919 + 13.4164i 0.762308 + 0.660178i
\(414\) 0 0
\(415\) 30.0000 1.47264
\(416\) −23.2379 −1.13933
\(417\) 0 0
\(418\) 25.9808i 1.27076i
\(419\) 15.4919 0.756830 0.378415 0.925636i \(-0.376469\pi\)
0.378415 + 0.925636i \(0.376469\pi\)
\(420\) 0 0
\(421\) 11.0000 0.536107 0.268054 0.963404i \(-0.413620\pi\)
0.268054 + 0.963404i \(0.413620\pi\)
\(422\) 17.8885i 0.870801i
\(423\) 0 0
\(424\) 20.0000 0.971286
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0000 13.8564i 0.580721 0.670559i
\(428\) 26.8328i 1.29701i
\(429\) 0 0
\(430\) 17.3205i 0.835269i
\(431\) 2.23607i 0.107708i −0.998549 0.0538538i \(-0.982850\pi\)
0.998549 0.0538538i \(-0.0171505\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i 0.968320 + 0.249711i \(0.0803357\pi\)
−0.968320 + 0.249711i \(0.919664\pi\)
\(434\) 7.74597 + 6.70820i 0.371818 + 0.322004i
\(435\) 0 0
\(436\) 21.0000 1.00572
\(437\) 11.6190 0.555810
\(438\) 0 0
\(439\) 38.1051i 1.81866i 0.416078 + 0.909329i \(0.363404\pi\)
−0.416078 + 0.909329i \(0.636596\pi\)
\(440\) 19.3649 0.923186
\(441\) 0 0
\(442\) 0 0
\(443\) 24.5967i 1.16863i 0.811528 + 0.584313i \(0.198636\pi\)
−0.811528 + 0.584313i \(0.801364\pi\)
\(444\) 0 0
\(445\) −45.0000 −2.13320
\(446\) 3.87298 0.183391
\(447\) 0 0
\(448\) −26.0000 22.5167i −1.22838 1.06381i
\(449\) 8.94427i 0.422106i −0.977475 0.211053i \(-0.932311\pi\)
0.977475 0.211053i \(-0.0676893\pi\)
\(450\) 0 0
\(451\) 8.66025i 0.407795i
\(452\) 26.8328i 1.26211i
\(453\) 0 0
\(454\) 34.6410i 1.62578i
\(455\) 23.2379 26.8328i 1.08941 1.25794i
\(456\) 0 0
\(457\) −25.0000 −1.16945 −0.584725 0.811231i \(-0.698798\pi\)
−0.584725 + 0.811231i \(0.698798\pi\)
\(458\) 30.9839 1.44778
\(459\) 0 0
\(460\) 25.9808i 1.21136i
\(461\) −3.87298 −0.180383 −0.0901914 0.995924i \(-0.528748\pi\)
−0.0901914 + 0.995924i \(0.528748\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 4.47214i 0.207614i
\(465\) 0 0
\(466\) 40.0000 1.85296
\(467\) 23.2379 1.07532 0.537661 0.843161i \(-0.319308\pi\)
0.537661 + 0.843161i \(0.319308\pi\)
\(468\) 0 0
\(469\) 20.0000 + 17.3205i 0.923514 + 0.799787i
\(470\) 67.0820i 3.09426i
\(471\) 0 0
\(472\) 17.3205i 0.797241i
\(473\) 4.47214i 0.205629i
\(474\) 0 0
\(475\) 51.9615i 2.38416i
\(476\) 0 0
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) 30.9839 1.41569 0.707845 0.706368i \(-0.249668\pi\)
0.707845 + 0.706368i \(0.249668\pi\)
\(480\) 0 0
\(481\) 3.46410i 0.157949i
\(482\) 38.7298 1.76410
\(483\) 0 0
\(484\) −18.0000 −0.818182
\(485\) 53.6656i 2.43683i
\(486\) 0 0
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) −15.4919 −0.701287
\(489\) 0 0
\(490\) 60.0000 8.66025i 2.71052 0.391230i
\(491\) 38.0132i 1.71551i 0.514059 + 0.857755i \(0.328141\pi\)
−0.514059 + 0.857755i \(0.671859\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 40.2492i 1.81090i
\(495\) 0 0
\(496\) 1.73205i 0.0777714i
\(497\) 19.3649 22.3607i 0.868635 1.00301i
\(498\) 0 0
\(499\) 38.0000 1.70111 0.850557 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) −58.0948 −2.59808
\(501\) 0 0
\(502\) 0 0
\(503\) −23.2379 −1.03613 −0.518063 0.855342i \(-0.673347\pi\)
−0.518063 + 0.855342i \(0.673347\pi\)
\(504\) 0 0
\(505\) −60.0000 −2.66996
\(506\) 11.1803i 0.497027i
\(507\) 0 0
\(508\) 30.0000 1.33103
\(509\) 15.4919 0.686668 0.343334 0.939213i \(-0.388444\pi\)
0.343334 + 0.939213i \(0.388444\pi\)
\(510\) 0 0
\(511\) 18.0000 20.7846i 0.796273 0.919457i
\(512\) 11.1803i 0.494106i
\(513\) 0 0
\(514\) 60.6218i 2.67391i
\(515\) 6.70820i 0.295599i
\(516\) 0 0
\(517\) 17.3205i 0.761755i
\(518\) 3.87298 4.47214i 0.170169 0.196494i
\(519\) 0 0
\(520\) −30.0000 −1.31559
\(521\) −34.8569 −1.52711 −0.763553 0.645745i \(-0.776547\pi\)
−0.763553 + 0.645745i \(0.776547\pi\)
\(522\) 0 0
\(523\) 15.5885i 0.681636i 0.940129 + 0.340818i \(0.110704\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 23.2379 1.01515
\(525\) 0 0
\(526\) −5.00000 −0.218010
\(527\) 0 0
\(528\) 0 0
\(529\) 18.0000 0.782609
\(530\) −77.4597 −3.36463
\(531\) 0 0
\(532\) −27.0000 + 31.1769i −1.17060 + 1.35169i
\(533\) 13.4164i 0.581129i
\(534\) 0 0
\(535\) 34.6410i 1.49766i
\(536\) 22.3607i 0.965834i
\(537\) 0 0
\(538\) 25.9808i 1.12011i
\(539\) 15.4919 2.23607i 0.667285 0.0963143i
\(540\) 0 0
\(541\) −1.00000 −0.0429934 −0.0214967 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) −23.2379 −0.998153
\(543\) 0 0
\(544\) 0 0
\(545\) −27.1109 −1.16130
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 67.0820i 2.86560i
\(549\) 0 0
\(550\) −50.0000 −2.13201
\(551\) −23.2379 −0.989968
\(552\) 0 0
\(553\) −4.00000 3.46410i −0.170097 0.147309i
\(554\) 24.5967i 1.04502i
\(555\) 0 0
\(556\) 10.3923i 0.440732i
\(557\) 8.94427i 0.378981i −0.981883 0.189490i \(-0.939316\pi\)
0.981883 0.189490i \(-0.0606836\pi\)
\(558\) 0 0
\(559\) 6.92820i 0.293032i
\(560\) 7.74597 + 6.70820i 0.327327 + 0.283473i
\(561\) 0 0
\(562\) −50.0000 −2.10912
\(563\) 38.7298 1.63227 0.816134 0.577863i \(-0.196113\pi\)
0.816134 + 0.577863i \(0.196113\pi\)
\(564\) 0 0
\(565\) 34.6410i 1.45736i
\(566\) 54.2218 2.27911
\(567\) 0 0
\(568\) −25.0000 −1.04898
\(569\) 17.8885i 0.749927i 0.927040 + 0.374963i \(0.122345\pi\)
−0.927040 + 0.374963i \(0.877655\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −23.2379 −0.971625
\(573\) 0 0
\(574\) −15.0000 + 17.3205i −0.626088 + 0.722944i
\(575\) 22.3607i 0.932505i
\(576\) 0 0
\(577\) 20.7846i 0.865275i −0.901568 0.432637i \(-0.857583\pi\)
0.901568 0.432637i \(-0.142417\pi\)
\(578\) 38.0132i 1.58114i
\(579\) 0 0
\(580\) 51.9615i 2.15758i
\(581\) −15.4919 13.4164i −0.642714 0.556606i
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) −23.2379 −0.961591
\(585\) 0 0
\(586\) 69.2820i 2.86201i
\(587\) 30.9839 1.27884 0.639421 0.768857i \(-0.279174\pi\)
0.639421 + 0.768857i \(0.279174\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 67.0820i 2.76172i
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −34.8569 −1.43140 −0.715700 0.698408i \(-0.753892\pi\)
−0.715700 + 0.698408i \(0.753892\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 26.8328i 1.09911i
\(597\) 0 0
\(598\) 17.3205i 0.708288i
\(599\) 29.0689i 1.18772i −0.804568 0.593861i \(-0.797603\pi\)
0.804568 0.593861i \(-0.202397\pi\)
\(600\) 0 0
\(601\) 24.2487i 0.989126i −0.869142 0.494563i \(-0.835328\pi\)
0.869142 0.494563i \(-0.164672\pi\)
\(602\) −7.74597 + 8.94427i −0.315702 + 0.364541i
\(603\) 0 0
\(604\) 48.0000 1.95309
\(605\) 23.2379 0.944755
\(606\) 0 0
\(607\) 38.1051i 1.54664i −0.634017 0.773320i \(-0.718595\pi\)
0.634017 0.773320i \(-0.281405\pi\)
\(608\) −34.8569 −1.41363
\(609\) 0 0
\(610\) 60.0000 2.42933
\(611\) 26.8328i 1.08554i
\(612\) 0 0
\(613\) 17.0000 0.686624 0.343312 0.939222i \(-0.388451\pi\)
0.343312 + 0.939222i \(0.388451\pi\)
\(614\) 11.6190 0.468903
\(615\) 0 0
\(616\) −10.0000 8.66025i −0.402911 0.348932i
\(617\) 22.3607i 0.900207i −0.892976 0.450104i \(-0.851387\pi\)
0.892976 0.450104i \(-0.148613\pi\)
\(618\) 0 0
\(619\) 22.5167i 0.905021i 0.891759 + 0.452510i \(0.149471\pi\)
−0.891759 + 0.452510i \(0.850529\pi\)
\(620\) 20.1246i 0.808224i
\(621\) 0 0
\(622\) 17.3205i 0.694489i
\(623\) 23.2379 + 20.1246i 0.931007 + 0.806276i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −7.74597 −0.309591
\(627\) 0 0
\(628\) 10.3923i 0.414698i
\(629\) 0 0
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 4.47214i 0.177892i
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) −38.7298 −1.53695
\(636\) 0 0
\(637\) −24.0000 + 3.46410i −0.950915 + 0.137253i
\(638\) 22.3607i 0.885268i
\(639\) 0 0
\(640\) 60.6218i 2.39629i
\(641\) 17.8885i 0.706555i 0.935519 + 0.353278i \(0.114933\pi\)
−0.935519 + 0.353278i \(0.885067\pi\)
\(642\) 0 0
\(643\) 39.8372i 1.57102i 0.618846 + 0.785512i \(0.287600\pi\)
−0.618846 + 0.785512i \(0.712400\pi\)
\(644\) 11.6190 13.4164i 0.457851 0.528681i
\(645\) 0 0
\(646\) 0 0
\(647\) −23.2379 −0.913576 −0.456788 0.889576i \(-0.651000\pi\)
−0.456788 + 0.889576i \(0.651000\pi\)
\(648\) 0 0
\(649\) 17.3205i 0.679889i
\(650\) 77.4597 3.03822
\(651\) 0 0
\(652\) 30.0000 1.17489
\(653\) 49.1935i 1.92509i −0.271122 0.962545i \(-0.587395\pi\)
0.271122 0.962545i \(-0.412605\pi\)
\(654\) 0 0
\(655\) −30.0000 −1.17220
\(656\) −3.87298 −0.151215
\(657\) 0 0
\(658\) −30.0000 + 34.6410i −1.16952 + 1.35045i
\(659\) 2.23607i 0.0871048i −0.999051 0.0435524i \(-0.986132\pi\)
0.999051 0.0435524i \(-0.0138676\pi\)
\(660\) 0 0
\(661\) 45.0333i 1.75159i 0.482680 + 0.875797i \(0.339663\pi\)
−0.482680 + 0.875797i \(0.660337\pi\)
\(662\) 8.94427i 0.347629i
\(663\) 0 0
\(664\) 17.3205i 0.672166i
\(665\) 34.8569 40.2492i 1.35169 1.56080i
\(666\) 0 0
\(667\) 10.0000 0.387202
\(668\) −46.4758 −1.79820
\(669\) 0 0
\(670\) 86.6025i 3.34575i
\(671\) 15.4919 0.598059
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 78.2624i 3.01455i
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) −27.1109 −1.04196 −0.520978 0.853570i \(-0.674433\pi\)
−0.520978 + 0.853570i \(0.674433\pi\)
\(678\) 0 0
\(679\) −24.0000 + 27.7128i −0.921035 + 1.06352i
\(680\) 0 0
\(681\) 0 0
\(682\) 8.66025i 0.331618i
\(683\) 29.0689i 1.11229i −0.831085 0.556145i \(-0.812280\pi\)
0.831085 0.556145i \(-0.187720\pi\)
\(684\) 0 0
\(685\) 86.6025i 3.30891i
\(686\) −34.8569 22.3607i −1.33084 0.853735i
\(687\) 0 0
\(688\) −2.00000 −0.0762493
\(689\) 30.9839 1.18039
\(690\) 0 0
\(691\) 17.3205i 0.658903i 0.944172 + 0.329452i \(0.106864\pi\)
−0.944172 + 0.329452i \(0.893136\pi\)
\(692\) 11.6190 0.441686
\(693\) 0 0
\(694\) 25.0000 0.948987
\(695\) 13.4164i 0.508913i
\(696\) 0 0
\(697\) 0 0
\(698\) −30.9839 −1.17276
\(699\) 0 0
\(700\) 60.0000 + 51.9615i 2.26779 + 1.96396i
\(701\) 31.3050i 1.18237i 0.806535 + 0.591186i \(0.201340\pi\)
−0.806535 + 0.591186i \(0.798660\pi\)
\(702\) 0 0
\(703\) 5.19615i 0.195977i
\(704\) 29.0689i 1.09557i
\(705\) 0 0
\(706\) 43.3013i 1.62966i
\(707\) 30.9839 + 26.8328i 1.16527 + 1.00915i
\(708\) 0 0
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) 96.8246 3.63376
\(711\) 0 0
\(712\) 25.9808i 0.973670i
\(713\) −3.87298 −0.145044
\(714\) 0 0
\(715\) 30.0000 1.12194
\(716\) 53.6656i 2.00558i
\(717\) 0 0
\(718\) −20.0000 −0.746393
\(719\) 23.2379 0.866627 0.433314 0.901243i \(-0.357344\pi\)
0.433314 + 0.901243i \(0.357344\pi\)
\(720\) 0 0
\(721\) −3.00000 + 3.46410i −0.111726 + 0.129010i
\(722\) 17.8885i 0.665743i
\(723\) 0 0
\(724\) 0 0
\(725\) 44.7214i 1.66091i
\(726\) 0 0
\(727\) 3.46410i 0.128476i −0.997935 0.0642382i \(-0.979538\pi\)
0.997935 0.0642382i \(-0.0204617\pi\)
\(728\) 15.4919 + 13.4164i 0.574169 + 0.497245i
\(729\) 0 0
\(730\) 90.0000 3.33105
\(731\) 0 0
\(732\) 0 0
\(733\) 38.1051i 1.40744i −0.710475 0.703722i \(-0.751520\pi\)
0.710475 0.703722i \(-0.248480\pi\)
\(734\) 3.87298 0.142954
\(735\) 0 0
\(736\) 15.0000 0.552907
\(737\) 22.3607i 0.823666i
\(738\) 0 0
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 11.6190 0.427121
\(741\) 0 0
\(742\) 40.0000 + 34.6410i 1.46845 + 1.27171i
\(743\) 11.1803i 0.410167i 0.978744 + 0.205083i \(0.0657466\pi\)
−0.978744 + 0.205083i \(0.934253\pi\)
\(744\) 0 0
\(745\) 34.6410i 1.26915i
\(746\) 38.0132i 1.39176i
\(747\) 0 0
\(748\) 0 0
\(749\) −15.4919 + 17.8885i −0.566063 + 0.653633i
\(750\) 0 0
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) −7.74597 −0.282466
\(753\) 0 0
\(754\) 34.6410i 1.26155i
\(755\) −61.9677 −2.25524
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 58.1378i 2.11166i
\(759\) 0 0
\(760\) −45.0000 −1.63232
\(761\) 15.4919 0.561582 0.280791 0.959769i \(-0.409403\pi\)
0.280791 + 0.959769i \(0.409403\pi\)
\(762\) 0 0
\(763\) 14.0000 + 12.1244i 0.506834 + 0.438931i
\(764\) 6.70820i 0.242694i
\(765\) 0 0
\(766\) 69.2820i 2.50326i
\(767\) 26.8328i 0.968877i
\(768\) 0 0
\(769\) 17.3205i 0.624593i −0.949985 0.312297i \(-0.898902\pi\)
0.949985 0.312297i \(-0.101098\pi\)
\(770\) 38.7298 + 33.5410i 1.39573 + 1.20873i
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) −11.6190 −0.417905 −0.208952 0.977926i \(-0.567005\pi\)
−0.208952 + 0.977926i \(0.567005\pi\)
\(774\) 0 0
\(775\) 17.3205i 0.622171i
\(776\) 30.9839 1.11226
\(777\) 0 0
\(778\) −80.0000 −2.86814
\(779\) 20.1246i 0.721039i
\(780\) 0 0
\(781\) 25.0000 0.894570
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 6.92820i −0.0357143 0.247436i
\(785\) 13.4164i 0.478852i
\(786\) 0 0
\(787\) 38.1051i 1.35830i −0.733999 0.679150i \(-0.762348\pi\)
0.733999 0.679150i \(-0.237652\pi\)
\(788\) 26.8328i 0.955879i
\(789\) 0 0
\(790\) 17.3205i 0.616236i
\(791\) −15.4919 + 17.8885i −0.550830 + 0.636043i
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) −46.4758 −1.64936
\(795\) 0 0
\(796\) 77.9423i 2.76259i
\(797\) 3.87298 0.137188 0.0685941 0.997645i \(-0.478149\pi\)
0.0685941 + 0.997645i \(0.478149\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 67.0820i 2.37171i
\(801\) 0 0
\(802\) 70.0000 2.47179
\(803\) 23.2379 0.820048
\(804\) 0 0
\(805\) −15.0000 + 17.3205i −0.528681 + 0.610468i
\(806\) 13.4164i 0.472573i
\(807\) 0 0
\(808\) 34.6410i 1.21867i
\(809\) 31.3050i 1.10062i 0.834959 + 0.550312i \(0.185491\pi\)
−0.834959 + 0.550312i \(0.814509\pi\)
\(810\) 0 0
\(811\) 36.3731i 1.27723i 0.769526 + 0.638616i \(0.220493\pi\)
−0.769526 + 0.638616i \(0.779507\pi\)
\(812\) −23.2379 + 26.8328i −0.815490 + 0.941647i
\(813\) 0 0
\(814\) 5.00000 0.175250
\(815\) −38.7298 −1.35665
\(816\) 0 0
\(817\) 10.3923i 0.363581i
\(818\) −38.7298 −1.35416
\(819\) 0 0
\(820\) −45.0000 −1.57147
\(821\) 22.3607i 0.780393i −0.920732 0.390197i \(-0.872407\pi\)
0.920732 0.390197i \(-0.127593\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 3.87298 0.134922
\(825\) 0 0
\(826\) 30.0000 34.6410i 1.04383 1.20532i
\(827\) 11.1803i 0.388779i 0.980924 + 0.194389i \(0.0622725\pi\)
−0.980924 + 0.194389i \(0.937728\pi\)
\(828\) 0 0
\(829\) 31.1769i 1.08282i 0.840759 + 0.541409i \(0.182109\pi\)
−0.840759 + 0.541409i \(0.817891\pi\)
\(830\) 67.0820i 2.32845i
\(831\) 0 0
\(832\) 45.0333i 1.56125i
\(833\) 0 0
\(834\) 0 0
\(835\) 60.0000 2.07639
\(836\) −34.8569 −1.20555
\(837\) 0 0
\(838\) 34.6410i 1.19665i
\(839\) −38.7298 −1.33710 −0.668551 0.743666i \(-0.733085\pi\)
−0.668551 + 0.743666i \(0.733085\pi\)
\(840\) 0 0
\(841\) 9.00000 0.310345
\(842\) 24.5967i 0.847660i
\(843\) 0 0
\(844\) −24.0000 −0.826114
\(845\) 3.87298 0.133235
\(846\) 0 0
\(847\) −12.0000 10.3923i −0.412325 0.357084i
\(848\) 8.94427i 0.307148i
\(849\) 0 0
\(850\) 0 0
\(851\) 2.23607i 0.0766514i
\(852\) 0 0
\(853\) 13.8564i 0.474434i −0.971457 0.237217i \(-0.923765\pi\)
0.971457 0.237217i \(-0.0762353\pi\)
\(854\) −30.9839 26.8328i −1.06025 0.918200i
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) −27.1109 −0.926090 −0.463045 0.886335i \(-0.653243\pi\)
−0.463045 + 0.886335i \(0.653243\pi\)
\(858\) 0 0
\(859\) 22.5167i 0.768259i −0.923279 0.384129i \(-0.874502\pi\)
0.923279 0.384129i \(-0.125498\pi\)
\(860\) −23.2379 −0.792406
\(861\) 0 0
\(862\) −5.00000 −0.170301
\(863\) 8.94427i 0.304467i −0.988345 0.152233i \(-0.951353\pi\)
0.988345 0.152233i \(-0.0486465\pi\)
\(864\) 0 0
\(865\) −15.0000 −0.510015
\(866\) 23.2379 0.789656
\(867\) 0 0
\(868\) 9.00000 10.3923i 0.305480 0.352738i
\(869\) 4.47214i 0.151707i
\(870\) 0 0
\(871\) 34.6410i 1.17377i
\(872\) 15.6525i 0.530060i
\(873\) 0 0
\(874\) 25.9808i 0.878812i
\(875\) −38.7298 33.5410i −1.30931 1.13389i
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 85.2056 2.87555
\(879\) 0 0
\(880\) 8.66025i 0.291937i
\(881\) 34.8569 1.17436 0.587179 0.809457i \(-0.300239\pi\)
0.587179 + 0.809457i \(0.300239\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 55.0000 1.84776
\(887\) −7.74597 −0.260084 −0.130042 0.991508i \(-0.541511\pi\)
−0.130042 + 0.991508i \(0.541511\pi\)
\(888\) 0 0
\(889\) 20.0000 + 17.3205i 0.670778 + 0.580911i
\(890\) 100.623i 3.37289i
\(891\) 0 0
\(892\) 5.19615i 0.173980i
\(893\) 40.2492i 1.34689i
\(894\) 0 0
\(895\) 69.2820i 2.31584i
\(896\) −27.1109 + 31.3050i −0.905711 + 1.04583i
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) 7.74597 0.258342
\(900\) 0 0
\(901\) 0 0
\(902\) −19.3649 −0.644781
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) 0 0
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 46.4758 1.54235
\(909\) 0 0
\(910\) −60.0000 51.9615i −1.98898 1.72251i
\(911\) 35.7771i 1.18535i −0.805443 0.592674i \(-0.798072\pi\)
0.805443 0.592674i \(-0.201928\pi\)
\(912\) 0 0
\(913\) 17.3205i 0.573225i
\(914\) 55.9017i 1.84906i
\(915\) 0 0
\(916\) 41.5692i 1.37349i
\(917\) 15.4919 + 13.4164i 0.511589 + 0.443049i
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) 19.3649 0.638442
\(921\) 0 0
\(922\) 8.66025i 0.285210i
\(923\) −38.7298 −1.27481
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 17.8885i 0.587854i
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) −15.4919 −0.508274 −0.254137 0.967168i \(-0.581791\pi\)
−0.254137 + 0.967168i \(0.581791\pi\)
\(930\) 0 0
\(931\) −36.0000 + 5.19615i −1.17985 + 0.170297i
\(932\) 53.6656i 1.75788i
\(933\) 0 0
\(934\) 51.9615i 1.70023i
\(935\) 0 0
\(936\) 0 0
\(937\) 20.7846i 0.679004i 0.940605 + 0.339502i \(0.110258\pi\)
−0.940605 + 0.339502i \(0.889742\pi\)
\(938\) 38.7298 44.7214i 1.26457 1.46020i
\(939\) 0 0
\(940\) −90.0000 −2.93548
\(941\) 3.87298 0.126256 0.0631278 0.998005i \(-0.479892\pi\)
0.0631278 + 0.998005i \(0.479892\pi\)
\(942\) 0 0
\(943\) 8.66025i 0.282017i
\(944\) 7.74597 0.252110
\(945\) 0 0
\(946\) −10.0000 −0.325128
\(947\) 15.6525i 0.508637i −0.967121 0.254319i \(-0.918149\pi\)
0.967121 0.254319i \(-0.0818512\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 116.190 3.76969
\(951\) 0 0
\(952\) 0 0
\(953\) 22.3607i 0.724333i −0.932113 0.362167i \(-0.882037\pi\)
0.932113 0.362167i \(-0.117963\pi\)
\(954\) 0 0
\(955\) 8.66025i 0.280239i
\(956\) 26.8328i 0.867835i
\(957\) 0 0
\(958\) 69.2820i 2.23840i
\(959\) −38.7298 + 44.7214i −1.25065 + 1.44413i
\(960\) 0 0
\(961\) 28.0000 0.903226
\(962\) −7.74597 −0.249740
\(963\) 0 0
\(964\) 51.9615i 1.67357i
\(965\) 7.74597 0.249351
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 13.4164i 0.431220i
\(969\) 0 0
\(970\) −120.000 −3.85297
\(971\) 23.2379 0.745740 0.372870 0.927884i \(-0.378374\pi\)
0.372870 + 0.927884i \(0.378374\pi\)
\(972\) 0 0
\(973\) 6.00000 6.92820i 0.192351 0.222108i
\(974\) 22.3607i 0.716482i
\(975\) 0 0
\(976\) 6.92820i 0.221766i
\(977\) 17.8885i 0.572305i 0.958184 + 0.286153i \(0.0923764\pi\)
−0.958184 + 0.286153i \(0.907624\pi\)
\(978\) 0 0
\(979\) 25.9808i 0.830349i
\(980\) −11.6190 80.4984i −0.371154 2.57143i
\(981\) 0 0
\(982\) 85.0000 2.71246
\(983\) 54.2218 1.72941 0.864703 0.502284i \(-0.167507\pi\)
0.864703 + 0.502284i \(0.167507\pi\)
\(984\) 0 0
\(985\) 34.6410i 1.10375i
\(986\) 0 0
\(987\) 0 0
\(988\) 54.0000 1.71797
\(989\) 4.47214i 0.142206i
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 11.6190 0.368902
\(993\) 0 0
\(994\) −50.0000 43.3013i −1.58590 1.37343i
\(995\) 100.623i 3.18997i
\(996\) 0 0
\(997\) 27.7128i 0.877674i 0.898567 + 0.438837i \(0.144609\pi\)
−0.898567 + 0.438837i \(0.855391\pi\)
\(998\) 84.9706i 2.68970i
\(999\) 0 0
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 189.2.c.b.188.2 yes 4
3.2 odd 2 inner 189.2.c.b.188.3 yes 4
4.3 odd 2 3024.2.k.i.1889.4 4
7.6 odd 2 inner 189.2.c.b.188.1 4
9.2 odd 6 567.2.o.c.377.2 4
9.4 even 3 567.2.o.d.188.2 4
9.5 odd 6 567.2.o.d.188.1 4
9.7 even 3 567.2.o.c.377.1 4
12.11 even 2 3024.2.k.i.1889.2 4
21.20 even 2 inner 189.2.c.b.188.4 yes 4
28.27 even 2 3024.2.k.i.1889.1 4
63.13 odd 6 567.2.o.c.188.2 4
63.20 even 6 567.2.o.d.377.2 4
63.34 odd 6 567.2.o.d.377.1 4
63.41 even 6 567.2.o.c.188.1 4
84.83 odd 2 3024.2.k.i.1889.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.c.b.188.1 4 7.6 odd 2 inner
189.2.c.b.188.2 yes 4 1.1 even 1 trivial
189.2.c.b.188.3 yes 4 3.2 odd 2 inner
189.2.c.b.188.4 yes 4 21.20 even 2 inner
567.2.o.c.188.1 4 63.41 even 6
567.2.o.c.188.2 4 63.13 odd 6
567.2.o.c.377.1 4 9.7 even 3
567.2.o.c.377.2 4 9.2 odd 6
567.2.o.d.188.1 4 9.5 odd 6
567.2.o.d.188.2 4 9.4 even 3
567.2.o.d.377.1 4 63.34 odd 6
567.2.o.d.377.2 4 63.20 even 6
3024.2.k.i.1889.1 4 28.27 even 2
3024.2.k.i.1889.2 4 12.11 even 2
3024.2.k.i.1889.3 4 84.83 odd 2
3024.2.k.i.1889.4 4 4.3 odd 2