Properties

Label 975.2.c.b.274.1
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(274,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("975.274");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not 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: \(7.78541419707\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.b.274.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} -5.00000i q^{17} +2.00000i q^{18} -2.00000 q^{19} -3.00000 q^{21} +10.0000i q^{22} -1.00000i q^{23} +2.00000 q^{26} -1.00000i q^{27} -6.00000i q^{28} -10.0000 q^{29} -2.00000 q^{31} +8.00000i q^{32} -5.00000i q^{33} -10.0000 q^{34} +2.00000 q^{36} +3.00000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -9.00000 q^{41} +6.00000i q^{42} -4.00000i q^{43} +10.0000 q^{44} -2.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +5.00000 q^{51} -2.00000i q^{52} +9.00000i q^{53} -2.00000 q^{54} -2.00000i q^{57} +20.0000i q^{58} -11.0000 q^{61} +4.00000i q^{62} -3.00000i q^{63} +8.00000 q^{64} -10.0000 q^{66} +4.00000i q^{67} +10.0000i q^{68} +1.00000 q^{69} +15.0000 q^{71} +6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -15.0000i q^{77} +2.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} +18.0000i q^{82} +8.00000i q^{83} +6.00000 q^{84} -8.00000 q^{86} -10.0000i q^{87} +11.0000 q^{89} -3.00000 q^{91} +2.00000i q^{92} -2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} +9.00000i q^{97} +4.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} - 10 q^{11} + 12 q^{14} - 8 q^{16} - 4 q^{19} - 6 q^{21} + 4 q^{26} - 20 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{36} - 2 q^{39} - 18 q^{41} + 20 q^{44} - 4 q^{46} - 4 q^{49} + 10 q^{51} - 4 q^{54} - 22 q^{61} + 16 q^{64} - 20 q^{66} + 2 q^{69} + 30 q^{71} + 12 q^{74} + 8 q^{76} + 22 q^{79} + 2 q^{81} + 12 q^{84} - 16 q^{86} + 22 q^{89} - 6 q^{91} - 40 q^{94} - 16 q^{96} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) 1.00000i 0.277350i
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 5.00000i − 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) 2.00000i 0.471405i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 10.0000i 2.13201i
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 6.00000i − 1.13389i
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 8.00000i 1.41421i
\(33\) − 5.00000i − 0.870388i
\(34\) −10.0000 −1.71499
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 6.00000i 0.925820i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 10.0000 1.50756
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) − 10.0000i − 1.45865i −0.684167 0.729325i \(-0.739834\pi\)
0.684167 0.729325i \(-0.260166\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) − 2.00000i − 0.277350i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 20.0000i 2.62613i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 4.00000i 0.508001i
\(63\) − 3.00000i − 0.377964i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) −10.0000 −1.23091
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 10.0000i 1.21268i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 15.0000i − 1.70941i
\(78\) 2.00000i 0.226455i
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 18.0000i 1.98777i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 6.00000 0.654654
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) − 10.0000i − 1.07211i
\(88\) 0 0
\(89\) 11.0000 1.16600 0.582999 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 2.00000i 0.208514i
\(93\) − 2.00000i − 0.207390i
\(94\) −20.0000 −2.06284
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 9.00000i 0.913812i 0.889515 + 0.456906i \(0.151042\pi\)
−0.889515 + 0.456906i \(0.848958\pi\)
\(98\) 4.00000i 0.404061i
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) − 10.0000i − 0.990148i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 2.00000i 0.192450i
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) − 12.0000i − 1.13389i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 20.0000 1.85695
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 22.0000i 1.99179i
\(123\) − 9.00000i − 0.811503i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) − 14.0000i − 1.24230i −0.783692 0.621150i \(-0.786666\pi\)
0.783692 0.621150i \(-0.213334\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 10.0000i 0.870388i
\(133\) − 6.00000i − 0.520266i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 2.00000i − 0.170251i
\(139\) 17.0000 1.44192 0.720961 0.692976i \(-0.243701\pi\)
0.720961 + 0.692976i \(0.243701\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) − 30.0000i − 2.51754i
\(143\) − 5.00000i − 0.418121i
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) − 2.00000i − 0.164957i
\(148\) − 6.00000i − 0.493197i
\(149\) 7.00000 0.573462 0.286731 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 5.00000i 0.404226i
\(154\) −30.0000 −2.41747
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 22.0000i − 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) − 22.0000i − 1.75023i
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) − 2.00000i − 0.157135i
\(163\) − 11.0000i − 0.861586i −0.902451 0.430793i \(-0.858234\pi\)
0.902451 0.430793i \(-0.141766\pi\)
\(164\) 18.0000 1.40556
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 8.00000i 0.609994i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) −20.0000 −1.51620
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) 0 0
\(178\) − 22.0000i − 1.64897i
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −23.0000 −1.70958 −0.854788 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) 6.00000i 0.444750i
\(183\) − 11.0000i − 0.813143i
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 25.0000i 1.82818i
\(188\) 20.0000i 1.45865i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 8.00000i 0.577350i
\(193\) 13.0000i 0.935760i 0.883792 + 0.467880i \(0.154982\pi\)
−0.883792 + 0.467880i \(0.845018\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) − 10.0000i − 0.710669i
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 24.0000i 1.68863i
\(203\) − 30.0000i − 2.10559i
\(204\) −10.0000 −0.700140
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 1.00000i 0.0695048i
\(208\) − 4.00000i − 0.277350i
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) − 18.0000i − 1.23625i
\(213\) 15.0000i 1.02778i
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) − 6.00000i − 0.407307i
\(218\) 32.0000i 2.16731i
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) 6.00000i 0.402694i
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −24.0000 −1.60357
\(225\) 0 0
\(226\) 4.00000 0.266076
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 0 0
\(233\) − 25.0000i − 1.63780i −0.573933 0.818902i \(-0.694583\pi\)
0.573933 0.818902i \(-0.305417\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 11.0000i 0.714527i
\(238\) − 30.0000i − 1.94461i
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 28.0000i − 1.79991i
\(243\) 1.00000i 0.0641500i
\(244\) 22.0000 1.40841
\(245\) 0 0
\(246\) −18.0000 −1.14764
\(247\) − 2.00000i − 0.127257i
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 6.00000i 0.377964i
\(253\) 5.00000i 0.314347i
\(254\) −28.0000 −1.75688
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) − 12.0000i − 0.741362i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −12.0000 −0.735767
\(267\) 11.0000i 0.673189i
\(268\) − 8.00000i − 0.488678i
\(269\) −32.0000 −1.95107 −0.975537 0.219834i \(-0.929448\pi\)
−0.975537 + 0.219834i \(0.929448\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 20.0000i 1.21268i
\(273\) − 3.00000i − 0.181568i
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) − 34.0000i − 2.03918i
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) − 20.0000i − 1.19098i
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) −30.0000 −1.78017
\(285\) 0 0
\(286\) −10.0000 −0.591312
\(287\) − 27.0000i − 1.59376i
\(288\) − 8.00000i − 0.471405i
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −9.00000 −0.527589
\(292\) − 12.0000i − 0.702247i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) −4.00000 −0.233285
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) − 14.0000i − 0.810998i
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 24.0000i 1.38104i
\(303\) − 12.0000i − 0.689382i
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 10.0000 0.571662
\(307\) 19.0000i 1.08439i 0.840254 + 0.542194i \(0.182406\pi\)
−0.840254 + 0.542194i \(0.817594\pi\)
\(308\) 30.0000i 1.70941i
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) −44.0000 −2.48306
\(315\) 0 0
\(316\) −22.0000 −1.23760
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 18.0000i 1.00939i
\(319\) 50.0000 2.79946
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) − 6.00000i − 0.334367i
\(323\) 10.0000i 0.556415i
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) −22.0000 −1.21847
\(327\) − 16.0000i − 0.884802i
\(328\) 0 0
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) − 3.00000i − 0.164399i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) − 4.00000i − 0.217894i −0.994048 0.108947i \(-0.965252\pi\)
0.994048 0.108947i \(-0.0347479\pi\)
\(338\) 2.00000i 0.108786i
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) − 4.00000i − 0.216295i
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) 1.00000i 0.0536828i 0.999640 + 0.0268414i \(0.00854491\pi\)
−0.999640 + 0.0268414i \(0.991455\pi\)
\(348\) 20.0000i 1.07211i
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) − 40.0000i − 2.13201i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −22.0000 −1.16600
\(357\) 15.0000i 0.793884i
\(358\) − 12.0000i − 0.634220i
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 46.0000i 2.41771i
\(363\) 14.0000i 0.734809i
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) −22.0000 −1.14996
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) −27.0000 −1.40177
\(372\) 4.00000i 0.207390i
\(373\) 16.0000i 0.828449i 0.910175 + 0.414224i \(0.135947\pi\)
−0.910175 + 0.414224i \(0.864053\pi\)
\(374\) 50.0000 2.58544
\(375\) 0 0
\(376\) 0 0
\(377\) − 10.0000i − 0.515026i
\(378\) − 6.00000i − 0.308607i
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) 40.0000i 2.04658i
\(383\) − 18.0000i − 0.919757i −0.887982 0.459879i \(-0.847893\pi\)
0.887982 0.459879i \(-0.152107\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) 4.00000i 0.203331i
\(388\) − 18.0000i − 0.913812i
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 24.0000 1.20910
\(395\) 0 0
\(396\) −10.0000 −0.502519
\(397\) 19.0000i 0.953583i 0.879017 + 0.476791i \(0.158200\pi\)
−0.879017 + 0.476791i \(0.841800\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 8.00000i 0.399004i
\(403\) − 2.00000i − 0.0996271i
\(404\) 24.0000 1.19404
\(405\) 0 0
\(406\) −60.0000 −2.97775
\(407\) − 15.0000i − 0.743522i
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) − 8.00000i − 0.394132i
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) −8.00000 −0.392232
\(417\) 17.0000i 0.832494i
\(418\) − 20.0000i − 0.978232i
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 8.00000i 0.389434i
\(423\) 10.0000i 0.486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 30.0000 1.45350
\(427\) − 33.0000i − 1.59698i
\(428\) 6.00000i 0.290021i
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 24.0000i 1.15337i 0.816968 + 0.576683i \(0.195653\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) 2.00000i 0.0956730i
\(438\) 12.0000i 0.573382i
\(439\) 33.0000 1.57500 0.787502 0.616312i \(-0.211374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) − 10.0000i − 0.475651i
\(443\) 35.0000i 1.66290i 0.555599 + 0.831450i \(0.312489\pi\)
−0.555599 + 0.831450i \(0.687511\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 0 0
\(447\) 7.00000i 0.331089i
\(448\) 24.0000i 1.13389i
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) − 4.00000i − 0.188144i
\(453\) − 12.0000i − 0.563809i
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) 13.0000i 0.608114i 0.952654 + 0.304057i \(0.0983414\pi\)
−0.952654 + 0.304057i \(0.901659\pi\)
\(458\) − 28.0000i − 1.30835i
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) − 30.0000i − 1.39573i
\(463\) 5.00000i 0.232370i 0.993228 + 0.116185i \(0.0370665\pi\)
−0.993228 + 0.116185i \(0.962933\pi\)
\(464\) 40.0000 1.85695
\(465\) 0 0
\(466\) −50.0000 −2.31621
\(467\) 29.0000i 1.34196i 0.741475 + 0.670980i \(0.234126\pi\)
−0.741475 + 0.670980i \(0.765874\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 20.0000i 0.919601i
\(474\) 22.0000 1.01049
\(475\) 0 0
\(476\) −30.0000 −1.37505
\(477\) − 9.00000i − 0.412082i
\(478\) 30.0000i 1.37217i
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) 0 0
\(481\) −3.00000 −0.136788
\(482\) 28.0000i 1.27537i
\(483\) 3.00000i 0.136505i
\(484\) −28.0000 −1.27273
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) − 7.00000i − 0.317200i −0.987343 0.158600i \(-0.949302\pi\)
0.987343 0.158600i \(-0.0506981\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 18.0000i 0.811503i
\(493\) 50.0000i 2.25189i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 45.0000i 2.01853i
\(498\) 16.0000i 0.716977i
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) − 40.0000i − 1.78529i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.0000 0.444554
\(507\) − 1.00000i − 0.0444116i
\(508\) 28.0000i 1.24230i
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) − 32.0000i − 1.41421i
\(513\) 2.00000i 0.0883022i
\(514\) −36.0000 −1.58789
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 50.0000i 2.19900i
\(518\) 18.0000i 0.790875i
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) − 20.0000i − 0.875376i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0000i 0.435607i
\(528\) 20.0000i 0.870388i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 12.0000i 0.520266i
\(533\) − 9.00000i − 0.389833i
\(534\) 22.0000 0.952033
\(535\) 0 0
\(536\) 0 0
\(537\) 6.00000i 0.258919i
\(538\) 64.0000i 2.75924i
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 4.00000i 0.171815i
\(543\) − 23.0000i − 0.987024i
\(544\) 40.0000 1.71499
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 11.0000 0.469469
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 33.0000i 1.40330i
\(554\) −52.0000 −2.20927
\(555\) 0 0
\(556\) −34.0000 −1.44192
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −25.0000 −1.05550
\(562\) 20.0000i 0.843649i
\(563\) − 41.0000i − 1.72794i −0.503540 0.863972i \(-0.667969\pi\)
0.503540 0.863972i \(-0.332031\pi\)
\(564\) −20.0000 −0.842152
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 3.00000i 0.125988i
\(568\) 0 0
\(569\) −16.0000 −0.670755 −0.335377 0.942084i \(-0.608864\pi\)
−0.335377 + 0.942084i \(0.608864\pi\)
\(570\) 0 0
\(571\) 17.0000 0.711428 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(572\) 10.0000i 0.418121i
\(573\) − 20.0000i − 0.835512i
\(574\) −54.0000 −2.25392
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 21.0000i 0.874241i 0.899403 + 0.437121i \(0.144002\pi\)
−0.899403 + 0.437121i \(0.855998\pi\)
\(578\) 16.0000i 0.665512i
\(579\) −13.0000 −0.540262
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 18.0000i 0.746124i
\(583\) − 45.0000i − 1.86371i
\(584\) 0 0
\(585\) 0 0
\(586\) 48.0000 1.98286
\(587\) − 42.0000i − 1.73353i −0.498721 0.866763i \(-0.666197\pi\)
0.498721 0.866763i \(-0.333803\pi\)
\(588\) 4.00000i 0.164957i
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) − 12.0000i − 0.493197i
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 10.0000 0.410305
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) − 4.00000i − 0.163709i
\(598\) − 2.00000i − 0.0817861i
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) − 24.0000i − 0.978167i
\(603\) − 4.00000i − 0.162893i
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) −24.0000 −0.974933
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) − 16.0000i − 0.648886i
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) 10.0000 0.404557
\(612\) − 10.0000i − 0.404226i
\(613\) 3.00000i 0.121169i 0.998163 + 0.0605844i \(0.0192964\pi\)
−0.998163 + 0.0605844i \(0.980704\pi\)
\(614\) 38.0000 1.53356
\(615\) 0 0
\(616\) 0 0
\(617\) − 22.0000i − 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 48.0000i − 1.92462i
\(623\) 33.0000i 1.32212i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) 10.0000i 0.399362i
\(628\) 44.0000i 1.75579i
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) 18.0000 0.713746
\(637\) − 2.00000i − 0.0792429i
\(638\) − 100.000i − 3.95904i
\(639\) −15.0000 −0.593391
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) − 6.00000i − 0.236801i
\(643\) 1.00000i 0.0394362i 0.999806 + 0.0197181i \(0.00627687\pi\)
−0.999806 + 0.0197181i \(0.993723\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) 20.0000 0.786889
\(647\) − 21.0000i − 0.825595i −0.910823 0.412798i \(-0.864552\pi\)
0.910823 0.412798i \(-0.135448\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 22.0000i 0.861586i
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) −32.0000 −1.25130
\(655\) 0 0
\(656\) 36.0000 1.40556
\(657\) − 6.00000i − 0.234082i
\(658\) − 60.0000i − 2.33904i
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) − 64.0000i − 2.48743i
\(663\) 5.00000i 0.194184i
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 10.0000i 0.387202i
\(668\) − 16.0000i − 0.619059i
\(669\) 0 0
\(670\) 0 0
\(671\) 55.0000 2.12325
\(672\) − 24.0000i − 0.925820i
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 2.00000 0.0769231
\(677\) 7.00000i 0.269032i 0.990911 + 0.134516i \(0.0429479\pi\)
−0.990911 + 0.134516i \(0.957052\pi\)
\(678\) 4.00000i 0.153619i
\(679\) −27.0000 −1.03616
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) − 20.0000i − 0.765840i
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 30.0000 1.14541
\(687\) 14.0000i 0.534133i
\(688\) 16.0000i 0.609994i
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) 4.00000i 0.152057i
\(693\) 15.0000i 0.569803i
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 0 0
\(697\) 45.0000i 1.70450i
\(698\) 40.0000i 1.51402i
\(699\) 25.0000 0.945587
\(700\) 0 0
\(701\) −4.00000 −0.151078 −0.0755390 0.997143i \(-0.524068\pi\)
−0.0755390 + 0.997143i \(0.524068\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) − 6.00000i − 0.226294i
\(704\) −40.0000 −1.50756
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) − 36.0000i − 1.35392i
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 0 0
\(713\) 2.00000i 0.0749006i
\(714\) 30.0000 1.12272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 15.0000i − 0.560185i
\(718\) 32.0000i 1.19423i
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 30.0000i 1.11648i
\(723\) − 14.0000i − 0.520666i
\(724\) 46.0000 1.70958
\(725\) 0 0
\(726\) 28.0000 1.03918
\(727\) − 6.00000i − 0.222528i −0.993791 0.111264i \(-0.964510\pi\)
0.993791 0.111264i \(-0.0354899\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 22.0000i 0.813143i
\(733\) − 15.0000i − 0.554038i −0.960864 0.277019i \(-0.910654\pi\)
0.960864 0.277019i \(-0.0893464\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) − 20.0000i − 0.736709i
\(738\) − 18.0000i − 0.662589i
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 54.0000i 1.98240i
\(743\) 6.00000i 0.220119i 0.993925 + 0.110059i \(0.0351041\pi\)
−0.993925 + 0.110059i \(0.964896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) − 8.00000i − 0.292705i
\(748\) − 50.0000i − 1.82818i
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) −45.0000 −1.64207 −0.821037 0.570875i \(-0.806604\pi\)
−0.821037 + 0.570875i \(0.806604\pi\)
\(752\) 40.0000i 1.45865i
\(753\) 20.0000i 0.728841i
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) −6.00000 −0.218218
\(757\) − 36.0000i − 1.30844i −0.756303 0.654221i \(-0.772997\pi\)
0.756303 0.654221i \(-0.227003\pi\)
\(758\) 12.0000i 0.435860i
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) − 28.0000i − 1.01433i
\(763\) − 48.0000i − 1.73772i
\(764\) 40.0000 1.44715
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 0 0
\(768\) 16.0000i 0.577350i
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 26.0000i − 0.935760i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.00000i − 0.322873i
\(778\) − 48.0000i − 1.72088i
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −75.0000 −2.68371
\(782\) 10.0000i 0.357599i
\(783\) 10.0000i 0.357371i
\(784\) 8.00000 0.285714
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) − 44.0000i − 1.56843i −0.620489 0.784215i \(-0.713066\pi\)
0.620489 0.784215i \(-0.286934\pi\)
\(788\) − 24.0000i − 0.854965i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) − 11.0000i − 0.390621i
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 5.00000i 0.177109i 0.996071 + 0.0885545i \(0.0282248\pi\)
−0.996071 + 0.0885545i \(0.971775\pi\)
\(798\) − 12.0000i − 0.424795i
\(799\) −50.0000 −1.76887
\(800\) 0 0
\(801\) −11.0000 −0.388666
\(802\) 36.0000i 1.27120i
\(803\) − 30.0000i − 1.05868i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) − 32.0000i − 1.12645i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 60.0000i 2.10559i
\(813\) − 2.00000i − 0.0701431i
\(814\) −30.0000 −1.05150
\(815\) 0 0
\(816\) −20.0000 −0.700140
\(817\) 8.00000i 0.279885i
\(818\) − 52.0000i − 1.81814i
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) −41.0000 −1.43091 −0.715455 0.698659i \(-0.753781\pi\)
−0.715455 + 0.698659i \(0.753781\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 48.0000i − 1.67317i −0.547833 0.836587i \(-0.684547\pi\)
0.547833 0.836587i \(-0.315453\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.0000i 1.46048i 0.683189 + 0.730242i \(0.260592\pi\)
−0.683189 + 0.730242i \(0.739408\pi\)
\(828\) − 2.00000i − 0.0695048i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 8.00000i 0.277350i
\(833\) 10.0000i 0.346479i
\(834\) 34.0000 1.17732
\(835\) 0 0
\(836\) −20.0000 −0.691714
\(837\) 2.00000i 0.0691301i
\(838\) 52.0000i 1.79631i
\(839\) −7.00000 −0.241667 −0.120833 0.992673i \(-0.538557\pi\)
−0.120833 + 0.992673i \(0.538557\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) − 64.0000i − 2.20559i
\(843\) − 10.0000i − 0.344418i
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 20.0000 0.687614
\(847\) 42.0000i 1.44314i
\(848\) − 36.0000i − 1.23625i
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) − 30.0000i − 1.02778i
\(853\) 51.0000i 1.74621i 0.487535 + 0.873103i \(0.337896\pi\)
−0.487535 + 0.873103i \(0.662104\pi\)
\(854\) −66.0000 −2.25847
\(855\) 0 0
\(856\) 0 0
\(857\) 17.0000i 0.580709i 0.956919 + 0.290354i \(0.0937732\pi\)
−0.956919 + 0.290354i \(0.906227\pi\)
\(858\) − 10.0000i − 0.341394i
\(859\) −35.0000 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(860\) 0 0
\(861\) 27.0000 0.920158
\(862\) 32.0000i 1.08992i
\(863\) − 22.0000i − 0.748889i −0.927249 0.374444i \(-0.877833\pi\)
0.927249 0.374444i \(-0.122167\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) 48.0000 1.63111
\(867\) − 8.00000i − 0.271694i
\(868\) 12.0000i 0.407307i
\(869\) −55.0000 −1.86575
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) − 9.00000i − 0.304604i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 66.0000i − 2.22739i
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) − 4.00000i − 0.134687i
\(883\) 12.0000i 0.403832i 0.979403 + 0.201916i \(0.0647168\pi\)
−0.979403 + 0.201916i \(0.935283\pi\)
\(884\) −10.0000 −0.336336
\(885\) 0 0
\(886\) 70.0000 2.35170
\(887\) 15.0000i 0.503651i 0.967773 + 0.251825i \(0.0810309\pi\)
−0.967773 + 0.251825i \(0.918969\pi\)
\(888\) 0 0
\(889\) 42.0000 1.40863
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 20.0000i 0.669274i
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) 1.00000i 0.0333890i
\(898\) 30.0000i 1.00111i
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 45.0000 1.49917
\(902\) − 90.0000i − 2.99667i
\(903\) 12.0000i 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) − 2.00000i − 0.0664089i −0.999449 0.0332045i \(-0.989429\pi\)
0.999449 0.0332045i \(-0.0105712\pi\)
\(908\) − 36.0000i − 1.19470i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 40.0000i − 1.32381i
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −28.0000 −0.925146
\(917\) 18.0000i 0.594412i
\(918\) 10.0000i 0.330049i
\(919\) −29.0000 −0.956622 −0.478311 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) −19.0000 −0.626071
\(922\) − 6.00000i − 0.197599i
\(923\) 15.0000i 0.493731i
\(924\) −30.0000 −0.986928
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) − 4.00000i − 0.131377i
\(928\) − 80.0000i − 2.62613i
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 50.0000i 1.63780i
\(933\) 24.0000i 0.785725i
\(934\) 58.0000 1.89782
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) 24.0000i 0.783628i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −23.0000 −0.749779 −0.374889 0.927070i \(-0.622319\pi\)
−0.374889 + 0.927070i \(0.622319\pi\)
\(942\) − 44.0000i − 1.43360i
\(943\) 9.00000i 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) 40.0000 1.30051
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) − 22.0000i − 0.714527i
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) − 11.0000i − 0.356325i −0.984001 0.178162i \(-0.942985\pi\)
0.984001 0.178162i \(-0.0570153\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) 30.0000 0.970269
\(957\) 50.0000i 1.61627i
\(958\) 10.0000i 0.323085i
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 6.00000i 0.193448i
\(963\) 3.00000i 0.0966736i
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) 6.00000 0.193047
\(967\) 40.0000i 1.28631i 0.765735 + 0.643157i \(0.222376\pi\)
−0.765735 + 0.643157i \(0.777624\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) 51.0000i 1.63498i
\(974\) −14.0000 −0.448589
\(975\) 0 0
\(976\) 44.0000 1.40841
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) − 22.0000i − 0.703482i
\(979\) −55.0000 −1.75781
\(980\) 0 0
\(981\) 16.0000 0.510841
\(982\) − 32.0000i − 1.02116i
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 100.000 3.18465
\(987\) 30.0000i 0.954911i
\(988\) 4.00000i 0.127257i
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 39.0000 1.23888 0.619438 0.785046i \(-0.287361\pi\)
0.619438 + 0.785046i \(0.287361\pi\)
\(992\) − 16.0000i − 0.508001i
\(993\) 32.0000i 1.01549i
\(994\) 90.0000 2.85463
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) 16.0000i 0.506725i 0.967371 + 0.253363i \(0.0815366\pi\)
−0.967371 + 0.253363i \(0.918463\pi\)
\(998\) 68.0000i 2.15250i
\(999\) 3.00000 0.0949158
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 975.2.c.b.274.1 2
3.2 odd 2 2925.2.c.d.2224.2 2
5.2 odd 4 195.2.a.d.1.1 1
5.3 odd 4 975.2.a.b.1.1 1
5.4 even 2 inner 975.2.c.b.274.2 2
15.2 even 4 585.2.a.a.1.1 1
15.8 even 4 2925.2.a.t.1.1 1
15.14 odd 2 2925.2.c.d.2224.1 2
20.7 even 4 3120.2.a.n.1.1 1
35.27 even 4 9555.2.a.t.1.1 1
60.47 odd 4 9360.2.a.w.1.1 1
65.12 odd 4 2535.2.a.b.1.1 1
195.77 even 4 7605.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.d.1.1 1 5.2 odd 4
585.2.a.a.1.1 1 15.2 even 4
975.2.a.b.1.1 1 5.3 odd 4
975.2.c.b.274.1 2 1.1 even 1 trivial
975.2.c.b.274.2 2 5.4 even 2 inner
2535.2.a.b.1.1 1 65.12 odd 4
2925.2.a.t.1.1 1 15.8 even 4
2925.2.c.d.2224.1 2 15.14 odd 2
2925.2.c.d.2224.2 2 3.2 odd 2
3120.2.a.n.1.1 1 20.7 even 4
7605.2.a.v.1.1 1 195.77 even 4
9360.2.a.w.1.1 1 60.47 odd 4
9555.2.a.t.1.1 1 35.27 even 4