Properties

Label 2925.2.c.v.2224.2
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.v.2224.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +O(q^{10})\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +1.26795 q^{11} -1.00000i q^{13} +3.46410 q^{14} -5.00000 q^{16} -3.46410i q^{17} -4.19615 q^{19} -2.19615i q^{22} +4.73205i q^{23} -1.73205 q^{26} -2.00000i q^{28} -9.46410 q^{29} -0.196152 q^{31} +5.19615i q^{32} -6.00000 q^{34} -4.00000i q^{37} +7.26795i q^{38} +3.46410 q^{41} -10.1962i q^{43} -1.26795 q^{44} +8.19615 q^{46} -6.00000i q^{47} +3.00000 q^{49} +1.00000i q^{52} -10.3923i q^{53} +3.46410 q^{56} +16.3923i q^{58} -15.1244 q^{59} +12.3923 q^{61} +0.339746i q^{62} -1.00000 q^{64} -14.3923i q^{67} +3.46410i q^{68} -1.26795 q^{71} +4.00000i q^{73} -6.92820 q^{74} +4.19615 q^{76} +2.53590i q^{77} -12.3923 q^{79} -6.00000i q^{82} -6.00000i q^{83} -17.6603 q^{86} -2.19615i q^{88} +0.928203 q^{89} +2.00000 q^{91} -4.73205i q^{92} -10.3923 q^{94} +2.00000i q^{97} -5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 12 q^{11} - 20 q^{16} + 4 q^{19} - 24 q^{29} + 20 q^{31} - 24 q^{34} - 12 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{59} + 8 q^{61} - 4 q^{64} - 12 q^{71} - 4 q^{76} - 8 q^{79} - 36 q^{86} - 24 q^{89} + 8 q^{91}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) − 3.46410i − 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) 0 0
\(19\) −4.19615 −0.962663 −0.481332 0.876539i \(-0.659847\pi\)
−0.481332 + 0.876539i \(0.659847\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.19615i − 0.468221i
\(23\) 4.73205i 0.986701i 0.869831 + 0.493350i \(0.164228\pi\)
−0.869831 + 0.493350i \(0.835772\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.73205 −0.339683
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) −9.46410 −1.75744 −0.878720 0.477338i \(-0.841602\pi\)
−0.878720 + 0.477338i \(0.841602\pi\)
\(30\) 0 0
\(31\) −0.196152 −0.0352300 −0.0176150 0.999845i \(-0.505607\pi\)
−0.0176150 + 0.999845i \(0.505607\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 7.26795i 1.17902i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) − 10.1962i − 1.55490i −0.628946 0.777449i \(-0.716513\pi\)
0.628946 0.777449i \(-0.283487\pi\)
\(44\) −1.26795 −0.191151
\(45\) 0 0
\(46\) 8.19615 1.20846
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) − 10.3923i − 1.42749i −0.700404 0.713746i \(-0.746997\pi\)
0.700404 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) 16.3923i 2.15242i
\(59\) −15.1244 −1.96902 −0.984512 0.175319i \(-0.943904\pi\)
−0.984512 + 0.175319i \(0.943904\pi\)
\(60\) 0 0
\(61\) 12.3923 1.58667 0.793336 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0.339746i 0.0431478i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 14.3923i − 1.75830i −0.476545 0.879150i \(-0.658111\pi\)
0.476545 0.879150i \(-0.341889\pi\)
\(68\) 3.46410i 0.420084i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.26795 −0.150478 −0.0752389 0.997166i \(-0.523972\pi\)
−0.0752389 + 0.997166i \(0.523972\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −6.92820 −0.805387
\(75\) 0 0
\(76\) 4.19615 0.481332
\(77\) 2.53590i 0.288992i
\(78\) 0 0
\(79\) −12.3923 −1.39424 −0.697122 0.716953i \(-0.745536\pi\)
−0.697122 + 0.716953i \(0.745536\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 6.00000i − 0.662589i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −17.6603 −1.90435
\(87\) 0 0
\(88\) − 2.19615i − 0.234111i
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 4.73205i − 0.493350i
\(93\) 0 0
\(94\) −10.3923 −1.07188
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 5.19615i − 0.524891i
\(99\) 0 0
\(100\) 0 0
\(101\) −12.9282 −1.28640 −0.643202 0.765696i \(-0.722395\pi\)
−0.643202 + 0.765696i \(0.722395\pi\)
\(102\) 0 0
\(103\) − 10.1962i − 1.00466i −0.864677 0.502328i \(-0.832477\pi\)
0.864677 0.502328i \(-0.167523\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) − 0.339746i − 0.0328445i −0.999865 0.0164222i \(-0.994772\pi\)
0.999865 0.0164222i \(-0.00522760\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 10.0000i − 0.944911i
\(113\) 15.4641i 1.45474i 0.686245 + 0.727370i \(0.259258\pi\)
−0.686245 + 0.727370i \(0.740742\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.46410 0.878720
\(117\) 0 0
\(118\) 26.1962i 2.41155i
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) − 21.4641i − 1.94327i
\(123\) 0 0
\(124\) 0.196152 0.0176150
\(125\) 0 0
\(126\) 0 0
\(127\) 5.80385i 0.515008i 0.966277 + 0.257504i \(0.0829001\pi\)
−0.966277 + 0.257504i \(0.917100\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) − 8.39230i − 0.727705i
\(134\) −24.9282 −2.15347
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 12.9282i 1.10453i 0.833668 + 0.552265i \(0.186237\pi\)
−0.833668 + 0.552265i \(0.813763\pi\)
\(138\) 0 0
\(139\) 8.39230 0.711826 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.19615i 0.184297i
\(143\) − 1.26795i − 0.106031i
\(144\) 0 0
\(145\) 0 0
\(146\) 6.92820 0.573382
\(147\) 0 0
\(148\) 4.00000i 0.328798i
\(149\) 19.8564 1.62670 0.813350 0.581775i \(-0.197641\pi\)
0.813350 + 0.581775i \(0.197641\pi\)
\(150\) 0 0
\(151\) −12.1962 −0.992509 −0.496254 0.868177i \(-0.665292\pi\)
−0.496254 + 0.868177i \(0.665292\pi\)
\(152\) 7.26795i 0.589509i
\(153\) 0 0
\(154\) 4.39230 0.353942
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 21.4641i 1.70759i
\(159\) 0 0
\(160\) 0 0
\(161\) −9.46410 −0.745876
\(162\) 0 0
\(163\) − 6.39230i − 0.500684i −0.968157 0.250342i \(-0.919457\pi\)
0.968157 0.250342i \(-0.0805431\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) −10.3923 −0.806599
\(167\) − 12.9282i − 1.00041i −0.865906 0.500207i \(-0.833257\pi\)
0.865906 0.500207i \(-0.166743\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 10.1962i 0.777449i
\(173\) 15.4641i 1.17571i 0.808965 + 0.587857i \(0.200028\pi\)
−0.808965 + 0.587857i \(0.799972\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.33975 −0.477876
\(177\) 0 0
\(178\) − 1.60770i − 0.120502i
\(179\) −5.07180 −0.379084 −0.189542 0.981873i \(-0.560700\pi\)
−0.189542 + 0.981873i \(0.560700\pi\)
\(180\) 0 0
\(181\) −20.3923 −1.51575 −0.757874 0.652401i \(-0.773762\pi\)
−0.757874 + 0.652401i \(0.773762\pi\)
\(182\) − 3.46410i − 0.256776i
\(183\) 0 0
\(184\) 8.19615 0.604228
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.39230i − 0.321197i
\(188\) 6.00000i 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 3.46410 0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 0.928203i − 0.0661317i −0.999453 0.0330659i \(-0.989473\pi\)
0.999453 0.0330659i \(-0.0105271\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 22.3923i 1.57552i
\(203\) − 18.9282i − 1.32850i
\(204\) 0 0
\(205\) 0 0
\(206\) −17.6603 −1.23045
\(207\) 0 0
\(208\) 5.00000i 0.346688i
\(209\) −5.32051 −0.368027
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 10.3923i 0.713746i
\(213\) 0 0
\(214\) −0.588457 −0.0402261
\(215\) 0 0
\(216\) 0 0
\(217\) − 0.392305i − 0.0266314i
\(218\) 3.46410i 0.234619i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.46410 −0.233021
\(222\) 0 0
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) 26.7846 1.78169
\(227\) − 3.46410i − 0.229920i −0.993370 0.114960i \(-0.963326\pi\)
0.993370 0.114960i \(-0.0366741\pi\)
\(228\) 0 0
\(229\) 14.3923 0.951070 0.475535 0.879697i \(-0.342254\pi\)
0.475535 + 0.879697i \(0.342254\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 16.3923i 1.07621i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 15.1244 0.984512
\(237\) 0 0
\(238\) − 12.0000i − 0.777844i
\(239\) −3.80385 −0.246050 −0.123025 0.992404i \(-0.539260\pi\)
−0.123025 + 0.992404i \(0.539260\pi\)
\(240\) 0 0
\(241\) 18.3923 1.18475 0.592376 0.805661i \(-0.298190\pi\)
0.592376 + 0.805661i \(0.298190\pi\)
\(242\) 16.2679i 1.04574i
\(243\) 0 0
\(244\) −12.3923 −0.793336
\(245\) 0 0
\(246\) 0 0
\(247\) 4.19615i 0.266995i
\(248\) 0.339746i 0.0215739i
\(249\) 0 0
\(250\) 0 0
\(251\) 14.5359 0.917498 0.458749 0.888566i \(-0.348298\pi\)
0.458749 + 0.888566i \(0.348298\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 10.0526 0.630754
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 7.85641i 0.490069i 0.969514 + 0.245035i \(0.0787993\pi\)
−0.969514 + 0.245035i \(0.921201\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.73205i 0.291791i 0.989300 + 0.145895i \(0.0466063\pi\)
−0.989300 + 0.145895i \(0.953394\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −14.5359 −0.891253
\(267\) 0 0
\(268\) 14.3923i 0.879150i
\(269\) 7.85641 0.479014 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(270\) 0 0
\(271\) −20.9808 −1.27449 −0.637245 0.770661i \(-0.719926\pi\)
−0.637245 + 0.770661i \(0.719926\pi\)
\(272\) 17.3205i 1.05021i
\(273\) 0 0
\(274\) 22.3923 1.35277
\(275\) 0 0
\(276\) 0 0
\(277\) − 5.60770i − 0.336934i −0.985707 0.168467i \(-0.946118\pi\)
0.985707 0.168467i \(-0.0538816\pi\)
\(278\) − 14.5359i − 0.871805i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.60770 −0.0959071 −0.0479535 0.998850i \(-0.515270\pi\)
−0.0479535 + 0.998850i \(0.515270\pi\)
\(282\) 0 0
\(283\) − 1.41154i − 0.0839075i −0.999120 0.0419538i \(-0.986642\pi\)
0.999120 0.0419538i \(-0.0133582\pi\)
\(284\) 1.26795 0.0752389
\(285\) 0 0
\(286\) −2.19615 −0.129861
\(287\) 6.92820i 0.408959i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) − 4.00000i − 0.234082i
\(293\) − 18.9282i − 1.10580i −0.833248 0.552899i \(-0.813522\pi\)
0.833248 0.552899i \(-0.186478\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.92820 −0.402694
\(297\) 0 0
\(298\) − 34.3923i − 1.99229i
\(299\) 4.73205 0.273662
\(300\) 0 0
\(301\) 20.3923 1.17539
\(302\) 21.1244i 1.21557i
\(303\) 0 0
\(304\) 20.9808 1.20333
\(305\) 0 0
\(306\) 0 0
\(307\) 22.7846i 1.30039i 0.759769 + 0.650193i \(0.225312\pi\)
−0.759769 + 0.650193i \(0.774688\pi\)
\(308\) − 2.53590i − 0.144496i
\(309\) 0 0
\(310\) 0 0
\(311\) −4.39230 −0.249065 −0.124532 0.992216i \(-0.539743\pi\)
−0.124532 + 0.992216i \(0.539743\pi\)
\(312\) 0 0
\(313\) − 6.39230i − 0.361314i −0.983546 0.180657i \(-0.942178\pi\)
0.983546 0.180657i \(-0.0578225\pi\)
\(314\) −17.3205 −0.977453
\(315\) 0 0
\(316\) 12.3923 0.697122
\(317\) − 24.0000i − 1.34797i −0.738743 0.673987i \(-0.764580\pi\)
0.738743 0.673987i \(-0.235420\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 16.3923i 0.913507i
\(323\) 14.5359i 0.808799i
\(324\) 0 0
\(325\) 0 0
\(326\) −11.0718 −0.613210
\(327\) 0 0
\(328\) − 6.00000i − 0.331295i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −28.5885 −1.57136 −0.785682 0.618631i \(-0.787688\pi\)
−0.785682 + 0.618631i \(0.787688\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 0 0
\(334\) −22.3923 −1.22525
\(335\) 0 0
\(336\) 0 0
\(337\) − 5.60770i − 0.305471i −0.988267 0.152735i \(-0.951192\pi\)
0.988267 0.152735i \(-0.0488082\pi\)
\(338\) 1.73205i 0.0942111i
\(339\) 0 0
\(340\) 0 0
\(341\) −0.248711 −0.0134685
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −17.6603 −0.952177
\(345\) 0 0
\(346\) 26.7846 1.43995
\(347\) − 11.6603i − 0.625955i −0.949761 0.312978i \(-0.898674\pi\)
0.949761 0.312978i \(-0.101326\pi\)
\(348\) 0 0
\(349\) −6.39230 −0.342172 −0.171086 0.985256i \(-0.554728\pi\)
−0.171086 + 0.985256i \(0.554728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.58846i 0.351166i
\(353\) 27.7128i 1.47500i 0.675345 + 0.737502i \(0.263995\pi\)
−0.675345 + 0.737502i \(0.736005\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.928203 −0.0491947
\(357\) 0 0
\(358\) 8.78461i 0.464281i
\(359\) 8.19615 0.432576 0.216288 0.976330i \(-0.430605\pi\)
0.216288 + 0.976330i \(0.430605\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) 35.3205i 1.85640i
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 22.1962i 1.15863i 0.815104 + 0.579315i \(0.196680\pi\)
−0.815104 + 0.579315i \(0.803320\pi\)
\(368\) − 23.6603i − 1.23338i
\(369\) 0 0
\(370\) 0 0
\(371\) 20.7846 1.07908
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) −7.60770 −0.393385
\(375\) 0 0
\(376\) −10.3923 −0.535942
\(377\) 9.46410i 0.487426i
\(378\) 0 0
\(379\) 32.9808 1.69411 0.847054 0.531507i \(-0.178374\pi\)
0.847054 + 0.531507i \(0.178374\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 32.7846i − 1.67741i
\(383\) − 0.928203i − 0.0474290i −0.999719 0.0237145i \(-0.992451\pi\)
0.999719 0.0237145i \(-0.00754926\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.3205 0.881591
\(387\) 0 0
\(388\) − 2.00000i − 0.101535i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 16.3923 0.828994
\(392\) − 5.19615i − 0.262445i
\(393\) 0 0
\(394\) −1.60770 −0.0809945
\(395\) 0 0
\(396\) 0 0
\(397\) − 12.7846i − 0.641641i −0.947140 0.320821i \(-0.896041\pi\)
0.947140 0.320821i \(-0.103959\pi\)
\(398\) 34.6410i 1.73640i
\(399\) 0 0
\(400\) 0 0
\(401\) 23.0718 1.15215 0.576075 0.817397i \(-0.304584\pi\)
0.576075 + 0.817397i \(0.304584\pi\)
\(402\) 0 0
\(403\) 0.196152i 0.00977105i
\(404\) 12.9282 0.643202
\(405\) 0 0
\(406\) −32.7846 −1.62707
\(407\) − 5.07180i − 0.251400i
\(408\) 0 0
\(409\) 38.3923 1.89838 0.949189 0.314708i \(-0.101906\pi\)
0.949189 + 0.314708i \(0.101906\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.1962i 0.502328i
\(413\) − 30.2487i − 1.48844i
\(414\) 0 0
\(415\) 0 0
\(416\) 5.19615 0.254762
\(417\) 0 0
\(418\) 9.21539i 0.450739i
\(419\) −9.46410 −0.462352 −0.231176 0.972912i \(-0.574257\pi\)
−0.231176 + 0.972912i \(0.574257\pi\)
\(420\) 0 0
\(421\) 10.7846 0.525610 0.262805 0.964849i \(-0.415352\pi\)
0.262805 + 0.964849i \(0.415352\pi\)
\(422\) − 13.8564i − 0.674519i
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 24.7846i 1.19941i
\(428\) 0.339746i 0.0164222i
\(429\) 0 0
\(430\) 0 0
\(431\) −19.5167 −0.940084 −0.470042 0.882644i \(-0.655761\pi\)
−0.470042 + 0.882644i \(0.655761\pi\)
\(432\) 0 0
\(433\) 6.78461i 0.326048i 0.986622 + 0.163024i \(0.0521247\pi\)
−0.986622 + 0.163024i \(0.947875\pi\)
\(434\) −0.679492 −0.0326167
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 19.8564i − 0.949861i
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000i 0.285391i
\(443\) 34.9808i 1.66199i 0.556283 + 0.830993i \(0.312227\pi\)
−0.556283 + 0.830993i \(0.687773\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.46410 −0.164030
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) 27.4641 1.29611 0.648056 0.761593i \(-0.275582\pi\)
0.648056 + 0.761593i \(0.275582\pi\)
\(450\) 0 0
\(451\) 4.39230 0.206826
\(452\) − 15.4641i − 0.727370i
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) − 30.7846i − 1.44004i −0.693952 0.720022i \(-0.744132\pi\)
0.693952 0.720022i \(-0.255868\pi\)
\(458\) − 24.9282i − 1.16482i
\(459\) 0 0
\(460\) 0 0
\(461\) −3.46410 −0.161339 −0.0806696 0.996741i \(-0.525706\pi\)
−0.0806696 + 0.996741i \(0.525706\pi\)
\(462\) 0 0
\(463\) − 18.3923i − 0.854763i −0.904071 0.427381i \(-0.859436\pi\)
0.904071 0.427381i \(-0.140564\pi\)
\(464\) 47.3205 2.19680
\(465\) 0 0
\(466\) −10.3923 −0.481414
\(467\) − 38.1962i − 1.76751i −0.467953 0.883754i \(-0.655008\pi\)
0.467953 0.883754i \(-0.344992\pi\)
\(468\) 0 0
\(469\) 28.7846 1.32915
\(470\) 0 0
\(471\) 0 0
\(472\) 26.1962i 1.20578i
\(473\) − 12.9282i − 0.594439i
\(474\) 0 0
\(475\) 0 0
\(476\) −6.92820 −0.317554
\(477\) 0 0
\(478\) 6.58846i 0.301349i
\(479\) 18.3397 0.837964 0.418982 0.907994i \(-0.362387\pi\)
0.418982 + 0.907994i \(0.362387\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) − 31.8564i − 1.45102i
\(483\) 0 0
\(484\) 9.39230 0.426923
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.60770i − 0.254109i −0.991896 0.127054i \(-0.959448\pi\)
0.991896 0.127054i \(-0.0405523\pi\)
\(488\) − 21.4641i − 0.971634i
\(489\) 0 0
\(490\) 0 0
\(491\) 9.46410 0.427109 0.213554 0.976931i \(-0.431496\pi\)
0.213554 + 0.976931i \(0.431496\pi\)
\(492\) 0 0
\(493\) 32.7846i 1.47654i
\(494\) 7.26795 0.327000
\(495\) 0 0
\(496\) 0.980762 0.0440375
\(497\) − 2.53590i − 0.113751i
\(498\) 0 0
\(499\) −12.9808 −0.581099 −0.290549 0.956860i \(-0.593838\pi\)
−0.290549 + 0.956860i \(0.593838\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 25.1769i − 1.12370i
\(503\) − 25.5167i − 1.13773i −0.822430 0.568866i \(-0.807382\pi\)
0.822430 0.568866i \(-0.192618\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.3923 0.461994
\(507\) 0 0
\(508\) − 5.80385i − 0.257504i
\(509\) 32.5359 1.44213 0.721064 0.692868i \(-0.243653\pi\)
0.721064 + 0.692868i \(0.243653\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) − 8.66025i − 0.382733i
\(513\) 0 0
\(514\) 13.6077 0.600210
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.60770i − 0.334586i
\(518\) − 13.8564i − 0.608816i
\(519\) 0 0
\(520\) 0 0
\(521\) 7.60770 0.333299 0.166650 0.986016i \(-0.446705\pi\)
0.166650 + 0.986016i \(0.446705\pi\)
\(522\) 0 0
\(523\) 13.8038i 0.603600i 0.953371 + 0.301800i \(0.0975875\pi\)
−0.953371 + 0.301800i \(0.902412\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 8.19615 0.357369
\(527\) 0.679492i 0.0295991i
\(528\) 0 0
\(529\) 0.607695 0.0264215
\(530\) 0 0
\(531\) 0 0
\(532\) 8.39230i 0.363853i
\(533\) − 3.46410i − 0.150047i
\(534\) 0 0
\(535\) 0 0
\(536\) −24.9282 −1.07673
\(537\) 0 0
\(538\) − 13.6077i − 0.586669i
\(539\) 3.80385 0.163843
\(540\) 0 0
\(541\) −5.60770 −0.241094 −0.120547 0.992708i \(-0.538465\pi\)
−0.120547 + 0.992708i \(0.538465\pi\)
\(542\) 36.3397i 1.56093i
\(543\) 0 0
\(544\) 18.0000 0.771744
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.80385i − 0.0771270i −0.999256 0.0385635i \(-0.987722\pi\)
0.999256 0.0385635i \(-0.0122782\pi\)
\(548\) − 12.9282i − 0.552265i
\(549\) 0 0
\(550\) 0 0
\(551\) 39.7128 1.69182
\(552\) 0 0
\(553\) − 24.7846i − 1.05395i
\(554\) −9.71281 −0.412658
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) 25.8564i 1.09557i 0.836619 + 0.547786i \(0.184529\pi\)
−0.836619 + 0.547786i \(0.815471\pi\)
\(558\) 0 0
\(559\) −10.1962 −0.431251
\(560\) 0 0
\(561\) 0 0
\(562\) 2.78461i 0.117462i
\(563\) 16.0526i 0.676535i 0.941050 + 0.338267i \(0.109841\pi\)
−0.941050 + 0.338267i \(0.890159\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.44486 −0.102765
\(567\) 0 0
\(568\) 2.19615i 0.0921485i
\(569\) −9.46410 −0.396756 −0.198378 0.980126i \(-0.563567\pi\)
−0.198378 + 0.980126i \(0.563567\pi\)
\(570\) 0 0
\(571\) 15.6077 0.653162 0.326581 0.945169i \(-0.394103\pi\)
0.326581 + 0.945169i \(0.394103\pi\)
\(572\) 1.26795i 0.0530156i
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) − 4.00000i − 0.166522i −0.996528 0.0832611i \(-0.973466\pi\)
0.996528 0.0832611i \(-0.0265335\pi\)
\(578\) − 8.66025i − 0.360219i
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) − 13.1769i − 0.545732i
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) −32.7846 −1.35432
\(587\) 15.4641i 0.638272i 0.947709 + 0.319136i \(0.103393\pi\)
−0.947709 + 0.319136i \(0.896607\pi\)
\(588\) 0 0
\(589\) 0.823085 0.0339146
\(590\) 0 0
\(591\) 0 0
\(592\) 20.0000i 0.821995i
\(593\) − 14.7846i − 0.607131i −0.952811 0.303566i \(-0.901823\pi\)
0.952811 0.303566i \(-0.0981771\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −19.8564 −0.813350
\(597\) 0 0
\(598\) − 8.19615i − 0.335166i
\(599\) −28.3923 −1.16008 −0.580039 0.814589i \(-0.696963\pi\)
−0.580039 + 0.814589i \(0.696963\pi\)
\(600\) 0 0
\(601\) −39.5692 −1.61406 −0.807031 0.590509i \(-0.798927\pi\)
−0.807031 + 0.590509i \(0.798927\pi\)
\(602\) − 35.3205i − 1.43956i
\(603\) 0 0
\(604\) 12.1962 0.496254
\(605\) 0 0
\(606\) 0 0
\(607\) − 26.9808i − 1.09512i −0.836768 0.547558i \(-0.815558\pi\)
0.836768 0.547558i \(-0.184442\pi\)
\(608\) − 21.8038i − 0.884263i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 39.4641 1.59264
\(615\) 0 0
\(616\) 4.39230 0.176971
\(617\) 21.7128i 0.874125i 0.899431 + 0.437062i \(0.143981\pi\)
−0.899431 + 0.437062i \(0.856019\pi\)
\(618\) 0 0
\(619\) 44.9808 1.80793 0.903965 0.427607i \(-0.140643\pi\)
0.903965 + 0.427607i \(0.140643\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 7.60770i 0.305041i
\(623\) 1.85641i 0.0743754i
\(624\) 0 0
\(625\) 0 0
\(626\) −11.0718 −0.442518
\(627\) 0 0
\(628\) 10.0000i 0.399043i
\(629\) −13.8564 −0.552491
\(630\) 0 0
\(631\) 16.1962 0.644759 0.322379 0.946611i \(-0.395517\pi\)
0.322379 + 0.946611i \(0.395517\pi\)
\(632\) 21.4641i 0.853796i
\(633\) 0 0
\(634\) −41.5692 −1.65092
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.00000i − 0.118864i
\(638\) 20.7846i 0.822871i
\(639\) 0 0
\(640\) 0 0
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) 0 0
\(643\) − 34.7846i − 1.37177i −0.727709 0.685886i \(-0.759415\pi\)
0.727709 0.685886i \(-0.240585\pi\)
\(644\) 9.46410 0.372938
\(645\) 0 0
\(646\) 25.1769 0.990572
\(647\) 16.0526i 0.631091i 0.948910 + 0.315546i \(0.102188\pi\)
−0.948910 + 0.315546i \(0.897812\pi\)
\(648\) 0 0
\(649\) −19.1769 −0.752760
\(650\) 0 0
\(651\) 0 0
\(652\) 6.39230i 0.250342i
\(653\) − 19.8564i − 0.777041i −0.921440 0.388521i \(-0.872986\pi\)
0.921440 0.388521i \(-0.127014\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −17.3205 −0.676252
\(657\) 0 0
\(658\) − 20.7846i − 0.810268i
\(659\) −14.5359 −0.566238 −0.283119 0.959085i \(-0.591369\pi\)
−0.283119 + 0.959085i \(0.591369\pi\)
\(660\) 0 0
\(661\) −30.7846 −1.19738 −0.598691 0.800980i \(-0.704312\pi\)
−0.598691 + 0.800980i \(0.704312\pi\)
\(662\) 49.5167i 1.92452i
\(663\) 0 0
\(664\) −10.3923 −0.403300
\(665\) 0 0
\(666\) 0 0
\(667\) − 44.7846i − 1.73407i
\(668\) 12.9282i 0.500207i
\(669\) 0 0
\(670\) 0 0
\(671\) 15.7128 0.606586
\(672\) 0 0
\(673\) − 6.39230i − 0.246405i −0.992382 0.123203i \(-0.960683\pi\)
0.992382 0.123203i \(-0.0393165\pi\)
\(674\) −9.71281 −0.374124
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 10.3923i 0.399409i 0.979856 + 0.199704i \(0.0639982\pi\)
−0.979856 + 0.199704i \(0.936002\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) 0.430781i 0.0164954i
\(683\) 39.4641i 1.51005i 0.655695 + 0.755026i \(0.272376\pi\)
−0.655695 + 0.755026i \(0.727624\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) 50.9808i 1.94362i
\(689\) −10.3923 −0.395915
\(690\) 0 0
\(691\) 45.7654 1.74100 0.870498 0.492171i \(-0.163797\pi\)
0.870498 + 0.492171i \(0.163797\pi\)
\(692\) − 15.4641i − 0.587857i
\(693\) 0 0
\(694\) −20.1962 −0.766635
\(695\) 0 0
\(696\) 0 0
\(697\) − 12.0000i − 0.454532i
\(698\) 11.0718i 0.419074i
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 16.7846i 0.633044i
\(704\) −1.26795 −0.0477876
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) − 25.8564i − 0.972430i
\(708\) 0 0
\(709\) −9.60770 −0.360825 −0.180412 0.983591i \(-0.557743\pi\)
−0.180412 + 0.983591i \(0.557743\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 1.60770i − 0.0602509i
\(713\) − 0.928203i − 0.0347615i
\(714\) 0 0
\(715\) 0 0
\(716\) 5.07180 0.189542
\(717\) 0 0
\(718\) − 14.1962i − 0.529796i
\(719\) 1.85641 0.0692323 0.0346161 0.999401i \(-0.488979\pi\)
0.0346161 + 0.999401i \(0.488979\pi\)
\(720\) 0 0
\(721\) 20.3923 0.759449
\(722\) 2.41154i 0.0897483i
\(723\) 0 0
\(724\) 20.3923 0.757874
\(725\) 0 0
\(726\) 0 0
\(727\) 13.4115i 0.497407i 0.968580 + 0.248703i \(0.0800044\pi\)
−0.968580 + 0.248703i \(0.919996\pi\)
\(728\) − 3.46410i − 0.128388i
\(729\) 0 0
\(730\) 0 0
\(731\) −35.3205 −1.30638
\(732\) 0 0
\(733\) − 38.0000i − 1.40356i −0.712393 0.701781i \(-0.752388\pi\)
0.712393 0.701781i \(-0.247612\pi\)
\(734\) 38.4449 1.41903
\(735\) 0 0
\(736\) −24.5885 −0.906343
\(737\) − 18.2487i − 0.672200i
\(738\) 0 0
\(739\) 7.80385 0.287069 0.143535 0.989645i \(-0.454153\pi\)
0.143535 + 0.989645i \(0.454153\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 36.0000i − 1.32160i
\(743\) − 43.8564i − 1.60894i −0.593996 0.804468i \(-0.702451\pi\)
0.593996 0.804468i \(-0.297549\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.3205 0.634149
\(747\) 0 0
\(748\) 4.39230i 0.160599i
\(749\) 0.679492 0.0248281
\(750\) 0 0
\(751\) 15.6077 0.569533 0.284766 0.958597i \(-0.408084\pi\)
0.284766 + 0.958597i \(0.408084\pi\)
\(752\) 30.0000i 1.09399i
\(753\) 0 0
\(754\) 16.3923 0.596973
\(755\) 0 0
\(756\) 0 0
\(757\) 18.3923i 0.668480i 0.942488 + 0.334240i \(0.108480\pi\)
−0.942488 + 0.334240i \(0.891520\pi\)
\(758\) − 57.1244i − 2.07485i
\(759\) 0 0
\(760\) 0 0
\(761\) −7.85641 −0.284795 −0.142397 0.989810i \(-0.545481\pi\)
−0.142397 + 0.989810i \(0.545481\pi\)
\(762\) 0 0
\(763\) − 4.00000i − 0.144810i
\(764\) −18.9282 −0.684798
\(765\) 0 0
\(766\) −1.60770 −0.0580884
\(767\) 15.1244i 0.546109i
\(768\) 0 0
\(769\) 6.78461 0.244659 0.122330 0.992490i \(-0.460963\pi\)
0.122330 + 0.992490i \(0.460963\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 10.0000i − 0.359908i
\(773\) − 6.92820i − 0.249190i −0.992208 0.124595i \(-0.960237\pi\)
0.992208 0.124595i \(-0.0397632\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.46410 0.124354
\(777\) 0 0
\(778\) − 10.3923i − 0.372582i
\(779\) −14.5359 −0.520803
\(780\) 0 0
\(781\) −1.60770 −0.0575279
\(782\) − 28.3923i − 1.01531i
\(783\) 0 0
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) − 51.5692i − 1.83824i −0.393973 0.919122i \(-0.628900\pi\)
0.393973 0.919122i \(-0.371100\pi\)
\(788\) 0.928203i 0.0330659i
\(789\) 0 0
\(790\) 0 0
\(791\) −30.9282 −1.09968
\(792\) 0 0
\(793\) − 12.3923i − 0.440064i
\(794\) −22.1436 −0.785847
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) − 28.6410i − 1.01452i −0.861794 0.507258i \(-0.830659\pi\)
0.861794 0.507258i \(-0.169341\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) 0 0
\(802\) − 39.9615i − 1.41109i
\(803\) 5.07180i 0.178980i
\(804\) 0 0
\(805\) 0 0
\(806\) 0.339746 0.0119670
\(807\) 0 0
\(808\) 22.3923i 0.787759i
\(809\) −9.46410 −0.332740 −0.166370 0.986063i \(-0.553205\pi\)
−0.166370 + 0.986063i \(0.553205\pi\)
\(810\) 0 0
\(811\) 28.1962 0.990101 0.495050 0.868864i \(-0.335150\pi\)
0.495050 + 0.868864i \(0.335150\pi\)
\(812\) 18.9282i 0.664250i
\(813\) 0 0
\(814\) −8.78461 −0.307900
\(815\) 0 0
\(816\) 0 0
\(817\) 42.7846i 1.49684i
\(818\) − 66.4974i − 2.32503i
\(819\) 0 0
\(820\) 0 0
\(821\) 40.6410 1.41838 0.709191 0.705017i \(-0.249061\pi\)
0.709191 + 0.705017i \(0.249061\pi\)
\(822\) 0 0
\(823\) 46.5885i 1.62397i 0.583677 + 0.811986i \(0.301613\pi\)
−0.583677 + 0.811986i \(0.698387\pi\)
\(824\) −17.6603 −0.615224
\(825\) 0 0
\(826\) −52.3923 −1.82296
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) 0 0
\(829\) 20.3923 0.708254 0.354127 0.935197i \(-0.384778\pi\)
0.354127 + 0.935197i \(0.384778\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) − 10.3923i − 0.360072i
\(834\) 0 0
\(835\) 0 0
\(836\) 5.32051 0.184014
\(837\) 0 0
\(838\) 16.3923i 0.566263i
\(839\) 17.6603 0.609700 0.304850 0.952400i \(-0.401394\pi\)
0.304850 + 0.952400i \(0.401394\pi\)
\(840\) 0 0
\(841\) 60.5692 2.08859
\(842\) − 18.6795i − 0.643738i
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) − 18.7846i − 0.645447i
\(848\) 51.9615i 1.78437i
\(849\) 0 0
\(850\) 0 0
\(851\) 18.9282 0.648850
\(852\) 0 0
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 42.9282 1.46897
\(855\) 0 0
\(856\) −0.588457 −0.0201131
\(857\) − 47.5692i − 1.62493i −0.583007 0.812467i \(-0.698124\pi\)
0.583007 0.812467i \(-0.301876\pi\)
\(858\) 0 0
\(859\) −45.1769 −1.54142 −0.770708 0.637188i \(-0.780097\pi\)
−0.770708 + 0.637188i \(0.780097\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 33.8038i 1.15136i
\(863\) 2.78461i 0.0947892i 0.998876 + 0.0473946i \(0.0150918\pi\)
−0.998876 + 0.0473946i \(0.984908\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 11.7513 0.399325
\(867\) 0 0
\(868\) 0.392305i 0.0133157i
\(869\) −15.7128 −0.533021
\(870\) 0 0
\(871\) −14.3923 −0.487665
\(872\) 3.46410i 0.117309i
\(873\) 0 0
\(874\) −34.3923 −1.16334
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 55.4256i 1.87052i
\(879\) 0 0
\(880\) 0 0
\(881\) 12.6795 0.427183 0.213591 0.976923i \(-0.431484\pi\)
0.213591 + 0.976923i \(0.431484\pi\)
\(882\) 0 0
\(883\) − 34.1962i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(884\) 3.46410 0.116510
\(885\) 0 0
\(886\) 60.5885 2.03551
\(887\) − 17.9090i − 0.601324i −0.953731 0.300662i \(-0.902792\pi\)
0.953731 0.300662i \(-0.0972076\pi\)
\(888\) 0 0
\(889\) −11.6077 −0.389310
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000i 0.0669650i
\(893\) 25.1769i 0.842513i
\(894\) 0 0
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) − 47.5692i − 1.58741i
\(899\) 1.85641 0.0619146
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) − 7.60770i − 0.253309i
\(903\) 0 0
\(904\) 26.7846 0.890843
\(905\) 0 0
\(906\) 0 0
\(907\) 39.7654i 1.32039i 0.751095 + 0.660194i \(0.229526\pi\)
−0.751095 + 0.660194i \(0.770474\pi\)
\(908\) 3.46410i 0.114960i
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) − 7.60770i − 0.251778i
\(914\) −53.3205 −1.76369
\(915\) 0 0
\(916\) −14.3923 −0.475535
\(917\) 0 0
\(918\) 0 0
\(919\) 53.1769 1.75414 0.877072 0.480358i \(-0.159493\pi\)
0.877072 + 0.480358i \(0.159493\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000i 0.197599i
\(923\) 1.26795i 0.0417351i
\(924\) 0 0
\(925\) 0 0
\(926\) −31.8564 −1.04687
\(927\) 0 0
\(928\) − 49.1769i − 1.61431i
\(929\) 51.4641 1.68848 0.844241 0.535963i \(-0.180052\pi\)
0.844241 + 0.535963i \(0.180052\pi\)
\(930\) 0 0
\(931\) −12.5885 −0.412570
\(932\) 6.00000i 0.196537i
\(933\) 0 0
\(934\) −66.1577 −2.16475
\(935\) 0 0
\(936\) 0 0
\(937\) − 6.78461i − 0.221644i −0.993840 0.110822i \(-0.964652\pi\)
0.993840 0.110822i \(-0.0353483\pi\)
\(938\) − 49.8564i − 1.62787i
\(939\) 0 0
\(940\) 0 0
\(941\) 31.1769 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(942\) 0 0
\(943\) 16.3923i 0.533807i
\(944\) 75.6218 2.46128
\(945\) 0 0
\(946\) −22.3923 −0.728037
\(947\) 28.6410i 0.930708i 0.885125 + 0.465354i \(0.154073\pi\)
−0.885125 + 0.465354i \(0.845927\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) − 12.0000i − 0.388922i
\(953\) − 12.9282i − 0.418786i −0.977832 0.209393i \(-0.932851\pi\)
0.977832 0.209393i \(-0.0671487\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.80385 0.123025
\(957\) 0 0
\(958\) − 31.7654i − 1.02629i
\(959\) −25.8564 −0.834947
\(960\) 0 0
\(961\) −30.9615 −0.998759
\(962\) 6.92820i 0.223374i
\(963\) 0 0
\(964\) −18.3923 −0.592376
\(965\) 0 0
\(966\) 0 0
\(967\) − 29.6077i − 0.952119i −0.879413 0.476060i \(-0.842065\pi\)
0.879413 0.476060i \(-0.157935\pi\)
\(968\) 16.2679i 0.522872i
\(969\) 0 0
\(970\) 0 0
\(971\) −5.07180 −0.162762 −0.0813809 0.996683i \(-0.525933\pi\)
−0.0813809 + 0.996683i \(0.525933\pi\)
\(972\) 0 0
\(973\) 16.7846i 0.538090i
\(974\) −9.71281 −0.311219
\(975\) 0 0
\(976\) −61.9615 −1.98334
\(977\) − 39.7128i − 1.27053i −0.772296 0.635263i \(-0.780892\pi\)
0.772296 0.635263i \(-0.219108\pi\)
\(978\) 0 0
\(979\) 1.17691 0.0376144
\(980\) 0 0
\(981\) 0 0
\(982\) − 16.3923i − 0.523099i
\(983\) − 13.6077i − 0.434018i −0.976170 0.217009i \(-0.930370\pi\)
0.976170 0.217009i \(-0.0696301\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 56.7846 1.80839
\(987\) 0 0
\(988\) − 4.19615i − 0.133497i
\(989\) 48.2487 1.53422
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) − 1.01924i − 0.0323608i
\(993\) 0 0
\(994\) −4.39230 −0.139315
\(995\) 0 0
\(996\) 0 0
\(997\) 54.3923i 1.72262i 0.508078 + 0.861311i \(0.330356\pi\)
−0.508078 + 0.861311i \(0.669644\pi\)
\(998\) 22.4833i 0.711698i
\(999\) 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.v.2224.2 4
3.2 odd 2 325.2.b.e.274.3 4
5.2 odd 4 2925.2.a.z.1.2 2
5.3 odd 4 585.2.a.k.1.1 2
5.4 even 2 inner 2925.2.c.v.2224.3 4
15.2 even 4 325.2.a.g.1.1 2
15.8 even 4 65.2.a.c.1.2 2
15.14 odd 2 325.2.b.e.274.2 4
20.3 even 4 9360.2.a.cm.1.2 2
60.23 odd 4 1040.2.a.h.1.2 2
60.47 odd 4 5200.2.a.ca.1.1 2
65.38 odd 4 7605.2.a.be.1.2 2
105.83 odd 4 3185.2.a.k.1.2 2
120.53 even 4 4160.2.a.y.1.2 2
120.83 odd 4 4160.2.a.bj.1.1 2
165.98 odd 4 7865.2.a.h.1.1 2
195.8 odd 4 845.2.c.e.506.3 4
195.23 even 12 845.2.e.f.191.2 4
195.38 even 4 845.2.a.d.1.1 2
195.68 even 12 845.2.e.e.191.1 4
195.77 even 4 4225.2.a.w.1.2 2
195.83 odd 4 845.2.c.e.506.1 4
195.98 odd 12 845.2.m.c.361.2 4
195.113 even 12 845.2.e.e.146.1 4
195.128 odd 12 845.2.m.a.316.2 4
195.158 odd 12 845.2.m.c.316.2 4
195.173 even 12 845.2.e.f.146.2 4
195.188 odd 12 845.2.m.a.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.2 2 15.8 even 4
325.2.a.g.1.1 2 15.2 even 4
325.2.b.e.274.2 4 15.14 odd 2
325.2.b.e.274.3 4 3.2 odd 2
585.2.a.k.1.1 2 5.3 odd 4
845.2.a.d.1.1 2 195.38 even 4
845.2.c.e.506.1 4 195.83 odd 4
845.2.c.e.506.3 4 195.8 odd 4
845.2.e.e.146.1 4 195.113 even 12
845.2.e.e.191.1 4 195.68 even 12
845.2.e.f.146.2 4 195.173 even 12
845.2.e.f.191.2 4 195.23 even 12
845.2.m.a.316.2 4 195.128 odd 12
845.2.m.a.361.2 4 195.188 odd 12
845.2.m.c.316.2 4 195.158 odd 12
845.2.m.c.361.2 4 195.98 odd 12
1040.2.a.h.1.2 2 60.23 odd 4
2925.2.a.z.1.2 2 5.2 odd 4
2925.2.c.v.2224.2 4 1.1 even 1 trivial
2925.2.c.v.2224.3 4 5.4 even 2 inner
3185.2.a.k.1.2 2 105.83 odd 4
4160.2.a.y.1.2 2 120.53 even 4
4160.2.a.bj.1.1 2 120.83 odd 4
4225.2.a.w.1.2 2 195.77 even 4
5200.2.a.ca.1.1 2 60.47 odd 4
7605.2.a.be.1.2 2 65.38 odd 4
7865.2.a.h.1.1 2 165.98 odd 4
9360.2.a.cm.1.2 2 20.3 even 4