Properties

Label 6525.2.a.by.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2175)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.57789\) 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)
 
\(f(q)\) \(=\) \(q-2.57789 q^{2} +4.64553 q^{4} +4.69867 q^{7} -6.81989 q^{8} +O(q^{10})\) \(q-2.57789 q^{2} +4.64553 q^{4} +4.69867 q^{7} -6.81989 q^{8} +3.11503 q^{11} -5.07945 q^{13} -12.1127 q^{14} +8.28989 q^{16} -1.40020 q^{17} -3.76514 q^{19} -8.03022 q^{22} -5.71483 q^{23} +13.0943 q^{26} +21.8278 q^{28} -1.00000 q^{29} -2.23218 q^{31} -7.73067 q^{32} +3.60957 q^{34} +5.79088 q^{37} +9.70614 q^{38} +10.6968 q^{41} -8.89527 q^{43} +14.4710 q^{44} +14.7322 q^{46} -3.62785 q^{47} +15.0775 q^{49} -23.5967 q^{52} +0.948260 q^{53} -32.0445 q^{56} +2.57789 q^{58} +8.53886 q^{59} -6.21467 q^{61} +5.75432 q^{62} +3.34905 q^{64} -13.9099 q^{67} -6.50468 q^{68} -5.88726 q^{71} +7.08398 q^{73} -14.9283 q^{74} -17.4911 q^{76} +14.6365 q^{77} +7.31672 q^{79} -27.5751 q^{82} -13.6579 q^{83} +22.9311 q^{86} -21.2442 q^{88} +7.98017 q^{89} -23.8667 q^{91} -26.5484 q^{92} +9.35221 q^{94} -13.8343 q^{97} -38.8683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.57789 −1.82285 −0.911423 0.411471i \(-0.865015\pi\)
−0.911423 + 0.411471i \(0.865015\pi\)
\(3\) 0 0
\(4\) 4.64553 2.32277
\(5\) 0 0
\(6\) 0 0
\(7\) 4.69867 1.77593 0.887966 0.459909i \(-0.152118\pi\)
0.887966 + 0.459909i \(0.152118\pi\)
\(8\) −6.81989 −2.41120
\(9\) 0 0
\(10\) 0 0
\(11\) 3.11503 0.939218 0.469609 0.882875i \(-0.344395\pi\)
0.469609 + 0.882875i \(0.344395\pi\)
\(12\) 0 0
\(13\) −5.07945 −1.40879 −0.704393 0.709810i \(-0.748781\pi\)
−0.704393 + 0.709810i \(0.748781\pi\)
\(14\) −12.1127 −3.23725
\(15\) 0 0
\(16\) 8.28989 2.07247
\(17\) −1.40020 −0.339599 −0.169799 0.985479i \(-0.554312\pi\)
−0.169799 + 0.985479i \(0.554312\pi\)
\(18\) 0 0
\(19\) −3.76514 −0.863783 −0.431892 0.901925i \(-0.642154\pi\)
−0.431892 + 0.901925i \(0.642154\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.03022 −1.71205
\(23\) −5.71483 −1.19162 −0.595812 0.803124i \(-0.703170\pi\)
−0.595812 + 0.803124i \(0.703170\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 13.0943 2.56800
\(27\) 0 0
\(28\) 21.8278 4.12507
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −2.23218 −0.400911 −0.200456 0.979703i \(-0.564242\pi\)
−0.200456 + 0.979703i \(0.564242\pi\)
\(32\) −7.73067 −1.36660
\(33\) 0 0
\(34\) 3.60957 0.619036
\(35\) 0 0
\(36\) 0 0
\(37\) 5.79088 0.952015 0.476008 0.879441i \(-0.342083\pi\)
0.476008 + 0.879441i \(0.342083\pi\)
\(38\) 9.70614 1.57454
\(39\) 0 0
\(40\) 0 0
\(41\) 10.6968 1.67055 0.835276 0.549830i \(-0.185308\pi\)
0.835276 + 0.549830i \(0.185308\pi\)
\(42\) 0 0
\(43\) −8.89527 −1.35652 −0.678258 0.734824i \(-0.737265\pi\)
−0.678258 + 0.734824i \(0.737265\pi\)
\(44\) 14.4710 2.18158
\(45\) 0 0
\(46\) 14.7322 2.17215
\(47\) −3.62785 −0.529176 −0.264588 0.964362i \(-0.585236\pi\)
−0.264588 + 0.964362i \(0.585236\pi\)
\(48\) 0 0
\(49\) 15.0775 2.15393
\(50\) 0 0
\(51\) 0 0
\(52\) −23.5967 −3.27228
\(53\) 0.948260 0.130254 0.0651268 0.997877i \(-0.479255\pi\)
0.0651268 + 0.997877i \(0.479255\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −32.0445 −4.28212
\(57\) 0 0
\(58\) 2.57789 0.338494
\(59\) 8.53886 1.11166 0.555832 0.831294i \(-0.312400\pi\)
0.555832 + 0.831294i \(0.312400\pi\)
\(60\) 0 0
\(61\) −6.21467 −0.795707 −0.397853 0.917449i \(-0.630245\pi\)
−0.397853 + 0.917449i \(0.630245\pi\)
\(62\) 5.75432 0.730799
\(63\) 0 0
\(64\) 3.34905 0.418631
\(65\) 0 0
\(66\) 0 0
\(67\) −13.9099 −1.69937 −0.849685 0.527291i \(-0.823208\pi\)
−0.849685 + 0.527291i \(0.823208\pi\)
\(68\) −6.50468 −0.788808
\(69\) 0 0
\(70\) 0 0
\(71\) −5.88726 −0.698690 −0.349345 0.936994i \(-0.613596\pi\)
−0.349345 + 0.936994i \(0.613596\pi\)
\(72\) 0 0
\(73\) 7.08398 0.829117 0.414559 0.910023i \(-0.363936\pi\)
0.414559 + 0.910023i \(0.363936\pi\)
\(74\) −14.9283 −1.73538
\(75\) 0 0
\(76\) −17.4911 −2.00637
\(77\) 14.6365 1.66799
\(78\) 0 0
\(79\) 7.31672 0.823195 0.411598 0.911366i \(-0.364971\pi\)
0.411598 + 0.911366i \(0.364971\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −27.5751 −3.04516
\(83\) −13.6579 −1.49915 −0.749577 0.661917i \(-0.769743\pi\)
−0.749577 + 0.661917i \(0.769743\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 22.9311 2.47272
\(87\) 0 0
\(88\) −21.2442 −2.26464
\(89\) 7.98017 0.845897 0.422948 0.906154i \(-0.360995\pi\)
0.422948 + 0.906154i \(0.360995\pi\)
\(90\) 0 0
\(91\) −23.8667 −2.50191
\(92\) −26.5484 −2.76786
\(93\) 0 0
\(94\) 9.35221 0.964607
\(95\) 0 0
\(96\) 0 0
\(97\) −13.8343 −1.40466 −0.702329 0.711853i \(-0.747856\pi\)
−0.702329 + 0.711853i \(0.747856\pi\)
\(98\) −38.8683 −3.92629
\(99\) 0 0
\(100\) 0 0
\(101\) −12.7461 −1.26829 −0.634144 0.773215i \(-0.718647\pi\)
−0.634144 + 0.773215i \(0.718647\pi\)
\(102\) 0 0
\(103\) 8.20406 0.808370 0.404185 0.914677i \(-0.367555\pi\)
0.404185 + 0.914677i \(0.367555\pi\)
\(104\) 34.6413 3.39686
\(105\) 0 0
\(106\) −2.44451 −0.237432
\(107\) −3.28873 −0.317933 −0.158967 0.987284i \(-0.550816\pi\)
−0.158967 + 0.987284i \(0.550816\pi\)
\(108\) 0 0
\(109\) −0.246732 −0.0236326 −0.0118163 0.999930i \(-0.503761\pi\)
−0.0118163 + 0.999930i \(0.503761\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 38.9515 3.68057
\(113\) −14.4273 −1.35721 −0.678603 0.734505i \(-0.737414\pi\)
−0.678603 + 0.734505i \(0.737414\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.64553 −0.431327
\(117\) 0 0
\(118\) −22.0123 −2.02639
\(119\) −6.57909 −0.603104
\(120\) 0 0
\(121\) −1.29657 −0.117870
\(122\) 16.0207 1.45045
\(123\) 0 0
\(124\) −10.3697 −0.931223
\(125\) 0 0
\(126\) 0 0
\(127\) 21.1707 1.87860 0.939299 0.343101i \(-0.111477\pi\)
0.939299 + 0.343101i \(0.111477\pi\)
\(128\) 6.82785 0.603503
\(129\) 0 0
\(130\) 0 0
\(131\) 8.18994 0.715559 0.357779 0.933806i \(-0.383534\pi\)
0.357779 + 0.933806i \(0.383534\pi\)
\(132\) 0 0
\(133\) −17.6912 −1.53402
\(134\) 35.8583 3.09769
\(135\) 0 0
\(136\) 9.54923 0.818840
\(137\) 17.4054 1.48704 0.743520 0.668714i \(-0.233155\pi\)
0.743520 + 0.668714i \(0.233155\pi\)
\(138\) 0 0
\(139\) −6.81900 −0.578380 −0.289190 0.957272i \(-0.593386\pi\)
−0.289190 + 0.957272i \(0.593386\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.1767 1.27360
\(143\) −15.8226 −1.32316
\(144\) 0 0
\(145\) 0 0
\(146\) −18.2617 −1.51135
\(147\) 0 0
\(148\) 26.9017 2.21131
\(149\) −23.7335 −1.94432 −0.972162 0.234311i \(-0.924717\pi\)
−0.972162 + 0.234311i \(0.924717\pi\)
\(150\) 0 0
\(151\) −17.3095 −1.40863 −0.704315 0.709888i \(-0.748746\pi\)
−0.704315 + 0.709888i \(0.748746\pi\)
\(152\) 25.6779 2.08275
\(153\) 0 0
\(154\) −37.7314 −3.04048
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0537 −0.802375 −0.401188 0.915996i \(-0.631402\pi\)
−0.401188 + 0.915996i \(0.631402\pi\)
\(158\) −18.8617 −1.50056
\(159\) 0 0
\(160\) 0 0
\(161\) −26.8521 −2.11624
\(162\) 0 0
\(163\) −12.2602 −0.960291 −0.480145 0.877189i \(-0.659416\pi\)
−0.480145 + 0.877189i \(0.659416\pi\)
\(164\) 49.6921 3.88030
\(165\) 0 0
\(166\) 35.2087 2.73273
\(167\) −13.5990 −1.05232 −0.526160 0.850386i \(-0.676369\pi\)
−0.526160 + 0.850386i \(0.676369\pi\)
\(168\) 0 0
\(169\) 12.8008 0.984677
\(170\) 0 0
\(171\) 0 0
\(172\) −41.3233 −3.15087
\(173\) 8.35226 0.635011 0.317505 0.948256i \(-0.397155\pi\)
0.317505 + 0.948256i \(0.397155\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 25.8233 1.94650
\(177\) 0 0
\(178\) −20.5720 −1.54194
\(179\) −26.5865 −1.98717 −0.993583 0.113109i \(-0.963919\pi\)
−0.993583 + 0.113109i \(0.963919\pi\)
\(180\) 0 0
\(181\) 3.72547 0.276912 0.138456 0.990369i \(-0.455786\pi\)
0.138456 + 0.990369i \(0.455786\pi\)
\(182\) 61.5257 4.56059
\(183\) 0 0
\(184\) 38.9745 2.87324
\(185\) 0 0
\(186\) 0 0
\(187\) −4.36167 −0.318957
\(188\) −16.8533 −1.22915
\(189\) 0 0
\(190\) 0 0
\(191\) −6.76433 −0.489450 −0.244725 0.969593i \(-0.578698\pi\)
−0.244725 + 0.969593i \(0.578698\pi\)
\(192\) 0 0
\(193\) −0.140715 −0.0101289 −0.00506444 0.999987i \(-0.501612\pi\)
−0.00506444 + 0.999987i \(0.501612\pi\)
\(194\) 35.6633 2.56047
\(195\) 0 0
\(196\) 70.0431 5.00308
\(197\) −1.48273 −0.105640 −0.0528200 0.998604i \(-0.516821\pi\)
−0.0528200 + 0.998604i \(0.516821\pi\)
\(198\) 0 0
\(199\) 25.8626 1.83335 0.916677 0.399628i \(-0.130861\pi\)
0.916677 + 0.399628i \(0.130861\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 32.8582 2.31189
\(203\) −4.69867 −0.329782
\(204\) 0 0
\(205\) 0 0
\(206\) −21.1492 −1.47353
\(207\) 0 0
\(208\) −42.1081 −2.91967
\(209\) −11.7285 −0.811281
\(210\) 0 0
\(211\) 23.3428 1.60698 0.803491 0.595317i \(-0.202974\pi\)
0.803491 + 0.595317i \(0.202974\pi\)
\(212\) 4.40517 0.302548
\(213\) 0 0
\(214\) 8.47799 0.579543
\(215\) 0 0
\(216\) 0 0
\(217\) −10.4883 −0.711991
\(218\) 0.636049 0.0430787
\(219\) 0 0
\(220\) 0 0
\(221\) 7.11225 0.478422
\(222\) 0 0
\(223\) −22.4256 −1.50173 −0.750865 0.660455i \(-0.770363\pi\)
−0.750865 + 0.660455i \(0.770363\pi\)
\(224\) −36.3239 −2.42699
\(225\) 0 0
\(226\) 37.1920 2.47398
\(227\) −4.07327 −0.270353 −0.135176 0.990822i \(-0.543160\pi\)
−0.135176 + 0.990822i \(0.543160\pi\)
\(228\) 0 0
\(229\) −15.7170 −1.03861 −0.519303 0.854590i \(-0.673808\pi\)
−0.519303 + 0.854590i \(0.673808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.81989 0.447748
\(233\) 16.4568 1.07812 0.539059 0.842268i \(-0.318780\pi\)
0.539059 + 0.842268i \(0.318780\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 39.6675 2.58214
\(237\) 0 0
\(238\) 16.9602 1.09937
\(239\) −3.18702 −0.206151 −0.103075 0.994674i \(-0.532868\pi\)
−0.103075 + 0.994674i \(0.532868\pi\)
\(240\) 0 0
\(241\) 11.1482 0.718118 0.359059 0.933315i \(-0.383098\pi\)
0.359059 + 0.933315i \(0.383098\pi\)
\(242\) 3.34243 0.214859
\(243\) 0 0
\(244\) −28.8704 −1.84824
\(245\) 0 0
\(246\) 0 0
\(247\) 19.1249 1.21689
\(248\) 15.2232 0.966676
\(249\) 0 0
\(250\) 0 0
\(251\) 3.23388 0.204121 0.102060 0.994778i \(-0.467457\pi\)
0.102060 + 0.994778i \(0.467457\pi\)
\(252\) 0 0
\(253\) −17.8019 −1.11919
\(254\) −54.5758 −3.42439
\(255\) 0 0
\(256\) −24.2996 −1.51872
\(257\) 3.96761 0.247493 0.123746 0.992314i \(-0.460509\pi\)
0.123746 + 0.992314i \(0.460509\pi\)
\(258\) 0 0
\(259\) 27.2095 1.69071
\(260\) 0 0
\(261\) 0 0
\(262\) −21.1128 −1.30435
\(263\) 16.6865 1.02893 0.514467 0.857510i \(-0.327990\pi\)
0.514467 + 0.857510i \(0.327990\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 45.6060 2.79628
\(267\) 0 0
\(268\) −64.6191 −3.94724
\(269\) −7.94329 −0.484311 −0.242155 0.970237i \(-0.577854\pi\)
−0.242155 + 0.970237i \(0.577854\pi\)
\(270\) 0 0
\(271\) −17.3087 −1.05143 −0.525714 0.850662i \(-0.676202\pi\)
−0.525714 + 0.850662i \(0.676202\pi\)
\(272\) −11.6075 −0.703810
\(273\) 0 0
\(274\) −44.8691 −2.71064
\(275\) 0 0
\(276\) 0 0
\(277\) −10.2791 −0.617612 −0.308806 0.951125i \(-0.599929\pi\)
−0.308806 + 0.951125i \(0.599929\pi\)
\(278\) 17.5786 1.05430
\(279\) 0 0
\(280\) 0 0
\(281\) −4.57869 −0.273142 −0.136571 0.990630i \(-0.543608\pi\)
−0.136571 + 0.990630i \(0.543608\pi\)
\(282\) 0 0
\(283\) −19.2431 −1.14388 −0.571941 0.820294i \(-0.693810\pi\)
−0.571941 + 0.820294i \(0.693810\pi\)
\(284\) −27.3495 −1.62289
\(285\) 0 0
\(286\) 40.7891 2.41191
\(287\) 50.2606 2.96679
\(288\) 0 0
\(289\) −15.0394 −0.884673
\(290\) 0 0
\(291\) 0 0
\(292\) 32.9089 1.92584
\(293\) 14.6045 0.853203 0.426602 0.904440i \(-0.359711\pi\)
0.426602 + 0.904440i \(0.359711\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −39.4932 −2.29550
\(297\) 0 0
\(298\) 61.1824 3.54420
\(299\) 29.0282 1.67874
\(300\) 0 0
\(301\) −41.7960 −2.40908
\(302\) 44.6221 2.56771
\(303\) 0 0
\(304\) −31.2127 −1.79017
\(305\) 0 0
\(306\) 0 0
\(307\) −12.3324 −0.703850 −0.351925 0.936028i \(-0.614473\pi\)
−0.351925 + 0.936028i \(0.614473\pi\)
\(308\) 67.9944 3.87434
\(309\) 0 0
\(310\) 0 0
\(311\) −12.3084 −0.697943 −0.348971 0.937133i \(-0.613469\pi\)
−0.348971 + 0.937133i \(0.613469\pi\)
\(312\) 0 0
\(313\) −16.7306 −0.945672 −0.472836 0.881150i \(-0.656770\pi\)
−0.472836 + 0.881150i \(0.656770\pi\)
\(314\) 25.9174 1.46261
\(315\) 0 0
\(316\) 33.9901 1.91209
\(317\) −10.9429 −0.614617 −0.307308 0.951610i \(-0.599428\pi\)
−0.307308 + 0.951610i \(0.599428\pi\)
\(318\) 0 0
\(319\) −3.11503 −0.174408
\(320\) 0 0
\(321\) 0 0
\(322\) 69.2219 3.85759
\(323\) 5.27196 0.293340
\(324\) 0 0
\(325\) 0 0
\(326\) 31.6054 1.75046
\(327\) 0 0
\(328\) −72.9508 −4.02803
\(329\) −17.0461 −0.939781
\(330\) 0 0
\(331\) −19.1431 −1.05220 −0.526099 0.850423i \(-0.676346\pi\)
−0.526099 + 0.850423i \(0.676346\pi\)
\(332\) −63.4484 −3.48218
\(333\) 0 0
\(334\) 35.0567 1.91822
\(335\) 0 0
\(336\) 0 0
\(337\) 17.0294 0.927650 0.463825 0.885927i \(-0.346477\pi\)
0.463825 + 0.885927i \(0.346477\pi\)
\(338\) −32.9991 −1.79491
\(339\) 0 0
\(340\) 0 0
\(341\) −6.95331 −0.376543
\(342\) 0 0
\(343\) 37.9537 2.04931
\(344\) 60.6648 3.27083
\(345\) 0 0
\(346\) −21.5312 −1.15753
\(347\) −17.2564 −0.926373 −0.463186 0.886261i \(-0.653294\pi\)
−0.463186 + 0.886261i \(0.653294\pi\)
\(348\) 0 0
\(349\) 13.3969 0.717118 0.358559 0.933507i \(-0.383268\pi\)
0.358559 + 0.933507i \(0.383268\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −24.0813 −1.28354
\(353\) 2.04594 0.108894 0.0544472 0.998517i \(-0.482660\pi\)
0.0544472 + 0.998517i \(0.482660\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 37.0721 1.96482
\(357\) 0 0
\(358\) 68.5370 3.62229
\(359\) 25.2336 1.33178 0.665889 0.746051i \(-0.268053\pi\)
0.665889 + 0.746051i \(0.268053\pi\)
\(360\) 0 0
\(361\) −4.82368 −0.253878
\(362\) −9.60387 −0.504768
\(363\) 0 0
\(364\) −110.873 −5.81134
\(365\) 0 0
\(366\) 0 0
\(367\) 9.13285 0.476731 0.238366 0.971176i \(-0.423388\pi\)
0.238366 + 0.971176i \(0.423388\pi\)
\(368\) −47.3753 −2.46961
\(369\) 0 0
\(370\) 0 0
\(371\) 4.45557 0.231321
\(372\) 0 0
\(373\) −5.41148 −0.280196 −0.140098 0.990138i \(-0.544742\pi\)
−0.140098 + 0.990138i \(0.544742\pi\)
\(374\) 11.2439 0.581410
\(375\) 0 0
\(376\) 24.7416 1.27595
\(377\) 5.07945 0.261605
\(378\) 0 0
\(379\) 19.8284 1.01852 0.509258 0.860614i \(-0.329920\pi\)
0.509258 + 0.860614i \(0.329920\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 17.4377 0.892191
\(383\) 14.8919 0.760941 0.380470 0.924793i \(-0.375762\pi\)
0.380470 + 0.924793i \(0.375762\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.362748 0.0184634
\(387\) 0 0
\(388\) −64.2675 −3.26269
\(389\) 15.7680 0.799470 0.399735 0.916631i \(-0.369102\pi\)
0.399735 + 0.916631i \(0.369102\pi\)
\(390\) 0 0
\(391\) 8.00192 0.404674
\(392\) −102.827 −5.19356
\(393\) 0 0
\(394\) 3.82232 0.192566
\(395\) 0 0
\(396\) 0 0
\(397\) 26.5318 1.33159 0.665796 0.746134i \(-0.268092\pi\)
0.665796 + 0.746134i \(0.268092\pi\)
\(398\) −66.6711 −3.34192
\(399\) 0 0
\(400\) 0 0
\(401\) −9.71205 −0.484997 −0.242498 0.970152i \(-0.577967\pi\)
−0.242498 + 0.970152i \(0.577967\pi\)
\(402\) 0 0
\(403\) 11.3382 0.564798
\(404\) −59.2126 −2.94593
\(405\) 0 0
\(406\) 12.1127 0.601142
\(407\) 18.0388 0.894150
\(408\) 0 0
\(409\) −17.4468 −0.862691 −0.431346 0.902187i \(-0.641961\pi\)
−0.431346 + 0.902187i \(0.641961\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 38.1122 1.87765
\(413\) 40.1213 1.97424
\(414\) 0 0
\(415\) 0 0
\(416\) 39.2675 1.92525
\(417\) 0 0
\(418\) 30.2349 1.47884
\(419\) −26.0588 −1.27306 −0.636529 0.771253i \(-0.719630\pi\)
−0.636529 + 0.771253i \(0.719630\pi\)
\(420\) 0 0
\(421\) 2.66465 0.129867 0.0649337 0.997890i \(-0.479316\pi\)
0.0649337 + 0.997890i \(0.479316\pi\)
\(422\) −60.1751 −2.92928
\(423\) 0 0
\(424\) −6.46704 −0.314067
\(425\) 0 0
\(426\) 0 0
\(427\) −29.2007 −1.41312
\(428\) −15.2779 −0.738485
\(429\) 0 0
\(430\) 0 0
\(431\) 0.150532 0.00725086 0.00362543 0.999993i \(-0.498846\pi\)
0.00362543 + 0.999993i \(0.498846\pi\)
\(432\) 0 0
\(433\) 33.2461 1.59770 0.798852 0.601527i \(-0.205441\pi\)
0.798852 + 0.601527i \(0.205441\pi\)
\(434\) 27.0377 1.29785
\(435\) 0 0
\(436\) −1.14620 −0.0548931
\(437\) 21.5172 1.02931
\(438\) 0 0
\(439\) 18.1320 0.865391 0.432695 0.901540i \(-0.357563\pi\)
0.432695 + 0.901540i \(0.357563\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −18.3346 −0.872089
\(443\) 34.8312 1.65488 0.827440 0.561554i \(-0.189796\pi\)
0.827440 + 0.561554i \(0.189796\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 57.8108 2.73742
\(447\) 0 0
\(448\) 15.7361 0.743460
\(449\) −36.1516 −1.70610 −0.853049 0.521830i \(-0.825249\pi\)
−0.853049 + 0.521830i \(0.825249\pi\)
\(450\) 0 0
\(451\) 33.3207 1.56901
\(452\) −67.0225 −3.15247
\(453\) 0 0
\(454\) 10.5005 0.492811
\(455\) 0 0
\(456\) 0 0
\(457\) 8.93775 0.418090 0.209045 0.977906i \(-0.432964\pi\)
0.209045 + 0.977906i \(0.432964\pi\)
\(458\) 40.5166 1.89322
\(459\) 0 0
\(460\) 0 0
\(461\) −34.6733 −1.61490 −0.807449 0.589937i \(-0.799152\pi\)
−0.807449 + 0.589937i \(0.799152\pi\)
\(462\) 0 0
\(463\) −30.9412 −1.43796 −0.718981 0.695030i \(-0.755391\pi\)
−0.718981 + 0.695030i \(0.755391\pi\)
\(464\) −8.28989 −0.384849
\(465\) 0 0
\(466\) −42.4237 −1.96524
\(467\) 5.40920 0.250308 0.125154 0.992137i \(-0.460057\pi\)
0.125154 + 0.992137i \(0.460057\pi\)
\(468\) 0 0
\(469\) −65.3583 −3.01797
\(470\) 0 0
\(471\) 0 0
\(472\) −58.2341 −2.68044
\(473\) −27.7091 −1.27406
\(474\) 0 0
\(475\) 0 0
\(476\) −30.5634 −1.40087
\(477\) 0 0
\(478\) 8.21578 0.375781
\(479\) −13.4270 −0.613494 −0.306747 0.951791i \(-0.599241\pi\)
−0.306747 + 0.951791i \(0.599241\pi\)
\(480\) 0 0
\(481\) −29.4145 −1.34119
\(482\) −28.7389 −1.30902
\(483\) 0 0
\(484\) −6.02327 −0.273785
\(485\) 0 0
\(486\) 0 0
\(487\) 24.8746 1.12718 0.563589 0.826055i \(-0.309420\pi\)
0.563589 + 0.826055i \(0.309420\pi\)
\(488\) 42.3834 1.91861
\(489\) 0 0
\(490\) 0 0
\(491\) −20.6717 −0.932902 −0.466451 0.884547i \(-0.654468\pi\)
−0.466451 + 0.884547i \(0.654468\pi\)
\(492\) 0 0
\(493\) 1.40020 0.0630619
\(494\) −49.3018 −2.21819
\(495\) 0 0
\(496\) −18.5045 −0.830878
\(497\) −27.6623 −1.24083
\(498\) 0 0
\(499\) 21.2182 0.949856 0.474928 0.880025i \(-0.342474\pi\)
0.474928 + 0.880025i \(0.342474\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −8.33660 −0.372081
\(503\) −35.9602 −1.60339 −0.801693 0.597736i \(-0.796067\pi\)
−0.801693 + 0.597736i \(0.796067\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 45.8913 2.04012
\(507\) 0 0
\(508\) 98.3492 4.36354
\(509\) −21.3861 −0.947924 −0.473962 0.880545i \(-0.657177\pi\)
−0.473962 + 0.880545i \(0.657177\pi\)
\(510\) 0 0
\(511\) 33.2853 1.47246
\(512\) 48.9860 2.16489
\(513\) 0 0
\(514\) −10.2281 −0.451141
\(515\) 0 0
\(516\) 0 0
\(517\) −11.3009 −0.497012
\(518\) −70.1431 −3.08191
\(519\) 0 0
\(520\) 0 0
\(521\) 6.56775 0.287738 0.143869 0.989597i \(-0.454046\pi\)
0.143869 + 0.989597i \(0.454046\pi\)
\(522\) 0 0
\(523\) 21.1811 0.926184 0.463092 0.886310i \(-0.346740\pi\)
0.463092 + 0.886310i \(0.346740\pi\)
\(524\) 38.0466 1.66207
\(525\) 0 0
\(526\) −43.0160 −1.87559
\(527\) 3.12550 0.136149
\(528\) 0 0
\(529\) 9.65929 0.419969
\(530\) 0 0
\(531\) 0 0
\(532\) −82.1850 −3.56317
\(533\) −54.3336 −2.35345
\(534\) 0 0
\(535\) 0 0
\(536\) 94.8643 4.09752
\(537\) 0 0
\(538\) 20.4769 0.882824
\(539\) 46.9670 2.02301
\(540\) 0 0
\(541\) −9.98542 −0.429307 −0.214653 0.976690i \(-0.568862\pi\)
−0.214653 + 0.976690i \(0.568862\pi\)
\(542\) 44.6199 1.91659
\(543\) 0 0
\(544\) 10.8245 0.464097
\(545\) 0 0
\(546\) 0 0
\(547\) 1.01143 0.0432457 0.0216228 0.999766i \(-0.493117\pi\)
0.0216228 + 0.999766i \(0.493117\pi\)
\(548\) 80.8571 3.45404
\(549\) 0 0
\(550\) 0 0
\(551\) 3.76514 0.160401
\(552\) 0 0
\(553\) 34.3789 1.46194
\(554\) 26.4985 1.12581
\(555\) 0 0
\(556\) −31.6779 −1.34344
\(557\) 30.4618 1.29071 0.645353 0.763885i \(-0.276710\pi\)
0.645353 + 0.763885i \(0.276710\pi\)
\(558\) 0 0
\(559\) 45.1831 1.91104
\(560\) 0 0
\(561\) 0 0
\(562\) 11.8034 0.497895
\(563\) 2.49954 0.105343 0.0526716 0.998612i \(-0.483226\pi\)
0.0526716 + 0.998612i \(0.483226\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 49.6066 2.08512
\(567\) 0 0
\(568\) 40.1505 1.68468
\(569\) 0.611166 0.0256214 0.0128107 0.999918i \(-0.495922\pi\)
0.0128107 + 0.999918i \(0.495922\pi\)
\(570\) 0 0
\(571\) −0.527376 −0.0220700 −0.0110350 0.999939i \(-0.503513\pi\)
−0.0110350 + 0.999939i \(0.503513\pi\)
\(572\) −73.5046 −3.07338
\(573\) 0 0
\(574\) −129.566 −5.40800
\(575\) 0 0
\(576\) 0 0
\(577\) −24.3938 −1.01553 −0.507764 0.861496i \(-0.669528\pi\)
−0.507764 + 0.861496i \(0.669528\pi\)
\(578\) 38.7700 1.61262
\(579\) 0 0
\(580\) 0 0
\(581\) −64.1742 −2.66239
\(582\) 0 0
\(583\) 2.95386 0.122336
\(584\) −48.3120 −1.99917
\(585\) 0 0
\(586\) −37.6488 −1.55526
\(587\) 28.4529 1.17438 0.587188 0.809450i \(-0.300235\pi\)
0.587188 + 0.809450i \(0.300235\pi\)
\(588\) 0 0
\(589\) 8.40448 0.346301
\(590\) 0 0
\(591\) 0 0
\(592\) 48.0058 1.97303
\(593\) −2.96753 −0.121862 −0.0609310 0.998142i \(-0.519407\pi\)
−0.0609310 + 0.998142i \(0.519407\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −110.255 −4.51621
\(597\) 0 0
\(598\) −74.8316 −3.06009
\(599\) −21.9546 −0.897040 −0.448520 0.893773i \(-0.648049\pi\)
−0.448520 + 0.893773i \(0.648049\pi\)
\(600\) 0 0
\(601\) −21.4446 −0.874745 −0.437372 0.899280i \(-0.644091\pi\)
−0.437372 + 0.899280i \(0.644091\pi\)
\(602\) 107.746 4.39138
\(603\) 0 0
\(604\) −80.4120 −3.27192
\(605\) 0 0
\(606\) 0 0
\(607\) 48.0114 1.94872 0.974361 0.224988i \(-0.0722344\pi\)
0.974361 + 0.224988i \(0.0722344\pi\)
\(608\) 29.1071 1.18045
\(609\) 0 0
\(610\) 0 0
\(611\) 18.4275 0.745496
\(612\) 0 0
\(613\) 37.9093 1.53114 0.765571 0.643351i \(-0.222456\pi\)
0.765571 + 0.643351i \(0.222456\pi\)
\(614\) 31.7917 1.28301
\(615\) 0 0
\(616\) −99.8195 −4.02184
\(617\) −8.29772 −0.334054 −0.167027 0.985952i \(-0.553417\pi\)
−0.167027 + 0.985952i \(0.553417\pi\)
\(618\) 0 0
\(619\) −15.6135 −0.627559 −0.313779 0.949496i \(-0.601595\pi\)
−0.313779 + 0.949496i \(0.601595\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 31.7296 1.27224
\(623\) 37.4962 1.50225
\(624\) 0 0
\(625\) 0 0
\(626\) 43.1298 1.72381
\(627\) 0 0
\(628\) −46.7049 −1.86373
\(629\) −8.10841 −0.323303
\(630\) 0 0
\(631\) 10.7848 0.429336 0.214668 0.976687i \(-0.431133\pi\)
0.214668 + 0.976687i \(0.431133\pi\)
\(632\) −49.8993 −1.98489
\(633\) 0 0
\(634\) 28.2097 1.12035
\(635\) 0 0
\(636\) 0 0
\(637\) −76.5856 −3.03443
\(638\) 8.03022 0.317919
\(639\) 0 0
\(640\) 0 0
\(641\) −16.1226 −0.636805 −0.318403 0.947956i \(-0.603146\pi\)
−0.318403 + 0.947956i \(0.603146\pi\)
\(642\) 0 0
\(643\) −25.0313 −0.987137 −0.493569 0.869707i \(-0.664308\pi\)
−0.493569 + 0.869707i \(0.664308\pi\)
\(644\) −124.742 −4.91554
\(645\) 0 0
\(646\) −13.5906 −0.534713
\(647\) −34.9798 −1.37520 −0.687599 0.726091i \(-0.741335\pi\)
−0.687599 + 0.726091i \(0.741335\pi\)
\(648\) 0 0
\(649\) 26.5988 1.04409
\(650\) 0 0
\(651\) 0 0
\(652\) −56.9550 −2.23053
\(653\) −22.0250 −0.861904 −0.430952 0.902375i \(-0.641822\pi\)
−0.430952 + 0.902375i \(0.641822\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 88.6750 3.46218
\(657\) 0 0
\(658\) 43.9430 1.71308
\(659\) 32.9279 1.28269 0.641345 0.767253i \(-0.278377\pi\)
0.641345 + 0.767253i \(0.278377\pi\)
\(660\) 0 0
\(661\) 9.78801 0.380710 0.190355 0.981715i \(-0.439036\pi\)
0.190355 + 0.981715i \(0.439036\pi\)
\(662\) 49.3488 1.91799
\(663\) 0 0
\(664\) 93.1457 3.61475
\(665\) 0 0
\(666\) 0 0
\(667\) 5.71483 0.221279
\(668\) −63.1745 −2.44429
\(669\) 0 0
\(670\) 0 0
\(671\) −19.3589 −0.747342
\(672\) 0 0
\(673\) −25.3634 −0.977689 −0.488844 0.872371i \(-0.662581\pi\)
−0.488844 + 0.872371i \(0.662581\pi\)
\(674\) −43.8999 −1.69096
\(675\) 0 0
\(676\) 59.4665 2.28717
\(677\) 19.7896 0.760578 0.380289 0.924868i \(-0.375825\pi\)
0.380289 + 0.924868i \(0.375825\pi\)
\(678\) 0 0
\(679\) −65.0027 −2.49458
\(680\) 0 0
\(681\) 0 0
\(682\) 17.9249 0.686380
\(683\) −44.9656 −1.72056 −0.860281 0.509820i \(-0.829712\pi\)
−0.860281 + 0.509820i \(0.829712\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −97.8406 −3.73557
\(687\) 0 0
\(688\) −73.7409 −2.81134
\(689\) −4.81664 −0.183499
\(690\) 0 0
\(691\) −30.7758 −1.17077 −0.585384 0.810756i \(-0.699056\pi\)
−0.585384 + 0.810756i \(0.699056\pi\)
\(692\) 38.8007 1.47498
\(693\) 0 0
\(694\) 44.4852 1.68863
\(695\) 0 0
\(696\) 0 0
\(697\) −14.9776 −0.567318
\(698\) −34.5357 −1.30720
\(699\) 0 0
\(700\) 0 0
\(701\) 23.2459 0.877984 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(702\) 0 0
\(703\) −21.8035 −0.822335
\(704\) 10.4324 0.393186
\(705\) 0 0
\(706\) −5.27422 −0.198498
\(707\) −59.8899 −2.25239
\(708\) 0 0
\(709\) 7.86251 0.295283 0.147641 0.989041i \(-0.452832\pi\)
0.147641 + 0.989041i \(0.452832\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −54.4239 −2.03962
\(713\) 12.7565 0.477736
\(714\) 0 0
\(715\) 0 0
\(716\) −123.508 −4.61572
\(717\) 0 0
\(718\) −65.0495 −2.42762
\(719\) −35.0267 −1.30628 −0.653138 0.757239i \(-0.726548\pi\)
−0.653138 + 0.757239i \(0.726548\pi\)
\(720\) 0 0
\(721\) 38.5482 1.43561
\(722\) 12.4349 0.462781
\(723\) 0 0
\(724\) 17.3068 0.643202
\(725\) 0 0
\(726\) 0 0
\(727\) 22.4518 0.832691 0.416346 0.909206i \(-0.363311\pi\)
0.416346 + 0.909206i \(0.363311\pi\)
\(728\) 162.768 6.03259
\(729\) 0 0
\(730\) 0 0
\(731\) 12.4552 0.460671
\(732\) 0 0
\(733\) −15.0028 −0.554140 −0.277070 0.960850i \(-0.589363\pi\)
−0.277070 + 0.960850i \(0.589363\pi\)
\(734\) −23.5435 −0.869007
\(735\) 0 0
\(736\) 44.1795 1.62848
\(737\) −43.3299 −1.59608
\(738\) 0 0
\(739\) −33.4801 −1.23159 −0.615793 0.787908i \(-0.711164\pi\)
−0.615793 + 0.787908i \(0.711164\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.4860 −0.421663
\(743\) −6.75081 −0.247663 −0.123832 0.992303i \(-0.539518\pi\)
−0.123832 + 0.992303i \(0.539518\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.9502 0.510754
\(747\) 0 0
\(748\) −20.2623 −0.740863
\(749\) −15.4527 −0.564628
\(750\) 0 0
\(751\) 28.0482 1.02349 0.511747 0.859136i \(-0.328999\pi\)
0.511747 + 0.859136i \(0.328999\pi\)
\(752\) −30.0745 −1.09670
\(753\) 0 0
\(754\) −13.0943 −0.476865
\(755\) 0 0
\(756\) 0 0
\(757\) 33.1698 1.20558 0.602789 0.797901i \(-0.294056\pi\)
0.602789 + 0.797901i \(0.294056\pi\)
\(758\) −51.1155 −1.85660
\(759\) 0 0
\(760\) 0 0
\(761\) 12.0982 0.438560 0.219280 0.975662i \(-0.429629\pi\)
0.219280 + 0.975662i \(0.429629\pi\)
\(762\) 0 0
\(763\) −1.15931 −0.0419700
\(764\) −31.4239 −1.13688
\(765\) 0 0
\(766\) −38.3897 −1.38708
\(767\) −43.3727 −1.56610
\(768\) 0 0
\(769\) −42.5811 −1.53551 −0.767756 0.640742i \(-0.778627\pi\)
−0.767756 + 0.640742i \(0.778627\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.653695 −0.0235270
\(773\) −6.87342 −0.247220 −0.123610 0.992331i \(-0.539447\pi\)
−0.123610 + 0.992331i \(0.539447\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 94.3483 3.38691
\(777\) 0 0
\(778\) −40.6482 −1.45731
\(779\) −40.2748 −1.44300
\(780\) 0 0
\(781\) −18.3390 −0.656222
\(782\) −20.6281 −0.737659
\(783\) 0 0
\(784\) 124.991 4.46397
\(785\) 0 0
\(786\) 0 0
\(787\) −4.51935 −0.161097 −0.0805487 0.996751i \(-0.525667\pi\)
−0.0805487 + 0.996751i \(0.525667\pi\)
\(788\) −6.88806 −0.245377
\(789\) 0 0
\(790\) 0 0
\(791\) −67.7892 −2.41031
\(792\) 0 0
\(793\) 31.5671 1.12098
\(794\) −68.3961 −2.42729
\(795\) 0 0
\(796\) 120.146 4.25845
\(797\) −13.9670 −0.494736 −0.247368 0.968922i \(-0.579566\pi\)
−0.247368 + 0.968922i \(0.579566\pi\)
\(798\) 0 0
\(799\) 5.07972 0.179708
\(800\) 0 0
\(801\) 0 0
\(802\) 25.0366 0.884074
\(803\) 22.0668 0.778722
\(804\) 0 0
\(805\) 0 0
\(806\) −29.2288 −1.02954
\(807\) 0 0
\(808\) 86.9273 3.05809
\(809\) −8.18148 −0.287646 −0.143823 0.989603i \(-0.545940\pi\)
−0.143823 + 0.989603i \(0.545940\pi\)
\(810\) 0 0
\(811\) −44.4011 −1.55913 −0.779566 0.626320i \(-0.784560\pi\)
−0.779566 + 0.626320i \(0.784560\pi\)
\(812\) −21.8278 −0.766007
\(813\) 0 0
\(814\) −46.5021 −1.62990
\(815\) 0 0
\(816\) 0 0
\(817\) 33.4920 1.17174
\(818\) 44.9761 1.57255
\(819\) 0 0
\(820\) 0 0
\(821\) −8.73119 −0.304721 −0.152360 0.988325i \(-0.548687\pi\)
−0.152360 + 0.988325i \(0.548687\pi\)
\(822\) 0 0
\(823\) −39.1844 −1.36588 −0.682942 0.730473i \(-0.739300\pi\)
−0.682942 + 0.730473i \(0.739300\pi\)
\(824\) −55.9508 −1.94914
\(825\) 0 0
\(826\) −103.428 −3.59873
\(827\) −6.33428 −0.220264 −0.110132 0.993917i \(-0.535127\pi\)
−0.110132 + 0.993917i \(0.535127\pi\)
\(828\) 0 0
\(829\) 1.36706 0.0474798 0.0237399 0.999718i \(-0.492443\pi\)
0.0237399 + 0.999718i \(0.492443\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −17.0113 −0.589762
\(833\) −21.1116 −0.731473
\(834\) 0 0
\(835\) 0 0
\(836\) −54.4853 −1.88441
\(837\) 0 0
\(838\) 67.1769 2.32059
\(839\) 55.7902 1.92609 0.963046 0.269339i \(-0.0868051\pi\)
0.963046 + 0.269339i \(0.0868051\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −6.86919 −0.236728
\(843\) 0 0
\(844\) 108.439 3.73264
\(845\) 0 0
\(846\) 0 0
\(847\) −6.09218 −0.209330
\(848\) 7.86098 0.269947
\(849\) 0 0
\(850\) 0 0
\(851\) −33.0939 −1.13444
\(852\) 0 0
\(853\) 6.42812 0.220095 0.110047 0.993926i \(-0.464900\pi\)
0.110047 + 0.993926i \(0.464900\pi\)
\(854\) 75.2762 2.57590
\(855\) 0 0
\(856\) 22.4288 0.766600
\(857\) −35.0274 −1.19651 −0.598257 0.801304i \(-0.704140\pi\)
−0.598257 + 0.801304i \(0.704140\pi\)
\(858\) 0 0
\(859\) 18.4412 0.629207 0.314603 0.949223i \(-0.398128\pi\)
0.314603 + 0.949223i \(0.398128\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −0.388055 −0.0132172
\(863\) −7.23911 −0.246422 −0.123211 0.992380i \(-0.539319\pi\)
−0.123211 + 0.992380i \(0.539319\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −85.7048 −2.91237
\(867\) 0 0
\(868\) −48.7237 −1.65379
\(869\) 22.7918 0.773160
\(870\) 0 0
\(871\) 70.6549 2.39405
\(872\) 1.68269 0.0569830
\(873\) 0 0
\(874\) −55.4689 −1.87626
\(875\) 0 0
\(876\) 0 0
\(877\) 24.6324 0.831777 0.415889 0.909416i \(-0.363471\pi\)
0.415889 + 0.909416i \(0.363471\pi\)
\(878\) −46.7422 −1.57747
\(879\) 0 0
\(880\) 0 0
\(881\) −26.2145 −0.883190 −0.441595 0.897214i \(-0.645587\pi\)
−0.441595 + 0.897214i \(0.645587\pi\)
\(882\) 0 0
\(883\) 10.7656 0.362290 0.181145 0.983456i \(-0.442020\pi\)
0.181145 + 0.983456i \(0.442020\pi\)
\(884\) 33.0402 1.11126
\(885\) 0 0
\(886\) −89.7911 −3.01659
\(887\) 6.27796 0.210793 0.105397 0.994430i \(-0.466389\pi\)
0.105397 + 0.994430i \(0.466389\pi\)
\(888\) 0 0
\(889\) 99.4743 3.33626
\(890\) 0 0
\(891\) 0 0
\(892\) −104.179 −3.48817
\(893\) 13.6594 0.457094
\(894\) 0 0
\(895\) 0 0
\(896\) 32.0818 1.07178
\(897\) 0 0
\(898\) 93.1949 3.10995
\(899\) 2.23218 0.0744474
\(900\) 0 0
\(901\) −1.32776 −0.0442340
\(902\) −85.8973 −2.86007
\(903\) 0 0
\(904\) 98.3927 3.27249
\(905\) 0 0
\(906\) 0 0
\(907\) −21.7954 −0.723704 −0.361852 0.932235i \(-0.617855\pi\)
−0.361852 + 0.932235i \(0.617855\pi\)
\(908\) −18.9225 −0.627966
\(909\) 0 0
\(910\) 0 0
\(911\) 5.56422 0.184351 0.0921753 0.995743i \(-0.470618\pi\)
0.0921753 + 0.995743i \(0.470618\pi\)
\(912\) 0 0
\(913\) −42.5449 −1.40803
\(914\) −23.0406 −0.762114
\(915\) 0 0
\(916\) −73.0136 −2.41244
\(917\) 38.4819 1.27078
\(918\) 0 0
\(919\) −7.76705 −0.256211 −0.128106 0.991761i \(-0.540890\pi\)
−0.128106 + 0.991761i \(0.540890\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 89.3841 2.94371
\(923\) 29.9041 0.984304
\(924\) 0 0
\(925\) 0 0
\(926\) 79.7632 2.62118
\(927\) 0 0
\(928\) 7.73067 0.253772
\(929\) 11.3588 0.372672 0.186336 0.982486i \(-0.440339\pi\)
0.186336 + 0.982486i \(0.440339\pi\)
\(930\) 0 0
\(931\) −56.7691 −1.86053
\(932\) 76.4503 2.50421
\(933\) 0 0
\(934\) −13.9443 −0.456273
\(935\) 0 0
\(936\) 0 0
\(937\) −32.7813 −1.07092 −0.535460 0.844561i \(-0.679862\pi\)
−0.535460 + 0.844561i \(0.679862\pi\)
\(938\) 168.487 5.50128
\(939\) 0 0
\(940\) 0 0
\(941\) 13.3074 0.433808 0.216904 0.976193i \(-0.430404\pi\)
0.216904 + 0.976193i \(0.430404\pi\)
\(942\) 0 0
\(943\) −61.1302 −1.99067
\(944\) 70.7862 2.30390
\(945\) 0 0
\(946\) 71.4310 2.32242
\(947\) 31.9463 1.03811 0.519057 0.854740i \(-0.326283\pi\)
0.519057 + 0.854740i \(0.326283\pi\)
\(948\) 0 0
\(949\) −35.9827 −1.16805
\(950\) 0 0
\(951\) 0 0
\(952\) 44.8687 1.45420
\(953\) −53.5304 −1.73402 −0.867009 0.498292i \(-0.833961\pi\)
−0.867009 + 0.498292i \(0.833961\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −14.8054 −0.478840
\(957\) 0 0
\(958\) 34.6133 1.11831
\(959\) 81.7821 2.64088
\(960\) 0 0
\(961\) −26.0174 −0.839270
\(962\) 75.8274 2.44477
\(963\) 0 0
\(964\) 51.7893 1.66802
\(965\) 0 0
\(966\) 0 0
\(967\) −16.9490 −0.545044 −0.272522 0.962150i \(-0.587858\pi\)
−0.272522 + 0.962150i \(0.587858\pi\)
\(968\) 8.84250 0.284209
\(969\) 0 0
\(970\) 0 0
\(971\) 28.5476 0.916135 0.458067 0.888917i \(-0.348542\pi\)
0.458067 + 0.888917i \(0.348542\pi\)
\(972\) 0 0
\(973\) −32.0402 −1.02716
\(974\) −64.1242 −2.05467
\(975\) 0 0
\(976\) −51.5189 −1.64908
\(977\) −30.8519 −0.987040 −0.493520 0.869734i \(-0.664290\pi\)
−0.493520 + 0.869734i \(0.664290\pi\)
\(978\) 0 0
\(979\) 24.8585 0.794481
\(980\) 0 0
\(981\) 0 0
\(982\) 53.2895 1.70054
\(983\) 25.9915 0.829000 0.414500 0.910049i \(-0.363956\pi\)
0.414500 + 0.910049i \(0.363956\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −3.60957 −0.114952
\(987\) 0 0
\(988\) 88.8451 2.82654
\(989\) 50.8350 1.61646
\(990\) 0 0
\(991\) −3.17860 −0.100972 −0.0504858 0.998725i \(-0.516077\pi\)
−0.0504858 + 0.998725i \(0.516077\pi\)
\(992\) 17.2562 0.547886
\(993\) 0 0
\(994\) 71.3105 2.26183
\(995\) 0 0
\(996\) 0 0
\(997\) −31.6331 −1.00183 −0.500916 0.865496i \(-0.667003\pi\)
−0.500916 + 0.865496i \(0.667003\pi\)
\(998\) −54.6982 −1.73144
\(999\) 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.by.1.2 8
3.2 odd 2 2175.2.a.bd.1.7 yes 8
5.4 even 2 6525.2.a.bz.1.7 8
15.2 even 4 2175.2.c.p.349.14 16
15.8 even 4 2175.2.c.p.349.3 16
15.14 odd 2 2175.2.a.bc.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.2 8 15.14 odd 2
2175.2.a.bd.1.7 yes 8 3.2 odd 2
2175.2.c.p.349.3 16 15.8 even 4
2175.2.c.p.349.14 16 15.2 even 4
6525.2.a.by.1.2 8 1.1 even 1 trivial
6525.2.a.bz.1.7 8 5.4 even 2