Properties

Label 2394.2.e.c.1063.11
Level $2394$
Weight $2$
Character 2394.1063
Analytic conductor $19.116$
Analytic rank $0$
Dimension $16$
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] = [2394,2,Mod(1063,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2394.1063");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.e (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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 44x^{14} + 708x^{12} + 5378x^{10} + 20592x^{8} + 38856x^{6} + 33265x^{4} + 10216x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 266)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1063.11
Root \(-2.30643i\) of defining polynomial
Character \(\chi\) \(=\) 2394.1063
Dual form 2394.2.e.c.1063.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -2.68862i q^{5} +(-0.592978 - 2.57844i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -2.68862i q^{5} +(-0.592978 - 2.57844i) q^{7} -1.00000i q^{8} +2.68862 q^{10} +1.22869 q^{11} +5.01856 q^{13} +(2.57844 - 0.592978i) q^{14} +1.00000 q^{16} +5.15689i q^{17} +(2.75779 + 3.37559i) q^{19} +2.68862i q^{20} +1.22869i q^{22} -3.18596 q^{23} -2.22869 q^{25} +5.01856i q^{26} +(0.592978 + 2.57844i) q^{28} -4.47509i q^{29} +6.61285 q^{31} +1.00000i q^{32} -5.15689 q^{34} +(-6.93246 + 1.59429i) q^{35} -3.74651i q^{37} +(-3.37559 + 2.75779i) q^{38} -2.68862 q^{40} -0.873974 q^{41} +8.11842 q^{43} -1.22869 q^{44} -3.18596i q^{46} -7.36603i q^{47} +(-6.29675 + 3.05792i) q^{49} -2.22869i q^{50} -5.01856 q^{52} -1.58536i q^{53} -3.30347i q^{55} +(-2.57844 + 0.592978i) q^{56} +4.47509 q^{58} -2.46827 q^{59} -9.69893i q^{61} +6.61285i q^{62} -1.00000 q^{64} -13.4930i q^{65} -0.728584i q^{67} -5.15689i q^{68} +(-1.59429 - 6.93246i) q^{70} -5.14323i q^{71} -6.11584i q^{73} +3.74651 q^{74} +(-2.75779 - 3.37559i) q^{76} +(-0.728584 - 3.16810i) q^{77} -9.74651i q^{79} -2.68862i q^{80} -0.873974i q^{82} +1.81465i q^{83} +13.8649 q^{85} +8.11842i q^{86} -1.22869i q^{88} +3.38845 q^{89} +(-2.97590 - 12.9401i) q^{91} +3.18596 q^{92} +7.36603 q^{94} +(9.07569 - 7.41464i) q^{95} -3.22440 q^{97} +(-3.05792 - 6.29675i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 6 q^{7} + 12 q^{11} + 16 q^{16} - 20 q^{23} - 28 q^{25} - 6 q^{28} - 8 q^{35} - 4 q^{43} - 12 q^{44} + 10 q^{49} - 16 q^{58} - 16 q^{64} - 12 q^{74} + 4 q^{77} + 16 q^{85} + 20 q^{92} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.68862i 1.20239i −0.799103 0.601194i \(-0.794692\pi\)
0.799103 0.601194i \(-0.205308\pi\)
\(6\) 0 0
\(7\) −0.592978 2.57844i −0.224125 0.974560i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.68862 0.850217
\(11\) 1.22869 0.370463 0.185231 0.982695i \(-0.440697\pi\)
0.185231 + 0.982695i \(0.440697\pi\)
\(12\) 0 0
\(13\) 5.01856 1.39190 0.695949 0.718091i \(-0.254984\pi\)
0.695949 + 0.718091i \(0.254984\pi\)
\(14\) 2.57844 0.592978i 0.689118 0.158480i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.15689i 1.25073i 0.780333 + 0.625365i \(0.215050\pi\)
−0.780333 + 0.625365i \(0.784950\pi\)
\(18\) 0 0
\(19\) 2.75779 + 3.37559i 0.632679 + 0.774414i
\(20\) 2.68862i 0.601194i
\(21\) 0 0
\(22\) 1.22869i 0.261957i
\(23\) −3.18596 −0.664318 −0.332159 0.943223i \(-0.607777\pi\)
−0.332159 + 0.943223i \(0.607777\pi\)
\(24\) 0 0
\(25\) −2.22869 −0.445737
\(26\) 5.01856i 0.984221i
\(27\) 0 0
\(28\) 0.592978 + 2.57844i 0.112062 + 0.487280i
\(29\) 4.47509i 0.831003i −0.909592 0.415502i \(-0.863606\pi\)
0.909592 0.415502i \(-0.136394\pi\)
\(30\) 0 0
\(31\) 6.61285 1.18770 0.593852 0.804574i \(-0.297607\pi\)
0.593852 + 0.804574i \(0.297607\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −5.15689 −0.884399
\(35\) −6.93246 + 1.59429i −1.17180 + 0.269485i
\(36\) 0 0
\(37\) 3.74651i 0.615922i −0.951399 0.307961i \(-0.900353\pi\)
0.951399 0.307961i \(-0.0996466\pi\)
\(38\) −3.37559 + 2.75779i −0.547593 + 0.447372i
\(39\) 0 0
\(40\) −2.68862 −0.425108
\(41\) −0.873974 −0.136492 −0.0682459 0.997669i \(-0.521740\pi\)
−0.0682459 + 0.997669i \(0.521740\pi\)
\(42\) 0 0
\(43\) 8.11842 1.23805 0.619024 0.785372i \(-0.287529\pi\)
0.619024 + 0.785372i \(0.287529\pi\)
\(44\) −1.22869 −0.185231
\(45\) 0 0
\(46\) 3.18596i 0.469744i
\(47\) 7.36603i 1.07445i −0.843440 0.537223i \(-0.819473\pi\)
0.843440 0.537223i \(-0.180527\pi\)
\(48\) 0 0
\(49\) −6.29675 + 3.05792i −0.899536 + 0.436846i
\(50\) 2.22869i 0.315184i
\(51\) 0 0
\(52\) −5.01856 −0.695949
\(53\) 1.58536i 0.217766i −0.994055 0.108883i \(-0.965273\pi\)
0.994055 0.108883i \(-0.0347273\pi\)
\(54\) 0 0
\(55\) 3.30347i 0.445440i
\(56\) −2.57844 + 0.592978i −0.344559 + 0.0792400i
\(57\) 0 0
\(58\) 4.47509 0.587608
\(59\) −2.46827 −0.321341 −0.160671 0.987008i \(-0.551366\pi\)
−0.160671 + 0.987008i \(0.551366\pi\)
\(60\) 0 0
\(61\) 9.69893i 1.24182i −0.783882 0.620910i \(-0.786763\pi\)
0.783882 0.620910i \(-0.213237\pi\)
\(62\) 6.61285i 0.839833i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 13.4930i 1.67360i
\(66\) 0 0
\(67\) 0.728584i 0.0890107i −0.999009 0.0445053i \(-0.985829\pi\)
0.999009 0.0445053i \(-0.0141712\pi\)
\(68\) 5.15689i 0.625365i
\(69\) 0 0
\(70\) −1.59429 6.93246i −0.190555 0.828588i
\(71\) 5.14323i 0.610389i −0.952290 0.305194i \(-0.901279\pi\)
0.952290 0.305194i \(-0.0987214\pi\)
\(72\) 0 0
\(73\) 6.11584i 0.715805i −0.933759 0.357903i \(-0.883492\pi\)
0.933759 0.357903i \(-0.116508\pi\)
\(74\) 3.74651 0.435522
\(75\) 0 0
\(76\) −2.75779 3.37559i −0.316340 0.387207i
\(77\) −0.728584 3.16810i −0.0830298 0.361038i
\(78\) 0 0
\(79\) 9.74651i 1.09657i −0.836292 0.548284i \(-0.815281\pi\)
0.836292 0.548284i \(-0.184719\pi\)
\(80\) 2.68862i 0.300597i
\(81\) 0 0
\(82\) 0.873974i 0.0965143i
\(83\) 1.81465i 0.199183i 0.995028 + 0.0995917i \(0.0317537\pi\)
−0.995028 + 0.0995917i \(0.968246\pi\)
\(84\) 0 0
\(85\) 13.8649 1.50386
\(86\) 8.11842i 0.875432i
\(87\) 0 0
\(88\) 1.22869i 0.130978i
\(89\) 3.38845 0.359175 0.179588 0.983742i \(-0.442524\pi\)
0.179588 + 0.983742i \(0.442524\pi\)
\(90\) 0 0
\(91\) −2.97590 12.9401i −0.311959 1.35649i
\(92\) 3.18596 0.332159
\(93\) 0 0
\(94\) 7.36603 0.759748
\(95\) 9.07569 7.41464i 0.931146 0.760726i
\(96\) 0 0
\(97\) −3.22440 −0.327388 −0.163694 0.986511i \(-0.552341\pi\)
−0.163694 + 0.986511i \(0.552341\pi\)
\(98\) −3.05792 6.29675i −0.308897 0.636068i
\(99\) 0 0
\(100\) 2.22869 0.222869
\(101\) 3.56260i 0.354492i −0.984167 0.177246i \(-0.943281\pi\)
0.984167 0.177246i \(-0.0567187\pi\)
\(102\) 0 0
\(103\) 8.28917 0.816757 0.408378 0.912813i \(-0.366094\pi\)
0.408378 + 0.912813i \(0.366094\pi\)
\(104\) 5.01856i 0.492110i
\(105\) 0 0
\(106\) 1.58536 0.153984
\(107\) 17.0509i 1.64837i 0.566320 + 0.824185i \(0.308367\pi\)
−0.566320 + 0.824185i \(0.691633\pi\)
\(108\) 0 0
\(109\) 1.58536i 0.151850i −0.997114 0.0759249i \(-0.975809\pi\)
0.997114 0.0759249i \(-0.0241909\pi\)
\(110\) 3.30347 0.314974
\(111\) 0 0
\(112\) −0.592978 2.57844i −0.0560312 0.243640i
\(113\) 10.3223i 0.971040i −0.874226 0.485520i \(-0.838630\pi\)
0.874226 0.485520i \(-0.161370\pi\)
\(114\) 0 0
\(115\) 8.56583i 0.798768i
\(116\) 4.47509i 0.415502i
\(117\) 0 0
\(118\) 2.46827i 0.227222i
\(119\) 13.2968 3.05792i 1.21891 0.280319i
\(120\) 0 0
\(121\) −9.49033 −0.862757
\(122\) 9.69893 0.878100
\(123\) 0 0
\(124\) −6.61285 −0.593852
\(125\) 7.45101i 0.666439i
\(126\) 0 0
\(127\) 9.80695i 0.870226i −0.900376 0.435113i \(-0.856708\pi\)
0.900376 0.435113i \(-0.143292\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 13.4930 1.18342
\(131\) 3.56260i 0.311265i 0.987815 + 0.155633i \(0.0497416\pi\)
−0.987815 + 0.155633i \(0.950258\pi\)
\(132\) 0 0
\(133\) 7.06847 9.11245i 0.612914 0.790150i
\(134\) 0.728584 0.0629401
\(135\) 0 0
\(136\) 5.15689 0.442200
\(137\) −9.38983 −0.802228 −0.401114 0.916028i \(-0.631377\pi\)
−0.401114 + 0.916028i \(0.631377\pi\)
\(138\) 0 0
\(139\) 18.4646i 1.56615i 0.621928 + 0.783075i \(0.286350\pi\)
−0.621928 + 0.783075i \(0.713650\pi\)
\(140\) 6.93246 1.59429i 0.585900 0.134742i
\(141\) 0 0
\(142\) 5.14323 0.431610
\(143\) 6.16624 0.515647
\(144\) 0 0
\(145\) −12.0318 −0.999189
\(146\) 6.11584 0.506151
\(147\) 0 0
\(148\) 3.74651i 0.307961i
\(149\) −19.8649 −1.62740 −0.813699 0.581286i \(-0.802550\pi\)
−0.813699 + 0.581286i \(0.802550\pi\)
\(150\) 0 0
\(151\) 11.7795i 0.958599i 0.877651 + 0.479300i \(0.159109\pi\)
−0.877651 + 0.479300i \(0.840891\pi\)
\(152\) 3.37559 2.75779i 0.273797 0.223686i
\(153\) 0 0
\(154\) 3.16810 0.728584i 0.255293 0.0587110i
\(155\) 17.7795i 1.42808i
\(156\) 0 0
\(157\) 13.3282i 1.06371i 0.846836 + 0.531854i \(0.178504\pi\)
−0.846836 + 0.531854i \(0.821496\pi\)
\(158\) 9.74651 0.775390
\(159\) 0 0
\(160\) 2.68862 0.212554
\(161\) 1.88920 + 8.21481i 0.148890 + 0.647418i
\(162\) 0 0
\(163\) −9.22869 −0.722846 −0.361423 0.932402i \(-0.617709\pi\)
−0.361423 + 0.932402i \(0.617709\pi\)
\(164\) 0.873974 0.0682459
\(165\) 0 0
\(166\) −1.81465 −0.140844
\(167\) −16.1317 −1.24831 −0.624155 0.781300i \(-0.714557\pi\)
−0.624155 + 0.781300i \(0.714557\pi\)
\(168\) 0 0
\(169\) 12.1860 0.937381
\(170\) 13.8649i 1.06339i
\(171\) 0 0
\(172\) −8.11842 −0.619024
\(173\) 4.00330 0.304366 0.152183 0.988352i \(-0.451370\pi\)
0.152183 + 0.988352i \(0.451370\pi\)
\(174\) 0 0
\(175\) 1.32156 + 5.74654i 0.0999007 + 0.434398i
\(176\) 1.22869 0.0926157
\(177\) 0 0
\(178\) 3.38845i 0.253975i
\(179\) 7.49301i 0.560054i −0.959992 0.280027i \(-0.909657\pi\)
0.959992 0.280027i \(-0.0903434\pi\)
\(180\) 0 0
\(181\) 1.09728 0.0815604 0.0407802 0.999168i \(-0.487016\pi\)
0.0407802 + 0.999168i \(0.487016\pi\)
\(182\) 12.9401 2.97590i 0.959183 0.220588i
\(183\) 0 0
\(184\) 3.18596i 0.234872i
\(185\) −10.0729 −0.740577
\(186\) 0 0
\(187\) 6.33620i 0.463349i
\(188\) 7.36603i 0.537223i
\(189\) 0 0
\(190\) 7.41464 + 9.07569i 0.537915 + 0.658420i
\(191\) −6.01524 −0.435248 −0.217624 0.976033i \(-0.569831\pi\)
−0.217624 + 0.976033i \(0.569831\pi\)
\(192\) 0 0
\(193\) 19.2725i 1.38726i −0.720330 0.693632i \(-0.756010\pi\)
0.720330 0.693632i \(-0.243990\pi\)
\(194\) 3.22440i 0.231499i
\(195\) 0 0
\(196\) 6.29675 3.05792i 0.449768 0.218423i
\(197\) 0.371912 0.0264976 0.0132488 0.999912i \(-0.495783\pi\)
0.0132488 + 0.999912i \(0.495783\pi\)
\(198\) 0 0
\(199\) 16.3446i 1.15864i 0.815100 + 0.579320i \(0.196682\pi\)
−0.815100 + 0.579320i \(0.803318\pi\)
\(200\) 2.22869i 0.157592i
\(201\) 0 0
\(202\) 3.56260 0.250663
\(203\) −11.5388 + 2.65363i −0.809863 + 0.186248i
\(204\) 0 0
\(205\) 2.34979i 0.164116i
\(206\) 8.28917i 0.577534i
\(207\) 0 0
\(208\) 5.01856 0.347975
\(209\) 3.38845 + 4.14754i 0.234384 + 0.286892i
\(210\) 0 0
\(211\) 2.44213i 0.168123i 0.996461 + 0.0840616i \(0.0267893\pi\)
−0.996461 + 0.0840616i \(0.973211\pi\)
\(212\) 1.58536i 0.108883i
\(213\) 0 0
\(214\) −17.0509 −1.16557
\(215\) 21.8274i 1.48861i
\(216\) 0 0
\(217\) −3.92128 17.0509i −0.266194 1.15749i
\(218\) 1.58536 0.107374
\(219\) 0 0
\(220\) 3.30347i 0.222720i
\(221\) 25.8802i 1.74089i
\(222\) 0 0
\(223\) −8.80742 −0.589789 −0.294894 0.955530i \(-0.595284\pi\)
−0.294894 + 0.955530i \(0.595284\pi\)
\(224\) 2.57844 0.592978i 0.172280 0.0396200i
\(225\) 0 0
\(226\) 10.3223 0.686629
\(227\) 25.8511 1.71580 0.857900 0.513816i \(-0.171769\pi\)
0.857900 + 0.513816i \(0.171769\pi\)
\(228\) 0 0
\(229\) 5.87721i 0.388377i −0.980964 0.194188i \(-0.937793\pi\)
0.980964 0.194188i \(-0.0622073\pi\)
\(230\) −8.56583 −0.564814
\(231\) 0 0
\(232\) −4.47509 −0.293804
\(233\) 9.09361 0.595742 0.297871 0.954606i \(-0.403723\pi\)
0.297871 + 0.954606i \(0.403723\pi\)
\(234\) 0 0
\(235\) −19.8045 −1.29190
\(236\) 2.46827 0.160671
\(237\) 0 0
\(238\) 3.05792 + 13.2968i 0.198216 + 0.861901i
\(239\) −5.64333 −0.365037 −0.182518 0.983202i \(-0.558425\pi\)
−0.182518 + 0.983202i \(0.558425\pi\)
\(240\) 0 0
\(241\) 22.2248 1.43163 0.715813 0.698292i \(-0.246056\pi\)
0.715813 + 0.698292i \(0.246056\pi\)
\(242\) 9.49033i 0.610062i
\(243\) 0 0
\(244\) 9.69893i 0.620910i
\(245\) 8.22160 + 16.9296i 0.525258 + 1.08159i
\(246\) 0 0
\(247\) 13.8401 + 16.9406i 0.880626 + 1.07791i
\(248\) 6.61285i 0.419917i
\(249\) 0 0
\(250\) 7.45101 0.471244
\(251\) 1.37394i 0.0867223i 0.999059 + 0.0433612i \(0.0138066\pi\)
−0.999059 + 0.0433612i \(0.986193\pi\)
\(252\) 0 0
\(253\) −3.91454 −0.246105
\(254\) 9.80695 0.615343
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.81051 0.362450 0.181225 0.983442i \(-0.441994\pi\)
0.181225 + 0.983442i \(0.441994\pi\)
\(258\) 0 0
\(259\) −9.66016 + 2.22160i −0.600253 + 0.138043i
\(260\) 13.4930i 0.836801i
\(261\) 0 0
\(262\) −3.56260 −0.220098
\(263\) 27.3579 1.68696 0.843481 0.537159i \(-0.180502\pi\)
0.843481 + 0.537159i \(0.180502\pi\)
\(264\) 0 0
\(265\) −4.26243 −0.261839
\(266\) 9.11245 + 7.06847i 0.558720 + 0.433396i
\(267\) 0 0
\(268\) 0.728584i 0.0445053i
\(269\) 27.5487 1.67967 0.839837 0.542839i \(-0.182650\pi\)
0.839837 + 0.542839i \(0.182650\pi\)
\(270\) 0 0
\(271\) 30.7284i 1.86662i −0.359073 0.933309i \(-0.616907\pi\)
0.359073 0.933309i \(-0.383093\pi\)
\(272\) 5.15689i 0.312682i
\(273\) 0 0
\(274\) 9.38983i 0.567261i
\(275\) −2.73836 −0.165129
\(276\) 0 0
\(277\) −7.03564 −0.422731 −0.211365 0.977407i \(-0.567791\pi\)
−0.211365 + 0.977407i \(0.567791\pi\)
\(278\) −18.4646 −1.10743
\(279\) 0 0
\(280\) 1.59429 + 6.93246i 0.0952773 + 0.414294i
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 18.2947i 1.08750i 0.839246 + 0.543752i \(0.182997\pi\)
−0.839246 + 0.543752i \(0.817003\pi\)
\(284\) 5.14323i 0.305194i
\(285\) 0 0
\(286\) 6.16624i 0.364617i
\(287\) 0.518248 + 2.25349i 0.0305912 + 0.133020i
\(288\) 0 0
\(289\) −9.59351 −0.564324
\(290\) 12.0318i 0.706533i
\(291\) 0 0
\(292\) 6.11584i 0.357903i
\(293\) 22.6626 1.32396 0.661980 0.749521i \(-0.269716\pi\)
0.661980 + 0.749521i \(0.269716\pi\)
\(294\) 0 0
\(295\) 6.63624i 0.386377i
\(296\) −3.74651 −0.217761
\(297\) 0 0
\(298\) 19.8649i 1.15074i
\(299\) −15.9889 −0.924663
\(300\) 0 0
\(301\) −4.81404 20.9329i −0.277477 1.20655i
\(302\) −11.7795 −0.677832
\(303\) 0 0
\(304\) 2.75779 + 3.37559i 0.158170 + 0.193603i
\(305\) −26.0767 −1.49315
\(306\) 0 0
\(307\) 3.48353 0.198815 0.0994077 0.995047i \(-0.468305\pi\)
0.0994077 + 0.995047i \(0.468305\pi\)
\(308\) 0.728584 + 3.16810i 0.0415149 + 0.180519i
\(309\) 0 0
\(310\) 17.7795 1.00981
\(311\) 14.7116i 0.834217i 0.908857 + 0.417109i \(0.136957\pi\)
−0.908857 + 0.417109i \(0.863043\pi\)
\(312\) 0 0
\(313\) 9.57518i 0.541221i −0.962689 0.270611i \(-0.912774\pi\)
0.962689 0.270611i \(-0.0872256\pi\)
\(314\) −13.3282 −0.752155
\(315\) 0 0
\(316\) 9.74651i 0.548284i
\(317\) 19.2439i 1.08085i −0.841393 0.540423i \(-0.818264\pi\)
0.841393 0.540423i \(-0.181736\pi\)
\(318\) 0 0
\(319\) 5.49848i 0.307856i
\(320\) 2.68862i 0.150299i
\(321\) 0 0
\(322\) −8.21481 + 1.88920i −0.457794 + 0.105281i
\(323\) −17.4076 + 14.2216i −0.968582 + 0.791311i
\(324\) 0 0
\(325\) −11.1848 −0.620421
\(326\) 9.22869i 0.511130i
\(327\) 0 0
\(328\) 0.873974i 0.0482572i
\(329\) −18.9929 + 4.36790i −1.04711 + 0.240810i
\(330\) 0 0
\(331\) 29.0509i 1.59678i 0.602140 + 0.798390i \(0.294315\pi\)
−0.602140 + 0.798390i \(0.705685\pi\)
\(332\) 1.81465i 0.0995917i
\(333\) 0 0
\(334\) 16.1317i 0.882689i
\(335\) −1.95889 −0.107025
\(336\) 0 0
\(337\) 23.7795i 1.29535i 0.761917 + 0.647675i \(0.224259\pi\)
−0.761917 + 0.647675i \(0.775741\pi\)
\(338\) 12.1860i 0.662829i
\(339\) 0 0
\(340\) −13.8649 −0.751931
\(341\) 8.12512 0.440000
\(342\) 0 0
\(343\) 11.6185 + 14.4226i 0.627341 + 0.778745i
\(344\) 8.11842i 0.437716i
\(345\) 0 0
\(346\) 4.00330i 0.215219i
\(347\) 28.4737 1.52855 0.764273 0.644892i \(-0.223098\pi\)
0.764273 + 0.644892i \(0.223098\pi\)
\(348\) 0 0
\(349\) 19.4236i 1.03972i −0.854251 0.519860i \(-0.825984\pi\)
0.854251 0.519860i \(-0.174016\pi\)
\(350\) −5.74654 + 1.32156i −0.307166 + 0.0706405i
\(351\) 0 0
\(352\) 1.22869i 0.0654892i
\(353\) 19.1000i 1.01659i 0.861183 + 0.508294i \(0.169724\pi\)
−0.861183 + 0.508294i \(0.830276\pi\)
\(354\) 0 0
\(355\) −13.8282 −0.733924
\(356\) −3.38845 −0.179588
\(357\) 0 0
\(358\) 7.49301 0.396018
\(359\) −2.59351 −0.136880 −0.0684401 0.997655i \(-0.521802\pi\)
−0.0684401 + 0.997655i \(0.521802\pi\)
\(360\) 0 0
\(361\) −3.78924 + 18.6183i −0.199433 + 0.979911i
\(362\) 1.09728i 0.0576719i
\(363\) 0 0
\(364\) 2.97590 + 12.9401i 0.155979 + 0.678245i
\(365\) −16.4432 −0.860676
\(366\) 0 0
\(367\) 28.6818i 1.49718i −0.663034 0.748589i \(-0.730732\pi\)
0.663034 0.748589i \(-0.269268\pi\)
\(368\) −3.18596 −0.166079
\(369\) 0 0
\(370\) 10.0729i 0.523667i
\(371\) −4.08776 + 0.940082i −0.212226 + 0.0488066i
\(372\) 0 0
\(373\) 4.47509i 0.231711i −0.993266 0.115856i \(-0.963039\pi\)
0.993266 0.115856i \(-0.0369610\pi\)
\(374\) −6.33620 −0.327637
\(375\) 0 0
\(376\) −7.36603 −0.379874
\(377\) 22.4585i 1.15667i
\(378\) 0 0
\(379\) 11.2714i 0.578974i −0.957182 0.289487i \(-0.906515\pi\)
0.957182 0.289487i \(-0.0934847\pi\)
\(380\) −9.07569 + 7.41464i −0.465573 + 0.380363i
\(381\) 0 0
\(382\) 6.01524i 0.307766i
\(383\) −3.58832 −0.183355 −0.0916773 0.995789i \(-0.529223\pi\)
−0.0916773 + 0.995789i \(0.529223\pi\)
\(384\) 0 0
\(385\) −8.51782 + 1.95889i −0.434108 + 0.0998341i
\(386\) 19.2725 0.980943
\(387\) 0 0
\(388\) 3.22440 0.163694
\(389\) −3.54263 −0.179618 −0.0898092 0.995959i \(-0.528626\pi\)
−0.0898092 + 0.995959i \(0.528626\pi\)
\(390\) 0 0
\(391\) 16.4296i 0.830882i
\(392\) 3.05792 + 6.29675i 0.154448 + 0.318034i
\(393\) 0 0
\(394\) 0.371912i 0.0187367i
\(395\) −26.2047 −1.31850
\(396\) 0 0
\(397\) 6.99202i 0.350920i −0.984487 0.175460i \(-0.943859\pi\)
0.984487 0.175460i \(-0.0561412\pi\)
\(398\) −16.3446 −0.819283
\(399\) 0 0
\(400\) −2.22869 −0.111434
\(401\) 28.1018i 1.40333i −0.712505 0.701667i \(-0.752439\pi\)
0.712505 0.701667i \(-0.247561\pi\)
\(402\) 0 0
\(403\) 33.1870 1.65316
\(404\) 3.56260i 0.177246i
\(405\) 0 0
\(406\) −2.65363 11.5388i −0.131697 0.572660i
\(407\) 4.60328i 0.228176i
\(408\) 0 0
\(409\) 24.2927 1.20120 0.600598 0.799551i \(-0.294929\pi\)
0.600598 + 0.799551i \(0.294929\pi\)
\(410\) −2.34979 −0.116048
\(411\) 0 0
\(412\) −8.28917 −0.408378
\(413\) 1.46363 + 6.36429i 0.0720205 + 0.313166i
\(414\) 0 0
\(415\) 4.87890 0.239496
\(416\) 5.01856i 0.246055i
\(417\) 0 0
\(418\) −4.14754 + 3.38845i −0.202863 + 0.165735i
\(419\) 27.1080i 1.32431i 0.749366 + 0.662156i \(0.230358\pi\)
−0.749366 + 0.662156i \(0.769642\pi\)
\(420\) 0 0
\(421\) 14.2216i 0.693118i −0.938028 0.346559i \(-0.887350\pi\)
0.938028 0.346559i \(-0.112650\pi\)
\(422\) −2.44213 −0.118881
\(423\) 0 0
\(424\) −1.58536 −0.0769918
\(425\) 11.4931i 0.557497i
\(426\) 0 0
\(427\) −25.0082 + 5.75125i −1.21023 + 0.278323i
\(428\) 17.0509i 0.824185i
\(429\) 0 0
\(430\) 21.8274 1.05261
\(431\) 32.0934i 1.54588i 0.634476 + 0.772942i \(0.281216\pi\)
−0.634476 + 0.772942i \(0.718784\pi\)
\(432\) 0 0
\(433\) −18.5896 −0.893359 −0.446680 0.894694i \(-0.647394\pi\)
−0.446680 + 0.894694i \(0.647394\pi\)
\(434\) 17.0509 3.92128i 0.818468 0.188227i
\(435\) 0 0
\(436\) 1.58536i 0.0759249i
\(437\) −8.78618 10.7545i −0.420300 0.514457i
\(438\) 0 0
\(439\) −22.0623 −1.05298 −0.526488 0.850183i \(-0.676491\pi\)
−0.526488 + 0.850183i \(0.676491\pi\)
\(440\) −3.30347 −0.157487
\(441\) 0 0
\(442\) −25.8802 −1.23099
\(443\) 8.62541 0.409805 0.204903 0.978782i \(-0.434312\pi\)
0.204903 + 0.978782i \(0.434312\pi\)
\(444\) 0 0
\(445\) 9.11027i 0.431868i
\(446\) 8.80742i 0.417043i
\(447\) 0 0
\(448\) 0.592978 + 2.57844i 0.0280156 + 0.121820i
\(449\) 22.1018i 1.04305i 0.853237 + 0.521523i \(0.174636\pi\)
−0.853237 + 0.521523i \(0.825364\pi\)
\(450\) 0 0
\(451\) −1.07384 −0.0505651
\(452\) 10.3223i 0.485520i
\(453\) 0 0
\(454\) 25.8511i 1.21325i
\(455\) −34.7910 + 8.00106i −1.63103 + 0.375095i
\(456\) 0 0
\(457\) −0.00688586 −0.000322107 −0.000161053 1.00000i \(-0.500051\pi\)
−0.000161053 1.00000i \(0.500051\pi\)
\(458\) 5.87721 0.274624
\(459\) 0 0
\(460\) 8.56583i 0.399384i
\(461\) 16.4758i 0.767356i 0.923467 + 0.383678i \(0.125343\pi\)
−0.923467 + 0.383678i \(0.874657\pi\)
\(462\) 0 0
\(463\) −24.7296 −1.14928 −0.574642 0.818405i \(-0.694859\pi\)
−0.574642 + 0.818405i \(0.694859\pi\)
\(464\) 4.47509i 0.207751i
\(465\) 0 0
\(466\) 9.09361i 0.421253i
\(467\) 5.00323i 0.231522i −0.993277 0.115761i \(-0.963069\pi\)
0.993277 0.115761i \(-0.0369307\pi\)
\(468\) 0 0
\(469\) −1.87861 + 0.432034i −0.0867463 + 0.0199495i
\(470\) 19.8045i 0.913512i
\(471\) 0 0
\(472\) 2.46827i 0.113611i
\(473\) 9.97499 0.458650
\(474\) 0 0
\(475\) −6.14624 7.52313i −0.282009 0.345185i
\(476\) −13.2968 + 3.05792i −0.609456 + 0.140160i
\(477\) 0 0
\(478\) 5.64333i 0.258120i
\(479\) 9.95805i 0.454995i −0.973779 0.227498i \(-0.926946\pi\)
0.973779 0.227498i \(-0.0730544\pi\)
\(480\) 0 0
\(481\) 18.8021i 0.857301i
\(482\) 22.2248i 1.01231i
\(483\) 0 0
\(484\) 9.49033 0.431379
\(485\) 8.66920i 0.393648i
\(486\) 0 0
\(487\) 39.1845i 1.77562i 0.460209 + 0.887810i \(0.347774\pi\)
−0.460209 + 0.887810i \(0.652226\pi\)
\(488\) −9.69893 −0.439050
\(489\) 0 0
\(490\) −16.9296 + 8.22160i −0.764801 + 0.371414i
\(491\) −28.4737 −1.28500 −0.642499 0.766286i \(-0.722102\pi\)
−0.642499 + 0.766286i \(0.722102\pi\)
\(492\) 0 0
\(493\) 23.0775 1.03936
\(494\) −16.9406 + 13.8401i −0.762194 + 0.622696i
\(495\) 0 0
\(496\) 6.61285 0.296926
\(497\) −13.2615 + 3.04982i −0.594861 + 0.136803i
\(498\) 0 0
\(499\) −8.52617 −0.381684 −0.190842 0.981621i \(-0.561122\pi\)
−0.190842 + 0.981621i \(0.561122\pi\)
\(500\) 7.45101i 0.333219i
\(501\) 0 0
\(502\) −1.37394 −0.0613219
\(503\) 0.653620i 0.0291435i −0.999894 0.0145717i \(-0.995362\pi\)
0.999894 0.0145717i \(-0.00463849\pi\)
\(504\) 0 0
\(505\) −9.57847 −0.426236
\(506\) 3.91454i 0.174023i
\(507\) 0 0
\(508\) 9.80695i 0.435113i
\(509\) 28.2602 1.25261 0.626305 0.779578i \(-0.284567\pi\)
0.626305 + 0.779578i \(0.284567\pi\)
\(510\) 0 0
\(511\) −15.7694 + 3.62656i −0.697596 + 0.160430i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 5.81051i 0.256291i
\(515\) 22.2865i 0.982058i
\(516\) 0 0
\(517\) 9.05054i 0.398042i
\(518\) −2.22160 9.66016i −0.0976113 0.424443i
\(519\) 0 0
\(520\) −13.4930 −0.591708
\(521\) −8.00661 −0.350776 −0.175388 0.984499i \(-0.556118\pi\)
−0.175388 + 0.984499i \(0.556118\pi\)
\(522\) 0 0
\(523\) 21.6685 0.947499 0.473750 0.880660i \(-0.342900\pi\)
0.473750 + 0.880660i \(0.342900\pi\)
\(524\) 3.56260i 0.155633i
\(525\) 0 0
\(526\) 27.3579i 1.19286i
\(527\) 34.1018i 1.48550i
\(528\) 0 0
\(529\) −12.8497 −0.558682
\(530\) 4.26243i 0.185148i
\(531\) 0 0
\(532\) −7.06847 + 9.11245i −0.306457 + 0.395075i
\(533\) −4.38609 −0.189983
\(534\) 0 0
\(535\) 45.8434 1.98198
\(536\) −0.728584 −0.0314700
\(537\) 0 0
\(538\) 27.5487i 1.18771i
\(539\) −7.73673 + 3.75723i −0.333245 + 0.161835i
\(540\) 0 0
\(541\) −11.2725 −0.484642 −0.242321 0.970196i \(-0.577909\pi\)
−0.242321 + 0.970196i \(0.577909\pi\)
\(542\) 30.7284 1.31990
\(543\) 0 0
\(544\) −5.15689 −0.221100
\(545\) −4.26243 −0.182582
\(546\) 0 0
\(547\) 17.6586i 0.755026i −0.926004 0.377513i \(-0.876779\pi\)
0.926004 0.377513i \(-0.123221\pi\)
\(548\) 9.38983 0.401114
\(549\) 0 0
\(550\) 2.73836i 0.116764i
\(551\) 15.1061 12.3413i 0.643540 0.525759i
\(552\) 0 0
\(553\) −25.1308 + 5.77946i −1.06867 + 0.245768i
\(554\) 7.03564i 0.298916i
\(555\) 0 0
\(556\) 18.4646i 0.783075i
\(557\) −28.0713 −1.18942 −0.594709 0.803941i \(-0.702733\pi\)
−0.594709 + 0.803941i \(0.702733\pi\)
\(558\) 0 0
\(559\) 40.7428 1.72324
\(560\) −6.93246 + 1.59429i −0.292950 + 0.0673712i
\(561\) 0 0
\(562\) 0 0
\(563\) 33.9148 1.42934 0.714669 0.699463i \(-0.246577\pi\)
0.714669 + 0.699463i \(0.246577\pi\)
\(564\) 0 0
\(565\) −27.7527 −1.16757
\(566\) −18.2947 −0.768982
\(567\) 0 0
\(568\) −5.14323 −0.215805
\(569\) 44.7296i 1.87516i −0.347764 0.937582i \(-0.613059\pi\)
0.347764 0.937582i \(-0.386941\pi\)
\(570\) 0 0
\(571\) 46.5316 1.94729 0.973644 0.228072i \(-0.0732421\pi\)
0.973644 + 0.228072i \(0.0732421\pi\)
\(572\) −6.16624 −0.257823
\(573\) 0 0
\(574\) −2.25349 + 0.518248i −0.0940590 + 0.0216312i
\(575\) 7.10050 0.296111
\(576\) 0 0
\(577\) 37.0982i 1.54442i 0.635370 + 0.772208i \(0.280848\pi\)
−0.635370 + 0.772208i \(0.719152\pi\)
\(578\) 9.59351i 0.399037i
\(579\) 0 0
\(580\) 12.0318 0.499594
\(581\) 4.67897 1.07605i 0.194116 0.0446419i
\(582\) 0 0
\(583\) 1.94791i 0.0806741i
\(584\) −6.11584 −0.253075
\(585\) 0 0
\(586\) 22.6626i 0.936182i
\(587\) 21.6532i 0.893724i −0.894603 0.446862i \(-0.852542\pi\)
0.894603 0.446862i \(-0.147458\pi\)
\(588\) 0 0
\(589\) 18.2368 + 22.3223i 0.751436 + 0.919774i
\(590\) −6.63624 −0.273210
\(591\) 0 0
\(592\) 3.74651i 0.153980i
\(593\) 31.3820i 1.28871i −0.764728 0.644353i \(-0.777127\pi\)
0.764728 0.644353i \(-0.222873\pi\)
\(594\) 0 0
\(595\) −8.22160 35.7499i −0.337053 1.46560i
\(596\) 19.8649 0.813699
\(597\) 0 0
\(598\) 15.9889i 0.653835i
\(599\) 18.8568i 0.770467i 0.922819 + 0.385233i \(0.125879\pi\)
−0.922819 + 0.385233i \(0.874121\pi\)
\(600\) 0 0
\(601\) 44.1670 1.80161 0.900806 0.434222i \(-0.142977\pi\)
0.900806 + 0.434222i \(0.142977\pi\)
\(602\) 20.9329 4.81404i 0.853161 0.196206i
\(603\) 0 0
\(604\) 11.7795i 0.479300i
\(605\) 25.5159i 1.03737i
\(606\) 0 0
\(607\) −18.3263 −0.743841 −0.371921 0.928265i \(-0.621301\pi\)
−0.371921 + 0.928265i \(0.621301\pi\)
\(608\) −3.37559 + 2.75779i −0.136898 + 0.111843i
\(609\) 0 0
\(610\) 26.0767i 1.05582i
\(611\) 36.9669i 1.49552i
\(612\) 0 0
\(613\) −9.82908 −0.396993 −0.198496 0.980102i \(-0.563606\pi\)
−0.198496 + 0.980102i \(0.563606\pi\)
\(614\) 3.48353i 0.140584i
\(615\) 0 0
\(616\) −3.16810 + 0.728584i −0.127646 + 0.0293555i
\(617\) −35.0495 −1.41104 −0.705519 0.708691i \(-0.749286\pi\)
−0.705519 + 0.708691i \(0.749286\pi\)
\(618\) 0 0
\(619\) 0.543970i 0.0218640i 0.999940 + 0.0109320i \(0.00347984\pi\)
−0.999940 + 0.0109320i \(0.996520\pi\)
\(620\) 17.7795i 0.714040i
\(621\) 0 0
\(622\) −14.7116 −0.589881
\(623\) −2.00928 8.73694i −0.0805000 0.350038i
\(624\) 0 0
\(625\) −31.1764 −1.24706
\(626\) 9.57518 0.382701
\(627\) 0 0
\(628\) 13.3282i 0.531854i
\(629\) 19.3203 0.770351
\(630\) 0 0
\(631\) 4.77967 0.190276 0.0951378 0.995464i \(-0.469671\pi\)
0.0951378 + 0.995464i \(0.469671\pi\)
\(632\) −9.74651 −0.387695
\(633\) 0 0
\(634\) 19.2439 0.764274
\(635\) −26.3672 −1.04635
\(636\) 0 0
\(637\) −31.6006 + 15.3464i −1.25206 + 0.608045i
\(638\) 5.49848 0.217687
\(639\) 0 0
\(640\) −2.68862 −0.106277
\(641\) 11.7795i 0.465261i −0.972565 0.232630i \(-0.925267\pi\)
0.972565 0.232630i \(-0.0747333\pi\)
\(642\) 0 0
\(643\) 36.0736i 1.42260i 0.702887 + 0.711301i \(0.251894\pi\)
−0.702887 + 0.711301i \(0.748106\pi\)
\(644\) −1.88920 8.21481i −0.0744450 0.323709i
\(645\) 0 0
\(646\) −14.2216 17.4076i −0.559541 0.684891i
\(647\) 6.20083i 0.243780i −0.992544 0.121890i \(-0.961105\pi\)
0.992544 0.121890i \(-0.0388955\pi\)
\(648\) 0 0
\(649\) −3.03273 −0.119045
\(650\) 11.1848i 0.438704i
\(651\) 0 0
\(652\) 9.22869 0.361423
\(653\) 33.6090 1.31522 0.657610 0.753359i \(-0.271567\pi\)
0.657610 + 0.753359i \(0.271567\pi\)
\(654\) 0 0
\(655\) 9.57847 0.374262
\(656\) −0.873974 −0.0341230
\(657\) 0 0
\(658\) −4.36790 18.9929i −0.170278 0.740421i
\(659\) 25.7593i 1.00344i 0.865030 + 0.501719i \(0.167299\pi\)
−0.865030 + 0.501719i \(0.832701\pi\)
\(660\) 0 0
\(661\) −46.1611 −1.79546 −0.897729 0.440549i \(-0.854784\pi\)
−0.897729 + 0.440549i \(0.854784\pi\)
\(662\) −29.0509 −1.12909
\(663\) 0 0
\(664\) 1.81465 0.0704220
\(665\) −24.4999 19.0044i −0.950066 0.736961i
\(666\) 0 0
\(667\) 14.2574i 0.552050i
\(668\) 16.1317 0.624155
\(669\) 0 0
\(670\) 1.95889i 0.0756784i
\(671\) 11.9169i 0.460048i
\(672\) 0 0
\(673\) 35.4380i 1.36604i 0.730402 + 0.683018i \(0.239333\pi\)
−0.730402 + 0.683018i \(0.760667\pi\)
\(674\) −23.7795 −0.915951
\(675\) 0 0
\(676\) −12.1860 −0.468691
\(677\) −12.1788 −0.468070 −0.234035 0.972228i \(-0.575193\pi\)
−0.234035 + 0.972228i \(0.575193\pi\)
\(678\) 0 0
\(679\) 1.91200 + 8.31394i 0.0733758 + 0.319060i
\(680\) 13.8649i 0.531696i
\(681\) 0 0
\(682\) 8.12512i 0.311127i
\(683\) 19.0519i 0.729002i −0.931203 0.364501i \(-0.881240\pi\)
0.931203 0.364501i \(-0.118760\pi\)
\(684\) 0 0
\(685\) 25.2457i 0.964589i
\(686\) −14.4226 + 11.6185i −0.550656 + 0.443597i
\(687\) 0 0
\(688\) 8.11842 0.309512
\(689\) 7.95621i 0.303108i
\(690\) 0 0
\(691\) 27.5487i 1.04800i 0.851718 + 0.524001i \(0.175561\pi\)
−0.851718 + 0.524001i \(0.824439\pi\)
\(692\) −4.00330 −0.152183
\(693\) 0 0
\(694\) 28.4737i 1.08085i
\(695\) 49.6444 1.88312
\(696\) 0 0
\(697\) 4.50699i 0.170714i
\(698\) 19.4236 0.735193
\(699\) 0 0
\(700\) −1.32156 5.74654i −0.0499503 0.217199i
\(701\) −9.50678 −0.359066 −0.179533 0.983752i \(-0.557459\pi\)
−0.179533 + 0.983752i \(0.557459\pi\)
\(702\) 0 0
\(703\) 12.6467 10.3321i 0.476978 0.389681i
\(704\) −1.22869 −0.0463079
\(705\) 0 0
\(706\) −19.1000 −0.718837
\(707\) −9.18596 + 2.11254i −0.345473 + 0.0794503i
\(708\) 0 0
\(709\) −28.7296 −1.07896 −0.539482 0.841997i \(-0.681380\pi\)
−0.539482 + 0.841997i \(0.681380\pi\)
\(710\) 13.8282i 0.518963i
\(711\) 0 0
\(712\) 3.38845i 0.126988i
\(713\) −21.0683 −0.789013
\(714\) 0 0
\(715\) 16.5787i 0.620007i
\(716\) 7.49301i 0.280027i
\(717\) 0 0
\(718\) 2.59351i 0.0967889i
\(719\) 14.2000i 0.529570i 0.964307 + 0.264785i \(0.0853010\pi\)
−0.964307 + 0.264785i \(0.914699\pi\)
\(720\) 0 0
\(721\) −4.91530 21.3732i −0.183055 0.795979i
\(722\) −18.6183 3.78924i −0.692902 0.141021i
\(723\) 0 0
\(724\) −1.09728 −0.0407802
\(725\) 9.97357i 0.370409i
\(726\) 0 0
\(727\) 7.54541i 0.279844i 0.990163 + 0.139922i \(0.0446852\pi\)
−0.990163 + 0.139922i \(0.955315\pi\)
\(728\) −12.9401 + 2.97590i −0.479591 + 0.110294i
\(729\) 0 0
\(730\) 16.4432i 0.608590i
\(731\) 41.8658i 1.54846i
\(732\) 0 0
\(733\) 35.8186i 1.32299i −0.749949 0.661495i \(-0.769922\pi\)
0.749949 0.661495i \(-0.230078\pi\)
\(734\) 28.6818 1.05866
\(735\) 0 0
\(736\) 3.18596i 0.117436i
\(737\) 0.895201i 0.0329751i
\(738\) 0 0
\(739\) −50.7272 −1.86603 −0.933014 0.359839i \(-0.882832\pi\)
−0.933014 + 0.359839i \(0.882832\pi\)
\(740\) 10.0729 0.370289
\(741\) 0 0
\(742\) −0.940082 4.08776i −0.0345115 0.150066i
\(743\) 12.9531i 0.475202i 0.971363 + 0.237601i \(0.0763611\pi\)
−0.971363 + 0.237601i \(0.923639\pi\)
\(744\) 0 0
\(745\) 53.4093i 1.95676i
\(746\) 4.47509 0.163845
\(747\) 0 0
\(748\) 6.33620i 0.231674i
\(749\) 43.9648 10.1108i 1.60644 0.369441i
\(750\) 0 0
\(751\) 2.41023i 0.0879507i −0.999033 0.0439753i \(-0.985998\pi\)
0.999033 0.0439753i \(-0.0140023\pi\)
\(752\) 7.36603i 0.268612i
\(753\) 0 0
\(754\) 22.4585 0.817891
\(755\) 31.6705 1.15261
\(756\) 0 0
\(757\) −36.2727 −1.31835 −0.659176 0.751988i \(-0.729095\pi\)
−0.659176 + 0.751988i \(0.729095\pi\)
\(758\) 11.2714 0.409396
\(759\) 0 0
\(760\) −7.41464 9.07569i −0.268957 0.329210i
\(761\) 25.0730i 0.908895i 0.890773 + 0.454448i \(0.150163\pi\)
−0.890773 + 0.454448i \(0.849837\pi\)
\(762\) 0 0
\(763\) −4.08776 + 0.940082i −0.147987 + 0.0340333i
\(764\) 6.01524 0.217624
\(765\) 0 0
\(766\) 3.58832i 0.129651i
\(767\) −12.3872 −0.447274
\(768\) 0 0
\(769\) 16.9479i 0.611156i −0.952167 0.305578i \(-0.901150\pi\)
0.952167 0.305578i \(-0.0988496\pi\)
\(770\) −1.95889 8.51782i −0.0705934 0.306961i
\(771\) 0 0
\(772\) 19.2725i 0.693632i
\(773\) −33.8578 −1.21778 −0.608889 0.793255i \(-0.708385\pi\)
−0.608889 + 0.793255i \(0.708385\pi\)
\(774\) 0 0
\(775\) −14.7380 −0.529404
\(776\) 3.22440i 0.115749i
\(777\) 0 0
\(778\) 3.54263i 0.127009i
\(779\) −2.41023 2.95018i −0.0863556 0.105701i
\(780\) 0 0
\(781\) 6.31941i 0.226126i
\(782\) 16.4296 0.587522
\(783\) 0 0
\(784\) −6.29675 + 3.05792i −0.224884 + 0.109212i
\(785\) 35.8345 1.27899
\(786\) 0 0
\(787\) 5.49280 0.195797 0.0978987 0.995196i \(-0.468788\pi\)
0.0978987 + 0.995196i \(0.468788\pi\)
\(788\) −0.371912 −0.0132488
\(789\) 0 0
\(790\) 26.2047i 0.932320i
\(791\) −26.6155 + 6.12090i −0.946337 + 0.217634i
\(792\) 0 0
\(793\) 48.6747i 1.72849i
\(794\) 6.99202 0.248138
\(795\) 0 0
\(796\) 16.3446i 0.579320i
\(797\) 25.8511 0.915695 0.457847 0.889031i \(-0.348621\pi\)
0.457847 + 0.889031i \(0.348621\pi\)
\(798\) 0 0
\(799\) 37.9858 1.34384
\(800\) 2.22869i 0.0787960i
\(801\) 0 0
\(802\) 28.1018 0.992308
\(803\) 7.51445i 0.265179i
\(804\) 0 0
\(805\) 22.0865 5.07935i 0.778448 0.179024i
\(806\) 33.1870i 1.16896i
\(807\) 0 0
\(808\) −3.56260 −0.125332
\(809\) 7.71213 0.271144 0.135572 0.990767i \(-0.456713\pi\)
0.135572 + 0.990767i \(0.456713\pi\)
\(810\) 0 0
\(811\) −20.0280 −0.703279 −0.351640 0.936135i \(-0.614376\pi\)
−0.351640 + 0.936135i \(0.614376\pi\)
\(812\) 11.5388 2.65363i 0.404932 0.0931242i
\(813\) 0 0
\(814\) 4.60328 0.161345
\(815\) 24.8124i 0.869142i
\(816\) 0 0
\(817\) 22.3889 + 27.4045i 0.783287 + 0.958761i
\(818\) 24.2927i 0.849373i
\(819\) 0 0
\(820\) 2.34979i 0.0820581i
\(821\) −28.8151 −1.00565 −0.502827 0.864387i \(-0.667707\pi\)
−0.502827 + 0.864387i \(0.667707\pi\)
\(822\) 0 0
\(823\) 7.15011 0.249237 0.124619 0.992205i \(-0.460229\pi\)
0.124619 + 0.992205i \(0.460229\pi\)
\(824\) 8.28917i 0.288767i
\(825\) 0 0
\(826\) −6.36429 + 1.46363i −0.221442 + 0.0509262i
\(827\) 55.7805i 1.93968i 0.243745 + 0.969839i \(0.421624\pi\)
−0.243745 + 0.969839i \(0.578376\pi\)
\(828\) 0 0
\(829\) 19.0274 0.660847 0.330424 0.943833i \(-0.392808\pi\)
0.330424 + 0.943833i \(0.392808\pi\)
\(830\) 4.87890i 0.169349i
\(831\) 0 0
\(832\) −5.01856 −0.173987
\(833\) −15.7694 32.4717i −0.546376 1.12508i
\(834\) 0 0
\(835\) 43.3721i 1.50095i
\(836\) −3.38845 4.14754i −0.117192 0.143446i
\(837\) 0 0
\(838\) −27.1080 −0.936430
\(839\) 38.2365 1.32007 0.660035 0.751235i \(-0.270541\pi\)
0.660035 + 0.751235i \(0.270541\pi\)
\(840\) 0 0
\(841\) 8.97357 0.309433
\(842\) 14.2216 0.490109
\(843\) 0 0
\(844\) 2.44213i 0.0840616i
\(845\) 32.7634i 1.12710i
\(846\) 0 0
\(847\) 5.62756 + 24.4703i 0.193365 + 0.840809i
\(848\) 1.58536i 0.0544414i
\(849\) 0 0
\(850\) 11.4931 0.394210
\(851\) 11.9362i 0.409168i
\(852\) 0 0
\(853\) 2.51866i 0.0862373i −0.999070 0.0431186i \(-0.986271\pi\)
0.999070 0.0431186i \(-0.0137293\pi\)
\(854\) −5.75125 25.0082i −0.196804 0.855761i
\(855\) 0 0
\(856\) 17.0509 0.582787
\(857\) −25.2933 −0.864004 −0.432002 0.901873i \(-0.642193\pi\)
−0.432002 + 0.901873i \(0.642193\pi\)
\(858\) 0 0
\(859\) 5.27398i 0.179946i 0.995944 + 0.0899729i \(0.0286781\pi\)
−0.995944 + 0.0899729i \(0.971322\pi\)
\(860\) 21.8274i 0.744307i
\(861\) 0 0
\(862\) −32.0934 −1.09311
\(863\) 24.7325i 0.841905i 0.907083 + 0.420953i \(0.138304\pi\)
−0.907083 + 0.420953i \(0.861696\pi\)
\(864\) 0 0
\(865\) 10.7634i 0.365965i
\(866\) 18.5896i 0.631700i
\(867\) 0 0
\(868\) 3.92128 + 17.0509i 0.133097 + 0.578745i
\(869\) 11.9754i 0.406238i
\(870\) 0 0
\(871\) 3.65644i 0.123894i
\(872\) −1.58536 −0.0536870
\(873\) 0 0
\(874\) 10.7545 8.78618i 0.363776 0.297197i
\(875\) −19.2120 + 4.41829i −0.649485 + 0.149365i
\(876\) 0 0
\(877\) 13.1443i 0.443851i 0.975064 + 0.221926i \(0.0712342\pi\)
−0.975064 + 0.221926i \(0.928766\pi\)
\(878\) 22.0623i 0.744566i
\(879\) 0 0
\(880\) 3.30347i 0.111360i
\(881\) 29.6341i 0.998398i 0.866487 + 0.499199i \(0.166372\pi\)
−0.866487 + 0.499199i \(0.833628\pi\)
\(882\) 0 0
\(883\) 38.9444 1.31058 0.655291 0.755377i \(-0.272546\pi\)
0.655291 + 0.755377i \(0.272546\pi\)
\(884\) 25.8802i 0.870444i
\(885\) 0 0
\(886\) 8.62541i 0.289776i
\(887\) 34.5297 1.15939 0.579696 0.814833i \(-0.303171\pi\)
0.579696 + 0.814833i \(0.303171\pi\)
\(888\) 0 0
\(889\) −25.2867 + 5.81531i −0.848088 + 0.195039i
\(890\) 9.11027 0.305377
\(891\) 0 0
\(892\) 8.80742 0.294894
\(893\) 24.8647 20.3139i 0.832066 0.679780i
\(894\) 0 0
\(895\) −20.1459 −0.673402
\(896\) −2.57844 + 0.592978i −0.0861398 + 0.0198100i
\(897\) 0 0
\(898\) −22.1018 −0.737545
\(899\) 29.5931i 0.986986i
\(900\) 0 0
\(901\) 8.17551 0.272366
\(902\) 1.07384i 0.0357550i
\(903\) 0 0
\(904\) −10.3223 −0.343314
\(905\) 2.95018i 0.0980673i
\(906\) 0 0
\(907\) 15.2164i 0.505254i −0.967564 0.252627i \(-0.918706\pi\)
0.967564 0.252627i \(-0.0812945\pi\)
\(908\) −25.8511 −0.857900
\(909\) 0 0
\(910\) −8.00106 34.7910i −0.265233 1.15331i
\(911\) 26.0330i 0.862510i 0.902230 + 0.431255i \(0.141929\pi\)
−0.902230 + 0.431255i \(0.858071\pi\)
\(912\) 0 0
\(913\) 2.22963i 0.0737901i
\(914\) 0.00688586i 0.000227764i
\(915\) 0 0
\(916\) 5.87721i 0.194188i
\(917\) 9.18596 2.11254i 0.303347 0.0697623i
\(918\) 0 0
\(919\) 15.0154 0.495314 0.247657 0.968848i \(-0.420339\pi\)
0.247657 + 0.968848i \(0.420339\pi\)
\(920\) 8.56583 0.282407
\(921\) 0 0
\(922\) −16.4758 −0.542603
\(923\) 25.8116i 0.849599i
\(924\) 0 0
\(925\) 8.34979i 0.274539i
\(926\) 24.7296i 0.812667i
\(927\) 0 0
\(928\) 4.47509 0.146902
\(929\) 19.8480i 0.651191i 0.945509 + 0.325596i \(0.105565\pi\)
−0.945509 + 0.325596i \(0.894435\pi\)
\(930\) 0 0
\(931\) −27.6874 12.8222i −0.907418 0.420230i
\(932\) −9.09361 −0.297871
\(933\) 0 0
\(934\) 5.00323 0.163711
\(935\) 17.0356 0.557125
\(936\) 0 0
\(937\) 37.4569i 1.22366i 0.790988 + 0.611832i \(0.209567\pi\)
−0.790988 + 0.611832i \(0.790433\pi\)
\(938\) −0.432034 1.87861i −0.0141064 0.0613389i
\(939\) 0 0
\(940\) 19.8045 0.645951
\(941\) 24.3097 0.792474 0.396237 0.918148i \(-0.370316\pi\)
0.396237 + 0.918148i \(0.370316\pi\)
\(942\) 0 0
\(943\) 2.78444 0.0906740
\(944\) −2.46827 −0.0803353
\(945\) 0 0
\(946\) 9.97499i 0.324315i
\(947\) 51.7736 1.68242 0.841209 0.540711i \(-0.181845\pi\)
0.841209 + 0.540711i \(0.181845\pi\)
\(948\) 0 0
\(949\) 30.6927i 0.996328i
\(950\) 7.52313 6.14624i 0.244083 0.199410i
\(951\) 0 0
\(952\) −3.05792 13.2968i −0.0991078 0.430950i
\(953\) 45.5398i 1.47518i 0.675250 + 0.737589i \(0.264036\pi\)
−0.675250 + 0.737589i \(0.735964\pi\)
\(954\) 0 0
\(955\) 16.1727i 0.523336i
\(956\) 5.64333 0.182518
\(957\) 0 0
\(958\) 9.95805 0.321730
\(959\) 5.56797 + 24.2112i 0.179799 + 0.781819i
\(960\) 0 0
\(961\) 12.7298 0.410640
\(962\) 18.8021 0.606203
\(963\) 0 0
\(964\) −22.2248 −0.715813
\(965\) −51.8164 −1.66803
\(966\) 0 0
\(967\) −19.9667 −0.642085 −0.321043 0.947065i \(-0.604033\pi\)
−0.321043 + 0.947065i \(0.604033\pi\)
\(968\) 9.49033i 0.305031i
\(969\) 0 0
\(970\) −8.66920 −0.278351
\(971\) −54.6758 −1.75463 −0.877314 0.479916i \(-0.840667\pi\)
−0.877314 + 0.479916i \(0.840667\pi\)
\(972\) 0 0
\(973\) 47.6100 10.9491i 1.52631 0.351013i
\(974\) −39.1845 −1.25555
\(975\) 0 0
\(976\) 9.69893i 0.310455i
\(977\) 7.11574i 0.227653i −0.993501 0.113826i \(-0.963689\pi\)
0.993501 0.113826i \(-0.0363107\pi\)
\(978\) 0 0
\(979\) 4.16335 0.133061
\(980\) −8.22160 16.9296i −0.262629 0.540796i
\(981\) 0 0
\(982\) 28.4737i 0.908631i
\(983\) 2.26620 0.0722804 0.0361402 0.999347i \(-0.488494\pi\)
0.0361402 + 0.999347i \(0.488494\pi\)
\(984\) 0 0
\(985\) 0.999931i 0.0318605i
\(986\) 23.0775i 0.734939i
\(987\) 0 0
\(988\) −13.8401 16.9406i −0.440313 0.538953i
\(989\) −25.8649 −0.822457
\(990\) 0 0
\(991\) 30.6004i 0.972054i 0.873944 + 0.486027i \(0.161554\pi\)
−0.873944 + 0.486027i \(0.838446\pi\)
\(992\) 6.61285i 0.209958i
\(993\) 0 0
\(994\) −3.04982 13.2615i −0.0967344 0.420630i
\(995\) 43.9446 1.39314
\(996\) 0 0
\(997\) 41.1959i 1.30469i 0.757924 + 0.652343i \(0.226214\pi\)
−0.757924 + 0.652343i \(0.773786\pi\)
\(998\) 8.52617i 0.269891i
\(999\) 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 2394.2.e.c.1063.11 16
3.2 odd 2 266.2.d.a.265.1 16
7.6 odd 2 inner 2394.2.e.c.1063.14 16
12.11 even 2 2128.2.m.f.1329.16 16
19.18 odd 2 inner 2394.2.e.c.1063.3 16
21.20 even 2 266.2.d.a.265.8 yes 16
57.56 even 2 266.2.d.a.265.16 yes 16
84.83 odd 2 2128.2.m.f.1329.1 16
133.132 even 2 inner 2394.2.e.c.1063.6 16
228.227 odd 2 2128.2.m.f.1329.2 16
399.398 odd 2 266.2.d.a.265.9 yes 16
1596.1595 even 2 2128.2.m.f.1329.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.d.a.265.1 16 3.2 odd 2
266.2.d.a.265.8 yes 16 21.20 even 2
266.2.d.a.265.9 yes 16 399.398 odd 2
266.2.d.a.265.16 yes 16 57.56 even 2
2128.2.m.f.1329.1 16 84.83 odd 2
2128.2.m.f.1329.2 16 228.227 odd 2
2128.2.m.f.1329.15 16 1596.1595 even 2
2128.2.m.f.1329.16 16 12.11 even 2
2394.2.e.c.1063.3 16 19.18 odd 2 inner
2394.2.e.c.1063.6 16 133.132 even 2 inner
2394.2.e.c.1063.11 16 1.1 even 1 trivial
2394.2.e.c.1063.14 16 7.6 odd 2 inner