Properties

Label 6525.2.a.be.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.a (trivial)

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: yes
Analytic conductor: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.311108 q^{2} -1.90321 q^{4} +0.903212 q^{7} -1.21432 q^{8} +O(q^{10})\) \(q+0.311108 q^{2} -1.90321 q^{4} +0.903212 q^{7} -1.21432 q^{8} +1.52543 q^{11} +0.622216 q^{13} +0.280996 q^{14} +3.42864 q^{16} -7.95407 q^{17} -1.09679 q^{19} +0.474572 q^{22} +7.52543 q^{23} +0.193576 q^{26} -1.71900 q^{28} +1.00000 q^{29} -6.90321 q^{31} +3.49532 q^{32} -2.47457 q^{34} -3.95407 q^{37} -0.341219 q^{38} -3.67307 q^{41} +10.5161 q^{43} -2.90321 q^{44} +2.34122 q^{46} +6.90321 q^{47} -6.18421 q^{49} -1.18421 q^{52} +6.42864 q^{53} -1.09679 q^{56} +0.311108 q^{58} +1.67307 q^{59} -1.86665 q^{61} -2.14764 q^{62} -5.76986 q^{64} -11.5254 q^{67} +15.1383 q^{68} -13.6731 q^{71} -10.1891 q^{73} -1.23014 q^{74} +2.08742 q^{76} +1.37778 q^{77} +9.13828 q^{79} -1.14272 q^{82} +10.7096 q^{83} +3.27163 q^{86} -1.85236 q^{88} +7.80642 q^{89} +0.561993 q^{91} -14.3225 q^{92} +2.14764 q^{94} +4.08742 q^{97} -1.92396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 4 q^{17} - 10 q^{19} + 8 q^{22} + 16 q^{23} + 14 q^{26} - 12 q^{28} + 3 q^{29} - 14 q^{31} - 3 q^{32} - 14 q^{34} + 8 q^{37} - 8 q^{38} + 2 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 14 q^{47} - 5 q^{49} + 10 q^{52} + 6 q^{53} - 10 q^{56} + q^{58} - 8 q^{59} - 6 q^{61} - 11 q^{64} - 28 q^{67} + 12 q^{68} - 28 q^{71} + 16 q^{73} - 10 q^{74} - 14 q^{76} + 4 q^{77} - 6 q^{79} - 30 q^{82} + 12 q^{83} - 24 q^{86} - 12 q^{88} + 10 q^{89} - 12 q^{91} + 4 q^{92} - 8 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108 0.219986 0.109993 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(3\) 0 0
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) 0 0
\(7\) 0.903212 0.341382 0.170691 0.985325i \(-0.445400\pi\)
0.170691 + 0.985325i \(0.445400\pi\)
\(8\) −1.21432 −0.429327
\(9\) 0 0
\(10\) 0 0
\(11\) 1.52543 0.459934 0.229967 0.973198i \(-0.426138\pi\)
0.229967 + 0.973198i \(0.426138\pi\)
\(12\) 0 0
\(13\) 0.622216 0.172572 0.0862858 0.996270i \(-0.472500\pi\)
0.0862858 + 0.996270i \(0.472500\pi\)
\(14\) 0.280996 0.0750994
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) −7.95407 −1.92914 −0.964572 0.263819i \(-0.915018\pi\)
−0.964572 + 0.263819i \(0.915018\pi\)
\(18\) 0 0
\(19\) −1.09679 −0.251620 −0.125810 0.992054i \(-0.540153\pi\)
−0.125810 + 0.992054i \(0.540153\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.474572 0.101179
\(23\) 7.52543 1.56916 0.784580 0.620028i \(-0.212879\pi\)
0.784580 + 0.620028i \(0.212879\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.193576 0.0379634
\(27\) 0 0
\(28\) −1.71900 −0.324861
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.90321 −1.23985 −0.619927 0.784660i \(-0.712838\pi\)
−0.619927 + 0.784660i \(0.712838\pi\)
\(32\) 3.49532 0.617890
\(33\) 0 0
\(34\) −2.47457 −0.424386
\(35\) 0 0
\(36\) 0 0
\(37\) −3.95407 −0.650045 −0.325022 0.945706i \(-0.605372\pi\)
−0.325022 + 0.945706i \(0.605372\pi\)
\(38\) −0.341219 −0.0553531
\(39\) 0 0
\(40\) 0 0
\(41\) −3.67307 −0.573637 −0.286819 0.957985i \(-0.592598\pi\)
−0.286819 + 0.957985i \(0.592598\pi\)
\(42\) 0 0
\(43\) 10.5161 1.60368 0.801842 0.597536i \(-0.203854\pi\)
0.801842 + 0.597536i \(0.203854\pi\)
\(44\) −2.90321 −0.437676
\(45\) 0 0
\(46\) 2.34122 0.345194
\(47\) 6.90321 1.00694 0.503468 0.864014i \(-0.332057\pi\)
0.503468 + 0.864014i \(0.332057\pi\)
\(48\) 0 0
\(49\) −6.18421 −0.883458
\(50\) 0 0
\(51\) 0 0
\(52\) −1.18421 −0.164220
\(53\) 6.42864 0.883042 0.441521 0.897251i \(-0.354439\pi\)
0.441521 + 0.897251i \(0.354439\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.09679 −0.146564
\(57\) 0 0
\(58\) 0.311108 0.0408505
\(59\) 1.67307 0.217815 0.108908 0.994052i \(-0.465265\pi\)
0.108908 + 0.994052i \(0.465265\pi\)
\(60\) 0 0
\(61\) −1.86665 −0.239000 −0.119500 0.992834i \(-0.538129\pi\)
−0.119500 + 0.992834i \(0.538129\pi\)
\(62\) −2.14764 −0.272751
\(63\) 0 0
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) 0 0
\(67\) −11.5254 −1.40806 −0.704028 0.710173i \(-0.748617\pi\)
−0.704028 + 0.710173i \(0.748617\pi\)
\(68\) 15.1383 1.83579
\(69\) 0 0
\(70\) 0 0
\(71\) −13.6731 −1.62269 −0.811347 0.584564i \(-0.801266\pi\)
−0.811347 + 0.584564i \(0.801266\pi\)
\(72\) 0 0
\(73\) −10.1891 −1.19255 −0.596274 0.802781i \(-0.703353\pi\)
−0.596274 + 0.802781i \(0.703353\pi\)
\(74\) −1.23014 −0.143001
\(75\) 0 0
\(76\) 2.08742 0.239444
\(77\) 1.37778 0.157013
\(78\) 0 0
\(79\) 9.13828 1.02814 0.514068 0.857749i \(-0.328138\pi\)
0.514068 + 0.857749i \(0.328138\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.14272 −0.126192
\(83\) 10.7096 1.17554 0.587768 0.809030i \(-0.300007\pi\)
0.587768 + 0.809030i \(0.300007\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.27163 0.352789
\(87\) 0 0
\(88\) −1.85236 −0.197462
\(89\) 7.80642 0.827479 0.413740 0.910395i \(-0.364222\pi\)
0.413740 + 0.910395i \(0.364222\pi\)
\(90\) 0 0
\(91\) 0.561993 0.0589128
\(92\) −14.3225 −1.49322
\(93\) 0 0
\(94\) 2.14764 0.221512
\(95\) 0 0
\(96\) 0 0
\(97\) 4.08742 0.415015 0.207507 0.978233i \(-0.433465\pi\)
0.207507 + 0.978233i \(0.433465\pi\)
\(98\) −1.92396 −0.194349
\(99\) 0 0
\(100\) 0 0
\(101\) −13.9081 −1.38391 −0.691956 0.721940i \(-0.743251\pi\)
−0.691956 + 0.721940i \(0.743251\pi\)
\(102\) 0 0
\(103\) 12.9447 1.27548 0.637740 0.770252i \(-0.279870\pi\)
0.637740 + 0.770252i \(0.279870\pi\)
\(104\) −0.755569 −0.0740896
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 11.0049 1.06389 0.531943 0.846780i \(-0.321462\pi\)
0.531943 + 0.846780i \(0.321462\pi\)
\(108\) 0 0
\(109\) −18.0415 −1.72806 −0.864031 0.503439i \(-0.832068\pi\)
−0.864031 + 0.503439i \(0.832068\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.09679 0.292619
\(113\) −10.2810 −0.967155 −0.483577 0.875302i \(-0.660663\pi\)
−0.483577 + 0.875302i \(0.660663\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.90321 −0.176709
\(117\) 0 0
\(118\) 0.520505 0.0479164
\(119\) −7.18421 −0.658575
\(120\) 0 0
\(121\) −8.67307 −0.788461
\(122\) −0.580728 −0.0525767
\(123\) 0 0
\(124\) 13.1383 1.17985
\(125\) 0 0
\(126\) 0 0
\(127\) −6.22077 −0.552004 −0.276002 0.961157i \(-0.589010\pi\)
−0.276002 + 0.961157i \(0.589010\pi\)
\(128\) −8.78568 −0.776552
\(129\) 0 0
\(130\) 0 0
\(131\) 11.7605 1.02752 0.513759 0.857934i \(-0.328252\pi\)
0.513759 + 0.857934i \(0.328252\pi\)
\(132\) 0 0
\(133\) −0.990632 −0.0858987
\(134\) −3.58565 −0.309753
\(135\) 0 0
\(136\) 9.65878 0.828234
\(137\) −3.56691 −0.304742 −0.152371 0.988323i \(-0.548691\pi\)
−0.152371 + 0.988323i \(0.548691\pi\)
\(138\) 0 0
\(139\) −8.56199 −0.726219 −0.363109 0.931747i \(-0.618285\pi\)
−0.363109 + 0.931747i \(0.618285\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.25380 −0.356971
\(143\) 0.949145 0.0793715
\(144\) 0 0
\(145\) 0 0
\(146\) −3.16992 −0.262344
\(147\) 0 0
\(148\) 7.52543 0.618586
\(149\) 5.61285 0.459822 0.229911 0.973212i \(-0.426156\pi\)
0.229911 + 0.973212i \(0.426156\pi\)
\(150\) 0 0
\(151\) 10.7971 0.878652 0.439326 0.898328i \(-0.355217\pi\)
0.439326 + 0.898328i \(0.355217\pi\)
\(152\) 1.33185 0.108027
\(153\) 0 0
\(154\) 0.428639 0.0345408
\(155\) 0 0
\(156\) 0 0
\(157\) −2.28100 −0.182043 −0.0910217 0.995849i \(-0.529013\pi\)
−0.0910217 + 0.995849i \(0.529013\pi\)
\(158\) 2.84299 0.226176
\(159\) 0 0
\(160\) 0 0
\(161\) 6.79706 0.535683
\(162\) 0 0
\(163\) −16.3225 −1.27848 −0.639238 0.769009i \(-0.720750\pi\)
−0.639238 + 0.769009i \(0.720750\pi\)
\(164\) 6.99063 0.545877
\(165\) 0 0
\(166\) 3.33185 0.258602
\(167\) −4.76986 −0.369103 −0.184551 0.982823i \(-0.559083\pi\)
−0.184551 + 0.982823i \(0.559083\pi\)
\(168\) 0 0
\(169\) −12.6128 −0.970219
\(170\) 0 0
\(171\) 0 0
\(172\) −20.0143 −1.52608
\(173\) 4.23506 0.321986 0.160993 0.986956i \(-0.448530\pi\)
0.160993 + 0.986956i \(0.448530\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.23014 0.394237
\(177\) 0 0
\(178\) 2.42864 0.182034
\(179\) −9.71456 −0.726100 −0.363050 0.931770i \(-0.618265\pi\)
−0.363050 + 0.931770i \(0.618265\pi\)
\(180\) 0 0
\(181\) 0.326929 0.0243005 0.0121502 0.999926i \(-0.496132\pi\)
0.0121502 + 0.999926i \(0.496132\pi\)
\(182\) 0.174840 0.0129600
\(183\) 0 0
\(184\) −9.13828 −0.673683
\(185\) 0 0
\(186\) 0 0
\(187\) −12.1334 −0.887279
\(188\) −13.1383 −0.958207
\(189\) 0 0
\(190\) 0 0
\(191\) 14.9447 1.08136 0.540680 0.841228i \(-0.318167\pi\)
0.540680 + 0.841228i \(0.318167\pi\)
\(192\) 0 0
\(193\) 14.1476 1.01837 0.509185 0.860657i \(-0.329947\pi\)
0.509185 + 0.860657i \(0.329947\pi\)
\(194\) 1.27163 0.0912976
\(195\) 0 0
\(196\) 11.7699 0.840704
\(197\) −5.70471 −0.406444 −0.203222 0.979133i \(-0.565141\pi\)
−0.203222 + 0.979133i \(0.565141\pi\)
\(198\) 0 0
\(199\) −22.1432 −1.56969 −0.784845 0.619692i \(-0.787257\pi\)
−0.784845 + 0.619692i \(0.787257\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4.32693 −0.304442
\(203\) 0.903212 0.0633930
\(204\) 0 0
\(205\) 0 0
\(206\) 4.02720 0.280588
\(207\) 0 0
\(208\) 2.13335 0.147921
\(209\) −1.67307 −0.115729
\(210\) 0 0
\(211\) −20.8430 −1.43489 −0.717445 0.696615i \(-0.754689\pi\)
−0.717445 + 0.696615i \(0.754689\pi\)
\(212\) −12.2351 −0.840308
\(213\) 0 0
\(214\) 3.42372 0.234040
\(215\) 0 0
\(216\) 0 0
\(217\) −6.23506 −0.423264
\(218\) −5.61285 −0.380150
\(219\) 0 0
\(220\) 0 0
\(221\) −4.94914 −0.332916
\(222\) 0 0
\(223\) −9.03657 −0.605133 −0.302567 0.953128i \(-0.597843\pi\)
−0.302567 + 0.953128i \(0.597843\pi\)
\(224\) 3.15701 0.210937
\(225\) 0 0
\(226\) −3.19850 −0.212761
\(227\) 19.4050 1.28795 0.643977 0.765045i \(-0.277283\pi\)
0.643977 + 0.765045i \(0.277283\pi\)
\(228\) 0 0
\(229\) −25.6128 −1.69254 −0.846272 0.532751i \(-0.821158\pi\)
−0.846272 + 0.532751i \(0.821158\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.21432 −0.0797240
\(233\) −3.12399 −0.204659 −0.102330 0.994751i \(-0.532630\pi\)
−0.102330 + 0.994751i \(0.532630\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.18421 −0.207274
\(237\) 0 0
\(238\) −2.23506 −0.144878
\(239\) −13.9398 −0.901689 −0.450845 0.892602i \(-0.648877\pi\)
−0.450845 + 0.892602i \(0.648877\pi\)
\(240\) 0 0
\(241\) −18.4701 −1.18977 −0.594883 0.803813i \(-0.702802\pi\)
−0.594883 + 0.803813i \(0.702802\pi\)
\(242\) −2.69826 −0.173451
\(243\) 0 0
\(244\) 3.55262 0.227433
\(245\) 0 0
\(246\) 0 0
\(247\) −0.682439 −0.0434225
\(248\) 8.38271 0.532302
\(249\) 0 0
\(250\) 0 0
\(251\) 13.7921 0.870552 0.435276 0.900297i \(-0.356651\pi\)
0.435276 + 0.900297i \(0.356651\pi\)
\(252\) 0 0
\(253\) 11.4795 0.721710
\(254\) −1.93533 −0.121433
\(255\) 0 0
\(256\) 8.80642 0.550401
\(257\) −1.47949 −0.0922883 −0.0461442 0.998935i \(-0.514693\pi\)
−0.0461442 + 0.998935i \(0.514693\pi\)
\(258\) 0 0
\(259\) −3.57136 −0.221914
\(260\) 0 0
\(261\) 0 0
\(262\) 3.65878 0.226040
\(263\) −0.442930 −0.0273122 −0.0136561 0.999907i \(-0.504347\pi\)
−0.0136561 + 0.999907i \(0.504347\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.308193 −0.0188965
\(267\) 0 0
\(268\) 21.9353 1.33991
\(269\) −3.93978 −0.240212 −0.120106 0.992761i \(-0.538324\pi\)
−0.120106 + 0.992761i \(0.538324\pi\)
\(270\) 0 0
\(271\) 6.20787 0.377101 0.188551 0.982063i \(-0.439621\pi\)
0.188551 + 0.982063i \(0.439621\pi\)
\(272\) −27.2716 −1.65359
\(273\) 0 0
\(274\) −1.10970 −0.0670391
\(275\) 0 0
\(276\) 0 0
\(277\) −5.57136 −0.334751 −0.167375 0.985893i \(-0.553529\pi\)
−0.167375 + 0.985893i \(0.553529\pi\)
\(278\) −2.66370 −0.159758
\(279\) 0 0
\(280\) 0 0
\(281\) −6.69535 −0.399411 −0.199705 0.979856i \(-0.563999\pi\)
−0.199705 + 0.979856i \(0.563999\pi\)
\(282\) 0 0
\(283\) −25.8020 −1.53377 −0.766884 0.641785i \(-0.778194\pi\)
−0.766884 + 0.641785i \(0.778194\pi\)
\(284\) 26.0228 1.54417
\(285\) 0 0
\(286\) 0.295286 0.0174607
\(287\) −3.31756 −0.195829
\(288\) 0 0
\(289\) 46.2672 2.72160
\(290\) 0 0
\(291\) 0 0
\(292\) 19.3921 1.13484
\(293\) −18.8430 −1.10082 −0.550410 0.834895i \(-0.685528\pi\)
−0.550410 + 0.834895i \(0.685528\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.80150 0.279082
\(297\) 0 0
\(298\) 1.74620 0.101155
\(299\) 4.68244 0.270792
\(300\) 0 0
\(301\) 9.49823 0.547469
\(302\) 3.35905 0.193292
\(303\) 0 0
\(304\) −3.76049 −0.215679
\(305\) 0 0
\(306\) 0 0
\(307\) 1.65878 0.0946716 0.0473358 0.998879i \(-0.484927\pi\)
0.0473358 + 0.998879i \(0.484927\pi\)
\(308\) −2.62222 −0.149415
\(309\) 0 0
\(310\) 0 0
\(311\) −21.3002 −1.20782 −0.603912 0.797051i \(-0.706392\pi\)
−0.603912 + 0.797051i \(0.706392\pi\)
\(312\) 0 0
\(313\) −8.62222 −0.487356 −0.243678 0.969856i \(-0.578354\pi\)
−0.243678 + 0.969856i \(0.578354\pi\)
\(314\) −0.709636 −0.0400471
\(315\) 0 0
\(316\) −17.3921 −0.978381
\(317\) −27.5955 −1.54992 −0.774959 0.632012i \(-0.782229\pi\)
−0.774959 + 0.632012i \(0.782229\pi\)
\(318\) 0 0
\(319\) 1.52543 0.0854075
\(320\) 0 0
\(321\) 0 0
\(322\) 2.11462 0.117843
\(323\) 8.72393 0.485412
\(324\) 0 0
\(325\) 0 0
\(326\) −5.07805 −0.281247
\(327\) 0 0
\(328\) 4.46028 0.246278
\(329\) 6.23506 0.343750
\(330\) 0 0
\(331\) 16.9131 0.929626 0.464813 0.885409i \(-0.346122\pi\)
0.464813 + 0.885409i \(0.346122\pi\)
\(332\) −20.3827 −1.11865
\(333\) 0 0
\(334\) −1.48394 −0.0811976
\(335\) 0 0
\(336\) 0 0
\(337\) −11.9956 −0.653439 −0.326720 0.945121i \(-0.605943\pi\)
−0.326720 + 0.945121i \(0.605943\pi\)
\(338\) −3.92396 −0.213435
\(339\) 0 0
\(340\) 0 0
\(341\) −10.5303 −0.570250
\(342\) 0 0
\(343\) −11.9081 −0.642979
\(344\) −12.7699 −0.688505
\(345\) 0 0
\(346\) 1.31756 0.0708325
\(347\) 6.14764 0.330023 0.165011 0.986292i \(-0.447234\pi\)
0.165011 + 0.986292i \(0.447234\pi\)
\(348\) 0 0
\(349\) −7.12399 −0.381338 −0.190669 0.981654i \(-0.561066\pi\)
−0.190669 + 0.981654i \(0.561066\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.33185 0.284189
\(353\) −16.9175 −0.900428 −0.450214 0.892921i \(-0.648652\pi\)
−0.450214 + 0.892921i \(0.648652\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.8573 −0.787434
\(357\) 0 0
\(358\) −3.02227 −0.159732
\(359\) −36.7096 −1.93746 −0.968730 0.248116i \(-0.920188\pi\)
−0.968730 + 0.248116i \(0.920188\pi\)
\(360\) 0 0
\(361\) −17.7971 −0.936687
\(362\) 0.101710 0.00534577
\(363\) 0 0
\(364\) −1.06959 −0.0560618
\(365\) 0 0
\(366\) 0 0
\(367\) −8.41435 −0.439225 −0.219613 0.975587i \(-0.570479\pi\)
−0.219613 + 0.975587i \(0.570479\pi\)
\(368\) 25.8020 1.34502
\(369\) 0 0
\(370\) 0 0
\(371\) 5.80642 0.301455
\(372\) 0 0
\(373\) 8.66370 0.448590 0.224295 0.974521i \(-0.427992\pi\)
0.224295 + 0.974521i \(0.427992\pi\)
\(374\) −3.77478 −0.195189
\(375\) 0 0
\(376\) −8.38271 −0.432305
\(377\) 0.622216 0.0320457
\(378\) 0 0
\(379\) −2.76986 −0.142278 −0.0711390 0.997466i \(-0.522663\pi\)
−0.0711390 + 0.997466i \(0.522663\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.64941 0.237885
\(383\) 1.67752 0.0857171 0.0428585 0.999081i \(-0.486354\pi\)
0.0428585 + 0.999081i \(0.486354\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.40144 0.224028
\(387\) 0 0
\(388\) −7.77923 −0.394930
\(389\) 5.77478 0.292793 0.146397 0.989226i \(-0.453232\pi\)
0.146397 + 0.989226i \(0.453232\pi\)
\(390\) 0 0
\(391\) −59.8578 −3.02714
\(392\) 7.50961 0.379292
\(393\) 0 0
\(394\) −1.77478 −0.0894122
\(395\) 0 0
\(396\) 0 0
\(397\) −29.9081 −1.50105 −0.750523 0.660844i \(-0.770198\pi\)
−0.750523 + 0.660844i \(0.770198\pi\)
\(398\) −6.88892 −0.345310
\(399\) 0 0
\(400\) 0 0
\(401\) −8.53035 −0.425985 −0.212993 0.977054i \(-0.568321\pi\)
−0.212993 + 0.977054i \(0.568321\pi\)
\(402\) 0 0
\(403\) −4.29529 −0.213963
\(404\) 26.4701 1.31694
\(405\) 0 0
\(406\) 0.280996 0.0139456
\(407\) −6.03164 −0.298977
\(408\) 0 0
\(409\) 5.09234 0.251800 0.125900 0.992043i \(-0.459818\pi\)
0.125900 + 0.992043i \(0.459818\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −24.6365 −1.21375
\(413\) 1.51114 0.0743582
\(414\) 0 0
\(415\) 0 0
\(416\) 2.17484 0.106630
\(417\) 0 0
\(418\) −0.520505 −0.0254588
\(419\) 24.3368 1.18893 0.594465 0.804122i \(-0.297364\pi\)
0.594465 + 0.804122i \(0.297364\pi\)
\(420\) 0 0
\(421\) 24.5018 1.19414 0.597072 0.802188i \(-0.296331\pi\)
0.597072 + 0.802188i \(0.296331\pi\)
\(422\) −6.48442 −0.315656
\(423\) 0 0
\(424\) −7.80642 −0.379113
\(425\) 0 0
\(426\) 0 0
\(427\) −1.68598 −0.0815902
\(428\) −20.9447 −1.01240
\(429\) 0 0
\(430\) 0 0
\(431\) −4.26671 −0.205520 −0.102760 0.994706i \(-0.532767\pi\)
−0.102760 + 0.994706i \(0.532767\pi\)
\(432\) 0 0
\(433\) −27.0049 −1.29777 −0.648887 0.760885i \(-0.724765\pi\)
−0.648887 + 0.760885i \(0.724765\pi\)
\(434\) −1.93978 −0.0931123
\(435\) 0 0
\(436\) 34.3368 1.64443
\(437\) −8.25380 −0.394833
\(438\) 0 0
\(439\) −2.03164 −0.0969650 −0.0484825 0.998824i \(-0.515439\pi\)
−0.0484825 + 0.998824i \(0.515439\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.53972 −0.0732369
\(443\) −3.46520 −0.164637 −0.0823184 0.996606i \(-0.526232\pi\)
−0.0823184 + 0.996606i \(0.526232\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.81135 −0.133121
\(447\) 0 0
\(448\) −5.21141 −0.246216
\(449\) −37.3590 −1.76308 −0.881541 0.472107i \(-0.843494\pi\)
−0.881541 + 0.472107i \(0.843494\pi\)
\(450\) 0 0
\(451\) −5.60300 −0.263835
\(452\) 19.5669 0.920350
\(453\) 0 0
\(454\) 6.03704 0.283332
\(455\) 0 0
\(456\) 0 0
\(457\) −13.4509 −0.629207 −0.314604 0.949223i \(-0.601872\pi\)
−0.314604 + 0.949223i \(0.601872\pi\)
\(458\) −7.96836 −0.372337
\(459\) 0 0
\(460\) 0 0
\(461\) 16.2766 0.758075 0.379037 0.925381i \(-0.376255\pi\)
0.379037 + 0.925381i \(0.376255\pi\)
\(462\) 0 0
\(463\) 30.3926 1.41246 0.706231 0.707982i \(-0.250394\pi\)
0.706231 + 0.707982i \(0.250394\pi\)
\(464\) 3.42864 0.159171
\(465\) 0 0
\(466\) −0.971896 −0.0450222
\(467\) −1.18865 −0.0550043 −0.0275022 0.999622i \(-0.508755\pi\)
−0.0275022 + 0.999622i \(0.508755\pi\)
\(468\) 0 0
\(469\) −10.4099 −0.480685
\(470\) 0 0
\(471\) 0 0
\(472\) −2.03164 −0.0935139
\(473\) 16.0415 0.737588
\(474\) 0 0
\(475\) 0 0
\(476\) 13.6731 0.626704
\(477\) 0 0
\(478\) −4.33677 −0.198359
\(479\) −41.0464 −1.87546 −0.937729 0.347367i \(-0.887076\pi\)
−0.937729 + 0.347367i \(0.887076\pi\)
\(480\) 0 0
\(481\) −2.46028 −0.112179
\(482\) −5.74620 −0.261732
\(483\) 0 0
\(484\) 16.5067 0.750304
\(485\) 0 0
\(486\) 0 0
\(487\) 10.1476 0.459834 0.229917 0.973210i \(-0.426155\pi\)
0.229917 + 0.973210i \(0.426155\pi\)
\(488\) 2.26671 0.102609
\(489\) 0 0
\(490\) 0 0
\(491\) 29.2083 1.31815 0.659077 0.752075i \(-0.270947\pi\)
0.659077 + 0.752075i \(0.270947\pi\)
\(492\) 0 0
\(493\) −7.95407 −0.358233
\(494\) −0.212312 −0.00955237
\(495\) 0 0
\(496\) −23.6686 −1.06275
\(497\) −12.3497 −0.553959
\(498\) 0 0
\(499\) 21.9813 0.984017 0.492008 0.870591i \(-0.336263\pi\)
0.492008 + 0.870591i \(0.336263\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.29084 0.191510
\(503\) 5.77923 0.257683 0.128841 0.991665i \(-0.458874\pi\)
0.128841 + 0.991665i \(0.458874\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.57136 0.158766
\(507\) 0 0
\(508\) 11.8394 0.525291
\(509\) −13.6543 −0.605218 −0.302609 0.953115i \(-0.597858\pi\)
−0.302609 + 0.953115i \(0.597858\pi\)
\(510\) 0 0
\(511\) −9.20294 −0.407114
\(512\) 20.3111 0.897633
\(513\) 0 0
\(514\) −0.460282 −0.0203022
\(515\) 0 0
\(516\) 0 0
\(517\) 10.5303 0.463124
\(518\) −1.11108 −0.0488180
\(519\) 0 0
\(520\) 0 0
\(521\) 19.6731 0.861893 0.430946 0.902378i \(-0.358180\pi\)
0.430946 + 0.902378i \(0.358180\pi\)
\(522\) 0 0
\(523\) 15.1383 0.661951 0.330975 0.943639i \(-0.392622\pi\)
0.330975 + 0.943639i \(0.392622\pi\)
\(524\) −22.3827 −0.977793
\(525\) 0 0
\(526\) −0.137799 −0.00600832
\(527\) 54.9086 2.39186
\(528\) 0 0
\(529\) 33.6321 1.46226
\(530\) 0 0
\(531\) 0 0
\(532\) 1.88538 0.0817417
\(533\) −2.28544 −0.0989935
\(534\) 0 0
\(535\) 0 0
\(536\) 13.9956 0.604516
\(537\) 0 0
\(538\) −1.22570 −0.0528435
\(539\) −9.43356 −0.406332
\(540\) 0 0
\(541\) 2.68244 0.115327 0.0576635 0.998336i \(-0.481635\pi\)
0.0576635 + 0.998336i \(0.481635\pi\)
\(542\) 1.93132 0.0829571
\(543\) 0 0
\(544\) −27.8020 −1.19200
\(545\) 0 0
\(546\) 0 0
\(547\) 15.3635 0.656896 0.328448 0.944522i \(-0.393475\pi\)
0.328448 + 0.944522i \(0.393475\pi\)
\(548\) 6.78859 0.289994
\(549\) 0 0
\(550\) 0 0
\(551\) −1.09679 −0.0467247
\(552\) 0 0
\(553\) 8.25380 0.350987
\(554\) −1.73329 −0.0736406
\(555\) 0 0
\(556\) 16.2953 0.691074
\(557\) −9.87955 −0.418610 −0.209305 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(558\) 0 0
\(559\) 6.54326 0.276750
\(560\) 0 0
\(561\) 0 0
\(562\) −2.08297 −0.0878650
\(563\) −27.4938 −1.15872 −0.579362 0.815070i \(-0.696698\pi\)
−0.579362 + 0.815070i \(0.696698\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.02720 −0.337408
\(567\) 0 0
\(568\) 16.6035 0.696667
\(569\) −17.3590 −0.727729 −0.363865 0.931452i \(-0.618543\pi\)
−0.363865 + 0.931452i \(0.618543\pi\)
\(570\) 0 0
\(571\) −25.4479 −1.06496 −0.532480 0.846443i \(-0.678740\pi\)
−0.532480 + 0.846443i \(0.678740\pi\)
\(572\) −1.80642 −0.0755304
\(573\) 0 0
\(574\) −1.03212 −0.0430798
\(575\) 0 0
\(576\) 0 0
\(577\) 10.6178 0.442024 0.221012 0.975271i \(-0.429064\pi\)
0.221012 + 0.975271i \(0.429064\pi\)
\(578\) 14.3941 0.598715
\(579\) 0 0
\(580\) 0 0
\(581\) 9.67307 0.401307
\(582\) 0 0
\(583\) 9.80642 0.406141
\(584\) 12.3729 0.511993
\(585\) 0 0
\(586\) −5.86220 −0.242165
\(587\) −8.94470 −0.369187 −0.184594 0.982815i \(-0.559097\pi\)
−0.184594 + 0.982815i \(0.559097\pi\)
\(588\) 0 0
\(589\) 7.57136 0.311972
\(590\) 0 0
\(591\) 0 0
\(592\) −13.5571 −0.557192
\(593\) 14.1619 0.581561 0.290780 0.956790i \(-0.406085\pi\)
0.290780 + 0.956790i \(0.406085\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.6824 −0.437570
\(597\) 0 0
\(598\) 1.45674 0.0595707
\(599\) 22.5575 0.921676 0.460838 0.887484i \(-0.347549\pi\)
0.460838 + 0.887484i \(0.347549\pi\)
\(600\) 0 0
\(601\) −40.6133 −1.65665 −0.828326 0.560246i \(-0.810706\pi\)
−0.828326 + 0.560246i \(0.810706\pi\)
\(602\) 2.95497 0.120436
\(603\) 0 0
\(604\) −20.5491 −0.836130
\(605\) 0 0
\(606\) 0 0
\(607\) −13.5955 −0.551824 −0.275912 0.961183i \(-0.588980\pi\)
−0.275912 + 0.961183i \(0.588980\pi\)
\(608\) −3.83362 −0.155474
\(609\) 0 0
\(610\) 0 0
\(611\) 4.29529 0.173769
\(612\) 0 0
\(613\) 42.0830 1.69972 0.849858 0.527012i \(-0.176688\pi\)
0.849858 + 0.527012i \(0.176688\pi\)
\(614\) 0.516060 0.0208265
\(615\) 0 0
\(616\) −1.67307 −0.0674099
\(617\) 33.5067 1.34893 0.674464 0.738307i \(-0.264375\pi\)
0.674464 + 0.738307i \(0.264375\pi\)
\(618\) 0 0
\(619\) −14.6780 −0.589958 −0.294979 0.955504i \(-0.595313\pi\)
−0.294979 + 0.955504i \(0.595313\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.62666 −0.265705
\(623\) 7.05086 0.282487
\(624\) 0 0
\(625\) 0 0
\(626\) −2.68244 −0.107212
\(627\) 0 0
\(628\) 4.34122 0.173234
\(629\) 31.4509 1.25403
\(630\) 0 0
\(631\) 11.3176 0.450545 0.225273 0.974296i \(-0.427673\pi\)
0.225273 + 0.974296i \(0.427673\pi\)
\(632\) −11.0968 −0.441407
\(633\) 0 0
\(634\) −8.58517 −0.340961
\(635\) 0 0
\(636\) 0 0
\(637\) −3.84791 −0.152460
\(638\) 0.474572 0.0187885
\(639\) 0 0
\(640\) 0 0
\(641\) −34.8988 −1.37842 −0.689209 0.724562i \(-0.742042\pi\)
−0.689209 + 0.724562i \(0.742042\pi\)
\(642\) 0 0
\(643\) 41.9768 1.65540 0.827702 0.561168i \(-0.189648\pi\)
0.827702 + 0.561168i \(0.189648\pi\)
\(644\) −12.9362 −0.509759
\(645\) 0 0
\(646\) 2.71408 0.106784
\(647\) 5.46520 0.214859 0.107430 0.994213i \(-0.465738\pi\)
0.107430 + 0.994213i \(0.465738\pi\)
\(648\) 0 0
\(649\) 2.55215 0.100181
\(650\) 0 0
\(651\) 0 0
\(652\) 31.0651 1.21660
\(653\) −8.76986 −0.343191 −0.171596 0.985167i \(-0.554892\pi\)
−0.171596 + 0.985167i \(0.554892\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −12.5936 −0.491699
\(657\) 0 0
\(658\) 1.93978 0.0756204
\(659\) −3.29036 −0.128174 −0.0640872 0.997944i \(-0.520414\pi\)
−0.0640872 + 0.997944i \(0.520414\pi\)
\(660\) 0 0
\(661\) 19.7560 0.768421 0.384211 0.923246i \(-0.374474\pi\)
0.384211 + 0.923246i \(0.374474\pi\)
\(662\) 5.26178 0.204505
\(663\) 0 0
\(664\) −13.0049 −0.504689
\(665\) 0 0
\(666\) 0 0
\(667\) 7.52543 0.291386
\(668\) 9.07805 0.351240
\(669\) 0 0
\(670\) 0 0
\(671\) −2.84743 −0.109924
\(672\) 0 0
\(673\) −44.3970 −1.71138 −0.855689 0.517490i \(-0.826866\pi\)
−0.855689 + 0.517490i \(0.826866\pi\)
\(674\) −3.73191 −0.143748
\(675\) 0 0
\(676\) 24.0049 0.923266
\(677\) 6.09726 0.234337 0.117168 0.993112i \(-0.462618\pi\)
0.117168 + 0.993112i \(0.462618\pi\)
\(678\) 0 0
\(679\) 3.69181 0.141679
\(680\) 0 0
\(681\) 0 0
\(682\) −3.27607 −0.125447
\(683\) 37.9224 1.45106 0.725531 0.688190i \(-0.241594\pi\)
0.725531 + 0.688190i \(0.241594\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −3.70471 −0.141447
\(687\) 0 0
\(688\) 36.0558 1.37461
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 13.3145 0.506507 0.253254 0.967400i \(-0.418499\pi\)
0.253254 + 0.967400i \(0.418499\pi\)
\(692\) −8.06022 −0.306404
\(693\) 0 0
\(694\) 1.91258 0.0726005
\(695\) 0 0
\(696\) 0 0
\(697\) 29.2159 1.10663
\(698\) −2.21633 −0.0838892
\(699\) 0 0
\(700\) 0 0
\(701\) 23.4893 0.887180 0.443590 0.896230i \(-0.353705\pi\)
0.443590 + 0.896230i \(0.353705\pi\)
\(702\) 0 0
\(703\) 4.33677 0.163565
\(704\) −8.80150 −0.331719
\(705\) 0 0
\(706\) −5.26317 −0.198082
\(707\) −12.5620 −0.472442
\(708\) 0 0
\(709\) 11.6731 0.438391 0.219196 0.975681i \(-0.429657\pi\)
0.219196 + 0.975681i \(0.429657\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −9.47949 −0.355259
\(713\) −51.9496 −1.94553
\(714\) 0 0
\(715\) 0 0
\(716\) 18.4889 0.690961
\(717\) 0 0
\(718\) −11.4207 −0.426215
\(719\) −29.5526 −1.10213 −0.551063 0.834463i \(-0.685778\pi\)
−0.551063 + 0.834463i \(0.685778\pi\)
\(720\) 0 0
\(721\) 11.6918 0.435426
\(722\) −5.53680 −0.206058
\(723\) 0 0
\(724\) −0.622216 −0.0231245
\(725\) 0 0
\(726\) 0 0
\(727\) −3.88094 −0.143936 −0.0719680 0.997407i \(-0.522928\pi\)
−0.0719680 + 0.997407i \(0.522928\pi\)
\(728\) −0.682439 −0.0252929
\(729\) 0 0
\(730\) 0 0
\(731\) −83.6454 −3.09374
\(732\) 0 0
\(733\) −14.8845 −0.549771 −0.274885 0.961477i \(-0.588640\pi\)
−0.274885 + 0.961477i \(0.588640\pi\)
\(734\) −2.61777 −0.0966236
\(735\) 0 0
\(736\) 26.3037 0.969569
\(737\) −17.5812 −0.647612
\(738\) 0 0
\(739\) 2.24935 0.0827438 0.0413719 0.999144i \(-0.486827\pi\)
0.0413719 + 0.999144i \(0.486827\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.80642 0.0663159
\(743\) −3.46520 −0.127126 −0.0635630 0.997978i \(-0.520246\pi\)
−0.0635630 + 0.997978i \(0.520246\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.69535 0.0986836
\(747\) 0 0
\(748\) 23.0923 0.844340
\(749\) 9.93978 0.363192
\(750\) 0 0
\(751\) 3.16992 0.115672 0.0578360 0.998326i \(-0.481580\pi\)
0.0578360 + 0.998326i \(0.481580\pi\)
\(752\) 23.6686 0.863106
\(753\) 0 0
\(754\) 0.193576 0.00704963
\(755\) 0 0
\(756\) 0 0
\(757\) 52.0785 1.89283 0.946413 0.322958i \(-0.104677\pi\)
0.946413 + 0.322958i \(0.104677\pi\)
\(758\) −0.861725 −0.0312993
\(759\) 0 0
\(760\) 0 0
\(761\) 14.9777 0.542942 0.271471 0.962447i \(-0.412490\pi\)
0.271471 + 0.962447i \(0.412490\pi\)
\(762\) 0 0
\(763\) −16.2953 −0.589929
\(764\) −28.4429 −1.02903
\(765\) 0 0
\(766\) 0.521889 0.0188566
\(767\) 1.04101 0.0375887
\(768\) 0 0
\(769\) −1.90813 −0.0688091 −0.0344045 0.999408i \(-0.510953\pi\)
−0.0344045 + 0.999408i \(0.510953\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −26.9260 −0.969087
\(773\) 21.7891 0.783698 0.391849 0.920029i \(-0.371835\pi\)
0.391849 + 0.920029i \(0.371835\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.96343 −0.178177
\(777\) 0 0
\(778\) 1.79658 0.0644105
\(779\) 4.02858 0.144339
\(780\) 0 0
\(781\) −20.8573 −0.746332
\(782\) −18.6222 −0.665929
\(783\) 0 0
\(784\) −21.2034 −0.757265
\(785\) 0 0
\(786\) 0 0
\(787\) 18.1388 0.646577 0.323288 0.946301i \(-0.395212\pi\)
0.323288 + 0.946301i \(0.395212\pi\)
\(788\) 10.8573 0.386775
\(789\) 0 0
\(790\) 0 0
\(791\) −9.28592 −0.330169
\(792\) 0 0
\(793\) −1.16146 −0.0412445
\(794\) −9.30465 −0.330210
\(795\) 0 0
\(796\) 42.1432 1.49373
\(797\) 2.96343 0.104970 0.0524851 0.998622i \(-0.483286\pi\)
0.0524851 + 0.998622i \(0.483286\pi\)
\(798\) 0 0
\(799\) −54.9086 −1.94253
\(800\) 0 0
\(801\) 0 0
\(802\) −2.65386 −0.0937110
\(803\) −15.5428 −0.548493
\(804\) 0 0
\(805\) 0 0
\(806\) −1.33630 −0.0470691
\(807\) 0 0
\(808\) 16.8889 0.594150
\(809\) 26.2953 0.924493 0.462247 0.886751i \(-0.347044\pi\)
0.462247 + 0.886751i \(0.347044\pi\)
\(810\) 0 0
\(811\) 24.3783 0.856037 0.428018 0.903770i \(-0.359212\pi\)
0.428018 + 0.903770i \(0.359212\pi\)
\(812\) −1.71900 −0.0603252
\(813\) 0 0
\(814\) −1.87649 −0.0657710
\(815\) 0 0
\(816\) 0 0
\(817\) −11.5339 −0.403520
\(818\) 1.58427 0.0553926
\(819\) 0 0
\(820\) 0 0
\(821\) −1.52987 −0.0533929 −0.0266965 0.999644i \(-0.508499\pi\)
−0.0266965 + 0.999644i \(0.508499\pi\)
\(822\) 0 0
\(823\) −46.7195 −1.62854 −0.814269 0.580487i \(-0.802862\pi\)
−0.814269 + 0.580487i \(0.802862\pi\)
\(824\) −15.7190 −0.547597
\(825\) 0 0
\(826\) 0.470127 0.0163578
\(827\) −29.6499 −1.03103 −0.515514 0.856881i \(-0.672399\pi\)
−0.515514 + 0.856881i \(0.672399\pi\)
\(828\) 0 0
\(829\) −8.79706 −0.305534 −0.152767 0.988262i \(-0.548818\pi\)
−0.152767 + 0.988262i \(0.548818\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.59010 −0.124464
\(833\) 49.1896 1.70432
\(834\) 0 0
\(835\) 0 0
\(836\) 3.18421 0.110128
\(837\) 0 0
\(838\) 7.57136 0.261548
\(839\) −11.3319 −0.391219 −0.195609 0.980682i \(-0.562668\pi\)
−0.195609 + 0.980682i \(0.562668\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 7.62269 0.262695
\(843\) 0 0
\(844\) 39.6686 1.36545
\(845\) 0 0
\(846\) 0 0
\(847\) −7.83362 −0.269166
\(848\) 22.0415 0.756908
\(849\) 0 0
\(850\) 0 0
\(851\) −29.7560 −1.02002
\(852\) 0 0
\(853\) −54.8845 −1.87921 −0.939604 0.342263i \(-0.888807\pi\)
−0.939604 + 0.342263i \(0.888807\pi\)
\(854\) −0.524521 −0.0179487
\(855\) 0 0
\(856\) −13.3635 −0.456755
\(857\) 36.4385 1.24471 0.622357 0.782733i \(-0.286175\pi\)
0.622357 + 0.782733i \(0.286175\pi\)
\(858\) 0 0
\(859\) 1.72885 0.0589875 0.0294938 0.999565i \(-0.490610\pi\)
0.0294938 + 0.999565i \(0.490610\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.32741 −0.0452116
\(863\) −9.40192 −0.320045 −0.160023 0.987113i \(-0.551157\pi\)
−0.160023 + 0.987113i \(0.551157\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −8.40144 −0.285493
\(867\) 0 0
\(868\) 11.8666 0.402780
\(869\) 13.9398 0.472875
\(870\) 0 0
\(871\) −7.17130 −0.242990
\(872\) 21.9081 0.741903
\(873\) 0 0
\(874\) −2.56782 −0.0868579
\(875\) 0 0
\(876\) 0 0
\(877\) −8.91750 −0.301123 −0.150561 0.988601i \(-0.548108\pi\)
−0.150561 + 0.988601i \(0.548108\pi\)
\(878\) −0.632060 −0.0213310
\(879\) 0 0
\(880\) 0 0
\(881\) 42.1245 1.41921 0.709605 0.704600i \(-0.248874\pi\)
0.709605 + 0.704600i \(0.248874\pi\)
\(882\) 0 0
\(883\) 38.4340 1.29341 0.646704 0.762741i \(-0.276147\pi\)
0.646704 + 0.762741i \(0.276147\pi\)
\(884\) 9.41927 0.316804
\(885\) 0 0
\(886\) −1.07805 −0.0362179
\(887\) −38.6365 −1.29729 −0.648643 0.761092i \(-0.724663\pi\)
−0.648643 + 0.761092i \(0.724663\pi\)
\(888\) 0 0
\(889\) −5.61868 −0.188444
\(890\) 0 0
\(891\) 0 0
\(892\) 17.1985 0.575848
\(893\) −7.57136 −0.253366
\(894\) 0 0
\(895\) 0 0
\(896\) −7.93533 −0.265101
\(897\) 0 0
\(898\) −11.6227 −0.387854
\(899\) −6.90321 −0.230235
\(900\) 0 0
\(901\) −51.1338 −1.70351
\(902\) −1.74314 −0.0580402
\(903\) 0 0
\(904\) 12.4844 0.415226
\(905\) 0 0
\(906\) 0 0
\(907\) −0.534795 −0.0177576 −0.00887880 0.999961i \(-0.502826\pi\)
−0.00887880 + 0.999961i \(0.502826\pi\)
\(908\) −36.9318 −1.22562
\(909\) 0 0
\(910\) 0 0
\(911\) 23.6686 0.784177 0.392088 0.919928i \(-0.371753\pi\)
0.392088 + 0.919928i \(0.371753\pi\)
\(912\) 0 0
\(913\) 16.3368 0.540668
\(914\) −4.18468 −0.138417
\(915\) 0 0
\(916\) 48.7467 1.61064
\(917\) 10.6222 0.350776
\(918\) 0 0
\(919\) 35.7748 1.18010 0.590051 0.807366i \(-0.299108\pi\)
0.590051 + 0.807366i \(0.299108\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 5.06376 0.166766
\(923\) −8.50760 −0.280031
\(924\) 0 0
\(925\) 0 0
\(926\) 9.45536 0.310722
\(927\) 0 0
\(928\) 3.49532 0.114739
\(929\) 52.7753 1.73150 0.865750 0.500477i \(-0.166842\pi\)
0.865750 + 0.500477i \(0.166842\pi\)
\(930\) 0 0
\(931\) 6.78277 0.222296
\(932\) 5.94561 0.194755
\(933\) 0 0
\(934\) −0.369800 −0.0121002
\(935\) 0 0
\(936\) 0 0
\(937\) −42.1245 −1.37615 −0.688073 0.725641i \(-0.741543\pi\)
−0.688073 + 0.725641i \(0.741543\pi\)
\(938\) −3.23860 −0.105744
\(939\) 0 0
\(940\) 0 0
\(941\) 3.89829 0.127081 0.0635403 0.997979i \(-0.479761\pi\)
0.0635403 + 0.997979i \(0.479761\pi\)
\(942\) 0 0
\(943\) −27.6414 −0.900129
\(944\) 5.73636 0.186703
\(945\) 0 0
\(946\) 4.99063 0.162259
\(947\) 9.56691 0.310883 0.155441 0.987845i \(-0.450320\pi\)
0.155441 + 0.987845i \(0.450320\pi\)
\(948\) 0 0
\(949\) −6.33984 −0.205800
\(950\) 0 0
\(951\) 0 0
\(952\) 8.72393 0.282744
\(953\) 27.2070 0.881320 0.440660 0.897674i \(-0.354744\pi\)
0.440660 + 0.897674i \(0.354744\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 26.5303 0.858053
\(957\) 0 0
\(958\) −12.7699 −0.412575
\(959\) −3.22168 −0.104033
\(960\) 0 0
\(961\) 16.6543 0.537237
\(962\) −0.765413 −0.0246779
\(963\) 0 0
\(964\) 35.1526 1.13219
\(965\) 0 0
\(966\) 0 0
\(967\) 16.8015 0.540300 0.270150 0.962818i \(-0.412927\pi\)
0.270150 + 0.962818i \(0.412927\pi\)
\(968\) 10.5319 0.338507
\(969\) 0 0
\(970\) 0 0
\(971\) 17.4465 0.559884 0.279942 0.960017i \(-0.409685\pi\)
0.279942 + 0.960017i \(0.409685\pi\)
\(972\) 0 0
\(973\) −7.73329 −0.247918
\(974\) 3.15701 0.101157
\(975\) 0 0
\(976\) −6.40006 −0.204861
\(977\) 32.0513 1.02541 0.512706 0.858564i \(-0.328643\pi\)
0.512706 + 0.858564i \(0.328643\pi\)
\(978\) 0 0
\(979\) 11.9081 0.380586
\(980\) 0 0
\(981\) 0 0
\(982\) 9.08694 0.289976
\(983\) −16.5259 −0.527094 −0.263547 0.964646i \(-0.584892\pi\)
−0.263547 + 0.964646i \(0.584892\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.47457 −0.0788064
\(987\) 0 0
\(988\) 1.29883 0.0413211
\(989\) 79.1378 2.51644
\(990\) 0 0
\(991\) −9.34920 −0.296987 −0.148494 0.988913i \(-0.547442\pi\)
−0.148494 + 0.988913i \(0.547442\pi\)
\(992\) −24.1289 −0.766094
\(993\) 0 0
\(994\) −3.84208 −0.121863
\(995\) 0 0
\(996\) 0 0
\(997\) 15.9956 0.506584 0.253292 0.967390i \(-0.418487\pi\)
0.253292 + 0.967390i \(0.418487\pi\)
\(998\) 6.83854 0.216470
\(999\) 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 6525.2.a.be.1.2 3
3.2 odd 2 725.2.a.e.1.2 3
5.4 even 2 1305.2.a.p.1.2 3
15.2 even 4 725.2.b.e.349.3 6
15.8 even 4 725.2.b.e.349.4 6
15.14 odd 2 145.2.a.c.1.2 3
60.59 even 2 2320.2.a.n.1.1 3
105.104 even 2 7105.2.a.o.1.2 3
120.29 odd 2 9280.2.a.bj.1.1 3
120.59 even 2 9280.2.a.br.1.3 3
435.434 odd 2 4205.2.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 15.14 odd 2
725.2.a.e.1.2 3 3.2 odd 2
725.2.b.e.349.3 6 15.2 even 4
725.2.b.e.349.4 6 15.8 even 4
1305.2.a.p.1.2 3 5.4 even 2
2320.2.a.n.1.1 3 60.59 even 2
4205.2.a.f.1.2 3 435.434 odd 2
6525.2.a.be.1.2 3 1.1 even 1 trivial
7105.2.a.o.1.2 3 105.104 even 2
9280.2.a.bj.1.1 3 120.29 odd 2
9280.2.a.br.1.3 3 120.59 even 2