Properties

Label 7938.2.a.bg.1.1
Level $7938$
Weight $2$
Character 7938.1
Self dual yes
Analytic conductor $63.385$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7938,2,Mod(1,7938)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7938, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7938.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7938.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.73205 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.73205 q^{5} -1.00000 q^{8} +3.73205 q^{10} -4.19615 q^{11} -0.464102 q^{13} +1.00000 q^{16} -7.00000 q^{17} +2.73205 q^{19} -3.73205 q^{20} +4.19615 q^{22} +6.19615 q^{23} +8.92820 q^{25} +0.464102 q^{26} +8.46410 q^{29} +2.19615 q^{31} -1.00000 q^{32} +7.00000 q^{34} -6.66025 q^{37} -2.73205 q^{38} +3.73205 q^{40} -9.46410 q^{41} +5.46410 q^{43} -4.19615 q^{44} -6.19615 q^{46} -1.26795 q^{47} -8.92820 q^{50} -0.464102 q^{52} -2.53590 q^{53} +15.6603 q^{55} -8.46410 q^{58} +6.19615 q^{59} +9.92820 q^{61} -2.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -3.26795 q^{67} -7.00000 q^{68} +13.4641 q^{71} -11.7321 q^{73} +6.66025 q^{74} +2.73205 q^{76} +15.1244 q^{79} -3.73205 q^{80} +9.46410 q^{82} +14.5885 q^{83} +26.1244 q^{85} -5.46410 q^{86} +4.19615 q^{88} +3.92820 q^{89} +6.19615 q^{92} +1.26795 q^{94} -10.1962 q^{95} +2.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34} + 4 q^{37} - 2 q^{38} + 4 q^{40} - 12 q^{41} + 4 q^{43} + 2 q^{44} - 2 q^{46} - 6 q^{47} - 4 q^{50} + 6 q^{52} - 12 q^{53} + 14 q^{55} - 10 q^{58} + 2 q^{59} + 6 q^{61} + 6 q^{62} + 2 q^{64} - 10 q^{67} - 14 q^{68} + 20 q^{71} - 20 q^{73} - 4 q^{74} + 2 q^{76} + 6 q^{79} - 4 q^{80} + 12 q^{82} - 2 q^{83} + 28 q^{85} - 4 q^{86} - 2 q^{88} - 6 q^{89} + 2 q^{92} + 6 q^{94} - 10 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.73205 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.73205 1.18018
\(11\) −4.19615 −1.26519 −0.632594 0.774484i \(-0.718010\pi\)
−0.632594 + 0.774484i \(0.718010\pi\)
\(12\) 0 0
\(13\) −0.464102 −0.128719 −0.0643593 0.997927i \(-0.520500\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 2.73205 0.626775 0.313388 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(20\) −3.73205 −0.834512
\(21\) 0 0
\(22\) 4.19615 0.894623
\(23\) 6.19615 1.29199 0.645994 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0.464102 0.0910178
\(27\) 0 0
\(28\) 0 0
\(29\) 8.46410 1.57174 0.785872 0.618389i \(-0.212214\pi\)
0.785872 + 0.618389i \(0.212214\pi\)
\(30\) 0 0
\(31\) 2.19615 0.394441 0.197220 0.980359i \(-0.436809\pi\)
0.197220 + 0.980359i \(0.436809\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 0 0
\(37\) −6.66025 −1.09494 −0.547470 0.836826i \(-0.684409\pi\)
−0.547470 + 0.836826i \(0.684409\pi\)
\(38\) −2.73205 −0.443197
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) −9.46410 −1.47804 −0.739022 0.673681i \(-0.764712\pi\)
−0.739022 + 0.673681i \(0.764712\pi\)
\(42\) 0 0
\(43\) 5.46410 0.833268 0.416634 0.909074i \(-0.363210\pi\)
0.416634 + 0.909074i \(0.363210\pi\)
\(44\) −4.19615 −0.632594
\(45\) 0 0
\(46\) −6.19615 −0.913573
\(47\) −1.26795 −0.184949 −0.0924747 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −8.92820 −1.26264
\(51\) 0 0
\(52\) −0.464102 −0.0643593
\(53\) −2.53590 −0.348332 −0.174166 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 0 0
\(55\) 15.6603 2.11163
\(56\) 0 0
\(57\) 0 0
\(58\) −8.46410 −1.11139
\(59\) 6.19615 0.806670 0.403335 0.915052i \(-0.367851\pi\)
0.403335 + 0.915052i \(0.367851\pi\)
\(60\) 0 0
\(61\) 9.92820 1.27118 0.635588 0.772028i \(-0.280758\pi\)
0.635588 + 0.772028i \(0.280758\pi\)
\(62\) −2.19615 −0.278912
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.73205 0.214834
\(66\) 0 0
\(67\) −3.26795 −0.399244 −0.199622 0.979873i \(-0.563971\pi\)
−0.199622 + 0.979873i \(0.563971\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4641 1.59789 0.798947 0.601401i \(-0.205391\pi\)
0.798947 + 0.601401i \(0.205391\pi\)
\(72\) 0 0
\(73\) −11.7321 −1.37313 −0.686566 0.727067i \(-0.740883\pi\)
−0.686566 + 0.727067i \(0.740883\pi\)
\(74\) 6.66025 0.774239
\(75\) 0 0
\(76\) 2.73205 0.313388
\(77\) 0 0
\(78\) 0 0
\(79\) 15.1244 1.70162 0.850811 0.525471i \(-0.176111\pi\)
0.850811 + 0.525471i \(0.176111\pi\)
\(80\) −3.73205 −0.417256
\(81\) 0 0
\(82\) 9.46410 1.04514
\(83\) 14.5885 1.60129 0.800646 0.599138i \(-0.204490\pi\)
0.800646 + 0.599138i \(0.204490\pi\)
\(84\) 0 0
\(85\) 26.1244 2.83358
\(86\) −5.46410 −0.589209
\(87\) 0 0
\(88\) 4.19615 0.447311
\(89\) 3.92820 0.416389 0.208194 0.978087i \(-0.433241\pi\)
0.208194 + 0.978087i \(0.433241\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.19615 0.645994
\(93\) 0 0
\(94\) 1.26795 0.130779
\(95\) −10.1962 −1.04610
\(96\) 0 0
\(97\) 2.92820 0.297314 0.148657 0.988889i \(-0.452505\pi\)
0.148657 + 0.988889i \(0.452505\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.92820 0.892820
\(101\) 4.92820 0.490375 0.245187 0.969476i \(-0.421151\pi\)
0.245187 + 0.969476i \(0.421151\pi\)
\(102\) 0 0
\(103\) −12.3923 −1.22105 −0.610525 0.791997i \(-0.709042\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(104\) 0.464102 0.0455089
\(105\) 0 0
\(106\) 2.53590 0.246308
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −7.19615 −0.689266 −0.344633 0.938737i \(-0.611997\pi\)
−0.344633 + 0.938737i \(0.611997\pi\)
\(110\) −15.6603 −1.49315
\(111\) 0 0
\(112\) 0 0
\(113\) −2.26795 −0.213351 −0.106675 0.994294i \(-0.534021\pi\)
−0.106675 + 0.994294i \(0.534021\pi\)
\(114\) 0 0
\(115\) −23.1244 −2.15636
\(116\) 8.46410 0.785872
\(117\) 0 0
\(118\) −6.19615 −0.570402
\(119\) 0 0
\(120\) 0 0
\(121\) 6.60770 0.600700
\(122\) −9.92820 −0.898857
\(123\) 0 0
\(124\) 2.19615 0.197220
\(125\) −14.6603 −1.31125
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.73205 −0.151911
\(131\) −17.4641 −1.52585 −0.762923 0.646490i \(-0.776236\pi\)
−0.762923 + 0.646490i \(0.776236\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.26795 0.282308
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) −11.7321 −1.00234 −0.501168 0.865350i \(-0.667096\pi\)
−0.501168 + 0.865350i \(0.667096\pi\)
\(138\) 0 0
\(139\) 6.73205 0.571005 0.285503 0.958378i \(-0.407839\pi\)
0.285503 + 0.958378i \(0.407839\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −13.4641 −1.12988
\(143\) 1.94744 0.162853
\(144\) 0 0
\(145\) −31.5885 −2.62328
\(146\) 11.7321 0.970951
\(147\) 0 0
\(148\) −6.66025 −0.547470
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −16.1962 −1.31802 −0.659012 0.752132i \(-0.729025\pi\)
−0.659012 + 0.752132i \(0.729025\pi\)
\(152\) −2.73205 −0.221599
\(153\) 0 0
\(154\) 0 0
\(155\) −8.19615 −0.658331
\(156\) 0 0
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) −15.1244 −1.20323
\(159\) 0 0
\(160\) 3.73205 0.295045
\(161\) 0 0
\(162\) 0 0
\(163\) −6.53590 −0.511931 −0.255966 0.966686i \(-0.582393\pi\)
−0.255966 + 0.966686i \(0.582393\pi\)
\(164\) −9.46410 −0.739022
\(165\) 0 0
\(166\) −14.5885 −1.13228
\(167\) 12.1962 0.943767 0.471883 0.881661i \(-0.343574\pi\)
0.471883 + 0.881661i \(0.343574\pi\)
\(168\) 0 0
\(169\) −12.7846 −0.983432
\(170\) −26.1244 −2.00365
\(171\) 0 0
\(172\) 5.46410 0.416634
\(173\) −9.73205 −0.739914 −0.369957 0.929049i \(-0.620628\pi\)
−0.369957 + 0.929049i \(0.620628\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.19615 −0.316297
\(177\) 0 0
\(178\) −3.92820 −0.294431
\(179\) 8.19615 0.612609 0.306305 0.951934i \(-0.400907\pi\)
0.306305 + 0.951934i \(0.400907\pi\)
\(180\) 0 0
\(181\) −4.39230 −0.326477 −0.163239 0.986587i \(-0.552194\pi\)
−0.163239 + 0.986587i \(0.552194\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.19615 −0.456786
\(185\) 24.8564 1.82748
\(186\) 0 0
\(187\) 29.3731 2.14797
\(188\) −1.26795 −0.0924747
\(189\) 0 0
\(190\) 10.1962 0.739707
\(191\) 11.6603 0.843706 0.421853 0.906664i \(-0.361380\pi\)
0.421853 + 0.906664i \(0.361380\pi\)
\(192\) 0 0
\(193\) −8.85641 −0.637498 −0.318749 0.947839i \(-0.603263\pi\)
−0.318749 + 0.947839i \(0.603263\pi\)
\(194\) −2.92820 −0.210233
\(195\) 0 0
\(196\) 0 0
\(197\) −25.7846 −1.83708 −0.918539 0.395331i \(-0.870630\pi\)
−0.918539 + 0.395331i \(0.870630\pi\)
\(198\) 0 0
\(199\) −5.12436 −0.363256 −0.181628 0.983367i \(-0.558137\pi\)
−0.181628 + 0.983367i \(0.558137\pi\)
\(200\) −8.92820 −0.631319
\(201\) 0 0
\(202\) −4.92820 −0.346747
\(203\) 0 0
\(204\) 0 0
\(205\) 35.3205 2.46689
\(206\) 12.3923 0.863413
\(207\) 0 0
\(208\) −0.464102 −0.0321797
\(209\) −11.4641 −0.792988
\(210\) 0 0
\(211\) −20.7321 −1.42725 −0.713627 0.700526i \(-0.752949\pi\)
−0.713627 + 0.700526i \(0.752949\pi\)
\(212\) −2.53590 −0.174166
\(213\) 0 0
\(214\) 0 0
\(215\) −20.3923 −1.39074
\(216\) 0 0
\(217\) 0 0
\(218\) 7.19615 0.487385
\(219\) 0 0
\(220\) 15.6603 1.05581
\(221\) 3.24871 0.218532
\(222\) 0 0
\(223\) 18.5359 1.24126 0.620628 0.784105i \(-0.286878\pi\)
0.620628 + 0.784105i \(0.286878\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.26795 0.150862
\(227\) 5.07180 0.336627 0.168313 0.985734i \(-0.446168\pi\)
0.168313 + 0.985734i \(0.446168\pi\)
\(228\) 0 0
\(229\) −4.46410 −0.294996 −0.147498 0.989062i \(-0.547122\pi\)
−0.147498 + 0.989062i \(0.547122\pi\)
\(230\) 23.1244 1.52477
\(231\) 0 0
\(232\) −8.46410 −0.555695
\(233\) −13.1962 −0.864509 −0.432254 0.901752i \(-0.642282\pi\)
−0.432254 + 0.901752i \(0.642282\pi\)
\(234\) 0 0
\(235\) 4.73205 0.308685
\(236\) 6.19615 0.403335
\(237\) 0 0
\(238\) 0 0
\(239\) 28.0526 1.81457 0.907285 0.420517i \(-0.138151\pi\)
0.907285 + 0.420517i \(0.138151\pi\)
\(240\) 0 0
\(241\) 17.7321 1.14222 0.571111 0.820873i \(-0.306513\pi\)
0.571111 + 0.820873i \(0.306513\pi\)
\(242\) −6.60770 −0.424759
\(243\) 0 0
\(244\) 9.92820 0.635588
\(245\) 0 0
\(246\) 0 0
\(247\) −1.26795 −0.0806777
\(248\) −2.19615 −0.139456
\(249\) 0 0
\(250\) 14.6603 0.927196
\(251\) −16.0526 −1.01323 −0.506614 0.862173i \(-0.669103\pi\)
−0.506614 + 0.862173i \(0.669103\pi\)
\(252\) 0 0
\(253\) −26.0000 −1.63461
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0.464102 0.0289499 0.0144749 0.999895i \(-0.495392\pi\)
0.0144749 + 0.999895i \(0.495392\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.73205 0.107417
\(261\) 0 0
\(262\) 17.4641 1.07894
\(263\) −23.6603 −1.45895 −0.729477 0.684005i \(-0.760236\pi\)
−0.729477 + 0.684005i \(0.760236\pi\)
\(264\) 0 0
\(265\) 9.46410 0.581375
\(266\) 0 0
\(267\) 0 0
\(268\) −3.26795 −0.199622
\(269\) 25.5885 1.56016 0.780078 0.625682i \(-0.215179\pi\)
0.780078 + 0.625682i \(0.215179\pi\)
\(270\) 0 0
\(271\) −25.5167 −1.55003 −0.775013 0.631945i \(-0.782257\pi\)
−0.775013 + 0.631945i \(0.782257\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) 11.7321 0.708759
\(275\) −37.4641 −2.25917
\(276\) 0 0
\(277\) 22.7846 1.36899 0.684497 0.729015i \(-0.260022\pi\)
0.684497 + 0.729015i \(0.260022\pi\)
\(278\) −6.73205 −0.403762
\(279\) 0 0
\(280\) 0 0
\(281\) −2.80385 −0.167264 −0.0836318 0.996497i \(-0.526652\pi\)
−0.0836318 + 0.996497i \(0.526652\pi\)
\(282\) 0 0
\(283\) 19.3205 1.14848 0.574242 0.818685i \(-0.305297\pi\)
0.574242 + 0.818685i \(0.305297\pi\)
\(284\) 13.4641 0.798947
\(285\) 0 0
\(286\) −1.94744 −0.115155
\(287\) 0 0
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 31.5885 1.85494
\(291\) 0 0
\(292\) −11.7321 −0.686566
\(293\) −20.6603 −1.20698 −0.603492 0.797369i \(-0.706225\pi\)
−0.603492 + 0.797369i \(0.706225\pi\)
\(294\) 0 0
\(295\) −23.1244 −1.34635
\(296\) 6.66025 0.387119
\(297\) 0 0
\(298\) −9.00000 −0.521356
\(299\) −2.87564 −0.166303
\(300\) 0 0
\(301\) 0 0
\(302\) 16.1962 0.931984
\(303\) 0 0
\(304\) 2.73205 0.156694
\(305\) −37.0526 −2.12162
\(306\) 0 0
\(307\) −5.85641 −0.334243 −0.167121 0.985936i \(-0.553447\pi\)
−0.167121 + 0.985936i \(0.553447\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.19615 0.465510
\(311\) 0.196152 0.0111228 0.00556139 0.999985i \(-0.498230\pi\)
0.00556139 + 0.999985i \(0.498230\pi\)
\(312\) 0 0
\(313\) 5.58846 0.315878 0.157939 0.987449i \(-0.449515\pi\)
0.157939 + 0.987449i \(0.449515\pi\)
\(314\) −1.00000 −0.0564333
\(315\) 0 0
\(316\) 15.1244 0.850811
\(317\) −10.6077 −0.595788 −0.297894 0.954599i \(-0.596284\pi\)
−0.297894 + 0.954599i \(0.596284\pi\)
\(318\) 0 0
\(319\) −35.5167 −1.98855
\(320\) −3.73205 −0.208628
\(321\) 0 0
\(322\) 0 0
\(323\) −19.1244 −1.06411
\(324\) 0 0
\(325\) −4.14359 −0.229845
\(326\) 6.53590 0.361990
\(327\) 0 0
\(328\) 9.46410 0.522568
\(329\) 0 0
\(330\) 0 0
\(331\) −8.39230 −0.461283 −0.230641 0.973039i \(-0.574082\pi\)
−0.230641 + 0.973039i \(0.574082\pi\)
\(332\) 14.5885 0.800646
\(333\) 0 0
\(334\) −12.1962 −0.667344
\(335\) 12.1962 0.666347
\(336\) 0 0
\(337\) 4.39230 0.239264 0.119632 0.992818i \(-0.461829\pi\)
0.119632 + 0.992818i \(0.461829\pi\)
\(338\) 12.7846 0.695391
\(339\) 0 0
\(340\) 26.1244 1.41679
\(341\) −9.21539 −0.499041
\(342\) 0 0
\(343\) 0 0
\(344\) −5.46410 −0.294605
\(345\) 0 0
\(346\) 9.73205 0.523198
\(347\) 14.5359 0.780328 0.390164 0.920745i \(-0.372418\pi\)
0.390164 + 0.920745i \(0.372418\pi\)
\(348\) 0 0
\(349\) 5.46410 0.292487 0.146243 0.989249i \(-0.453282\pi\)
0.146243 + 0.989249i \(0.453282\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.19615 0.223656
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −50.2487 −2.66692
\(356\) 3.92820 0.208194
\(357\) 0 0
\(358\) −8.19615 −0.433180
\(359\) 2.92820 0.154545 0.0772723 0.997010i \(-0.475379\pi\)
0.0772723 + 0.997010i \(0.475379\pi\)
\(360\) 0 0
\(361\) −11.5359 −0.607153
\(362\) 4.39230 0.230854
\(363\) 0 0
\(364\) 0 0
\(365\) 43.7846 2.29179
\(366\) 0 0
\(367\) 13.1244 0.685086 0.342543 0.939502i \(-0.388712\pi\)
0.342543 + 0.939502i \(0.388712\pi\)
\(368\) 6.19615 0.322997
\(369\) 0 0
\(370\) −24.8564 −1.29222
\(371\) 0 0
\(372\) 0 0
\(373\) −33.8564 −1.75302 −0.876509 0.481385i \(-0.840134\pi\)
−0.876509 + 0.481385i \(0.840134\pi\)
\(374\) −29.3731 −1.51885
\(375\) 0 0
\(376\) 1.26795 0.0653895
\(377\) −3.92820 −0.202313
\(378\) 0 0
\(379\) 17.5167 0.899770 0.449885 0.893086i \(-0.351465\pi\)
0.449885 + 0.893086i \(0.351465\pi\)
\(380\) −10.1962 −0.523052
\(381\) 0 0
\(382\) −11.6603 −0.596590
\(383\) 35.7128 1.82484 0.912420 0.409256i \(-0.134212\pi\)
0.912420 + 0.409256i \(0.134212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.85641 0.450779
\(387\) 0 0
\(388\) 2.92820 0.148657
\(389\) −6.53590 −0.331383 −0.165692 0.986178i \(-0.552986\pi\)
−0.165692 + 0.986178i \(0.552986\pi\)
\(390\) 0 0
\(391\) −43.3731 −2.19347
\(392\) 0 0
\(393\) 0 0
\(394\) 25.7846 1.29901
\(395\) −56.4449 −2.84005
\(396\) 0 0
\(397\) 21.0000 1.05396 0.526980 0.849878i \(-0.323324\pi\)
0.526980 + 0.849878i \(0.323324\pi\)
\(398\) 5.12436 0.256861
\(399\) 0 0
\(400\) 8.92820 0.446410
\(401\) −34.5167 −1.72368 −0.861840 0.507180i \(-0.830688\pi\)
−0.861840 + 0.507180i \(0.830688\pi\)
\(402\) 0 0
\(403\) −1.01924 −0.0507719
\(404\) 4.92820 0.245187
\(405\) 0 0
\(406\) 0 0
\(407\) 27.9474 1.38530
\(408\) 0 0
\(409\) −34.6603 −1.71384 −0.856920 0.515450i \(-0.827625\pi\)
−0.856920 + 0.515450i \(0.827625\pi\)
\(410\) −35.3205 −1.74436
\(411\) 0 0
\(412\) −12.3923 −0.610525
\(413\) 0 0
\(414\) 0 0
\(415\) −54.4449 −2.67259
\(416\) 0.464102 0.0227545
\(417\) 0 0
\(418\) 11.4641 0.560728
\(419\) 2.53590 0.123887 0.0619434 0.998080i \(-0.480270\pi\)
0.0619434 + 0.998080i \(0.480270\pi\)
\(420\) 0 0
\(421\) −24.1244 −1.17575 −0.587875 0.808952i \(-0.700035\pi\)
−0.587875 + 0.808952i \(0.700035\pi\)
\(422\) 20.7321 1.00922
\(423\) 0 0
\(424\) 2.53590 0.123154
\(425\) −62.4974 −3.03157
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 20.3923 0.983404
\(431\) −21.4641 −1.03389 −0.516945 0.856019i \(-0.672931\pi\)
−0.516945 + 0.856019i \(0.672931\pi\)
\(432\) 0 0
\(433\) −12.2679 −0.589560 −0.294780 0.955565i \(-0.595246\pi\)
−0.294780 + 0.955565i \(0.595246\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7.19615 −0.344633
\(437\) 16.9282 0.809786
\(438\) 0 0
\(439\) 11.3205 0.540298 0.270149 0.962818i \(-0.412927\pi\)
0.270149 + 0.962818i \(0.412927\pi\)
\(440\) −15.6603 −0.746573
\(441\) 0 0
\(442\) −3.24871 −0.154525
\(443\) −18.7321 −0.889987 −0.444993 0.895534i \(-0.646794\pi\)
−0.444993 + 0.895534i \(0.646794\pi\)
\(444\) 0 0
\(445\) −14.6603 −0.694963
\(446\) −18.5359 −0.877700
\(447\) 0 0
\(448\) 0 0
\(449\) −11.8564 −0.559538 −0.279769 0.960067i \(-0.590258\pi\)
−0.279769 + 0.960067i \(0.590258\pi\)
\(450\) 0 0
\(451\) 39.7128 1.87000
\(452\) −2.26795 −0.106675
\(453\) 0 0
\(454\) −5.07180 −0.238031
\(455\) 0 0
\(456\) 0 0
\(457\) 20.8564 0.975622 0.487811 0.872949i \(-0.337796\pi\)
0.487811 + 0.872949i \(0.337796\pi\)
\(458\) 4.46410 0.208594
\(459\) 0 0
\(460\) −23.1244 −1.07818
\(461\) −34.7846 −1.62008 −0.810040 0.586374i \(-0.800555\pi\)
−0.810040 + 0.586374i \(0.800555\pi\)
\(462\) 0 0
\(463\) −32.5885 −1.51451 −0.757257 0.653117i \(-0.773461\pi\)
−0.757257 + 0.653117i \(0.773461\pi\)
\(464\) 8.46410 0.392936
\(465\) 0 0
\(466\) 13.1962 0.611300
\(467\) −14.5885 −0.675073 −0.337537 0.941312i \(-0.609594\pi\)
−0.337537 + 0.941312i \(0.609594\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.73205 −0.218273
\(471\) 0 0
\(472\) −6.19615 −0.285201
\(473\) −22.9282 −1.05424
\(474\) 0 0
\(475\) 24.3923 1.11920
\(476\) 0 0
\(477\) 0 0
\(478\) −28.0526 −1.28309
\(479\) −23.5167 −1.07450 −0.537252 0.843422i \(-0.680538\pi\)
−0.537252 + 0.843422i \(0.680538\pi\)
\(480\) 0 0
\(481\) 3.09103 0.140939
\(482\) −17.7321 −0.807673
\(483\) 0 0
\(484\) 6.60770 0.300350
\(485\) −10.9282 −0.496224
\(486\) 0 0
\(487\) −28.5885 −1.29547 −0.647733 0.761867i \(-0.724283\pi\)
−0.647733 + 0.761867i \(0.724283\pi\)
\(488\) −9.92820 −0.449429
\(489\) 0 0
\(490\) 0 0
\(491\) 33.4641 1.51021 0.755107 0.655602i \(-0.227585\pi\)
0.755107 + 0.655602i \(0.227585\pi\)
\(492\) 0 0
\(493\) −59.2487 −2.66843
\(494\) 1.26795 0.0570477
\(495\) 0 0
\(496\) 2.19615 0.0986102
\(497\) 0 0
\(498\) 0 0
\(499\) 30.1962 1.35177 0.675883 0.737009i \(-0.263763\pi\)
0.675883 + 0.737009i \(0.263763\pi\)
\(500\) −14.6603 −0.655626
\(501\) 0 0
\(502\) 16.0526 0.716461
\(503\) −1.94744 −0.0868321 −0.0434161 0.999057i \(-0.513824\pi\)
−0.0434161 + 0.999057i \(0.513824\pi\)
\(504\) 0 0
\(505\) −18.3923 −0.818447
\(506\) 26.0000 1.15584
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) −4.14359 −0.183662 −0.0918308 0.995775i \(-0.529272\pi\)
−0.0918308 + 0.995775i \(0.529272\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −0.464102 −0.0204706
\(515\) 46.2487 2.03796
\(516\) 0 0
\(517\) 5.32051 0.233996
\(518\) 0 0
\(519\) 0 0
\(520\) −1.73205 −0.0759555
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 29.1769 1.27582 0.637909 0.770112i \(-0.279800\pi\)
0.637909 + 0.770112i \(0.279800\pi\)
\(524\) −17.4641 −0.762923
\(525\) 0 0
\(526\) 23.6603 1.03164
\(527\) −15.3731 −0.669661
\(528\) 0 0
\(529\) 15.3923 0.669231
\(530\) −9.46410 −0.411094
\(531\) 0 0
\(532\) 0 0
\(533\) 4.39230 0.190252
\(534\) 0 0
\(535\) 0 0
\(536\) 3.26795 0.141154
\(537\) 0 0
\(538\) −25.5885 −1.10320
\(539\) 0 0
\(540\) 0 0
\(541\) 20.6603 0.888254 0.444127 0.895964i \(-0.353514\pi\)
0.444127 + 0.895964i \(0.353514\pi\)
\(542\) 25.5167 1.09603
\(543\) 0 0
\(544\) 7.00000 0.300123
\(545\) 26.8564 1.15040
\(546\) 0 0
\(547\) −19.2679 −0.823838 −0.411919 0.911220i \(-0.635141\pi\)
−0.411919 + 0.911220i \(0.635141\pi\)
\(548\) −11.7321 −0.501168
\(549\) 0 0
\(550\) 37.4641 1.59747
\(551\) 23.1244 0.985131
\(552\) 0 0
\(553\) 0 0
\(554\) −22.7846 −0.968025
\(555\) 0 0
\(556\) 6.73205 0.285503
\(557\) 10.0718 0.426756 0.213378 0.976970i \(-0.431553\pi\)
0.213378 + 0.976970i \(0.431553\pi\)
\(558\) 0 0
\(559\) −2.53590 −0.107257
\(560\) 0 0
\(561\) 0 0
\(562\) 2.80385 0.118273
\(563\) 35.7128 1.50512 0.752558 0.658526i \(-0.228820\pi\)
0.752558 + 0.658526i \(0.228820\pi\)
\(564\) 0 0
\(565\) 8.46410 0.356087
\(566\) −19.3205 −0.812102
\(567\) 0 0
\(568\) −13.4641 −0.564941
\(569\) 12.8038 0.536765 0.268383 0.963312i \(-0.413511\pi\)
0.268383 + 0.963312i \(0.413511\pi\)
\(570\) 0 0
\(571\) −19.2679 −0.806339 −0.403169 0.915125i \(-0.632091\pi\)
−0.403169 + 0.915125i \(0.632091\pi\)
\(572\) 1.94744 0.0814266
\(573\) 0 0
\(574\) 0 0
\(575\) 55.3205 2.30702
\(576\) 0 0
\(577\) −7.33975 −0.305558 −0.152779 0.988260i \(-0.548822\pi\)
−0.152779 + 0.988260i \(0.548822\pi\)
\(578\) −32.0000 −1.33102
\(579\) 0 0
\(580\) −31.5885 −1.31164
\(581\) 0 0
\(582\) 0 0
\(583\) 10.6410 0.440706
\(584\) 11.7321 0.485476
\(585\) 0 0
\(586\) 20.6603 0.853467
\(587\) −11.2679 −0.465078 −0.232539 0.972587i \(-0.574703\pi\)
−0.232539 + 0.972587i \(0.574703\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 23.1244 0.952015
\(591\) 0 0
\(592\) −6.66025 −0.273735
\(593\) −40.1769 −1.64987 −0.824934 0.565229i \(-0.808788\pi\)
−0.824934 + 0.565229i \(0.808788\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.00000 0.368654
\(597\) 0 0
\(598\) 2.87564 0.117594
\(599\) 9.12436 0.372811 0.186406 0.982473i \(-0.440316\pi\)
0.186406 + 0.982473i \(0.440316\pi\)
\(600\) 0 0
\(601\) −8.80385 −0.359116 −0.179558 0.983747i \(-0.557467\pi\)
−0.179558 + 0.983747i \(0.557467\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.1962 −0.659012
\(605\) −24.6603 −1.00258
\(606\) 0 0
\(607\) −6.58846 −0.267417 −0.133709 0.991021i \(-0.542689\pi\)
−0.133709 + 0.991021i \(0.542689\pi\)
\(608\) −2.73205 −0.110799
\(609\) 0 0
\(610\) 37.0526 1.50021
\(611\) 0.588457 0.0238064
\(612\) 0 0
\(613\) −14.7846 −0.597145 −0.298572 0.954387i \(-0.596510\pi\)
−0.298572 + 0.954387i \(0.596510\pi\)
\(614\) 5.85641 0.236345
\(615\) 0 0
\(616\) 0 0
\(617\) −39.9808 −1.60956 −0.804782 0.593570i \(-0.797718\pi\)
−0.804782 + 0.593570i \(0.797718\pi\)
\(618\) 0 0
\(619\) −31.7128 −1.27465 −0.637323 0.770597i \(-0.719958\pi\)
−0.637323 + 0.770597i \(0.719958\pi\)
\(620\) −8.19615 −0.329165
\(621\) 0 0
\(622\) −0.196152 −0.00786500
\(623\) 0 0
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) −5.58846 −0.223360
\(627\) 0 0
\(628\) 1.00000 0.0399043
\(629\) 46.6218 1.85893
\(630\) 0 0
\(631\) 13.6603 0.543806 0.271903 0.962325i \(-0.412347\pi\)
0.271903 + 0.962325i \(0.412347\pi\)
\(632\) −15.1244 −0.601615
\(633\) 0 0
\(634\) 10.6077 0.421285
\(635\) −44.7846 −1.77722
\(636\) 0 0
\(637\) 0 0
\(638\) 35.5167 1.40612
\(639\) 0 0
\(640\) 3.73205 0.147522
\(641\) −19.4449 −0.768026 −0.384013 0.923328i \(-0.625458\pi\)
−0.384013 + 0.923328i \(0.625458\pi\)
\(642\) 0 0
\(643\) 40.5885 1.60065 0.800326 0.599565i \(-0.204660\pi\)
0.800326 + 0.599565i \(0.204660\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 19.1244 0.752438
\(647\) −16.3923 −0.644448 −0.322224 0.946663i \(-0.604430\pi\)
−0.322224 + 0.946663i \(0.604430\pi\)
\(648\) 0 0
\(649\) −26.0000 −1.02059
\(650\) 4.14359 0.162525
\(651\) 0 0
\(652\) −6.53590 −0.255966
\(653\) 18.2487 0.714127 0.357064 0.934080i \(-0.383778\pi\)
0.357064 + 0.934080i \(0.383778\pi\)
\(654\) 0 0
\(655\) 65.1769 2.54667
\(656\) −9.46410 −0.369511
\(657\) 0 0
\(658\) 0 0
\(659\) −15.6077 −0.607989 −0.303995 0.952674i \(-0.598321\pi\)
−0.303995 + 0.952674i \(0.598321\pi\)
\(660\) 0 0
\(661\) −14.8564 −0.577847 −0.288924 0.957352i \(-0.593297\pi\)
−0.288924 + 0.957352i \(0.593297\pi\)
\(662\) 8.39230 0.326176
\(663\) 0 0
\(664\) −14.5885 −0.566142
\(665\) 0 0
\(666\) 0 0
\(667\) 52.4449 2.03067
\(668\) 12.1962 0.471883
\(669\) 0 0
\(670\) −12.1962 −0.471178
\(671\) −41.6603 −1.60828
\(672\) 0 0
\(673\) 16.3205 0.629109 0.314555 0.949239i \(-0.398145\pi\)
0.314555 + 0.949239i \(0.398145\pi\)
\(674\) −4.39230 −0.169185
\(675\) 0 0
\(676\) −12.7846 −0.491716
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −26.1244 −1.00182
\(681\) 0 0
\(682\) 9.21539 0.352876
\(683\) 25.8564 0.989368 0.494684 0.869073i \(-0.335284\pi\)
0.494684 + 0.869073i \(0.335284\pi\)
\(684\) 0 0
\(685\) 43.7846 1.67292
\(686\) 0 0
\(687\) 0 0
\(688\) 5.46410 0.208317
\(689\) 1.17691 0.0448369
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −9.73205 −0.369957
\(693\) 0 0
\(694\) −14.5359 −0.551775
\(695\) −25.1244 −0.953021
\(696\) 0 0
\(697\) 66.2487 2.50935
\(698\) −5.46410 −0.206819
\(699\) 0 0
\(700\) 0 0
\(701\) 6.60770 0.249569 0.124785 0.992184i \(-0.460176\pi\)
0.124785 + 0.992184i \(0.460176\pi\)
\(702\) 0 0
\(703\) −18.1962 −0.686281
\(704\) −4.19615 −0.158148
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −28.1244 −1.05623 −0.528116 0.849172i \(-0.677101\pi\)
−0.528116 + 0.849172i \(0.677101\pi\)
\(710\) 50.2487 1.88580
\(711\) 0 0
\(712\) −3.92820 −0.147216
\(713\) 13.6077 0.509612
\(714\) 0 0
\(715\) −7.26795 −0.271806
\(716\) 8.19615 0.306305
\(717\) 0 0
\(718\) −2.92820 −0.109280
\(719\) 2.53590 0.0945731 0.0472865 0.998881i \(-0.484943\pi\)
0.0472865 + 0.998881i \(0.484943\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 11.5359 0.429322
\(723\) 0 0
\(724\) −4.39230 −0.163239
\(725\) 75.5692 2.80657
\(726\) 0 0
\(727\) −16.6795 −0.618608 −0.309304 0.950963i \(-0.600096\pi\)
−0.309304 + 0.950963i \(0.600096\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −43.7846 −1.62054
\(731\) −38.2487 −1.41468
\(732\) 0 0
\(733\) −12.6795 −0.468328 −0.234164 0.972197i \(-0.575235\pi\)
−0.234164 + 0.972197i \(0.575235\pi\)
\(734\) −13.1244 −0.484429
\(735\) 0 0
\(736\) −6.19615 −0.228393
\(737\) 13.7128 0.505118
\(738\) 0 0
\(739\) −16.7321 −0.615498 −0.307749 0.951468i \(-0.599576\pi\)
−0.307749 + 0.951468i \(0.599576\pi\)
\(740\) 24.8564 0.913740
\(741\) 0 0
\(742\) 0 0
\(743\) −19.6077 −0.719337 −0.359668 0.933080i \(-0.617110\pi\)
−0.359668 + 0.933080i \(0.617110\pi\)
\(744\) 0 0
\(745\) −33.5885 −1.23059
\(746\) 33.8564 1.23957
\(747\) 0 0
\(748\) 29.3731 1.07399
\(749\) 0 0
\(750\) 0 0
\(751\) −49.8564 −1.81929 −0.909643 0.415391i \(-0.863645\pi\)
−0.909643 + 0.415391i \(0.863645\pi\)
\(752\) −1.26795 −0.0462373
\(753\) 0 0
\(754\) 3.92820 0.143057
\(755\) 60.4449 2.19981
\(756\) 0 0
\(757\) −20.7846 −0.755429 −0.377715 0.925922i \(-0.623290\pi\)
−0.377715 + 0.925922i \(0.623290\pi\)
\(758\) −17.5167 −0.636234
\(759\) 0 0
\(760\) 10.1962 0.369853
\(761\) −37.0000 −1.34125 −0.670624 0.741797i \(-0.733974\pi\)
−0.670624 + 0.741797i \(0.733974\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 11.6603 0.421853
\(765\) 0 0
\(766\) −35.7128 −1.29036
\(767\) −2.87564 −0.103833
\(768\) 0 0
\(769\) 35.5885 1.28335 0.641676 0.766976i \(-0.278239\pi\)
0.641676 + 0.766976i \(0.278239\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.85641 −0.318749
\(773\) −20.1244 −0.723823 −0.361911 0.932213i \(-0.617876\pi\)
−0.361911 + 0.932213i \(0.617876\pi\)
\(774\) 0 0
\(775\) 19.6077 0.704329
\(776\) −2.92820 −0.105116
\(777\) 0 0
\(778\) 6.53590 0.234323
\(779\) −25.8564 −0.926402
\(780\) 0 0
\(781\) −56.4974 −2.02164
\(782\) 43.3731 1.55102
\(783\) 0 0
\(784\) 0 0
\(785\) −3.73205 −0.133203
\(786\) 0 0
\(787\) 7.60770 0.271185 0.135593 0.990765i \(-0.456706\pi\)
0.135593 + 0.990765i \(0.456706\pi\)
\(788\) −25.7846 −0.918539
\(789\) 0 0
\(790\) 56.4449 2.00822
\(791\) 0 0
\(792\) 0 0
\(793\) −4.60770 −0.163624
\(794\) −21.0000 −0.745262
\(795\) 0 0
\(796\) −5.12436 −0.181628
\(797\) −29.4449 −1.04299 −0.521495 0.853254i \(-0.674626\pi\)
−0.521495 + 0.853254i \(0.674626\pi\)
\(798\) 0 0
\(799\) 8.87564 0.313998
\(800\) −8.92820 −0.315660
\(801\) 0 0
\(802\) 34.5167 1.21883
\(803\) 49.2295 1.73727
\(804\) 0 0
\(805\) 0 0
\(806\) 1.01924 0.0359011
\(807\) 0 0
\(808\) −4.92820 −0.173374
\(809\) 7.87564 0.276893 0.138446 0.990370i \(-0.455789\pi\)
0.138446 + 0.990370i \(0.455789\pi\)
\(810\) 0 0
\(811\) −7.80385 −0.274030 −0.137015 0.990569i \(-0.543751\pi\)
−0.137015 + 0.990569i \(0.543751\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −27.9474 −0.979557
\(815\) 24.3923 0.854425
\(816\) 0 0
\(817\) 14.9282 0.522272
\(818\) 34.6603 1.21187
\(819\) 0 0
\(820\) 35.3205 1.23345
\(821\) 12.0718 0.421309 0.210654 0.977561i \(-0.432441\pi\)
0.210654 + 0.977561i \(0.432441\pi\)
\(822\) 0 0
\(823\) −40.7846 −1.42166 −0.710831 0.703363i \(-0.751681\pi\)
−0.710831 + 0.703363i \(0.751681\pi\)
\(824\) 12.3923 0.431706
\(825\) 0 0
\(826\) 0 0
\(827\) 11.3205 0.393653 0.196826 0.980438i \(-0.436936\pi\)
0.196826 + 0.980438i \(0.436936\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 54.4449 1.88981
\(831\) 0 0
\(832\) −0.464102 −0.0160898
\(833\) 0 0
\(834\) 0 0
\(835\) −45.5167 −1.57517
\(836\) −11.4641 −0.396494
\(837\) 0 0
\(838\) −2.53590 −0.0876012
\(839\) 5.46410 0.188642 0.0943209 0.995542i \(-0.469932\pi\)
0.0943209 + 0.995542i \(0.469932\pi\)
\(840\) 0 0
\(841\) 42.6410 1.47038
\(842\) 24.1244 0.831380
\(843\) 0 0
\(844\) −20.7321 −0.713627
\(845\) 47.7128 1.64137
\(846\) 0 0
\(847\) 0 0
\(848\) −2.53590 −0.0870831
\(849\) 0 0
\(850\) 62.4974 2.14364
\(851\) −41.2679 −1.41465
\(852\) 0 0
\(853\) −5.71281 −0.195603 −0.0978015 0.995206i \(-0.531181\pi\)
−0.0978015 + 0.995206i \(0.531181\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.8564 −0.370848 −0.185424 0.982659i \(-0.559366\pi\)
−0.185424 + 0.982659i \(0.559366\pi\)
\(858\) 0 0
\(859\) −3.60770 −0.123093 −0.0615465 0.998104i \(-0.519603\pi\)
−0.0615465 + 0.998104i \(0.519603\pi\)
\(860\) −20.3923 −0.695372
\(861\) 0 0
\(862\) 21.4641 0.731070
\(863\) 17.1244 0.582920 0.291460 0.956583i \(-0.405859\pi\)
0.291460 + 0.956583i \(0.405859\pi\)
\(864\) 0 0
\(865\) 36.3205 1.23493
\(866\) 12.2679 0.416882
\(867\) 0 0
\(868\) 0 0
\(869\) −63.4641 −2.15287
\(870\) 0 0
\(871\) 1.51666 0.0513901
\(872\) 7.19615 0.243692
\(873\) 0 0
\(874\) −16.9282 −0.572605
\(875\) 0 0
\(876\) 0 0
\(877\) 8.51666 0.287587 0.143794 0.989608i \(-0.454070\pi\)
0.143794 + 0.989608i \(0.454070\pi\)
\(878\) −11.3205 −0.382049
\(879\) 0 0
\(880\) 15.6603 0.527907
\(881\) 18.2487 0.614815 0.307407 0.951578i \(-0.400539\pi\)
0.307407 + 0.951578i \(0.400539\pi\)
\(882\) 0 0
\(883\) 7.66025 0.257788 0.128894 0.991658i \(-0.458857\pi\)
0.128894 + 0.991658i \(0.458857\pi\)
\(884\) 3.24871 0.109266
\(885\) 0 0
\(886\) 18.7321 0.629316
\(887\) −2.44486 −0.0820905 −0.0410452 0.999157i \(-0.513069\pi\)
−0.0410452 + 0.999157i \(0.513069\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 14.6603 0.491413
\(891\) 0 0
\(892\) 18.5359 0.620628
\(893\) −3.46410 −0.115922
\(894\) 0 0
\(895\) −30.5885 −1.02246
\(896\) 0 0
\(897\) 0 0
\(898\) 11.8564 0.395653
\(899\) 18.5885 0.619960
\(900\) 0 0
\(901\) 17.7513 0.591381
\(902\) −39.7128 −1.32229
\(903\) 0 0
\(904\) 2.26795 0.0754309
\(905\) 16.3923 0.544899
\(906\) 0 0
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 5.07180 0.168313
\(909\) 0 0
\(910\) 0 0
\(911\) −6.24871 −0.207029 −0.103515 0.994628i \(-0.533009\pi\)
−0.103515 + 0.994628i \(0.533009\pi\)
\(912\) 0 0
\(913\) −61.2154 −2.02593
\(914\) −20.8564 −0.689869
\(915\) 0 0
\(916\) −4.46410 −0.147498
\(917\) 0 0
\(918\) 0 0
\(919\) −24.9808 −0.824039 −0.412020 0.911175i \(-0.635176\pi\)
−0.412020 + 0.911175i \(0.635176\pi\)
\(920\) 23.1244 0.762387
\(921\) 0 0
\(922\) 34.7846 1.14557
\(923\) −6.24871 −0.205679
\(924\) 0 0
\(925\) −59.4641 −1.95517
\(926\) 32.5885 1.07092
\(927\) 0 0
\(928\) −8.46410 −0.277848
\(929\) −45.4974 −1.49272 −0.746361 0.665541i \(-0.768201\pi\)
−0.746361 + 0.665541i \(0.768201\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −13.1962 −0.432254
\(933\) 0 0
\(934\) 14.5885 0.477349
\(935\) −109.622 −3.58502
\(936\) 0 0
\(937\) −53.8372 −1.75878 −0.879392 0.476099i \(-0.842050\pi\)
−0.879392 + 0.476099i \(0.842050\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 4.73205 0.154342
\(941\) 9.87564 0.321937 0.160968 0.986960i \(-0.448538\pi\)
0.160968 + 0.986960i \(0.448538\pi\)
\(942\) 0 0
\(943\) −58.6410 −1.90961
\(944\) 6.19615 0.201668
\(945\) 0 0
\(946\) 22.9282 0.745460
\(947\) 2.24871 0.0730733 0.0365366 0.999332i \(-0.488367\pi\)
0.0365366 + 0.999332i \(0.488367\pi\)
\(948\) 0 0
\(949\) 5.44486 0.176748
\(950\) −24.3923 −0.791391
\(951\) 0 0
\(952\) 0 0
\(953\) −10.4115 −0.337263 −0.168631 0.985679i \(-0.553935\pi\)
−0.168631 + 0.985679i \(0.553935\pi\)
\(954\) 0 0
\(955\) −43.5167 −1.40817
\(956\) 28.0526 0.907285
\(957\) 0 0
\(958\) 23.5167 0.759789
\(959\) 0 0
\(960\) 0 0
\(961\) −26.1769 −0.844417
\(962\) −3.09103 −0.0996590
\(963\) 0 0
\(964\) 17.7321 0.571111
\(965\) 33.0526 1.06400
\(966\) 0 0
\(967\) 13.6603 0.439284 0.219642 0.975581i \(-0.429511\pi\)
0.219642 + 0.975581i \(0.429511\pi\)
\(968\) −6.60770 −0.212379
\(969\) 0 0
\(970\) 10.9282 0.350883
\(971\) −33.1244 −1.06301 −0.531506 0.847055i \(-0.678374\pi\)
−0.531506 + 0.847055i \(0.678374\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 28.5885 0.916033
\(975\) 0 0
\(976\) 9.92820 0.317794
\(977\) 2.28719 0.0731736 0.0365868 0.999330i \(-0.488351\pi\)
0.0365868 + 0.999330i \(0.488351\pi\)
\(978\) 0 0
\(979\) −16.4833 −0.526810
\(980\) 0 0
\(981\) 0 0
\(982\) −33.4641 −1.06788
\(983\) 10.6410 0.339396 0.169698 0.985496i \(-0.445721\pi\)
0.169698 + 0.985496i \(0.445721\pi\)
\(984\) 0 0
\(985\) 96.2295 3.06613
\(986\) 59.2487 1.88686
\(987\) 0 0
\(988\) −1.26795 −0.0403388
\(989\) 33.8564 1.07657
\(990\) 0 0
\(991\) −10.3397 −0.328453 −0.164226 0.986423i \(-0.552513\pi\)
−0.164226 + 0.986423i \(0.552513\pi\)
\(992\) −2.19615 −0.0697279
\(993\) 0 0
\(994\) 0 0
\(995\) 19.1244 0.606283
\(996\) 0 0
\(997\) 7.24871 0.229569 0.114784 0.993390i \(-0.463382\pi\)
0.114784 + 0.993390i \(0.463382\pi\)
\(998\) −30.1962 −0.955843
\(999\) 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 7938.2.a.bg.1.1 2
3.2 odd 2 7938.2.a.bt.1.2 2
7.6 odd 2 1134.2.a.l.1.2 2
21.20 even 2 1134.2.a.m.1.1 yes 2
28.27 even 2 9072.2.a.bp.1.2 2
63.13 odd 6 1134.2.f.s.379.1 4
63.20 even 6 1134.2.f.r.757.2 4
63.34 odd 6 1134.2.f.s.757.1 4
63.41 even 6 1134.2.f.r.379.2 4
84.83 odd 2 9072.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 2 7.6 odd 2
1134.2.a.m.1.1 yes 2 21.20 even 2
1134.2.f.r.379.2 4 63.41 even 6
1134.2.f.r.757.2 4 63.20 even 6
1134.2.f.s.379.1 4 63.13 odd 6
1134.2.f.s.757.1 4 63.34 odd 6
7938.2.a.bg.1.1 2 1.1 even 1 trivial
7938.2.a.bt.1.2 2 3.2 odd 2
9072.2.a.y.1.1 2 84.83 odd 2
9072.2.a.bp.1.2 2 28.27 even 2