Properties

Label 2925.2.c.d.2224.1
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
Analytic rank $0$
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] = [2925,2,Mod(2224,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, 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("2925.2224");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.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: \(23.3562425912\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.d.2224.2

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(-0.250000\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 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 \(-0.207363\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 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
\(27\) 0 0
\(28\) 6.00000i 1.13389i
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −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 \(-0.260166\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) − 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
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 10.0000i 1.21268i
\(69\) 0 0
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 15.0000i − 1.70941i
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 18.0000i − 1.98777i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 2.00000i 0.208514i
\(93\) 0 0
\(94\) −20.0000 −2.06284
\(95\) 0 0
\(96\) 0 0
\(97\) − 9.00000i − 0.913812i −0.889515 0.456906i \(-0.848958\pi\)
0.889515 0.456906i \(-0.151042\pi\)
\(98\) 4.00000i 0.404061i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 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 \(-0.0463216\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12.0000i 1.13389i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −20.0000 −1.85695
\(117\) 0 0
\(118\) 0 0
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 22.0000i 1.99179i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 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 \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 17.0000 1.44192 0.720961 0.692976i \(-0.243701\pi\)
0.720961 + 0.692976i \(0.243701\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 30.0000i 2.51754i
\(143\) − 5.00000i − 0.418121i
\(144\) 0 0
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) 6.00000i 0.493197i
\(149\) −7.00000 −0.573462 −0.286731 0.958011i \(-0.592569\pi\)
−0.286731 + 0.958011i \(0.592569\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −30.0000 −2.41747
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) − 22.0000i − 1.75023i
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) −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
\(171\) 0 0
\(172\) − 8.00000i − 0.609994i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.0000 −1.50756
\(177\) 0 0
\(178\) 22.0000i 1.64897i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 25.0000i − 1.82818i
\(188\) 20.0000i 1.45865i
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) − 13.0000i − 0.935760i −0.883792 0.467880i \(-0.845018\pi\)
0.883792 0.467880i \(-0.154982\pi\)
\(194\) −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 \(0.859401\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 24.0000i − 1.68863i
\(203\) − 30.0000i − 2.10559i
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000i 0.407307i
\(218\) 32.0000i 2.16731i
\(219\) 0 0
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) 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 \(0.796220\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 25.0000i − 1.63780i −0.573933 0.818902i \(-0.694583\pi\)
0.573933 0.818902i \(-0.305417\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 30.0000i 1.94461i
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 22.0000 1.40841
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) − 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 \(-0.189739\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) 32.0000 1.95107 0.975537 0.219834i \(-0.0705517\pi\)
0.975537 + 0.219834i \(0.0705517\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) − 34.0000i − 2.03918i
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 8.00000i 0.475551i 0.971320 + 0.237775i \(0.0764182\pi\)
−0.971320 + 0.237775i \(0.923582\pi\)
\(284\) 30.0000 1.78017
\(285\) 0 0
\(286\) −10.0000 −0.591312
\(287\) − 27.0000i − 1.59376i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 12.0000i 0.702247i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 14.0000i 0.810998i
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 24.0000i 1.38104i
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) − 19.0000i − 1.08439i −0.840254 0.542194i \(-0.817594\pi\)
0.840254 0.542194i \(-0.182406\pi\)
\(308\) 30.0000i 1.70941i
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 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 \(-0.109410\pi\)
\(318\) 0 0
\(319\) 50.0000 2.79946
\(320\) 0 0
\(321\) 0 0
\(322\) 6.00000i 0.334367i
\(323\) 10.0000i 0.556415i
\(324\) 0 0
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) 4.00000i 0.217894i 0.994048 + 0.108947i \(0.0347479\pi\)
−0.994048 + 0.108947i \(0.965252\pi\)
\(338\) 2.00000i 0.108786i
\(339\) 0 0
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) − 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 \(0.991455\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 46.0000i 2.41771i
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 27.0000 1.40177
\(372\) 0 0
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) −50.0000 −2.58544
\(375\) 0 0
\(376\) 0 0
\(377\) − 10.0000i − 0.515026i
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 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
\(387\) 0 0
\(388\) 18.0000i 0.913812i
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 24.0000 1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) − 19.0000i − 0.953583i −0.879017 0.476791i \(-0.841800\pi\)
0.879017 0.476791i \(-0.158200\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 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
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) 20.0000i 0.978232i
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 33.0000i 1.59698i
\(428\) 6.00000i 0.290021i
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) − 24.0000i − 1.15337i −0.816968 0.576683i \(-0.804347\pi\)
0.816968 0.576683i \(-0.195653\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) 2.00000i 0.0956730i
\(438\) 0 0
\(439\) 33.0000 1.57500 0.787502 0.616312i \(-0.211374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.0000i 0.475651i
\(443\) 35.0000i 1.66290i 0.555599 + 0.831450i \(0.312489\pi\)
−0.555599 + 0.831450i \(0.687511\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) − 24.0000i − 1.13389i
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) − 4.00000i − 0.188144i
\(453\) 0 0
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) − 13.0000i − 0.608114i −0.952654 0.304057i \(-0.901659\pi\)
0.952654 0.304057i \(-0.0983414\pi\)
\(458\) − 28.0000i − 1.30835i
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) − 5.00000i − 0.232370i −0.993228 0.116185i \(-0.962933\pi\)
0.993228 0.116185i \(-0.0370665\pi\)
\(464\) −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 \(0.765874\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20.0000i 0.919601i
\(474\) 0 0
\(475\) 0 0
\(476\) 30.0000 1.37505
\(477\) 0 0
\(478\) − 30.0000i − 1.37217i
\(479\) 5.00000 0.228456 0.114228 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(480\) 0 0
\(481\) −3.00000 −0.136788
\(482\) 28.0000i 1.27537i
\(483\) 0 0
\(484\) −28.0000 −1.27273
\(485\) 0 0
\(486\) 0 0
\(487\) 7.00000i 0.317200i 0.987343 + 0.158600i \(0.0506981\pi\)
−0.987343 + 0.158600i \(0.949302\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) − 50.0000i − 2.25189i
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 45.0000i 2.01853i
\(498\) 0 0
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) − 28.0000i − 1.24230i
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) − 32.0000i − 1.41421i
\(513\) 0 0
\(514\) −36.0000 −1.58789
\(515\) 0 0
\(516\) 0 0
\(517\) − 50.0000i − 2.19900i
\(518\) 18.0000i 0.790875i
\(519\) 0 0
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0000i 0.435607i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) − 12.0000i − 0.520266i
\(533\) − 9.00000i − 0.389833i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) − 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
\(543\) 0 0
\(544\) 40.0000 1.71499
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 0 0
\(550\) 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 \(0.780765\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) − 20.0000i − 0.843649i
\(563\) − 41.0000i − 1.72794i −0.503540 0.863972i \(-0.667969\pi\)
0.503540 0.863972i \(-0.332031\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 0 0
\(569\) 16.0000 0.670755 0.335377 0.942084i \(-0.391136\pi\)
0.335377 + 0.942084i \(0.391136\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −54.0000 −2.25392
\(575\) 0 0
\(576\) 0 0
\(577\) − 21.0000i − 0.874241i −0.899403 0.437121i \(-0.855998\pi\)
0.899403 0.437121i \(-0.144002\pi\)
\(578\) 16.0000i 0.665512i
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 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 \(-0.333803\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 12.0000i 0.493197i
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) 2.00000i 0.0817861i
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) − 16.0000i − 0.648886i
\(609\) 0 0
\(610\) 0 0
\(611\) −10.0000 −0.404557
\(612\) 0 0
\(613\) − 3.00000i − 0.121169i −0.998163 0.0605844i \(-0.980704\pi\)
0.998163 0.0605844i \(-0.0192964\pi\)
\(614\) −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 \(-0.146030\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 48.0000i 1.92462i
\(623\) 33.0000i 1.32212i
\(624\) 0 0
\(625\) 0 0
\(626\) 20.0000 0.799361
\(627\) 0 0
\(628\) − 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
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) − 100.000i − 3.95904i
\(639\) 0 0
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0 0
\(643\) − 1.00000i − 0.0394362i −0.999806 0.0197181i \(-0.993723\pi\)
0.999806 0.0197181i \(-0.00627687\pi\)
\(644\) 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
\(651\) 0 0
\(652\) − 22.0000i − 0.861586i
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36.0000 −1.40556
\(657\) 0 0
\(658\) 60.0000i 2.33904i
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 10.0000i − 0.387202i
\(668\) − 16.0000i − 0.619059i
\(669\) 0 0
\(670\) 0 0
\(671\) −55.0000 −2.12325
\(672\) 0 0
\(673\) − 22.0000i − 0.848038i −0.905653 0.424019i \(-0.860619\pi\)
0.905653 0.424019i \(-0.139381\pi\)
\(674\) 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 \(0.957052\pi\)
\(678\) 0 0
\(679\) −27.0000 −1.03616
\(680\) 0 0
\(681\) 0 0
\(682\) 20.0000i 0.765840i
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −30.0000 −1.14541
\(687\) 0 0
\(688\) − 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
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 0 0
\(697\) − 45.0000i − 1.70450i
\(698\) 40.0000i 1.51402i
\(699\) 0 0
\(700\) 0 0
\(701\) 4.00000 0.151078 0.0755390 0.997143i \(-0.475932\pi\)
0.0755390 + 0.997143i \(0.475932\pi\)
\(702\) 0 0
\(703\) 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
\(711\) 0 0
\(712\) 0 0
\(713\) 2.00000i 0.0749006i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) − 32.0000i − 1.19423i
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 30.0000i 1.11648i
\(723\) 0 0
\(724\) 46.0000 1.70958
\(725\) 0 0
\(726\) 0 0
\(727\) 6.00000i 0.222528i 0.993791 + 0.111264i \(0.0354899\pi\)
−0.993791 + 0.111264i \(0.964510\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 15.0000i 0.554038i 0.960864 + 0.277019i \(0.0893464\pi\)
−0.960864 + 0.277019i \(0.910654\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) − 20.0000i − 0.736709i
\(738\) 0 0
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 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
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) 36.0000i 1.30844i 0.756303 + 0.654221i \(0.227003\pi\)
−0.756303 + 0.654221i \(0.772997\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 0 0
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) 48.0000i 1.73772i
\(764\) −40.0000 −1.44715
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 0 0
\(768\) 0 0
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.0000i 0.935760i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 48.0000i 1.72088i
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) −75.0000 −2.68371
\(782\) 10.0000i 0.357599i
\(783\) 0 0
\(784\) 8.00000 0.285714
\(785\) 0 0
\(786\) 0 0
\(787\) 44.0000i 1.56843i 0.620489 + 0.784215i \(0.286934\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(788\) − 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 \(0.971775\pi\)
\(798\) 0 0
\(799\) −50.0000 −1.76887
\(800\) 0 0
\(801\) 0 0
\(802\) − 36.0000i − 1.27120i
\(803\) − 30.0000i − 1.05868i
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −30.0000 −1.05150
\(815\) 0 0
\(816\) 0 0
\(817\) − 8.00000i − 0.279885i
\(818\) − 52.0000i − 1.81814i
\(819\) 0 0
\(820\) 0 0
\(821\) 41.0000 1.43091 0.715455 0.698659i \(-0.246219\pi\)
0.715455 + 0.698659i \(0.246219\pi\)
\(822\) 0 0
\(823\) 48.0000i 1.67317i 0.547833 + 0.836587i \(0.315453\pi\)
−0.547833 + 0.836587i \(0.684547\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.0000i 1.46048i 0.683189 + 0.730242i \(0.260592\pi\)
−0.683189 + 0.730242i \(0.739408\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 8.00000i − 0.277350i
\(833\) 10.0000i 0.346479i
\(834\) 0 0
\(835\) 0 0
\(836\) 20.0000 0.691714
\(837\) 0 0
\(838\) − 52.0000i − 1.79631i
\(839\) 7.00000 0.241667 0.120833 0.992673i \(-0.461443\pi\)
0.120833 + 0.992673i \(0.461443\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) − 64.0000i − 2.20559i
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) − 42.0000i − 1.44314i
\(848\) − 36.0000i − 1.23625i
\(849\) 0 0
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) − 51.0000i − 1.74621i −0.487535 0.873103i \(-0.662104\pi\)
0.487535 0.873103i \(-0.337896\pi\)
\(854\) 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 \(0.906227\pi\)
\(858\) 0 0
\(859\) −35.0000 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 32.0000i − 1.08992i
\(863\) − 22.0000i − 0.748889i −0.927249 0.374444i \(-0.877833\pi\)
0.927249 0.374444i \(-0.122167\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −48.0000 −1.63111
\(867\) 0 0
\(868\) − 12.0000i − 0.407307i
\(869\) 55.0000 1.86575
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 0 0
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 66.0000i − 2.22739i
\(879\) 0 0
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) − 12.0000i − 0.403832i −0.979403 0.201916i \(-0.935283\pi\)
0.979403 0.201916i \(-0.0647168\pi\)
\(884\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) 20.0000i 0.669274i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 30.0000i − 1.00111i
\(899\) −20.0000 −0.667037
\(900\) 0 0
\(901\) 45.0000 1.49917
\(902\) − 90.0000i − 2.99667i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.00000i 0.0664089i 0.999449 + 0.0332045i \(0.0105712\pi\)
−0.999449 + 0.0332045i \(0.989429\pi\)
\(908\) − 36.0000i − 1.19470i
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 40.0000i 1.32381i
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −28.0000 −0.925146
\(917\) 18.0000i 0.594412i
\(918\) 0 0
\(919\) −29.0000 −0.956622 −0.478311 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000i 0.197599i
\(923\) 15.0000i 0.493731i
\(924\) 0 0
\(925\) 0 0
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) 80.0000i 2.62613i
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 50.0000i 1.63780i
\(933\) 0 0
\(934\) 58.0000 1.89782
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 24.0000i 0.783628i
\(939\) 0 0
\(940\) 0 0
\(941\) 23.0000 0.749779 0.374889 0.927070i \(-0.377681\pi\)
0.374889 + 0.927070i \(0.377681\pi\)
\(942\) 0 0
\(943\) − 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 \(-0.198875\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 11.0000i − 0.356325i −0.984001 0.178162i \(-0.942985\pi\)
0.984001 0.178162i \(-0.0570153\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −30.0000 −0.970269
\(957\) 0 0
\(958\) − 10.0000i − 0.323085i
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 6.00000i 0.193448i
\(963\) 0 0
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) − 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 \(-0.0614834\pi\)
\(978\) 0 0
\(979\) −55.0000 −1.75781
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(987\) 0 0
\(988\) − 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
\(993\) 0 0
\(994\) 90.0000 2.85463
\(995\) 0 0
\(996\) 0 0
\(997\) − 16.0000i − 0.506725i −0.967371 0.253363i \(-0.918463\pi\)
0.967371 0.253363i \(-0.0815366\pi\)
\(998\) 68.0000i 2.15250i
\(999\) 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 2925.2.c.d.2224.1 2
3.2 odd 2 975.2.c.b.274.2 2
5.2 odd 4 2925.2.a.t.1.1 1
5.3 odd 4 585.2.a.a.1.1 1
5.4 even 2 inner 2925.2.c.d.2224.2 2
15.2 even 4 975.2.a.b.1.1 1
15.8 even 4 195.2.a.d.1.1 1
15.14 odd 2 975.2.c.b.274.1 2
20.3 even 4 9360.2.a.w.1.1 1
60.23 odd 4 3120.2.a.n.1.1 1
65.38 odd 4 7605.2.a.v.1.1 1
105.83 odd 4 9555.2.a.t.1.1 1
195.38 even 4 2535.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.d.1.1 1 15.8 even 4
585.2.a.a.1.1 1 5.3 odd 4
975.2.a.b.1.1 1 15.2 even 4
975.2.c.b.274.1 2 15.14 odd 2
975.2.c.b.274.2 2 3.2 odd 2
2535.2.a.b.1.1 1 195.38 even 4
2925.2.a.t.1.1 1 5.2 odd 4
2925.2.c.d.2224.1 2 1.1 even 1 trivial
2925.2.c.d.2224.2 2 5.4 even 2 inner
3120.2.a.n.1.1 1 60.23 odd 4
7605.2.a.v.1.1 1 65.38 odd 4
9360.2.a.w.1.1 1 20.3 even 4
9555.2.a.t.1.1 1 105.83 odd 4