Properties

Label 4140.2.s.b.737.8
Level $4140$
Weight $2$
Character 4140.737
Analytic conductor $33.058$
Analytic rank $0$
Dimension $44$
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] = [4140,2,Mod(737,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.s (of order \(4\), degree \(2\), 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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 737.8
Character \(\chi\) \(=\) 4140.737
Dual form 4140.2.s.b.2393.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.07726 - 1.95947i) q^{5} +(2.36049 + 2.36049i) q^{7} +O(q^{10})\) \(q+(-1.07726 - 1.95947i) q^{5} +(2.36049 + 2.36049i) q^{7} -1.43372i q^{11} +(-4.27764 + 4.27764i) q^{13} +(0.739991 - 0.739991i) q^{17} -3.16505i q^{19} +(0.707107 + 0.707107i) q^{23} +(-2.67901 + 4.22172i) q^{25} -8.01732 q^{29} +7.15041 q^{31} +(2.08243 - 7.16817i) q^{35} +(-6.14782 - 6.14782i) q^{37} -10.2995i q^{41} +(-1.42388 + 1.42388i) q^{43} +(5.93386 - 5.93386i) q^{47} +4.14383i q^{49} +(-4.27375 - 4.27375i) q^{53} +(-2.80933 + 1.54450i) q^{55} +11.8478 q^{59} +10.6358 q^{61} +(12.9900 + 3.77375i) q^{65} +(-2.37607 - 2.37607i) q^{67} +2.71584i q^{71} +(-9.82434 + 9.82434i) q^{73} +(3.38429 - 3.38429i) q^{77} -10.7612i q^{79} +(-5.86966 - 5.86966i) q^{83} +(-2.24715 - 0.652822i) q^{85} +5.70441 q^{89} -20.1947 q^{91} +(-6.20180 + 3.40959i) q^{95} +(-9.11436 - 9.11436i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −1.07726 1.95947i −0.481767 0.876300i
\(6\) 0 0
\(7\) 2.36049 + 2.36049i 0.892182 + 0.892182i 0.994728 0.102547i \(-0.0326991\pi\)
−0.102547 + 0.994728i \(0.532699\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.43372i 0.432284i −0.976362 0.216142i \(-0.930653\pi\)
0.976362 0.216142i \(-0.0693474\pi\)
\(12\) 0 0
\(13\) −4.27764 + 4.27764i −1.18640 + 1.18640i −0.208350 + 0.978054i \(0.566809\pi\)
−0.978054 + 0.208350i \(0.933191\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.739991 0.739991i 0.179474 0.179474i −0.611652 0.791127i \(-0.709495\pi\)
0.791127 + 0.611652i \(0.209495\pi\)
\(18\) 0 0
\(19\) 3.16505i 0.726112i −0.931767 0.363056i \(-0.881733\pi\)
0.931767 0.363056i \(-0.118267\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.707107 + 0.707107i 0.147442 + 0.147442i
\(24\) 0 0
\(25\) −2.67901 + 4.22172i −0.535802 + 0.844344i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.01732 −1.48878 −0.744389 0.667746i \(-0.767259\pi\)
−0.744389 + 0.667746i \(0.767259\pi\)
\(30\) 0 0
\(31\) 7.15041 1.28425 0.642126 0.766599i \(-0.278053\pi\)
0.642126 + 0.766599i \(0.278053\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.08243 7.16817i 0.351995 1.21164i
\(36\) 0 0
\(37\) −6.14782 6.14782i −1.01070 1.01070i −0.999942 0.0107528i \(-0.996577\pi\)
−0.0107528 0.999942i \(-0.503423\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.2995i 1.60851i −0.594283 0.804256i \(-0.702564\pi\)
0.594283 0.804256i \(-0.297436\pi\)
\(42\) 0 0
\(43\) −1.42388 + 1.42388i −0.217140 + 0.217140i −0.807292 0.590152i \(-0.799068\pi\)
0.590152 + 0.807292i \(0.299068\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.93386 5.93386i 0.865542 0.865542i −0.126433 0.991975i \(-0.540353\pi\)
0.991975 + 0.126433i \(0.0403530\pi\)
\(48\) 0 0
\(49\) 4.14383i 0.591976i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.27375 4.27375i −0.587045 0.587045i 0.349785 0.936830i \(-0.386255\pi\)
−0.936830 + 0.349785i \(0.886255\pi\)
\(54\) 0 0
\(55\) −2.80933 + 1.54450i −0.378810 + 0.208260i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.8478 1.54245 0.771226 0.636562i \(-0.219644\pi\)
0.771226 + 0.636562i \(0.219644\pi\)
\(60\) 0 0
\(61\) 10.6358 1.36177 0.680886 0.732390i \(-0.261595\pi\)
0.680886 + 0.732390i \(0.261595\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.9900 + 3.77375i 1.61122 + 0.468076i
\(66\) 0 0
\(67\) −2.37607 2.37607i −0.290284 0.290284i 0.546909 0.837192i \(-0.315804\pi\)
−0.837192 + 0.546909i \(0.815804\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.71584i 0.322311i 0.986929 + 0.161155i \(0.0515220\pi\)
−0.986929 + 0.161155i \(0.948478\pi\)
\(72\) 0 0
\(73\) −9.82434 + 9.82434i −1.14985 + 1.14985i −0.163270 + 0.986581i \(0.552204\pi\)
−0.986581 + 0.163270i \(0.947796\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.38429 3.38429i 0.385676 0.385676i
\(78\) 0 0
\(79\) 10.7612i 1.21073i −0.795948 0.605365i \(-0.793027\pi\)
0.795948 0.605365i \(-0.206973\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.86966 5.86966i −0.644279 0.644279i 0.307325 0.951605i \(-0.400566\pi\)
−0.951605 + 0.307325i \(0.900566\pi\)
\(84\) 0 0
\(85\) −2.24715 0.652822i −0.243738 0.0708085i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.70441 0.604666 0.302333 0.953202i \(-0.402235\pi\)
0.302333 + 0.953202i \(0.402235\pi\)
\(90\) 0 0
\(91\) −20.1947 −2.11698
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.20180 + 3.40959i −0.636291 + 0.349816i
\(96\) 0 0
\(97\) −9.11436 9.11436i −0.925423 0.925423i 0.0719825 0.997406i \(-0.477067\pi\)
−0.997406 + 0.0719825i \(0.977067\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.3613i 1.52851i −0.644914 0.764255i \(-0.723107\pi\)
0.644914 0.764255i \(-0.276893\pi\)
\(102\) 0 0
\(103\) 5.37979 5.37979i 0.530087 0.530087i −0.390511 0.920598i \(-0.627702\pi\)
0.920598 + 0.390511i \(0.127702\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.5770 + 11.5770i −1.11919 + 1.11919i −0.127329 + 0.991861i \(0.540640\pi\)
−0.991861 + 0.127329i \(0.959360\pi\)
\(108\) 0 0
\(109\) 4.96184i 0.475259i −0.971356 0.237629i \(-0.923630\pi\)
0.971356 0.237629i \(-0.0763703\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.02913 + 7.02913i 0.661245 + 0.661245i 0.955674 0.294428i \(-0.0951292\pi\)
−0.294428 + 0.955674i \(0.595129\pi\)
\(114\) 0 0
\(115\) 0.623811 2.14729i 0.0581707 0.200236i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.49348 0.320247
\(120\) 0 0
\(121\) 8.94444 0.813131
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1583 + 0.701523i 0.998030 + 0.0627461i
\(126\) 0 0
\(127\) −11.2616 11.2616i −0.999308 0.999308i 0.000691828 1.00000i \(-0.499780\pi\)
−1.00000 0.000691828i \(0.999780\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.0880i 1.31825i −0.752034 0.659124i \(-0.770927\pi\)
0.752034 0.659124i \(-0.229073\pi\)
\(132\) 0 0
\(133\) 7.47107 7.47107i 0.647824 0.647824i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.6557 + 13.6557i −1.16668 + 1.16668i −0.183702 + 0.982982i \(0.558808\pi\)
−0.982982 + 0.183702i \(0.941192\pi\)
\(138\) 0 0
\(139\) 17.7226i 1.50321i 0.659615 + 0.751604i \(0.270719\pi\)
−0.659615 + 0.751604i \(0.729281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.13296 + 6.13296i 0.512864 + 0.512864i
\(144\) 0 0
\(145\) 8.63676 + 15.7097i 0.717244 + 1.30462i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.47737 −0.448723 −0.224362 0.974506i \(-0.572030\pi\)
−0.224362 + 0.974506i \(0.572030\pi\)
\(150\) 0 0
\(151\) 19.5207 1.58857 0.794285 0.607546i \(-0.207846\pi\)
0.794285 + 0.607546i \(0.207846\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.70287 14.0110i −0.618710 1.12539i
\(156\) 0 0
\(157\) −8.60289 8.60289i −0.686586 0.686586i 0.274890 0.961476i \(-0.411359\pi\)
−0.961476 + 0.274890i \(0.911359\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.33824i 0.263090i
\(162\) 0 0
\(163\) 7.10226 7.10226i 0.556292 0.556292i −0.371958 0.928250i \(-0.621313\pi\)
0.928250 + 0.371958i \(0.121313\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.27820 9.27820i 0.717969 0.717969i −0.250220 0.968189i \(-0.580503\pi\)
0.968189 + 0.250220i \(0.0805030\pi\)
\(168\) 0 0
\(169\) 23.5965i 1.81511i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.5243 + 10.5243i 0.800146 + 0.800146i 0.983118 0.182972i \(-0.0585717\pi\)
−0.182972 + 0.983118i \(0.558572\pi\)
\(174\) 0 0
\(175\) −16.2891 + 3.64155i −1.23134 + 0.275276i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.28105 −0.693698 −0.346849 0.937921i \(-0.612748\pi\)
−0.346849 + 0.937921i \(0.612748\pi\)
\(180\) 0 0
\(181\) −18.6502 −1.38626 −0.693129 0.720814i \(-0.743768\pi\)
−0.693129 + 0.720814i \(0.743768\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.42362 + 18.6693i −0.398752 + 1.37259i
\(186\) 0 0
\(187\) −1.06094 1.06094i −0.0775838 0.0775838i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.3782i 1.54687i −0.633875 0.773436i \(-0.718536\pi\)
0.633875 0.773436i \(-0.281464\pi\)
\(192\) 0 0
\(193\) 0.487819 0.487819i 0.0351140 0.0351140i −0.689332 0.724446i \(-0.742096\pi\)
0.724446 + 0.689332i \(0.242096\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8271 13.8271i 0.985139 0.985139i −0.0147517 0.999891i \(-0.504696\pi\)
0.999891 + 0.0147517i \(0.00469579\pi\)
\(198\) 0 0
\(199\) 4.28566i 0.303803i −0.988396 0.151901i \(-0.951460\pi\)
0.988396 0.151901i \(-0.0485396\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −18.9248 18.9248i −1.32826 1.32826i
\(204\) 0 0
\(205\) −20.1815 + 11.0953i −1.40954 + 0.774927i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.53780 −0.313886
\(210\) 0 0
\(211\) −20.5870 −1.41727 −0.708635 0.705575i \(-0.750689\pi\)
−0.708635 + 0.705575i \(0.750689\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.32394 + 1.25615i 0.294890 + 0.0856688i
\(216\) 0 0
\(217\) 16.8785 + 16.8785i 1.14579 + 1.14579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.33084i 0.425858i
\(222\) 0 0
\(223\) −12.6753 + 12.6753i −0.848804 + 0.848804i −0.989984 0.141180i \(-0.954910\pi\)
0.141180 + 0.989984i \(0.454910\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.36572 5.36572i 0.356135 0.356135i −0.506251 0.862386i \(-0.668969\pi\)
0.862386 + 0.506251i \(0.168969\pi\)
\(228\) 0 0
\(229\) 7.36542i 0.486721i −0.969936 0.243360i \(-0.921750\pi\)
0.969936 0.243360i \(-0.0782498\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.29910 4.29910i −0.281644 0.281644i 0.552121 0.833764i \(-0.313819\pi\)
−0.833764 + 0.552121i \(0.813819\pi\)
\(234\) 0 0
\(235\) −18.0195 5.23486i −1.17546 0.341485i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.87398 −0.444641 −0.222320 0.974974i \(-0.571363\pi\)
−0.222320 + 0.974974i \(0.571363\pi\)
\(240\) 0 0
\(241\) −9.52062 −0.613277 −0.306638 0.951826i \(-0.599204\pi\)
−0.306638 + 0.951826i \(0.599204\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.11970 4.46400i 0.518748 0.285194i
\(246\) 0 0
\(247\) 13.5389 + 13.5389i 0.861462 + 0.861462i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.30367i 0.271645i −0.990733 0.135822i \(-0.956632\pi\)
0.990733 0.135822i \(-0.0433677\pi\)
\(252\) 0 0
\(253\) 1.01380 1.01380i 0.0637368 0.0637368i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.15928 + 7.15928i −0.446584 + 0.446584i −0.894217 0.447634i \(-0.852267\pi\)
0.447634 + 0.894217i \(0.352267\pi\)
\(258\) 0 0
\(259\) 29.0237i 1.80345i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −19.1505 19.1505i −1.18087 1.18087i −0.979519 0.201354i \(-0.935466\pi\)
−0.201354 0.979519i \(-0.564534\pi\)
\(264\) 0 0
\(265\) −3.77031 + 12.9782i −0.231609 + 0.797246i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.4817 0.821993 0.410997 0.911637i \(-0.365181\pi\)
0.410997 + 0.911637i \(0.365181\pi\)
\(270\) 0 0
\(271\) 1.05275 0.0639498 0.0319749 0.999489i \(-0.489820\pi\)
0.0319749 + 0.999489i \(0.489820\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.05278 + 3.84096i 0.364996 + 0.231618i
\(276\) 0 0
\(277\) −9.67191 9.67191i −0.581129 0.581129i 0.354085 0.935213i \(-0.384792\pi\)
−0.935213 + 0.354085i \(0.884792\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.3868i 0.679278i −0.940556 0.339639i \(-0.889695\pi\)
0.940556 0.339639i \(-0.110305\pi\)
\(282\) 0 0
\(283\) −9.97976 + 9.97976i −0.593235 + 0.593235i −0.938504 0.345269i \(-0.887788\pi\)
0.345269 + 0.938504i \(0.387788\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.3119 24.3119i 1.43508 1.43508i
\(288\) 0 0
\(289\) 15.9048i 0.935578i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.79618 4.79618i −0.280196 0.280196i 0.552991 0.833187i \(-0.313486\pi\)
−0.833187 + 0.552991i \(0.813486\pi\)
\(294\) 0 0
\(295\) −12.7632 23.2153i −0.743102 1.35165i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.04950 −0.349852
\(300\) 0 0
\(301\) −6.72211 −0.387456
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.4575 20.8404i −0.656056 1.19332i
\(306\) 0 0
\(307\) 8.60169 + 8.60169i 0.490925 + 0.490925i 0.908597 0.417673i \(-0.137154\pi\)
−0.417673 + 0.908597i \(0.637154\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.7964i 1.51949i 0.650223 + 0.759743i \(0.274675\pi\)
−0.650223 + 0.759743i \(0.725325\pi\)
\(312\) 0 0
\(313\) 21.9766 21.9766i 1.24219 1.24219i 0.283103 0.959089i \(-0.408636\pi\)
0.959089 0.283103i \(-0.0913639\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.273414 0.273414i 0.0153565 0.0153565i −0.699387 0.714743i \(-0.746544\pi\)
0.714743 + 0.699387i \(0.246544\pi\)
\(318\) 0 0
\(319\) 11.4946i 0.643575i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.34211 2.34211i −0.130318 0.130318i
\(324\) 0 0
\(325\) −6.59916 29.5188i −0.366056 1.63741i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 28.0136 1.54444
\(330\) 0 0
\(331\) 6.82665 0.375227 0.187613 0.982243i \(-0.439925\pi\)
0.187613 + 0.982243i \(0.439925\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.09618 + 7.21549i −0.114526 + 0.394224i
\(336\) 0 0
\(337\) −8.47778 8.47778i −0.461814 0.461814i 0.437436 0.899250i \(-0.355887\pi\)
−0.899250 + 0.437436i \(0.855887\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.2517i 0.555161i
\(342\) 0 0
\(343\) 6.74195 6.74195i 0.364031 0.364031i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.2564 20.2564i 1.08742 1.08742i 0.0916254 0.995794i \(-0.470794\pi\)
0.995794 0.0916254i \(-0.0292062\pi\)
\(348\) 0 0
\(349\) 25.8771i 1.38517i −0.721336 0.692585i \(-0.756472\pi\)
0.721336 0.692585i \(-0.243528\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.52342 2.52342i −0.134308 0.134308i 0.636757 0.771065i \(-0.280276\pi\)
−0.771065 + 0.636757i \(0.780276\pi\)
\(354\) 0 0
\(355\) 5.32159 2.92567i 0.282441 0.155278i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.5358 −1.03106 −0.515530 0.856872i \(-0.672405\pi\)
−0.515530 + 0.856872i \(0.672405\pi\)
\(360\) 0 0
\(361\) 8.98247 0.472762
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 29.8338 + 8.66705i 1.56157 + 0.453654i
\(366\) 0 0
\(367\) 10.4560 + 10.4560i 0.545800 + 0.545800i 0.925223 0.379424i \(-0.123878\pi\)
−0.379424 + 0.925223i \(0.623878\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.1763i 1.04750i
\(372\) 0 0
\(373\) −26.7247 + 26.7247i −1.38375 + 1.38375i −0.545911 + 0.837843i \(0.683817\pi\)
−0.837843 + 0.545911i \(0.816183\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 34.2952 34.2952i 1.76629 1.76629i
\(378\) 0 0
\(379\) 25.5576i 1.31281i 0.754410 + 0.656404i \(0.227923\pi\)
−0.754410 + 0.656404i \(0.772077\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.77385 5.77385i −0.295030 0.295030i 0.544034 0.839063i \(-0.316896\pi\)
−0.839063 + 0.544034i \(0.816896\pi\)
\(384\) 0 0
\(385\) −10.2772 2.98563i −0.523773 0.152162i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.37661 −0.120499 −0.0602495 0.998183i \(-0.519190\pi\)
−0.0602495 + 0.998183i \(0.519190\pi\)
\(390\) 0 0
\(391\) 1.04651 0.0529241
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −21.0862 + 11.5926i −1.06096 + 0.583289i
\(396\) 0 0
\(397\) −3.01481 3.01481i −0.151309 0.151309i 0.627393 0.778702i \(-0.284122\pi\)
−0.778702 + 0.627393i \(0.784122\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.6096i 0.529816i −0.964274 0.264908i \(-0.914658\pi\)
0.964274 0.264908i \(-0.0853416\pi\)
\(402\) 0 0
\(403\) −30.5869 + 30.5869i −1.52364 + 1.52364i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.81427 + 8.81427i −0.436907 + 0.436907i
\(408\) 0 0
\(409\) 1.68918i 0.0835243i 0.999128 + 0.0417622i \(0.0132972\pi\)
−0.999128 + 0.0417622i \(0.986703\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 27.9666 + 27.9666i 1.37615 + 1.37615i
\(414\) 0 0
\(415\) −5.17823 + 17.8246i −0.254189 + 0.874974i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −37.4445 −1.82928 −0.914641 0.404266i \(-0.867527\pi\)
−0.914641 + 0.404266i \(0.867527\pi\)
\(420\) 0 0
\(421\) 23.9493 1.16722 0.583609 0.812035i \(-0.301640\pi\)
0.583609 + 0.812035i \(0.301640\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.14159 + 5.10648i 0.0553754 + 0.247701i
\(426\) 0 0
\(427\) 25.1057 + 25.1057i 1.21495 + 1.21495i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.53602i 0.411166i 0.978640 + 0.205583i \(0.0659090\pi\)
−0.978640 + 0.205583i \(0.934091\pi\)
\(432\) 0 0
\(433\) −4.04078 + 4.04078i −0.194187 + 0.194187i −0.797503 0.603315i \(-0.793846\pi\)
0.603315 + 0.797503i \(0.293846\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.23803 2.23803i 0.107059 0.107059i
\(438\) 0 0
\(439\) 17.6291i 0.841389i 0.907202 + 0.420695i \(0.138214\pi\)
−0.907202 + 0.420695i \(0.861786\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.60747 8.60747i −0.408953 0.408953i 0.472420 0.881373i \(-0.343380\pi\)
−0.881373 + 0.472420i \(0.843380\pi\)
\(444\) 0 0
\(445\) −6.14515 11.1776i −0.291308 0.529868i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.16266 −0.432413 −0.216206 0.976348i \(-0.569368\pi\)
−0.216206 + 0.976348i \(0.569368\pi\)
\(450\) 0 0
\(451\) −14.7666 −0.695334
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 21.7550 + 39.5708i 1.01989 + 1.85511i
\(456\) 0 0
\(457\) 22.3611 + 22.3611i 1.04601 + 1.04601i 0.998889 + 0.0471209i \(0.0150046\pi\)
0.0471209 + 0.998889i \(0.484995\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 33.8833i 1.57810i −0.614328 0.789051i \(-0.710573\pi\)
0.614328 0.789051i \(-0.289427\pi\)
\(462\) 0 0
\(463\) 28.7582 28.7582i 1.33651 1.33651i 0.437090 0.899418i \(-0.356009\pi\)
0.899418 0.437090i \(-0.143991\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.54232 + 7.54232i −0.349017 + 0.349017i −0.859743 0.510726i \(-0.829376\pi\)
0.510726 + 0.859743i \(0.329376\pi\)
\(468\) 0 0
\(469\) 11.2174i 0.517971i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.04145 + 2.04145i 0.0938660 + 0.0938660i
\(474\) 0 0
\(475\) 13.3619 + 8.47919i 0.613088 + 0.389052i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −42.6425 −1.94839 −0.974193 0.225716i \(-0.927528\pi\)
−0.974193 + 0.225716i \(0.927528\pi\)
\(480\) 0 0
\(481\) 52.5963 2.39819
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.04071 + 27.6778i −0.365110 + 1.25679i
\(486\) 0 0
\(487\) 16.7925 + 16.7925i 0.760939 + 0.760939i 0.976492 0.215553i \(-0.0691554\pi\)
−0.215553 + 0.976492i \(0.569155\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.6111i 0.704519i 0.935902 + 0.352259i \(0.114587\pi\)
−0.935902 + 0.352259i \(0.885413\pi\)
\(492\) 0 0
\(493\) −5.93274 + 5.93274i −0.267197 + 0.267197i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.41071 + 6.41071i −0.287560 + 0.287560i
\(498\) 0 0
\(499\) 28.5955i 1.28011i −0.768328 0.640056i \(-0.778911\pi\)
0.768328 0.640056i \(-0.221089\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −13.9948 13.9948i −0.623999 0.623999i 0.322552 0.946552i \(-0.395459\pi\)
−0.946552 + 0.322552i \(0.895459\pi\)
\(504\) 0 0
\(505\) −30.1000 + 16.5482i −1.33943 + 0.736385i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.22752 −0.0544087 −0.0272043 0.999630i \(-0.508660\pi\)
−0.0272043 + 0.999630i \(0.508660\pi\)
\(510\) 0 0
\(511\) −46.3805 −2.05175
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.3370 4.74607i −0.719893 0.209137i
\(516\) 0 0
\(517\) −8.50751 8.50751i −0.374160 0.374160i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.59240i 0.420251i 0.977674 + 0.210125i \(0.0673872\pi\)
−0.977674 + 0.210125i \(0.932613\pi\)
\(522\) 0 0
\(523\) −5.36681 + 5.36681i −0.234674 + 0.234674i −0.814641 0.579966i \(-0.803066\pi\)
0.579966 + 0.814641i \(0.303066\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.29124 5.29124i 0.230490 0.230490i
\(528\) 0 0
\(529\) 1.00000i 0.0434783i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 44.0576 + 44.0576i 1.90835 + 1.90835i
\(534\) 0 0
\(535\) 35.1562 + 10.2132i 1.51993 + 0.441557i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.94111 0.255902
\(540\) 0 0
\(541\) 43.4522 1.86815 0.934077 0.357071i \(-0.116225\pi\)
0.934077 + 0.357071i \(0.116225\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.72256 + 5.34521i −0.416469 + 0.228964i
\(546\) 0 0
\(547\) 26.5763 + 26.5763i 1.13632 + 1.13632i 0.989104 + 0.147216i \(0.0470313\pi\)
0.147216 + 0.989104i \(0.452969\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 25.3752i 1.08102i
\(552\) 0 0
\(553\) 25.4017 25.4017i 1.08019 1.08019i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.59700 2.59700i 0.110038 0.110038i −0.649944 0.759982i \(-0.725208\pi\)
0.759982 + 0.649944i \(0.225208\pi\)
\(558\) 0 0
\(559\) 12.1817i 0.515231i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.12684 + 8.12684i 0.342506 + 0.342506i 0.857309 0.514803i \(-0.172135\pi\)
−0.514803 + 0.857309i \(0.672135\pi\)
\(564\) 0 0
\(565\) 6.20112 21.3456i 0.260883 0.898015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.4711 1.36126 0.680630 0.732627i \(-0.261706\pi\)
0.680630 + 0.732627i \(0.261706\pi\)
\(570\) 0 0
\(571\) 31.3156 1.31052 0.655258 0.755405i \(-0.272560\pi\)
0.655258 + 0.755405i \(0.272560\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.87955 + 1.09086i −0.203491 + 0.0454920i
\(576\) 0 0
\(577\) 15.0525 + 15.0525i 0.626645 + 0.626645i 0.947222 0.320578i \(-0.103877\pi\)
−0.320578 + 0.947222i \(0.603877\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 27.7106i 1.14963i
\(582\) 0 0
\(583\) −6.12738 + 6.12738i −0.253770 + 0.253770i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.09099 + 5.09099i −0.210128 + 0.210128i −0.804322 0.594194i \(-0.797471\pi\)
0.594194 + 0.804322i \(0.297471\pi\)
\(588\) 0 0
\(589\) 22.6314i 0.932510i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.1780 + 20.1780i 0.828611 + 0.828611i 0.987325 0.158714i \(-0.0507348\pi\)
−0.158714 + 0.987325i \(0.550735\pi\)
\(594\) 0 0
\(595\) −3.76340 6.84536i −0.154284 0.280632i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.99628 0.285860 0.142930 0.989733i \(-0.454348\pi\)
0.142930 + 0.989733i \(0.454348\pi\)
\(600\) 0 0
\(601\) 3.66796 0.149619 0.0748096 0.997198i \(-0.476165\pi\)
0.0748096 + 0.997198i \(0.476165\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.63551 17.5263i −0.391739 0.712546i
\(606\) 0 0
\(607\) −0.438034 0.438034i −0.0177793 0.0177793i 0.698161 0.715941i \(-0.254002\pi\)
−0.715941 + 0.698161i \(0.754002\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.7658i 2.05377i
\(612\) 0 0
\(613\) −3.34350 + 3.34350i −0.135043 + 0.135043i −0.771397 0.636354i \(-0.780442\pi\)
0.636354 + 0.771397i \(0.280442\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.22022 + 8.22022i −0.330934 + 0.330934i −0.852941 0.522007i \(-0.825183\pi\)
0.522007 + 0.852941i \(0.325183\pi\)
\(618\) 0 0
\(619\) 17.5378i 0.704904i 0.935830 + 0.352452i \(0.114652\pi\)
−0.935830 + 0.352452i \(0.885348\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.4652 + 13.4652i 0.539472 + 0.539472i
\(624\) 0 0
\(625\) −10.6458 22.6200i −0.425833 0.904802i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.09866 −0.362787
\(630\) 0 0
\(631\) 1.04232 0.0414943 0.0207471 0.999785i \(-0.493396\pi\)
0.0207471 + 0.999785i \(0.493396\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.93503 + 34.1985i −0.394260 + 1.35713i
\(636\) 0 0
\(637\) −17.7258 17.7258i −0.702323 0.702323i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21.4308i 0.846465i −0.906021 0.423232i \(-0.860895\pi\)
0.906021 0.423232i \(-0.139105\pi\)
\(642\) 0 0
\(643\) −7.38419 + 7.38419i −0.291204 + 0.291204i −0.837556 0.546352i \(-0.816016\pi\)
0.546352 + 0.837556i \(0.316016\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.475908 + 0.475908i −0.0187099 + 0.0187099i −0.716400 0.697690i \(-0.754211\pi\)
0.697690 + 0.716400i \(0.254211\pi\)
\(648\) 0 0
\(649\) 16.9865i 0.666777i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.8215 18.8215i −0.736543 0.736543i 0.235364 0.971907i \(-0.424372\pi\)
−0.971907 + 0.235364i \(0.924372\pi\)
\(654\) 0 0
\(655\) −29.5645 + 16.2538i −1.15518 + 0.635088i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.08299 −0.0421874 −0.0210937 0.999778i \(-0.506715\pi\)
−0.0210937 + 0.999778i \(0.506715\pi\)
\(660\) 0 0
\(661\) −34.8950 −1.35726 −0.678628 0.734482i \(-0.737425\pi\)
−0.678628 + 0.734482i \(0.737425\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −22.6876 6.59099i −0.879787 0.255588i
\(666\) 0 0
\(667\) −5.66910 5.66910i −0.219508 0.219508i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.2488i 0.588672i
\(672\) 0 0
\(673\) 31.2082 31.2082i 1.20299 1.20299i 0.229734 0.973254i \(-0.426215\pi\)
0.973254 0.229734i \(-0.0737855\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.73333 3.73333i 0.143484 0.143484i −0.631716 0.775200i \(-0.717649\pi\)
0.775200 + 0.631716i \(0.217649\pi\)
\(678\) 0 0
\(679\) 43.0287i 1.65129i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.1671 + 26.1671i 1.00126 + 1.00126i 0.999999 + 0.00125685i \(0.000400068\pi\)
0.00125685 + 0.999999i \(0.499600\pi\)
\(684\) 0 0
\(685\) 41.4686 + 12.0471i 1.58443 + 0.460295i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 36.5632 1.39295
\(690\) 0 0
\(691\) −44.3255 −1.68622 −0.843111 0.537739i \(-0.819279\pi\)
−0.843111 + 0.537739i \(0.819279\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 34.7267 19.0919i 1.31726 0.724195i
\(696\) 0 0
\(697\) −7.62154 7.62154i −0.288686 0.288686i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 20.8805i 0.788647i 0.918972 + 0.394323i \(0.129021\pi\)
−0.918972 + 0.394323i \(0.870979\pi\)
\(702\) 0 0
\(703\) −19.4581 + 19.4581i −0.733878 + 0.733878i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.2603 36.2603i 1.36371 1.36371i
\(708\) 0 0
\(709\) 13.5595i 0.509239i −0.967041 0.254619i \(-0.918050\pi\)
0.967041 0.254619i \(-0.0819502\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.05610 + 5.05610i 0.189353 + 0.189353i
\(714\) 0 0
\(715\) 5.41051 18.6241i 0.202342 0.696503i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.2048 −0.716217 −0.358108 0.933680i \(-0.616578\pi\)
−0.358108 + 0.933680i \(0.616578\pi\)
\(720\) 0 0
\(721\) 25.3979 0.945867
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.4785 33.8469i 0.797690 1.25704i
\(726\) 0 0
\(727\) −0.0313047 0.0313047i −0.00116103 0.00116103i 0.706526 0.707687i \(-0.250261\pi\)
−0.707687 + 0.706526i \(0.750261\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.10732i 0.0779420i
\(732\) 0 0
\(733\) −29.8085 + 29.8085i −1.10100 + 1.10100i −0.106713 + 0.994290i \(0.534033\pi\)
−0.994290 + 0.106713i \(0.965967\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.40663 + 3.40663i −0.125485 + 0.125485i
\(738\) 0 0
\(739\) 2.55333i 0.0939256i 0.998897 + 0.0469628i \(0.0149542\pi\)
−0.998897 + 0.0469628i \(0.985046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.5941 + 23.5941i 0.865583 + 0.865583i 0.991980 0.126397i \(-0.0403413\pi\)
−0.126397 + 0.991980i \(0.540341\pi\)
\(744\) 0 0
\(745\) 5.90056 + 10.7327i 0.216180 + 0.393216i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −54.6547 −1.99704
\(750\) 0 0
\(751\) 14.4024 0.525553 0.262776 0.964857i \(-0.415362\pi\)
0.262776 + 0.964857i \(0.415362\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.0289 38.2501i −0.765320 1.39206i
\(756\) 0 0
\(757\) 11.5333 + 11.5333i 0.419185 + 0.419185i 0.884923 0.465738i \(-0.154211\pi\)
−0.465738 + 0.884923i \(0.654211\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.1691i 1.63738i 0.574237 + 0.818689i \(0.305299\pi\)
−0.574237 + 0.818689i \(0.694701\pi\)
\(762\) 0 0
\(763\) 11.7124 11.7124i 0.424017 0.424017i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −50.6806 + 50.6806i −1.82997 + 1.82997i
\(768\) 0 0
\(769\) 37.6458i 1.35754i −0.734350 0.678771i \(-0.762513\pi\)
0.734350 0.678771i \(-0.237487\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.4598 + 14.4598i 0.520082 + 0.520082i 0.917596 0.397514i \(-0.130127\pi\)
−0.397514 + 0.917596i \(0.630127\pi\)
\(774\) 0 0
\(775\) −19.1560 + 30.1870i −0.688104 + 1.08435i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −32.5984 −1.16796
\(780\) 0 0
\(781\) 3.89376 0.139330
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.58949 + 26.1247i −0.270881 + 0.932429i
\(786\) 0 0
\(787\) 31.6312 + 31.6312i 1.12753 + 1.12753i 0.990578 + 0.136953i \(0.0437309\pi\)
0.136953 + 0.990578i \(0.456269\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 33.1844i 1.17990i
\(792\) 0 0
\(793\) −45.4961 + 45.4961i −1.61561 + 1.61561i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.2875 + 19.2875i −0.683199 + 0.683199i −0.960720 0.277521i \(-0.910487\pi\)
0.277521 + 0.960720i \(0.410487\pi\)
\(798\) 0 0
\(799\) 8.78200i 0.310685i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.0854 + 14.0854i 0.497062 + 0.497062i
\(804\) 0 0
\(805\) 6.54116 3.59616i 0.230546 0.126748i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.2075 −0.569825 −0.284913 0.958553i \(-0.591965\pi\)
−0.284913 + 0.958553i \(0.591965\pi\)
\(810\) 0 0
\(811\) 26.6360 0.935318 0.467659 0.883909i \(-0.345098\pi\)
0.467659 + 0.883909i \(0.345098\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −21.5676 6.26563i −0.755481 0.219475i
\(816\) 0 0
\(817\) 4.50665 + 4.50665i 0.157668 + 0.157668i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.3930i 0.432517i 0.976336 + 0.216259i \(0.0693854\pi\)
−0.976336 + 0.216259i \(0.930615\pi\)
\(822\) 0 0
\(823\) −1.33177 + 1.33177i −0.0464227 + 0.0464227i −0.729937 0.683514i \(-0.760451\pi\)
0.683514 + 0.729937i \(0.260451\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.44923 + 9.44923i −0.328582 + 0.328582i −0.852047 0.523465i \(-0.824639\pi\)
0.523465 + 0.852047i \(0.324639\pi\)
\(828\) 0 0
\(829\) 21.7068i 0.753910i −0.926232 0.376955i \(-0.876971\pi\)
0.926232 0.376955i \(-0.123029\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.06640 + 3.06640i 0.106244 + 0.106244i
\(834\) 0 0
\(835\) −28.1754 8.18525i −0.975049 0.283262i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.65951 −0.0918166 −0.0459083 0.998946i \(-0.514618\pi\)
−0.0459083 + 0.998946i \(0.514618\pi\)
\(840\) 0 0
\(841\) 35.2774 1.21646
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −46.2364 + 25.4196i −1.59058 + 0.874460i
\(846\) 0 0
\(847\) 21.1133 + 21.1133i 0.725460 + 0.725460i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.69433i 0.298038i
\(852\) 0 0
\(853\) −25.9284 + 25.9284i −0.887773 + 0.887773i −0.994309 0.106536i \(-0.966024\pi\)
0.106536 + 0.994309i \(0.466024\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25.3097 + 25.3097i −0.864562 + 0.864562i −0.991864 0.127302i \(-0.959368\pi\)
0.127302 + 0.991864i \(0.459368\pi\)
\(858\) 0 0
\(859\) 5.26398i 0.179605i 0.995960 + 0.0898024i \(0.0286235\pi\)
−0.995960 + 0.0898024i \(0.971376\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.41629 8.41629i −0.286494 0.286494i 0.549198 0.835692i \(-0.314933\pi\)
−0.835692 + 0.549198i \(0.814933\pi\)
\(864\) 0 0
\(865\) 9.28454 31.9594i 0.315684 1.08665i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −15.4286 −0.523379
\(870\) 0 0
\(871\) 20.3280 0.688788
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.6831 + 27.9950i 0.834443 + 0.946405i
\(876\) 0 0
\(877\) −7.62691 7.62691i −0.257542 0.257542i 0.566511 0.824054i \(-0.308293\pi\)
−0.824054 + 0.566511i \(0.808293\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 11.0296i 0.371598i −0.982588 0.185799i \(-0.940513\pi\)
0.982588 0.185799i \(-0.0594873\pi\)
\(882\) 0 0
\(883\) −34.0579 + 34.0579i −1.14614 + 1.14614i −0.158835 + 0.987305i \(0.550774\pi\)
−0.987305 + 0.158835i \(0.949226\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.44680 7.44680i 0.250039 0.250039i −0.570948 0.820987i \(-0.693424\pi\)
0.820987 + 0.570948i \(0.193424\pi\)
\(888\) 0 0
\(889\) 53.1659i 1.78313i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −18.7809 18.7809i −0.628480 0.628480i
\(894\) 0 0
\(895\) 9.99813 + 18.1859i 0.334201 + 0.607887i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −57.3271 −1.91197
\(900\) 0 0
\(901\) −6.32508 −0.210719
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.0912 + 36.5444i 0.667852 + 1.21478i
\(906\) 0 0
\(907\) −6.91758 6.91758i −0.229694 0.229694i 0.582871 0.812565i \(-0.301929\pi\)
−0.812565 + 0.582871i \(0.801929\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 40.8344i 1.35290i −0.736487 0.676452i \(-0.763517\pi\)
0.736487 0.676452i \(-0.236483\pi\)
\(912\) 0 0
\(913\) −8.41548 + 8.41548i −0.278512 + 0.278512i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 35.6152 35.6152i 1.17612 1.17612i
\(918\) 0 0
\(919\) 1.62017i 0.0534445i 0.999643 + 0.0267223i \(0.00850697\pi\)
−0.999643 + 0.0267223i \(0.991493\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11.6174 11.6174i −0.382391 0.382391i
\(924\) 0 0
\(925\) 42.4244 9.48430i 1.39491 0.311842i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.1310 −0.726093 −0.363047 0.931771i \(-0.618263\pi\)
−0.363047 + 0.931771i \(0.618263\pi\)
\(930\) 0 0
\(931\) 13.1154 0.429841
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.935966 + 3.22180i −0.0306094 + 0.105364i
\(936\) 0 0
\(937\) 22.2383 + 22.2383i 0.726494 + 0.726494i 0.969920 0.243425i \(-0.0782711\pi\)
−0.243425 + 0.969920i \(0.578271\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.2838i 0.628633i 0.949318 + 0.314316i \(0.101775\pi\)
−0.949318 + 0.314316i \(0.898225\pi\)
\(942\) 0 0
\(943\) 7.28285 7.28285i 0.237162 0.237162i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.564137 0.564137i 0.0183320 0.0183320i −0.697881 0.716213i \(-0.745874\pi\)
0.716213 + 0.697881i \(0.245874\pi\)
\(948\) 0 0
\(949\) 84.0500i 2.72838i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.91380 5.91380i −0.191567 0.191567i 0.604806 0.796373i \(-0.293251\pi\)
−0.796373 + 0.604806i \(0.793251\pi\)
\(954\) 0 0
\(955\) −41.8898 + 23.0299i −1.35552 + 0.745231i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −64.4683 −2.08179
\(960\) 0 0
\(961\) 20.1284 0.649302
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.48137 0.430355i −0.0476871 0.0138536i
\(966\) 0 0
\(967\) 3.14135 + 3.14135i 0.101019 + 0.101019i 0.755810 0.654791i \(-0.227243\pi\)
−0.654791 + 0.755810i \(0.727243\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 50.8627i 1.63226i −0.577867 0.816131i \(-0.696115\pi\)
0.577867 0.816131i \(-0.303885\pi\)
\(972\) 0 0
\(973\) −41.8339 + 41.8339i −1.34113 + 1.34113i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.54880 + 5.54880i −0.177522 + 0.177522i −0.790275 0.612753i \(-0.790062\pi\)
0.612753 + 0.790275i \(0.290062\pi\)
\(978\) 0 0
\(979\) 8.17854i 0.261387i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.36966 + 7.36966i 0.235056 + 0.235056i 0.814799 0.579743i \(-0.196847\pi\)
−0.579743 + 0.814799i \(0.696847\pi\)
\(984\) 0 0
\(985\) −41.9891 12.1983i −1.33788 0.388670i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.01367 −0.0640310
\(990\) 0 0
\(991\) 14.9125 0.473711 0.236855 0.971545i \(-0.423883\pi\)
0.236855 + 0.971545i \(0.423883\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.39761 + 4.61679i −0.266222 + 0.146362i
\(996\) 0 0
\(997\) −2.25187 2.25187i −0.0713175 0.0713175i 0.670548 0.741866i \(-0.266059\pi\)
−0.741866 + 0.670548i \(0.766059\pi\)
\(998\) 0 0
\(999\) 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 4140.2.s.b.737.8 44
3.2 odd 2 inner 4140.2.s.b.737.15 yes 44
5.3 odd 4 inner 4140.2.s.b.2393.15 yes 44
15.8 even 4 inner 4140.2.s.b.2393.8 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.s.b.737.8 44 1.1 even 1 trivial
4140.2.s.b.737.15 yes 44 3.2 odd 2 inner
4140.2.s.b.2393.8 yes 44 15.8 even 4 inner
4140.2.s.b.2393.15 yes 44 5.3 odd 4 inner