Properties

Label 4225.2.a.bg.1.1
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
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] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.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: \(33.7367948540\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.801938 q^{2} +2.24698 q^{3} -1.35690 q^{4} -1.80194 q^{6} +2.35690 q^{7} +2.69202 q^{8} +2.04892 q^{9} +O(q^{10})\) \(q-0.801938 q^{2} +2.24698 q^{3} -1.35690 q^{4} -1.80194 q^{6} +2.35690 q^{7} +2.69202 q^{8} +2.04892 q^{9} -4.24698 q^{11} -3.04892 q^{12} -1.89008 q^{14} +0.554958 q^{16} -2.15883 q^{17} -1.64310 q^{18} -0.0881460 q^{19} +5.29590 q^{21} +3.40581 q^{22} -1.49396 q^{23} +6.04892 q^{24} -2.13706 q^{27} -3.19806 q^{28} +4.63102 q^{29} -6.63102 q^{31} -5.82908 q^{32} -9.54288 q^{33} +1.73125 q^{34} -2.78017 q^{36} -5.69202 q^{37} +0.0706876 q^{38} -11.5918 q^{41} -4.24698 q^{42} +0.295897 q^{43} +5.76271 q^{44} +1.19806 q^{46} +7.35690 q^{47} +1.24698 q^{48} -1.44504 q^{49} -4.85086 q^{51} +10.3937 q^{53} +1.71379 q^{54} +6.34481 q^{56} -0.198062 q^{57} -3.71379 q^{58} -6.78017 q^{59} +3.47219 q^{61} +5.31767 q^{62} +4.82908 q^{63} +3.56465 q^{64} +7.65279 q^{66} -7.67994 q^{67} +2.92931 q^{68} -3.35690 q^{69} -8.66487 q^{71} +5.51573 q^{72} -6.73556 q^{73} +4.56465 q^{74} +0.119605 q^{76} -10.0097 q^{77} +9.97046 q^{79} -10.9487 q^{81} +9.29590 q^{82} -1.60925 q^{83} -7.18598 q^{84} -0.237291 q^{86} +10.4058 q^{87} -11.4330 q^{88} -2.88471 q^{89} +2.02715 q^{92} -14.8998 q^{93} -5.89977 q^{94} -13.0978 q^{96} +8.05861 q^{97} +1.15883 q^{98} -8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{19} + 2 q^{21} - 3 q^{22} + 5 q^{23} + 9 q^{24} - q^{27} - 14 q^{28} - q^{29} - 5 q^{31} - 7 q^{32} - 10 q^{33} + 13 q^{34} - 7 q^{36} - 12 q^{37} - 12 q^{38} - 7 q^{41} - 8 q^{42} - 13 q^{43} + 8 q^{46} + 18 q^{47} - q^{48} - 4 q^{49} - q^{51} - q^{53} - 3 q^{54} - 4 q^{56} - 5 q^{57} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} + 4 q^{63} - 11 q^{64} + 5 q^{66} + q^{67} + 21 q^{68} - 6 q^{69} - 27 q^{71} + 4 q^{72} - 9 q^{73} - 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} - q^{81} + 14 q^{82} + 7 q^{83} - 7 q^{84} - 18 q^{86} + 18 q^{87} - 15 q^{88} - 11 q^{89} - 22 q^{93} + 5 q^{94} - 21 q^{96} - 7 q^{97} - 5 q^{98} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.801938 −0.567056 −0.283528 0.958964i \(-0.591505\pi\)
−0.283528 + 0.958964i \(0.591505\pi\)
\(3\) 2.24698 1.29729 0.648647 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(4\) −1.35690 −0.678448
\(5\) 0 0
\(6\) −1.80194 −0.735638
\(7\) 2.35690 0.890823 0.445411 0.895326i \(-0.353057\pi\)
0.445411 + 0.895326i \(0.353057\pi\)
\(8\) 2.69202 0.951773
\(9\) 2.04892 0.682972
\(10\) 0 0
\(11\) −4.24698 −1.28051 −0.640256 0.768161i \(-0.721172\pi\)
−0.640256 + 0.768161i \(0.721172\pi\)
\(12\) −3.04892 −0.880147
\(13\) 0 0
\(14\) −1.89008 −0.505146
\(15\) 0 0
\(16\) 0.554958 0.138740
\(17\) −2.15883 −0.523594 −0.261797 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(18\) −1.64310 −0.387283
\(19\) −0.0881460 −0.0202221 −0.0101110 0.999949i \(-0.503218\pi\)
−0.0101110 + 0.999949i \(0.503218\pi\)
\(20\) 0 0
\(21\) 5.29590 1.15566
\(22\) 3.40581 0.726122
\(23\) −1.49396 −0.311512 −0.155756 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(24\) 6.04892 1.23473
\(25\) 0 0
\(26\) 0 0
\(27\) −2.13706 −0.411278
\(28\) −3.19806 −0.604377
\(29\) 4.63102 0.859959 0.429980 0.902839i \(-0.358521\pi\)
0.429980 + 0.902839i \(0.358521\pi\)
\(30\) 0 0
\(31\) −6.63102 −1.19097 −0.595483 0.803368i \(-0.703039\pi\)
−0.595483 + 0.803368i \(0.703039\pi\)
\(32\) −5.82908 −1.03045
\(33\) −9.54288 −1.66120
\(34\) 1.73125 0.296907
\(35\) 0 0
\(36\) −2.78017 −0.463361
\(37\) −5.69202 −0.935763 −0.467881 0.883791i \(-0.654983\pi\)
−0.467881 + 0.883791i \(0.654983\pi\)
\(38\) 0.0706876 0.0114670
\(39\) 0 0
\(40\) 0 0
\(41\) −11.5918 −1.81033 −0.905167 0.425056i \(-0.860254\pi\)
−0.905167 + 0.425056i \(0.860254\pi\)
\(42\) −4.24698 −0.655323
\(43\) 0.295897 0.0451239 0.0225619 0.999745i \(-0.492818\pi\)
0.0225619 + 0.999745i \(0.492818\pi\)
\(44\) 5.76271 0.868761
\(45\) 0 0
\(46\) 1.19806 0.176645
\(47\) 7.35690 1.07311 0.536557 0.843864i \(-0.319725\pi\)
0.536557 + 0.843864i \(0.319725\pi\)
\(48\) 1.24698 0.179986
\(49\) −1.44504 −0.206435
\(50\) 0 0
\(51\) −4.85086 −0.679256
\(52\) 0 0
\(53\) 10.3937 1.42769 0.713844 0.700304i \(-0.246952\pi\)
0.713844 + 0.700304i \(0.246952\pi\)
\(54\) 1.71379 0.233218
\(55\) 0 0
\(56\) 6.34481 0.847861
\(57\) −0.198062 −0.0262340
\(58\) −3.71379 −0.487645
\(59\) −6.78017 −0.882703 −0.441351 0.897334i \(-0.645501\pi\)
−0.441351 + 0.897334i \(0.645501\pi\)
\(60\) 0 0
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) 5.31767 0.675344
\(63\) 4.82908 0.608407
\(64\) 3.56465 0.445581
\(65\) 0 0
\(66\) 7.65279 0.941994
\(67\) −7.67994 −0.938254 −0.469127 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(68\) 2.92931 0.355231
\(69\) −3.35690 −0.404123
\(70\) 0 0
\(71\) −8.66487 −1.02833 −0.514166 0.857691i \(-0.671898\pi\)
−0.514166 + 0.857691i \(0.671898\pi\)
\(72\) 5.51573 0.650035
\(73\) −6.73556 −0.788338 −0.394169 0.919038i \(-0.628968\pi\)
−0.394169 + 0.919038i \(0.628968\pi\)
\(74\) 4.56465 0.530629
\(75\) 0 0
\(76\) 0.119605 0.0137196
\(77\) −10.0097 −1.14071
\(78\) 0 0
\(79\) 9.97046 1.12176 0.560882 0.827896i \(-0.310462\pi\)
0.560882 + 0.827896i \(0.310462\pi\)
\(80\) 0 0
\(81\) −10.9487 −1.21652
\(82\) 9.29590 1.02656
\(83\) −1.60925 −0.176638 −0.0883192 0.996092i \(-0.528150\pi\)
−0.0883192 + 0.996092i \(0.528150\pi\)
\(84\) −7.18598 −0.784055
\(85\) 0 0
\(86\) −0.237291 −0.0255877
\(87\) 10.4058 1.11562
\(88\) −11.4330 −1.21876
\(89\) −2.88471 −0.305778 −0.152889 0.988243i \(-0.548858\pi\)
−0.152889 + 0.988243i \(0.548858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.02715 0.211345
\(93\) −14.8998 −1.54503
\(94\) −5.89977 −0.608515
\(95\) 0 0
\(96\) −13.0978 −1.33679
\(97\) 8.05861 0.818227 0.409114 0.912483i \(-0.365838\pi\)
0.409114 + 0.912483i \(0.365838\pi\)
\(98\) 1.15883 0.117060
\(99\) −8.70171 −0.874555
\(100\) 0 0
\(101\) −13.3545 −1.32882 −0.664411 0.747367i \(-0.731318\pi\)
−0.664411 + 0.747367i \(0.731318\pi\)
\(102\) 3.89008 0.385176
\(103\) −1.36227 −0.134229 −0.0671144 0.997745i \(-0.521379\pi\)
−0.0671144 + 0.997745i \(0.521379\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −8.33513 −0.809579
\(107\) −3.26875 −0.316002 −0.158001 0.987439i \(-0.550505\pi\)
−0.158001 + 0.987439i \(0.550505\pi\)
\(108\) 2.89977 0.279031
\(109\) 15.7017 1.50395 0.751976 0.659191i \(-0.229101\pi\)
0.751976 + 0.659191i \(0.229101\pi\)
\(110\) 0 0
\(111\) −12.7899 −1.21396
\(112\) 1.30798 0.123592
\(113\) −12.0489 −1.13347 −0.566733 0.823901i \(-0.691793\pi\)
−0.566733 + 0.823901i \(0.691793\pi\)
\(114\) 0.158834 0.0148761
\(115\) 0 0
\(116\) −6.28382 −0.583438
\(117\) 0 0
\(118\) 5.43727 0.500541
\(119\) −5.08815 −0.466430
\(120\) 0 0
\(121\) 7.03684 0.639712
\(122\) −2.78448 −0.252095
\(123\) −26.0465 −2.34854
\(124\) 8.99761 0.808009
\(125\) 0 0
\(126\) −3.87263 −0.345001
\(127\) 9.80731 0.870258 0.435129 0.900368i \(-0.356703\pi\)
0.435129 + 0.900368i \(0.356703\pi\)
\(128\) 8.79954 0.777777
\(129\) 0.664874 0.0585389
\(130\) 0 0
\(131\) −6.57673 −0.574611 −0.287306 0.957839i \(-0.592760\pi\)
−0.287306 + 0.957839i \(0.592760\pi\)
\(132\) 12.9487 1.12704
\(133\) −0.207751 −0.0180143
\(134\) 6.15883 0.532042
\(135\) 0 0
\(136\) −5.81163 −0.498343
\(137\) −6.21983 −0.531396 −0.265698 0.964056i \(-0.585602\pi\)
−0.265698 + 0.964056i \(0.585602\pi\)
\(138\) 2.69202 0.229160
\(139\) −14.7071 −1.24744 −0.623719 0.781648i \(-0.714379\pi\)
−0.623719 + 0.781648i \(0.714379\pi\)
\(140\) 0 0
\(141\) 16.5308 1.39214
\(142\) 6.94869 0.583121
\(143\) 0 0
\(144\) 1.13706 0.0947553
\(145\) 0 0
\(146\) 5.40150 0.447031
\(147\) −3.24698 −0.267806
\(148\) 7.72348 0.634866
\(149\) 4.33513 0.355147 0.177574 0.984108i \(-0.443175\pi\)
0.177574 + 0.984108i \(0.443175\pi\)
\(150\) 0 0
\(151\) 3.94438 0.320989 0.160494 0.987037i \(-0.448691\pi\)
0.160494 + 0.987037i \(0.448691\pi\)
\(152\) −0.237291 −0.0192468
\(153\) −4.42327 −0.357600
\(154\) 8.02715 0.646846
\(155\) 0 0
\(156\) 0 0
\(157\) −4.45473 −0.355526 −0.177763 0.984073i \(-0.556886\pi\)
−0.177763 + 0.984073i \(0.556886\pi\)
\(158\) −7.99569 −0.636103
\(159\) 23.3545 1.85213
\(160\) 0 0
\(161\) −3.52111 −0.277502
\(162\) 8.78017 0.689835
\(163\) −16.1588 −1.26566 −0.632829 0.774292i \(-0.718106\pi\)
−0.632829 + 0.774292i \(0.718106\pi\)
\(164\) 15.7289 1.22822
\(165\) 0 0
\(166\) 1.29052 0.100164
\(167\) −16.1172 −1.24719 −0.623594 0.781749i \(-0.714328\pi\)
−0.623594 + 0.781749i \(0.714328\pi\)
\(168\) 14.2567 1.09993
\(169\) 0 0
\(170\) 0 0
\(171\) −0.180604 −0.0138111
\(172\) −0.401501 −0.0306142
\(173\) 21.5362 1.63736 0.818682 0.574247i \(-0.194705\pi\)
0.818682 + 0.574247i \(0.194705\pi\)
\(174\) −8.34481 −0.632619
\(175\) 0 0
\(176\) −2.35690 −0.177658
\(177\) −15.2349 −1.14513
\(178\) 2.31336 0.173393
\(179\) 11.4330 0.854540 0.427270 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(180\) 0 0
\(181\) 20.9705 1.55872 0.779361 0.626575i \(-0.215544\pi\)
0.779361 + 0.626575i \(0.215544\pi\)
\(182\) 0 0
\(183\) 7.80194 0.576736
\(184\) −4.02177 −0.296489
\(185\) 0 0
\(186\) 11.9487 0.876120
\(187\) 9.16852 0.670469
\(188\) −9.98254 −0.728052
\(189\) −5.03684 −0.366376
\(190\) 0 0
\(191\) −14.4373 −1.04464 −0.522322 0.852748i \(-0.674934\pi\)
−0.522322 + 0.852748i \(0.674934\pi\)
\(192\) 8.00969 0.578049
\(193\) 13.5797 0.977489 0.488745 0.872427i \(-0.337455\pi\)
0.488745 + 0.872427i \(0.337455\pi\)
\(194\) −6.46250 −0.463980
\(195\) 0 0
\(196\) 1.96077 0.140055
\(197\) 0.560335 0.0399222 0.0199611 0.999801i \(-0.493646\pi\)
0.0199611 + 0.999801i \(0.493646\pi\)
\(198\) 6.97823 0.495921
\(199\) 11.4916 0.814616 0.407308 0.913291i \(-0.366468\pi\)
0.407308 + 0.913291i \(0.366468\pi\)
\(200\) 0 0
\(201\) −17.2567 −1.21719
\(202\) 10.7095 0.753516
\(203\) 10.9148 0.766071
\(204\) 6.58211 0.460840
\(205\) 0 0
\(206\) 1.09246 0.0761151
\(207\) −3.06100 −0.212754
\(208\) 0 0
\(209\) 0.374354 0.0258946
\(210\) 0 0
\(211\) 8.78448 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(212\) −14.1032 −0.968613
\(213\) −19.4698 −1.33405
\(214\) 2.62133 0.179191
\(215\) 0 0
\(216\) −5.75302 −0.391443
\(217\) −15.6286 −1.06094
\(218\) −12.5918 −0.852824
\(219\) −15.1347 −1.02271
\(220\) 0 0
\(221\) 0 0
\(222\) 10.2567 0.688383
\(223\) 2.25906 0.151278 0.0756390 0.997135i \(-0.475900\pi\)
0.0756390 + 0.997135i \(0.475900\pi\)
\(224\) −13.7385 −0.917945
\(225\) 0 0
\(226\) 9.66248 0.642739
\(227\) 6.96615 0.462359 0.231180 0.972911i \(-0.425741\pi\)
0.231180 + 0.972911i \(0.425741\pi\)
\(228\) 0.268750 0.0177984
\(229\) −24.1739 −1.59746 −0.798728 0.601692i \(-0.794493\pi\)
−0.798728 + 0.601692i \(0.794493\pi\)
\(230\) 0 0
\(231\) −22.4916 −1.47984
\(232\) 12.4668 0.818486
\(233\) 3.06100 0.200533 0.100266 0.994961i \(-0.468031\pi\)
0.100266 + 0.994961i \(0.468031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 9.19998 0.598868
\(237\) 22.4034 1.45526
\(238\) 4.08038 0.264492
\(239\) −25.1468 −1.62661 −0.813304 0.581839i \(-0.802333\pi\)
−0.813304 + 0.581839i \(0.802333\pi\)
\(240\) 0 0
\(241\) −20.2664 −1.30547 −0.652735 0.757586i \(-0.726379\pi\)
−0.652735 + 0.757586i \(0.726379\pi\)
\(242\) −5.64310 −0.362752
\(243\) −18.1903 −1.16691
\(244\) −4.71140 −0.301616
\(245\) 0 0
\(246\) 20.8877 1.33175
\(247\) 0 0
\(248\) −17.8509 −1.13353
\(249\) −3.61596 −0.229152
\(250\) 0 0
\(251\) −23.7211 −1.49726 −0.748631 0.662987i \(-0.769288\pi\)
−0.748631 + 0.662987i \(0.769288\pi\)
\(252\) −6.55257 −0.412773
\(253\) 6.34481 0.398895
\(254\) −7.86486 −0.493485
\(255\) 0 0
\(256\) −14.1860 −0.886624
\(257\) −14.2241 −0.887278 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(258\) −0.533188 −0.0331948
\(259\) −13.4155 −0.833599
\(260\) 0 0
\(261\) 9.48858 0.587329
\(262\) 5.27413 0.325837
\(263\) 17.0954 1.05415 0.527075 0.849819i \(-0.323289\pi\)
0.527075 + 0.849819i \(0.323289\pi\)
\(264\) −25.6896 −1.58109
\(265\) 0 0
\(266\) 0.166603 0.0102151
\(267\) −6.48188 −0.396684
\(268\) 10.4209 0.636556
\(269\) −6.46681 −0.394288 −0.197144 0.980374i \(-0.563167\pi\)
−0.197144 + 0.980374i \(0.563167\pi\)
\(270\) 0 0
\(271\) 6.44803 0.391690 0.195845 0.980635i \(-0.437255\pi\)
0.195845 + 0.980635i \(0.437255\pi\)
\(272\) −1.19806 −0.0726432
\(273\) 0 0
\(274\) 4.98792 0.301331
\(275\) 0 0
\(276\) 4.55496 0.274176
\(277\) −13.4601 −0.808739 −0.404370 0.914596i \(-0.632509\pi\)
−0.404370 + 0.914596i \(0.632509\pi\)
\(278\) 11.7942 0.707367
\(279\) −13.5864 −0.813398
\(280\) 0 0
\(281\) 5.03684 0.300472 0.150236 0.988650i \(-0.451997\pi\)
0.150236 + 0.988650i \(0.451997\pi\)
\(282\) −13.2567 −0.789423
\(283\) −22.1280 −1.31537 −0.657686 0.753293i \(-0.728464\pi\)
−0.657686 + 0.753293i \(0.728464\pi\)
\(284\) 11.7573 0.697669
\(285\) 0 0
\(286\) 0 0
\(287\) −27.3207 −1.61269
\(288\) −11.9433 −0.703766
\(289\) −12.3394 −0.725849
\(290\) 0 0
\(291\) 18.1075 1.06148
\(292\) 9.13946 0.534846
\(293\) −14.9463 −0.873172 −0.436586 0.899663i \(-0.643813\pi\)
−0.436586 + 0.899663i \(0.643813\pi\)
\(294\) 2.60388 0.151861
\(295\) 0 0
\(296\) −15.3230 −0.890634
\(297\) 9.07606 0.526647
\(298\) −3.47650 −0.201388
\(299\) 0 0
\(300\) 0 0
\(301\) 0.697398 0.0401974
\(302\) −3.16315 −0.182019
\(303\) −30.0073 −1.72387
\(304\) −0.0489173 −0.00280560
\(305\) 0 0
\(306\) 3.54719 0.202779
\(307\) −19.1293 −1.09177 −0.545883 0.837861i \(-0.683806\pi\)
−0.545883 + 0.837861i \(0.683806\pi\)
\(308\) 13.5821 0.773912
\(309\) −3.06100 −0.174134
\(310\) 0 0
\(311\) −0.269815 −0.0152998 −0.00764990 0.999971i \(-0.502435\pi\)
−0.00764990 + 0.999971i \(0.502435\pi\)
\(312\) 0 0
\(313\) 23.3937 1.32229 0.661146 0.750257i \(-0.270070\pi\)
0.661146 + 0.750257i \(0.270070\pi\)
\(314\) 3.57242 0.201603
\(315\) 0 0
\(316\) −13.5289 −0.761059
\(317\) 13.9952 0.786050 0.393025 0.919528i \(-0.371429\pi\)
0.393025 + 0.919528i \(0.371429\pi\)
\(318\) −18.7289 −1.05026
\(319\) −19.6679 −1.10119
\(320\) 0 0
\(321\) −7.34481 −0.409948
\(322\) 2.82371 0.157359
\(323\) 0.190293 0.0105882
\(324\) 14.8562 0.825346
\(325\) 0 0
\(326\) 12.9584 0.717698
\(327\) 35.2814 1.95107
\(328\) −31.2054 −1.72303
\(329\) 17.3394 0.955954
\(330\) 0 0
\(331\) 17.8213 0.979548 0.489774 0.871849i \(-0.337079\pi\)
0.489774 + 0.871849i \(0.337079\pi\)
\(332\) 2.18359 0.119840
\(333\) −11.6625 −0.639100
\(334\) 12.9250 0.707225
\(335\) 0 0
\(336\) 2.93900 0.160336
\(337\) 27.8485 1.51700 0.758501 0.651672i \(-0.225932\pi\)
0.758501 + 0.651672i \(0.225932\pi\)
\(338\) 0 0
\(339\) −27.0737 −1.47044
\(340\) 0 0
\(341\) 28.1618 1.52505
\(342\) 0.144833 0.00783167
\(343\) −19.9041 −1.07472
\(344\) 0.796561 0.0429477
\(345\) 0 0
\(346\) −17.2707 −0.928477
\(347\) −1.50365 −0.0807200 −0.0403600 0.999185i \(-0.512850\pi\)
−0.0403600 + 0.999185i \(0.512850\pi\)
\(348\) −14.1196 −0.756890
\(349\) −14.1860 −0.759358 −0.379679 0.925118i \(-0.623966\pi\)
−0.379679 + 0.925118i \(0.623966\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 24.7560 1.31950
\(353\) 7.16852 0.381542 0.190771 0.981635i \(-0.438901\pi\)
0.190771 + 0.981635i \(0.438901\pi\)
\(354\) 12.2174 0.649350
\(355\) 0 0
\(356\) 3.91425 0.207455
\(357\) −11.4330 −0.605096
\(358\) −9.16852 −0.484571
\(359\) 19.8853 1.04951 0.524753 0.851255i \(-0.324158\pi\)
0.524753 + 0.851255i \(0.324158\pi\)
\(360\) 0 0
\(361\) −18.9922 −0.999591
\(362\) −16.8170 −0.883882
\(363\) 15.8116 0.829895
\(364\) 0 0
\(365\) 0 0
\(366\) −6.25667 −0.327041
\(367\) −1.08383 −0.0565757 −0.0282878 0.999600i \(-0.509006\pi\)
−0.0282878 + 0.999600i \(0.509006\pi\)
\(368\) −0.829085 −0.0432190
\(369\) −23.7506 −1.23641
\(370\) 0 0
\(371\) 24.4969 1.27182
\(372\) 20.2174 1.04823
\(373\) 6.13036 0.317418 0.158709 0.987325i \(-0.449267\pi\)
0.158709 + 0.987325i \(0.449267\pi\)
\(374\) −7.35258 −0.380193
\(375\) 0 0
\(376\) 19.8049 1.02136
\(377\) 0 0
\(378\) 4.03923 0.207756
\(379\) −2.40880 −0.123732 −0.0618658 0.998084i \(-0.519705\pi\)
−0.0618658 + 0.998084i \(0.519705\pi\)
\(380\) 0 0
\(381\) 22.0368 1.12898
\(382\) 11.5778 0.592371
\(383\) 30.3913 1.55292 0.776462 0.630164i \(-0.217012\pi\)
0.776462 + 0.630164i \(0.217012\pi\)
\(384\) 19.7724 1.00901
\(385\) 0 0
\(386\) −10.8901 −0.554291
\(387\) 0.606268 0.0308184
\(388\) −10.9347 −0.555125
\(389\) −15.9409 −0.808237 −0.404118 0.914707i \(-0.632422\pi\)
−0.404118 + 0.914707i \(0.632422\pi\)
\(390\) 0 0
\(391\) 3.22521 0.163106
\(392\) −3.89008 −0.196479
\(393\) −14.7778 −0.745440
\(394\) −0.449354 −0.0226381
\(395\) 0 0
\(396\) 11.8073 0.593340
\(397\) −16.9148 −0.848931 −0.424466 0.905444i \(-0.639538\pi\)
−0.424466 + 0.905444i \(0.639538\pi\)
\(398\) −9.21552 −0.461932
\(399\) −0.466812 −0.0233698
\(400\) 0 0
\(401\) 26.6625 1.33146 0.665730 0.746192i \(-0.268120\pi\)
0.665730 + 0.746192i \(0.268120\pi\)
\(402\) 13.8388 0.690215
\(403\) 0 0
\(404\) 18.1207 0.901537
\(405\) 0 0
\(406\) −8.75302 −0.434405
\(407\) 24.1739 1.19826
\(408\) −13.0586 −0.646497
\(409\) 28.5163 1.41004 0.705021 0.709187i \(-0.250938\pi\)
0.705021 + 0.709187i \(0.250938\pi\)
\(410\) 0 0
\(411\) −13.9758 −0.689377
\(412\) 1.84846 0.0910672
\(413\) −15.9801 −0.786332
\(414\) 2.45473 0.120643
\(415\) 0 0
\(416\) 0 0
\(417\) −33.0465 −1.61830
\(418\) −0.300209 −0.0146837
\(419\) −29.6093 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(420\) 0 0
\(421\) −11.6606 −0.568301 −0.284151 0.958780i \(-0.591712\pi\)
−0.284151 + 0.958780i \(0.591712\pi\)
\(422\) −7.04461 −0.342926
\(423\) 15.0737 0.732907
\(424\) 27.9801 1.35884
\(425\) 0 0
\(426\) 15.6136 0.756480
\(427\) 8.18359 0.396032
\(428\) 4.43535 0.214391
\(429\) 0 0
\(430\) 0 0
\(431\) 4.34913 0.209490 0.104745 0.994499i \(-0.466597\pi\)
0.104745 + 0.994499i \(0.466597\pi\)
\(432\) −1.18598 −0.0570605
\(433\) 14.3884 0.691460 0.345730 0.938334i \(-0.387631\pi\)
0.345730 + 0.938334i \(0.387631\pi\)
\(434\) 12.5332 0.601612
\(435\) 0 0
\(436\) −21.3056 −1.02035
\(437\) 0.131687 0.00629942
\(438\) 12.1371 0.579931
\(439\) −20.2325 −0.965645 −0.482822 0.875718i \(-0.660388\pi\)
−0.482822 + 0.875718i \(0.660388\pi\)
\(440\) 0 0
\(441\) −2.96077 −0.140989
\(442\) 0 0
\(443\) −8.12200 −0.385888 −0.192944 0.981210i \(-0.561804\pi\)
−0.192944 + 0.981210i \(0.561804\pi\)
\(444\) 17.3545 0.823608
\(445\) 0 0
\(446\) −1.81163 −0.0857830
\(447\) 9.74094 0.460731
\(448\) 8.40150 0.396934
\(449\) 12.4916 0.589513 0.294757 0.955572i \(-0.404761\pi\)
0.294757 + 0.955572i \(0.404761\pi\)
\(450\) 0 0
\(451\) 49.2301 2.31816
\(452\) 16.3491 0.768998
\(453\) 8.86294 0.416417
\(454\) −5.58642 −0.262184
\(455\) 0 0
\(456\) −0.533188 −0.0249688
\(457\) −5.98121 −0.279789 −0.139895 0.990166i \(-0.544676\pi\)
−0.139895 + 0.990166i \(0.544676\pi\)
\(458\) 19.3860 0.905847
\(459\) 4.61356 0.215343
\(460\) 0 0
\(461\) −2.05669 −0.0957895 −0.0478947 0.998852i \(-0.515251\pi\)
−0.0478947 + 0.998852i \(0.515251\pi\)
\(462\) 18.0368 0.839150
\(463\) 8.44935 0.392675 0.196337 0.980536i \(-0.437095\pi\)
0.196337 + 0.980536i \(0.437095\pi\)
\(464\) 2.57002 0.119310
\(465\) 0 0
\(466\) −2.45473 −0.113713
\(467\) −33.5139 −1.55084 −0.775420 0.631446i \(-0.782462\pi\)
−0.775420 + 0.631446i \(0.782462\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) 0 0
\(471\) −10.0097 −0.461222
\(472\) −18.2524 −0.840133
\(473\) −1.25667 −0.0577817
\(474\) −17.9661 −0.825213
\(475\) 0 0
\(476\) 6.90408 0.316448
\(477\) 21.2959 0.975072
\(478\) 20.1661 0.922377
\(479\) −24.7313 −1.13000 −0.565000 0.825091i \(-0.691124\pi\)
−0.565000 + 0.825091i \(0.691124\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 16.2524 0.740275
\(483\) −7.91185 −0.360002
\(484\) −9.54825 −0.434012
\(485\) 0 0
\(486\) 14.5875 0.661702
\(487\) 37.7555 1.71087 0.855433 0.517913i \(-0.173291\pi\)
0.855433 + 0.517913i \(0.173291\pi\)
\(488\) 9.34721 0.423128
\(489\) −36.3086 −1.64193
\(490\) 0 0
\(491\) 31.3110 1.41304 0.706522 0.707691i \(-0.250263\pi\)
0.706522 + 0.707691i \(0.250263\pi\)
\(492\) 35.3424 1.59336
\(493\) −9.99761 −0.450270
\(494\) 0 0
\(495\) 0 0
\(496\) −3.67994 −0.165234
\(497\) −20.4222 −0.916061
\(498\) 2.89977 0.129942
\(499\) 21.4873 0.961902 0.480951 0.876748i \(-0.340292\pi\)
0.480951 + 0.876748i \(0.340292\pi\)
\(500\) 0 0
\(501\) −36.2150 −1.61797
\(502\) 19.0228 0.849031
\(503\) −37.5924 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(504\) 13.0000 0.579066
\(505\) 0 0
\(506\) −5.08815 −0.226196
\(507\) 0 0
\(508\) −13.3075 −0.590425
\(509\) −17.1075 −0.758278 −0.379139 0.925340i \(-0.623780\pi\)
−0.379139 + 0.925340i \(0.623780\pi\)
\(510\) 0 0
\(511\) −15.8750 −0.702269
\(512\) −6.22282 −0.275012
\(513\) 0.188374 0.00831690
\(514\) 11.4069 0.503136
\(515\) 0 0
\(516\) −0.902165 −0.0397156
\(517\) −31.2446 −1.37414
\(518\) 10.7584 0.472697
\(519\) 48.3913 2.12414
\(520\) 0 0
\(521\) −19.8465 −0.869493 −0.434746 0.900553i \(-0.643162\pi\)
−0.434746 + 0.900553i \(0.643162\pi\)
\(522\) −7.60925 −0.333048
\(523\) 11.4300 0.499798 0.249899 0.968272i \(-0.419603\pi\)
0.249899 + 0.968272i \(0.419603\pi\)
\(524\) 8.92394 0.389844
\(525\) 0 0
\(526\) −13.7095 −0.597762
\(527\) 14.3153 0.623583
\(528\) −5.29590 −0.230474
\(529\) −20.7681 −0.902960
\(530\) 0 0
\(531\) −13.8920 −0.602862
\(532\) 0.281896 0.0122218
\(533\) 0 0
\(534\) 5.19806 0.224942
\(535\) 0 0
\(536\) −20.6746 −0.893005
\(537\) 25.6896 1.10859
\(538\) 5.18598 0.223584
\(539\) 6.13706 0.264342
\(540\) 0 0
\(541\) 16.1884 0.695993 0.347996 0.937496i \(-0.386862\pi\)
0.347996 + 0.937496i \(0.386862\pi\)
\(542\) −5.17092 −0.222110
\(543\) 47.1202 2.02212
\(544\) 12.5840 0.539536
\(545\) 0 0
\(546\) 0 0
\(547\) −5.33081 −0.227929 −0.113965 0.993485i \(-0.536355\pi\)
−0.113965 + 0.993485i \(0.536355\pi\)
\(548\) 8.43967 0.360525
\(549\) 7.11423 0.303628
\(550\) 0 0
\(551\) −0.408206 −0.0173902
\(552\) −9.03684 −0.384633
\(553\) 23.4993 0.999293
\(554\) 10.7942 0.458600
\(555\) 0 0
\(556\) 19.9560 0.846322
\(557\) −7.39075 −0.313156 −0.156578 0.987666i \(-0.550046\pi\)
−0.156578 + 0.987666i \(0.550046\pi\)
\(558\) 10.8955 0.461242
\(559\) 0 0
\(560\) 0 0
\(561\) 20.6015 0.869795
\(562\) −4.03923 −0.170385
\(563\) 9.47889 0.399488 0.199744 0.979848i \(-0.435989\pi\)
0.199744 + 0.979848i \(0.435989\pi\)
\(564\) −22.4306 −0.944497
\(565\) 0 0
\(566\) 17.7453 0.745889
\(567\) −25.8049 −1.08370
\(568\) −23.3260 −0.978738
\(569\) −10.1438 −0.425249 −0.212624 0.977134i \(-0.568201\pi\)
−0.212624 + 0.977134i \(0.568201\pi\)
\(570\) 0 0
\(571\) −14.0925 −0.589751 −0.294876 0.955536i \(-0.595278\pi\)
−0.294876 + 0.955536i \(0.595278\pi\)
\(572\) 0 0
\(573\) −32.4403 −1.35521
\(574\) 21.9095 0.914483
\(575\) 0 0
\(576\) 7.30367 0.304319
\(577\) −25.1545 −1.04720 −0.523598 0.851965i \(-0.675411\pi\)
−0.523598 + 0.851965i \(0.675411\pi\)
\(578\) 9.89546 0.411597
\(579\) 30.5133 1.26809
\(580\) 0 0
\(581\) −3.79284 −0.157354
\(582\) −14.5211 −0.601919
\(583\) −44.1420 −1.82817
\(584\) −18.1323 −0.750319
\(585\) 0 0
\(586\) 11.9860 0.495137
\(587\) −43.8353 −1.80928 −0.904639 0.426180i \(-0.859859\pi\)
−0.904639 + 0.426180i \(0.859859\pi\)
\(588\) 4.40581 0.181693
\(589\) 0.584498 0.0240838
\(590\) 0 0
\(591\) 1.25906 0.0517909
\(592\) −3.15883 −0.129827
\(593\) 24.9965 1.02648 0.513242 0.858244i \(-0.328444\pi\)
0.513242 + 0.858244i \(0.328444\pi\)
\(594\) −7.27844 −0.298638
\(595\) 0 0
\(596\) −5.88231 −0.240949
\(597\) 25.8213 1.05680
\(598\) 0 0
\(599\) −6.24027 −0.254971 −0.127485 0.991840i \(-0.540691\pi\)
−0.127485 + 0.991840i \(0.540691\pi\)
\(600\) 0 0
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) −0.559270 −0.0227941
\(603\) −15.7356 −0.640802
\(604\) −5.35211 −0.217774
\(605\) 0 0
\(606\) 24.0640 0.977532
\(607\) 43.6480 1.77162 0.885809 0.464050i \(-0.153604\pi\)
0.885809 + 0.464050i \(0.153604\pi\)
\(608\) 0.513811 0.0208378
\(609\) 24.5254 0.993820
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00192 0.242613
\(613\) 25.9541 1.04827 0.524137 0.851634i \(-0.324388\pi\)
0.524137 + 0.851634i \(0.324388\pi\)
\(614\) 15.3405 0.619092
\(615\) 0 0
\(616\) −26.9463 −1.08570
\(617\) −45.9396 −1.84946 −0.924729 0.380626i \(-0.875709\pi\)
−0.924729 + 0.380626i \(0.875709\pi\)
\(618\) 2.45473 0.0987437
\(619\) 6.73556 0.270725 0.135363 0.990796i \(-0.456780\pi\)
0.135363 + 0.990796i \(0.456780\pi\)
\(620\) 0 0
\(621\) 3.19269 0.128118
\(622\) 0.216375 0.00867583
\(623\) −6.79895 −0.272394
\(624\) 0 0
\(625\) 0 0
\(626\) −18.7603 −0.749813
\(627\) 0.841166 0.0335930
\(628\) 6.04461 0.241206
\(629\) 12.2881 0.489960
\(630\) 0 0
\(631\) 45.0998 1.79539 0.897696 0.440614i \(-0.145239\pi\)
0.897696 + 0.440614i \(0.145239\pi\)
\(632\) 26.8407 1.06767
\(633\) 19.7385 0.784537
\(634\) −11.2233 −0.445734
\(635\) 0 0
\(636\) −31.6896 −1.25658
\(637\) 0 0
\(638\) 15.7724 0.624435
\(639\) −17.7536 −0.702322
\(640\) 0 0
\(641\) 32.5821 1.28692 0.643458 0.765482i \(-0.277499\pi\)
0.643458 + 0.765482i \(0.277499\pi\)
\(642\) 5.89008 0.232463
\(643\) −25.5754 −1.00860 −0.504298 0.863530i \(-0.668249\pi\)
−0.504298 + 0.863530i \(0.668249\pi\)
\(644\) 4.77777 0.188271
\(645\) 0 0
\(646\) −0.152603 −0.00600408
\(647\) 30.1715 1.18616 0.593082 0.805142i \(-0.297911\pi\)
0.593082 + 0.805142i \(0.297911\pi\)
\(648\) −29.4741 −1.15785
\(649\) 28.7952 1.13031
\(650\) 0 0
\(651\) −35.1172 −1.37635
\(652\) 21.9259 0.858683
\(653\) −36.9028 −1.44412 −0.722058 0.691832i \(-0.756804\pi\)
−0.722058 + 0.691832i \(0.756804\pi\)
\(654\) −28.2935 −1.10636
\(655\) 0 0
\(656\) −6.43296 −0.251165
\(657\) −13.8006 −0.538413
\(658\) −13.9051 −0.542079
\(659\) 23.6866 0.922701 0.461350 0.887218i \(-0.347365\pi\)
0.461350 + 0.887218i \(0.347365\pi\)
\(660\) 0 0
\(661\) −31.7590 −1.23528 −0.617641 0.786460i \(-0.711911\pi\)
−0.617641 + 0.786460i \(0.711911\pi\)
\(662\) −14.2916 −0.555458
\(663\) 0 0
\(664\) −4.33214 −0.168120
\(665\) 0 0
\(666\) 9.35258 0.362405
\(667\) −6.91856 −0.267888
\(668\) 21.8694 0.846152
\(669\) 5.07606 0.196252
\(670\) 0 0
\(671\) −14.7463 −0.569275
\(672\) −30.8702 −1.19085
\(673\) 7.50232 0.289193 0.144597 0.989491i \(-0.453812\pi\)
0.144597 + 0.989491i \(0.453812\pi\)
\(674\) −22.3327 −0.860225
\(675\) 0 0
\(676\) 0 0
\(677\) 35.0315 1.34637 0.673184 0.739475i \(-0.264926\pi\)
0.673184 + 0.739475i \(0.264926\pi\)
\(678\) 21.7114 0.833821
\(679\) 18.9933 0.728896
\(680\) 0 0
\(681\) 15.6528 0.599816
\(682\) −22.5840 −0.864787
\(683\) 24.0834 0.921524 0.460762 0.887524i \(-0.347576\pi\)
0.460762 + 0.887524i \(0.347576\pi\)
\(684\) 0.245061 0.00937013
\(685\) 0 0
\(686\) 15.9618 0.609426
\(687\) −54.3183 −2.07237
\(688\) 0.164210 0.00626046
\(689\) 0 0
\(690\) 0 0
\(691\) 2.01447 0.0766342 0.0383171 0.999266i \(-0.487800\pi\)
0.0383171 + 0.999266i \(0.487800\pi\)
\(692\) −29.2223 −1.11087
\(693\) −20.5090 −0.779073
\(694\) 1.20583 0.0457728
\(695\) 0 0
\(696\) 28.0127 1.06182
\(697\) 25.0248 0.947880
\(698\) 11.3763 0.430598
\(699\) 6.87800 0.260150
\(700\) 0 0
\(701\) −48.8189 −1.84387 −0.921933 0.387350i \(-0.873390\pi\)
−0.921933 + 0.387350i \(0.873390\pi\)
\(702\) 0 0
\(703\) 0.501729 0.0189231
\(704\) −15.1390 −0.570572
\(705\) 0 0
\(706\) −5.74871 −0.216355
\(707\) −31.4752 −1.18375
\(708\) 20.6722 0.776908
\(709\) −20.8060 −0.781385 −0.390693 0.920521i \(-0.627764\pi\)
−0.390693 + 0.920521i \(0.627764\pi\)
\(710\) 0 0
\(711\) 20.4286 0.766134
\(712\) −7.76569 −0.291032
\(713\) 9.90648 0.371000
\(714\) 9.16852 0.343123
\(715\) 0 0
\(716\) −15.5133 −0.579761
\(717\) −56.5042 −2.11019
\(718\) −15.9468 −0.595128
\(719\) 21.4306 0.799225 0.399613 0.916684i \(-0.369145\pi\)
0.399613 + 0.916684i \(0.369145\pi\)
\(720\) 0 0
\(721\) −3.21073 −0.119574
\(722\) 15.2306 0.566824
\(723\) −45.5381 −1.69358
\(724\) −28.4547 −1.05751
\(725\) 0 0
\(726\) −12.6799 −0.470597
\(727\) −13.4862 −0.500175 −0.250088 0.968223i \(-0.580459\pi\)
−0.250088 + 0.968223i \(0.580459\pi\)
\(728\) 0 0
\(729\) −8.02715 −0.297302
\(730\) 0 0
\(731\) −0.638792 −0.0236266
\(732\) −10.5864 −0.391285
\(733\) 43.5424 1.60828 0.804138 0.594443i \(-0.202627\pi\)
0.804138 + 0.594443i \(0.202627\pi\)
\(734\) 0.869167 0.0320816
\(735\) 0 0
\(736\) 8.70841 0.320996
\(737\) 32.6165 1.20145
\(738\) 19.0465 0.701112
\(739\) 20.0543 0.737709 0.368855 0.929487i \(-0.379750\pi\)
0.368855 + 0.929487i \(0.379750\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −19.6450 −0.721191
\(743\) 33.1685 1.21684 0.608418 0.793617i \(-0.291805\pi\)
0.608418 + 0.793617i \(0.291805\pi\)
\(744\) −40.1105 −1.47052
\(745\) 0 0
\(746\) −4.91617 −0.179994
\(747\) −3.29722 −0.120639
\(748\) −12.4407 −0.454878
\(749\) −7.70410 −0.281502
\(750\) 0 0
\(751\) 39.2814 1.43340 0.716700 0.697382i \(-0.245652\pi\)
0.716700 + 0.697382i \(0.245652\pi\)
\(752\) 4.08277 0.148883
\(753\) −53.3008 −1.94239
\(754\) 0 0
\(755\) 0 0
\(756\) 6.83446 0.248567
\(757\) 46.6426 1.69526 0.847628 0.530592i \(-0.178030\pi\)
0.847628 + 0.530592i \(0.178030\pi\)
\(758\) 1.93171 0.0701627
\(759\) 14.2567 0.517484
\(760\) 0 0
\(761\) −21.8984 −0.793818 −0.396909 0.917858i \(-0.629917\pi\)
−0.396909 + 0.917858i \(0.629917\pi\)
\(762\) −17.6722 −0.640195
\(763\) 37.0073 1.33975
\(764\) 19.5899 0.708737
\(765\) 0 0
\(766\) −24.3720 −0.880595
\(767\) 0 0
\(768\) −31.8756 −1.15021
\(769\) 46.7096 1.68439 0.842196 0.539172i \(-0.181263\pi\)
0.842196 + 0.539172i \(0.181263\pi\)
\(770\) 0 0
\(771\) −31.9614 −1.15106
\(772\) −18.4263 −0.663175
\(773\) 30.2416 1.08771 0.543857 0.839178i \(-0.316963\pi\)
0.543857 + 0.839178i \(0.316963\pi\)
\(774\) −0.486189 −0.0174757
\(775\) 0 0
\(776\) 21.6939 0.778767
\(777\) −30.1444 −1.08142
\(778\) 12.7836 0.458315
\(779\) 1.02177 0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) −2.58642 −0.0924901
\(783\) −9.89679 −0.353682
\(784\) −0.801938 −0.0286406
\(785\) 0 0
\(786\) 11.8509 0.422706
\(787\) −28.7023 −1.02313 −0.511563 0.859246i \(-0.670933\pi\)
−0.511563 + 0.859246i \(0.670933\pi\)
\(788\) −0.760316 −0.0270851
\(789\) 38.4131 1.36754
\(790\) 0 0
\(791\) −28.3980 −1.00972
\(792\) −23.4252 −0.832378
\(793\) 0 0
\(794\) 13.5646 0.481391
\(795\) 0 0
\(796\) −15.5929 −0.552674
\(797\) 18.5418 0.656785 0.328392 0.944541i \(-0.393493\pi\)
0.328392 + 0.944541i \(0.393493\pi\)
\(798\) 0.374354 0.0132520
\(799\) −15.8823 −0.561876
\(800\) 0 0
\(801\) −5.91053 −0.208838
\(802\) −21.3817 −0.755012
\(803\) 28.6058 1.00948
\(804\) 23.4155 0.825801
\(805\) 0 0
\(806\) 0 0
\(807\) −14.5308 −0.511508
\(808\) −35.9506 −1.26474
\(809\) −10.0677 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(810\) 0 0
\(811\) −10.0285 −0.352147 −0.176074 0.984377i \(-0.556340\pi\)
−0.176074 + 0.984377i \(0.556340\pi\)
\(812\) −14.8103 −0.519740
\(813\) 14.4886 0.508137
\(814\) −19.3860 −0.679478
\(815\) 0 0
\(816\) −2.69202 −0.0942396
\(817\) −0.0260821 −0.000912498 0
\(818\) −22.8683 −0.799572
\(819\) 0 0
\(820\) 0 0
\(821\) −26.1704 −0.913355 −0.456677 0.889632i \(-0.650961\pi\)
−0.456677 + 0.889632i \(0.650961\pi\)
\(822\) 11.2078 0.390915
\(823\) −1.82238 −0.0635242 −0.0317621 0.999495i \(-0.510112\pi\)
−0.0317621 + 0.999495i \(0.510112\pi\)
\(824\) −3.66727 −0.127755
\(825\) 0 0
\(826\) 12.8151 0.445894
\(827\) 32.2941 1.12298 0.561488 0.827485i \(-0.310229\pi\)
0.561488 + 0.827485i \(0.310229\pi\)
\(828\) 4.15346 0.144343
\(829\) 15.1002 0.524453 0.262226 0.965006i \(-0.415543\pi\)
0.262226 + 0.965006i \(0.415543\pi\)
\(830\) 0 0
\(831\) −30.2446 −1.04917
\(832\) 0 0
\(833\) 3.11960 0.108088
\(834\) 26.5013 0.917663
\(835\) 0 0
\(836\) −0.507960 −0.0175682
\(837\) 14.1709 0.489818
\(838\) 23.7448 0.820250
\(839\) −32.9965 −1.13917 −0.569584 0.821933i \(-0.692896\pi\)
−0.569584 + 0.821933i \(0.692896\pi\)
\(840\) 0 0
\(841\) −7.55363 −0.260470
\(842\) 9.35105 0.322258
\(843\) 11.3177 0.389801
\(844\) −11.9196 −0.410290
\(845\) 0 0
\(846\) −12.0881 −0.415599
\(847\) 16.5851 0.569870
\(848\) 5.76809 0.198077
\(849\) −49.7211 −1.70642
\(850\) 0 0
\(851\) 8.50365 0.291501
\(852\) 26.4185 0.905082
\(853\) 37.7802 1.29357 0.646784 0.762673i \(-0.276113\pi\)
0.646784 + 0.762673i \(0.276113\pi\)
\(854\) −6.56273 −0.224572
\(855\) 0 0
\(856\) −8.79954 −0.300762
\(857\) −27.3623 −0.934677 −0.467339 0.884078i \(-0.654787\pi\)
−0.467339 + 0.884078i \(0.654787\pi\)
\(858\) 0 0
\(859\) −20.0629 −0.684538 −0.342269 0.939602i \(-0.611195\pi\)
−0.342269 + 0.939602i \(0.611195\pi\)
\(860\) 0 0
\(861\) −61.3889 −2.09213
\(862\) −3.48773 −0.118792
\(863\) 6.14483 0.209173 0.104586 0.994516i \(-0.466648\pi\)
0.104586 + 0.994516i \(0.466648\pi\)
\(864\) 12.4571 0.423800
\(865\) 0 0
\(866\) −11.5386 −0.392096
\(867\) −27.7265 −0.941640
\(868\) 21.2064 0.719793
\(869\) −42.3443 −1.43643
\(870\) 0 0
\(871\) 0 0
\(872\) 42.2693 1.43142
\(873\) 16.5114 0.558827
\(874\) −0.105604 −0.00357212
\(875\) 0 0
\(876\) 20.5362 0.693853
\(877\) 13.5077 0.456123 0.228061 0.973647i \(-0.426761\pi\)
0.228061 + 0.973647i \(0.426761\pi\)
\(878\) 16.2252 0.547574
\(879\) −33.5840 −1.13276
\(880\) 0 0
\(881\) −5.23431 −0.176348 −0.0881741 0.996105i \(-0.528103\pi\)
−0.0881741 + 0.996105i \(0.528103\pi\)
\(882\) 2.37435 0.0799487
\(883\) 4.57301 0.153894 0.0769470 0.997035i \(-0.475483\pi\)
0.0769470 + 0.997035i \(0.475483\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6.51334 0.218820
\(887\) 1.64071 0.0550897 0.0275448 0.999621i \(-0.491231\pi\)
0.0275448 + 0.999621i \(0.491231\pi\)
\(888\) −34.4306 −1.15541
\(889\) 23.1148 0.775246
\(890\) 0 0
\(891\) 46.4989 1.55777
\(892\) −3.06531 −0.102634
\(893\) −0.648481 −0.0217006
\(894\) −7.81163 −0.261260
\(895\) 0 0
\(896\) 20.7396 0.692862
\(897\) 0 0
\(898\) −10.0175 −0.334287
\(899\) −30.7084 −1.02418
\(900\) 0 0
\(901\) −22.4383 −0.747529
\(902\) −39.4795 −1.31452
\(903\) 1.56704 0.0521478
\(904\) −32.4359 −1.07880
\(905\) 0 0
\(906\) −7.10752 −0.236132
\(907\) −8.10215 −0.269027 −0.134514 0.990912i \(-0.542947\pi\)
−0.134514 + 0.990912i \(0.542947\pi\)
\(908\) −9.45234 −0.313687
\(909\) −27.3623 −0.907549
\(910\) 0 0
\(911\) −9.18119 −0.304187 −0.152093 0.988366i \(-0.548601\pi\)
−0.152093 + 0.988366i \(0.548601\pi\)
\(912\) −0.109916 −0.00363969
\(913\) 6.83446 0.226188
\(914\) 4.79656 0.158656
\(915\) 0 0
\(916\) 32.8015 1.08379
\(917\) −15.5007 −0.511877
\(918\) −3.69979 −0.122111
\(919\) 27.5036 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(920\) 0 0
\(921\) −42.9831 −1.41634
\(922\) 1.64933 0.0543180
\(923\) 0 0
\(924\) 30.5187 1.00399
\(925\) 0 0
\(926\) −6.77586 −0.222668
\(927\) −2.79118 −0.0916745
\(928\) −26.9946 −0.886142
\(929\) 24.2131 0.794407 0.397203 0.917731i \(-0.369981\pi\)
0.397203 + 0.917731i \(0.369981\pi\)
\(930\) 0 0
\(931\) 0.127375 0.00417454
\(932\) −4.15346 −0.136051
\(933\) −0.606268 −0.0198483
\(934\) 26.8761 0.879412
\(935\) 0 0
\(936\) 0 0
\(937\) −11.1830 −0.365333 −0.182666 0.983175i \(-0.558473\pi\)
−0.182666 + 0.983175i \(0.558473\pi\)
\(938\) 14.5157 0.473955
\(939\) 52.5652 1.71540
\(940\) 0 0
\(941\) −15.9638 −0.520404 −0.260202 0.965554i \(-0.583789\pi\)
−0.260202 + 0.965554i \(0.583789\pi\)
\(942\) 8.02715 0.261539
\(943\) 17.3177 0.563941
\(944\) −3.76271 −0.122466
\(945\) 0 0
\(946\) 1.00777 0.0327654
\(947\) −6.51466 −0.211698 −0.105849 0.994382i \(-0.533756\pi\)
−0.105849 + 0.994382i \(0.533756\pi\)
\(948\) −30.3991 −0.987317
\(949\) 0 0
\(950\) 0 0
\(951\) 31.4470 1.01974
\(952\) −13.6974 −0.443935
\(953\) −47.6469 −1.54344 −0.771718 0.635965i \(-0.780602\pi\)
−0.771718 + 0.635965i \(0.780602\pi\)
\(954\) −17.0780 −0.552920
\(955\) 0 0
\(956\) 34.1215 1.10357
\(957\) −44.1933 −1.42857
\(958\) 19.8329 0.640773
\(959\) −14.6595 −0.473380
\(960\) 0 0
\(961\) 12.9705 0.418402
\(962\) 0 0
\(963\) −6.69740 −0.215821
\(964\) 27.4993 0.885694
\(965\) 0 0
\(966\) 6.34481 0.204141
\(967\) 43.8122 1.40891 0.704453 0.709751i \(-0.251192\pi\)
0.704453 + 0.709751i \(0.251192\pi\)
\(968\) 18.9433 0.608861
\(969\) 0.427583 0.0137360
\(970\) 0 0
\(971\) 4.29483 0.137828 0.0689139 0.997623i \(-0.478047\pi\)
0.0689139 + 0.997623i \(0.478047\pi\)
\(972\) 24.6823 0.791686
\(973\) −34.6631 −1.11125
\(974\) −30.2776 −0.970156
\(975\) 0 0
\(976\) 1.92692 0.0616792
\(977\) −26.8019 −0.857470 −0.428735 0.903430i \(-0.641041\pi\)
−0.428735 + 0.903430i \(0.641041\pi\)
\(978\) 29.1172 0.931066
\(979\) 12.2513 0.391553
\(980\) 0 0
\(981\) 32.1715 1.02716
\(982\) −25.1094 −0.801275
\(983\) −27.2495 −0.869124 −0.434562 0.900642i \(-0.643097\pi\)
−0.434562 + 0.900642i \(0.643097\pi\)
\(984\) −70.1178 −2.23527
\(985\) 0 0
\(986\) 8.01746 0.255328
\(987\) 38.9614 1.24015
\(988\) 0 0
\(989\) −0.442058 −0.0140566
\(990\) 0 0
\(991\) 24.3889 0.774740 0.387370 0.921924i \(-0.373384\pi\)
0.387370 + 0.921924i \(0.373384\pi\)
\(992\) 38.6528 1.22723
\(993\) 40.0441 1.27076
\(994\) 16.3773 0.519458
\(995\) 0 0
\(996\) 4.90648 0.155468
\(997\) −31.3207 −0.991935 −0.495967 0.868341i \(-0.665186\pi\)
−0.495967 + 0.868341i \(0.665186\pi\)
\(998\) −17.2314 −0.545452
\(999\) 12.1642 0.384859
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 4225.2.a.bg.1.1 3
5.4 even 2 169.2.a.b.1.3 3
13.12 even 2 4225.2.a.bb.1.3 3
15.14 odd 2 1521.2.a.r.1.1 3
20.19 odd 2 2704.2.a.z.1.3 3
35.34 odd 2 8281.2.a.bf.1.3 3
65.4 even 6 169.2.c.b.146.3 6
65.9 even 6 169.2.c.c.146.1 6
65.19 odd 12 169.2.e.b.23.2 12
65.24 odd 12 169.2.e.b.147.2 12
65.29 even 6 169.2.c.c.22.1 6
65.34 odd 4 169.2.b.b.168.5 6
65.44 odd 4 169.2.b.b.168.2 6
65.49 even 6 169.2.c.b.22.3 6
65.54 odd 12 169.2.e.b.147.5 12
65.59 odd 12 169.2.e.b.23.5 12
65.64 even 2 169.2.a.c.1.1 yes 3
195.44 even 4 1521.2.b.l.1351.5 6
195.164 even 4 1521.2.b.l.1351.2 6
195.194 odd 2 1521.2.a.o.1.3 3
260.99 even 4 2704.2.f.o.337.6 6
260.239 even 4 2704.2.f.o.337.5 6
260.259 odd 2 2704.2.a.ba.1.3 3
455.454 odd 2 8281.2.a.bj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 5.4 even 2
169.2.a.c.1.1 yes 3 65.64 even 2
169.2.b.b.168.2 6 65.44 odd 4
169.2.b.b.168.5 6 65.34 odd 4
169.2.c.b.22.3 6 65.49 even 6
169.2.c.b.146.3 6 65.4 even 6
169.2.c.c.22.1 6 65.29 even 6
169.2.c.c.146.1 6 65.9 even 6
169.2.e.b.23.2 12 65.19 odd 12
169.2.e.b.23.5 12 65.59 odd 12
169.2.e.b.147.2 12 65.24 odd 12
169.2.e.b.147.5 12 65.54 odd 12
1521.2.a.o.1.3 3 195.194 odd 2
1521.2.a.r.1.1 3 15.14 odd 2
1521.2.b.l.1351.2 6 195.164 even 4
1521.2.b.l.1351.5 6 195.44 even 4
2704.2.a.z.1.3 3 20.19 odd 2
2704.2.a.ba.1.3 3 260.259 odd 2
2704.2.f.o.337.5 6 260.239 even 4
2704.2.f.o.337.6 6 260.99 even 4
4225.2.a.bb.1.3 3 13.12 even 2
4225.2.a.bg.1.1 3 1.1 even 1 trivial
8281.2.a.bf.1.3 3 35.34 odd 2
8281.2.a.bj.1.1 3 455.454 odd 2