Properties

Label 4225.2.a.be.1.3
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.51414\) 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)
 
\(f(q)\) \(=\) \(q+2.51414 q^{2} -1.51414 q^{3} +4.32088 q^{4} -3.80675 q^{6} +3.32088 q^{7} +5.83502 q^{8} -0.707389 q^{9} +O(q^{10})\) \(q+2.51414 q^{2} -1.51414 q^{3} +4.32088 q^{4} -3.80675 q^{6} +3.32088 q^{7} +5.83502 q^{8} -0.707389 q^{9} +2.83502 q^{11} -6.54241 q^{12} +8.34916 q^{14} +6.02827 q^{16} +6.64177 q^{17} -1.77847 q^{18} -2.19325 q^{19} -5.02827 q^{21} +7.12763 q^{22} +0.485863 q^{23} -8.83502 q^{24} +5.61350 q^{27} +14.3492 q^{28} -3.32088 q^{29} -3.80675 q^{31} +3.48586 q^{32} -4.29261 q^{33} +16.6983 q^{34} -3.05655 q^{36} +9.32088 q^{37} -5.51414 q^{38} +1.61350 q^{41} -12.6418 q^{42} +0.872368 q^{43} +12.2498 q^{44} +1.22153 q^{46} -3.32088 q^{47} -9.12763 q^{48} +4.02827 q^{49} -10.0565 q^{51} -11.6700 q^{53} +14.1131 q^{54} +19.3774 q^{56} +3.32088 q^{57} -8.34916 q^{58} +8.83502 q^{59} -3.70739 q^{61} -9.57068 q^{62} -2.34916 q^{63} -3.29261 q^{64} -10.7922 q^{66} +4.29261 q^{67} +28.6983 q^{68} -0.735663 q^{69} -2.19325 q^{71} -4.12763 q^{72} +12.7357 q^{73} +23.4340 q^{74} -9.47679 q^{76} +9.41478 q^{77} +0.585221 q^{79} -6.37743 q^{81} +4.05655 q^{82} +7.70739 q^{83} -21.7266 q^{84} +2.19325 q^{86} +5.02827 q^{87} +16.5424 q^{88} -3.41478 q^{89} +2.09936 q^{92} +5.76394 q^{93} -8.34916 q^{94} -5.27807 q^{96} +0.641769 q^{97} +10.1276 q^{98} -2.00546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{3} + 5 q^{4} - 10 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{3} + 5 q^{4} - 10 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - 6 q^{11} + 4 q^{14} + 5 q^{16} + 4 q^{17} - 17 q^{18} - 8 q^{19} - 2 q^{21} + 12 q^{22} + 8 q^{23} - 12 q^{24} + 14 q^{27} + 22 q^{28} - 2 q^{29} - 10 q^{31} + 17 q^{32} - 18 q^{33} + 8 q^{34} + 17 q^{36} + 20 q^{37} - 10 q^{38} + 2 q^{41} - 22 q^{42} + 12 q^{43} + 12 q^{44} - 8 q^{46} - 2 q^{47} - 18 q^{48} - q^{49} - 4 q^{51} - 6 q^{53} - 10 q^{54} + 24 q^{56} + 2 q^{57} - 4 q^{58} + 12 q^{59} - 6 q^{61} + 4 q^{62} + 14 q^{63} - 15 q^{64} + 12 q^{66} + 18 q^{67} + 44 q^{68} + 16 q^{69} - 8 q^{71} - 3 q^{72} + 20 q^{73} + 10 q^{74} + 2 q^{76} + 18 q^{77} + 12 q^{79} + 15 q^{81} - 14 q^{82} + 18 q^{83} - 10 q^{84} + 8 q^{86} + 2 q^{87} + 30 q^{88} + 10 q^{92} - 14 q^{93} - 4 q^{94} + 22 q^{96} - 14 q^{97} + 21 q^{98} - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51414 1.77776 0.888882 0.458137i \(-0.151483\pi\)
0.888882 + 0.458137i \(0.151483\pi\)
\(3\) −1.51414 −0.874187 −0.437094 0.899416i \(-0.643992\pi\)
−0.437094 + 0.899416i \(0.643992\pi\)
\(4\) 4.32088 2.16044
\(5\) 0 0
\(6\) −3.80675 −1.55410
\(7\) 3.32088 1.25518 0.627588 0.778545i \(-0.284042\pi\)
0.627588 + 0.778545i \(0.284042\pi\)
\(8\) 5.83502 2.06299
\(9\) −0.707389 −0.235796
\(10\) 0 0
\(11\) 2.83502 0.854791 0.427396 0.904065i \(-0.359431\pi\)
0.427396 + 0.904065i \(0.359431\pi\)
\(12\) −6.54241 −1.88863
\(13\) 0 0
\(14\) 8.34916 2.23141
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) 6.64177 1.61087 0.805433 0.592687i \(-0.201933\pi\)
0.805433 + 0.592687i \(0.201933\pi\)
\(18\) −1.77847 −0.419190
\(19\) −2.19325 −0.503167 −0.251583 0.967836i \(-0.580951\pi\)
−0.251583 + 0.967836i \(0.580951\pi\)
\(20\) 0 0
\(21\) −5.02827 −1.09726
\(22\) 7.12763 1.51962
\(23\) 0.485863 0.101309 0.0506547 0.998716i \(-0.483869\pi\)
0.0506547 + 0.998716i \(0.483869\pi\)
\(24\) −8.83502 −1.80344
\(25\) 0 0
\(26\) 0 0
\(27\) 5.61350 1.08032
\(28\) 14.3492 2.71174
\(29\) −3.32088 −0.616673 −0.308336 0.951277i \(-0.599772\pi\)
−0.308336 + 0.951277i \(0.599772\pi\)
\(30\) 0 0
\(31\) −3.80675 −0.683712 −0.341856 0.939752i \(-0.611056\pi\)
−0.341856 + 0.939752i \(0.611056\pi\)
\(32\) 3.48586 0.616219
\(33\) −4.29261 −0.747248
\(34\) 16.6983 2.86374
\(35\) 0 0
\(36\) −3.05655 −0.509425
\(37\) 9.32088 1.53234 0.766172 0.642636i \(-0.222159\pi\)
0.766172 + 0.642636i \(0.222159\pi\)
\(38\) −5.51414 −0.894511
\(39\) 0 0
\(40\) 0 0
\(41\) 1.61350 0.251986 0.125993 0.992031i \(-0.459788\pi\)
0.125993 + 0.992031i \(0.459788\pi\)
\(42\) −12.6418 −1.95067
\(43\) 0.872368 0.133035 0.0665174 0.997785i \(-0.478811\pi\)
0.0665174 + 0.997785i \(0.478811\pi\)
\(44\) 12.2498 1.84673
\(45\) 0 0
\(46\) 1.22153 0.180104
\(47\) −3.32088 −0.484401 −0.242200 0.970226i \(-0.577869\pi\)
−0.242200 + 0.970226i \(0.577869\pi\)
\(48\) −9.12763 −1.31746
\(49\) 4.02827 0.575468
\(50\) 0 0
\(51\) −10.0565 −1.40820
\(52\) 0 0
\(53\) −11.6700 −1.60300 −0.801502 0.597992i \(-0.795965\pi\)
−0.801502 + 0.597992i \(0.795965\pi\)
\(54\) 14.1131 1.92055
\(55\) 0 0
\(56\) 19.3774 2.58942
\(57\) 3.32088 0.439862
\(58\) −8.34916 −1.09630
\(59\) 8.83502 1.15022 0.575111 0.818075i \(-0.304959\pi\)
0.575111 + 0.818075i \(0.304959\pi\)
\(60\) 0 0
\(61\) −3.70739 −0.474683 −0.237341 0.971426i \(-0.576276\pi\)
−0.237341 + 0.971426i \(0.576276\pi\)
\(62\) −9.57068 −1.21548
\(63\) −2.34916 −0.295966
\(64\) −3.29261 −0.411576
\(65\) 0 0
\(66\) −10.7922 −1.32843
\(67\) 4.29261 0.524426 0.262213 0.965010i \(-0.415548\pi\)
0.262213 + 0.965010i \(0.415548\pi\)
\(68\) 28.6983 3.48018
\(69\) −0.735663 −0.0885634
\(70\) 0 0
\(71\) −2.19325 −0.260291 −0.130146 0.991495i \(-0.541544\pi\)
−0.130146 + 0.991495i \(0.541544\pi\)
\(72\) −4.12763 −0.486446
\(73\) 12.7357 1.49060 0.745298 0.666731i \(-0.232307\pi\)
0.745298 + 0.666731i \(0.232307\pi\)
\(74\) 23.4340 2.72414
\(75\) 0 0
\(76\) −9.47679 −1.08706
\(77\) 9.41478 1.07291
\(78\) 0 0
\(79\) 0.585221 0.0658425 0.0329213 0.999458i \(-0.489519\pi\)
0.0329213 + 0.999458i \(0.489519\pi\)
\(80\) 0 0
\(81\) −6.37743 −0.708604
\(82\) 4.05655 0.447971
\(83\) 7.70739 0.845996 0.422998 0.906131i \(-0.360978\pi\)
0.422998 + 0.906131i \(0.360978\pi\)
\(84\) −21.7266 −2.37057
\(85\) 0 0
\(86\) 2.19325 0.236504
\(87\) 5.02827 0.539088
\(88\) 16.5424 1.76343
\(89\) −3.41478 −0.361966 −0.180983 0.983486i \(-0.557928\pi\)
−0.180983 + 0.983486i \(0.557928\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.09936 0.218873
\(93\) 5.76394 0.597692
\(94\) −8.34916 −0.861150
\(95\) 0 0
\(96\) −5.27807 −0.538691
\(97\) 0.641769 0.0651618 0.0325809 0.999469i \(-0.489627\pi\)
0.0325809 + 0.999469i \(0.489627\pi\)
\(98\) 10.1276 1.02305
\(99\) −2.00546 −0.201557
\(100\) 0 0
\(101\) 6.97173 0.693713 0.346856 0.937918i \(-0.387249\pi\)
0.346856 + 0.937918i \(0.387249\pi\)
\(102\) −25.2835 −2.50344
\(103\) 7.18418 0.707878 0.353939 0.935268i \(-0.384842\pi\)
0.353939 + 0.935268i \(0.384842\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −29.3401 −2.84976
\(107\) 9.57068 0.925233 0.462617 0.886558i \(-0.346911\pi\)
0.462617 + 0.886558i \(0.346911\pi\)
\(108\) 24.2553 2.33396
\(109\) 10.6983 1.02471 0.512356 0.858773i \(-0.328773\pi\)
0.512356 + 0.858773i \(0.328773\pi\)
\(110\) 0 0
\(111\) −14.1131 −1.33956
\(112\) 20.0192 1.89164
\(113\) 3.22699 0.303570 0.151785 0.988414i \(-0.451498\pi\)
0.151785 + 0.988414i \(0.451498\pi\)
\(114\) 8.34916 0.781970
\(115\) 0 0
\(116\) −14.3492 −1.33229
\(117\) 0 0
\(118\) 22.2125 2.04482
\(119\) 22.0565 2.02192
\(120\) 0 0
\(121\) −2.96265 −0.269332
\(122\) −9.32088 −0.843873
\(123\) −2.44305 −0.220283
\(124\) −16.4485 −1.47712
\(125\) 0 0
\(126\) −5.90611 −0.526158
\(127\) −5.90064 −0.523597 −0.261799 0.965123i \(-0.584316\pi\)
−0.261799 + 0.965123i \(0.584316\pi\)
\(128\) −15.2498 −1.34790
\(129\) −1.32088 −0.116297
\(130\) 0 0
\(131\) 3.41478 0.298351 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(132\) −18.5479 −1.61439
\(133\) −7.28354 −0.631563
\(134\) 10.7922 0.932305
\(135\) 0 0
\(136\) 38.7549 3.32320
\(137\) −15.0848 −1.28878 −0.644392 0.764696i \(-0.722889\pi\)
−0.644392 + 0.764696i \(0.722889\pi\)
\(138\) −1.84956 −0.157445
\(139\) −20.6983 −1.75561 −0.877804 0.479020i \(-0.840992\pi\)
−0.877804 + 0.479020i \(0.840992\pi\)
\(140\) 0 0
\(141\) 5.02827 0.423457
\(142\) −5.51414 −0.462736
\(143\) 0 0
\(144\) −4.26434 −0.355361
\(145\) 0 0
\(146\) 32.0192 2.64993
\(147\) −6.09936 −0.503067
\(148\) 40.2745 3.31054
\(149\) −0.641769 −0.0525758 −0.0262879 0.999654i \(-0.508369\pi\)
−0.0262879 + 0.999654i \(0.508369\pi\)
\(150\) 0 0
\(151\) −16.6363 −1.35384 −0.676922 0.736055i \(-0.736686\pi\)
−0.676922 + 0.736055i \(0.736686\pi\)
\(152\) −12.7977 −1.03803
\(153\) −4.69832 −0.379836
\(154\) 23.6700 1.90739
\(155\) 0 0
\(156\) 0 0
\(157\) −10.2553 −0.818459 −0.409230 0.912431i \(-0.634202\pi\)
−0.409230 + 0.912431i \(0.634202\pi\)
\(158\) 1.47133 0.117052
\(159\) 17.6700 1.40133
\(160\) 0 0
\(161\) 1.61350 0.127161
\(162\) −16.0337 −1.25973
\(163\) −1.37743 −0.107889 −0.0539444 0.998544i \(-0.517179\pi\)
−0.0539444 + 0.998544i \(0.517179\pi\)
\(164\) 6.97173 0.544400
\(165\) 0 0
\(166\) 19.3774 1.50398
\(167\) 17.5761 1.36008 0.680042 0.733174i \(-0.261962\pi\)
0.680042 + 0.733174i \(0.261962\pi\)
\(168\) −29.3401 −2.26364
\(169\) 0 0
\(170\) 0 0
\(171\) 1.55148 0.118645
\(172\) 3.76940 0.287414
\(173\) −6.31181 −0.479878 −0.239939 0.970788i \(-0.577127\pi\)
−0.239939 + 0.970788i \(0.577127\pi\)
\(174\) 12.6418 0.958370
\(175\) 0 0
\(176\) 17.0903 1.28823
\(177\) −13.3774 −1.00551
\(178\) −8.58522 −0.643490
\(179\) −9.08482 −0.679031 −0.339516 0.940600i \(-0.610263\pi\)
−0.339516 + 0.940600i \(0.610263\pi\)
\(180\) 0 0
\(181\) 12.0192 0.893380 0.446690 0.894689i \(-0.352603\pi\)
0.446690 + 0.894689i \(0.352603\pi\)
\(182\) 0 0
\(183\) 5.61350 0.414962
\(184\) 2.83502 0.209001
\(185\) 0 0
\(186\) 14.4913 1.06256
\(187\) 18.8296 1.37695
\(188\) −14.3492 −1.04652
\(189\) 18.6418 1.35599
\(190\) 0 0
\(191\) 13.2835 0.961163 0.480582 0.876950i \(-0.340426\pi\)
0.480582 + 0.876950i \(0.340426\pi\)
\(192\) 4.98546 0.359795
\(193\) 16.3684 1.17822 0.589110 0.808053i \(-0.299478\pi\)
0.589110 + 0.808053i \(0.299478\pi\)
\(194\) 1.61350 0.115842
\(195\) 0 0
\(196\) 17.4057 1.24326
\(197\) 14.8970 1.06137 0.530685 0.847569i \(-0.321935\pi\)
0.530685 + 0.847569i \(0.321935\pi\)
\(198\) −5.04201 −0.358320
\(199\) −16.3118 −1.15631 −0.578157 0.815926i \(-0.696228\pi\)
−0.578157 + 0.815926i \(0.696228\pi\)
\(200\) 0 0
\(201\) −6.49960 −0.458446
\(202\) 17.5279 1.23326
\(203\) −11.0283 −0.774033
\(204\) −43.4532 −3.04233
\(205\) 0 0
\(206\) 18.0620 1.25844
\(207\) −0.343694 −0.0238884
\(208\) 0 0
\(209\) −6.21792 −0.430102
\(210\) 0 0
\(211\) 11.8013 0.812434 0.406217 0.913777i \(-0.366848\pi\)
0.406217 + 0.913777i \(0.366848\pi\)
\(212\) −50.4249 −3.46320
\(213\) 3.32088 0.227543
\(214\) 24.0620 1.64485
\(215\) 0 0
\(216\) 32.7549 2.22869
\(217\) −12.6418 −0.858179
\(218\) 26.8970 1.82170
\(219\) −19.2835 −1.30306
\(220\) 0 0
\(221\) 0 0
\(222\) −35.4823 −2.38141
\(223\) −20.0192 −1.34058 −0.670292 0.742097i \(-0.733831\pi\)
−0.670292 + 0.742097i \(0.733831\pi\)
\(224\) 11.5761 0.773464
\(225\) 0 0
\(226\) 8.11310 0.539675
\(227\) −22.4623 −1.49087 −0.745436 0.666577i \(-0.767759\pi\)
−0.745436 + 0.666577i \(0.767759\pi\)
\(228\) 14.3492 0.950296
\(229\) 10.3684 0.685160 0.342580 0.939489i \(-0.388699\pi\)
0.342580 + 0.939489i \(0.388699\pi\)
\(230\) 0 0
\(231\) −14.2553 −0.937928
\(232\) −19.3774 −1.27219
\(233\) −11.3582 −0.744102 −0.372051 0.928212i \(-0.621345\pi\)
−0.372051 + 0.928212i \(0.621345\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 38.1751 2.48499
\(237\) −0.886105 −0.0575587
\(238\) 55.4532 3.59450
\(239\) −28.1186 −1.81884 −0.909419 0.415881i \(-0.863473\pi\)
−0.909419 + 0.415881i \(0.863473\pi\)
\(240\) 0 0
\(241\) 15.0848 0.971699 0.485849 0.874043i \(-0.338510\pi\)
0.485849 + 0.874043i \(0.338510\pi\)
\(242\) −7.44852 −0.478809
\(243\) −7.18418 −0.460865
\(244\) −16.0192 −1.02552
\(245\) 0 0
\(246\) −6.14217 −0.391610
\(247\) 0 0
\(248\) −22.2125 −1.41049
\(249\) −11.6700 −0.739559
\(250\) 0 0
\(251\) 1.15951 0.0731879 0.0365940 0.999330i \(-0.488349\pi\)
0.0365940 + 0.999330i \(0.488349\pi\)
\(252\) −10.1504 −0.639418
\(253\) 1.37743 0.0865984
\(254\) −14.8350 −0.930832
\(255\) 0 0
\(256\) −31.7549 −1.98468
\(257\) 0.829557 0.0517464 0.0258732 0.999665i \(-0.491763\pi\)
0.0258732 + 0.999665i \(0.491763\pi\)
\(258\) −3.32088 −0.206749
\(259\) 30.9536 1.92336
\(260\) 0 0
\(261\) 2.34916 0.145409
\(262\) 8.58522 0.530397
\(263\) 25.6272 1.58024 0.790121 0.612950i \(-0.210017\pi\)
0.790121 + 0.612950i \(0.210017\pi\)
\(264\) −25.0475 −1.54157
\(265\) 0 0
\(266\) −18.3118 −1.12277
\(267\) 5.17044 0.316426
\(268\) 18.5479 1.13299
\(269\) 5.02827 0.306579 0.153290 0.988181i \(-0.451013\pi\)
0.153290 + 0.988181i \(0.451013\pi\)
\(270\) 0 0
\(271\) 18.8916 1.14758 0.573791 0.819002i \(-0.305472\pi\)
0.573791 + 0.819002i \(0.305472\pi\)
\(272\) 40.0384 2.42768
\(273\) 0 0
\(274\) −37.9253 −2.29115
\(275\) 0 0
\(276\) −3.17872 −0.191336
\(277\) −23.7831 −1.42899 −0.714495 0.699640i \(-0.753344\pi\)
−0.714495 + 0.699640i \(0.753344\pi\)
\(278\) −52.0384 −3.12106
\(279\) 2.69285 0.161217
\(280\) 0 0
\(281\) −31.5953 −1.88482 −0.942410 0.334459i \(-0.891446\pi\)
−0.942410 + 0.334459i \(0.891446\pi\)
\(282\) 12.6418 0.752806
\(283\) −6.35462 −0.377743 −0.188872 0.982002i \(-0.560483\pi\)
−0.188872 + 0.982002i \(0.560483\pi\)
\(284\) −9.47679 −0.562344
\(285\) 0 0
\(286\) 0 0
\(287\) 5.35823 0.316286
\(288\) −2.46586 −0.145302
\(289\) 27.1131 1.59489
\(290\) 0 0
\(291\) −0.971726 −0.0569636
\(292\) 55.0293 3.22035
\(293\) −8.03735 −0.469547 −0.234773 0.972050i \(-0.575435\pi\)
−0.234773 + 0.972050i \(0.575435\pi\)
\(294\) −15.3346 −0.894333
\(295\) 0 0
\(296\) 54.3876 3.16121
\(297\) 15.9144 0.923446
\(298\) −1.61350 −0.0934673
\(299\) 0 0
\(300\) 0 0
\(301\) 2.89703 0.166982
\(302\) −41.8259 −2.40681
\(303\) −10.5561 −0.606435
\(304\) −13.2215 −0.758307
\(305\) 0 0
\(306\) −11.8122 −0.675259
\(307\) −15.8205 −0.902923 −0.451461 0.892291i \(-0.649097\pi\)
−0.451461 + 0.892291i \(0.649097\pi\)
\(308\) 40.6802 2.31797
\(309\) −10.8778 −0.618818
\(310\) 0 0
\(311\) 24.3118 1.37860 0.689298 0.724478i \(-0.257919\pi\)
0.689298 + 0.724478i \(0.257919\pi\)
\(312\) 0 0
\(313\) 11.2270 0.634587 0.317294 0.948327i \(-0.397226\pi\)
0.317294 + 0.948327i \(0.397226\pi\)
\(314\) −25.7831 −1.45503
\(315\) 0 0
\(316\) 2.52867 0.142249
\(317\) −16.1504 −0.907099 −0.453550 0.891231i \(-0.649843\pi\)
−0.453550 + 0.891231i \(0.649843\pi\)
\(318\) 44.4249 2.49123
\(319\) −9.41478 −0.527126
\(320\) 0 0
\(321\) −14.4913 −0.808827
\(322\) 4.05655 0.226063
\(323\) −14.5671 −0.810534
\(324\) −27.5561 −1.53090
\(325\) 0 0
\(326\) −3.46305 −0.191801
\(327\) −16.1987 −0.895791
\(328\) 9.41478 0.519844
\(329\) −11.0283 −0.608008
\(330\) 0 0
\(331\) −30.0620 −1.65236 −0.826179 0.563408i \(-0.809490\pi\)
−0.826179 + 0.563408i \(0.809490\pi\)
\(332\) 33.3027 1.82773
\(333\) −6.59349 −0.361321
\(334\) 44.1888 2.41791
\(335\) 0 0
\(336\) −30.3118 −1.65364
\(337\) 17.4713 0.951724 0.475862 0.879520i \(-0.342136\pi\)
0.475862 + 0.879520i \(0.342136\pi\)
\(338\) 0 0
\(339\) −4.88611 −0.265377
\(340\) 0 0
\(341\) −10.7922 −0.584431
\(342\) 3.90064 0.210923
\(343\) −9.86876 −0.532863
\(344\) 5.09029 0.274450
\(345\) 0 0
\(346\) −15.8688 −0.853110
\(347\) −33.2407 −1.78446 −0.892228 0.451585i \(-0.850859\pi\)
−0.892228 + 0.451585i \(0.850859\pi\)
\(348\) 21.7266 1.16467
\(349\) 20.8970 1.11859 0.559296 0.828968i \(-0.311071\pi\)
0.559296 + 0.828968i \(0.311071\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 9.88250 0.526739
\(353\) 12.5479 0.667856 0.333928 0.942599i \(-0.391626\pi\)
0.333928 + 0.942599i \(0.391626\pi\)
\(354\) −33.6327 −1.78756
\(355\) 0 0
\(356\) −14.7549 −0.782006
\(357\) −33.3966 −1.76754
\(358\) −22.8405 −1.20716
\(359\) 2.97719 0.157130 0.0785650 0.996909i \(-0.474966\pi\)
0.0785650 + 0.996909i \(0.474966\pi\)
\(360\) 0 0
\(361\) −14.1896 −0.746823
\(362\) 30.2179 1.58822
\(363\) 4.48586 0.235447
\(364\) 0 0
\(365\) 0 0
\(366\) 14.1131 0.737703
\(367\) 30.5990 1.59725 0.798626 0.601827i \(-0.205560\pi\)
0.798626 + 0.601827i \(0.205560\pi\)
\(368\) 2.92892 0.152680
\(369\) −1.14137 −0.0594173
\(370\) 0 0
\(371\) −38.7549 −2.01205
\(372\) 24.9053 1.29128
\(373\) −18.2553 −0.945222 −0.472611 0.881271i \(-0.656688\pi\)
−0.472611 + 0.881271i \(0.656688\pi\)
\(374\) 47.3401 2.44790
\(375\) 0 0
\(376\) −19.3774 −0.999315
\(377\) 0 0
\(378\) 46.8680 2.41063
\(379\) −15.9945 −0.821584 −0.410792 0.911729i \(-0.634748\pi\)
−0.410792 + 0.911729i \(0.634748\pi\)
\(380\) 0 0
\(381\) 8.93438 0.457722
\(382\) 33.3966 1.70872
\(383\) −2.16137 −0.110441 −0.0552204 0.998474i \(-0.517586\pi\)
−0.0552204 + 0.998474i \(0.517586\pi\)
\(384\) 23.0903 1.17832
\(385\) 0 0
\(386\) 41.1523 2.09460
\(387\) −0.617104 −0.0313691
\(388\) 2.77301 0.140778
\(389\) −0.453981 −0.0230177 −0.0115089 0.999934i \(-0.503663\pi\)
−0.0115089 + 0.999934i \(0.503663\pi\)
\(390\) 0 0
\(391\) 3.22699 0.163196
\(392\) 23.5051 1.18719
\(393\) −5.17044 −0.260814
\(394\) 37.4532 1.88686
\(395\) 0 0
\(396\) −8.66538 −0.435452
\(397\) −29.2462 −1.46782 −0.733912 0.679244i \(-0.762308\pi\)
−0.733912 + 0.679244i \(0.762308\pi\)
\(398\) −41.0101 −2.05565
\(399\) 11.0283 0.552104
\(400\) 0 0
\(401\) −15.2270 −0.760400 −0.380200 0.924904i \(-0.624145\pi\)
−0.380200 + 0.924904i \(0.624145\pi\)
\(402\) −16.3409 −0.815009
\(403\) 0 0
\(404\) 30.1240 1.49873
\(405\) 0 0
\(406\) −27.7266 −1.37605
\(407\) 26.4249 1.30983
\(408\) −58.6802 −2.90510
\(409\) 23.6700 1.17041 0.585204 0.810886i \(-0.301014\pi\)
0.585204 + 0.810886i \(0.301014\pi\)
\(410\) 0 0
\(411\) 22.8405 1.12664
\(412\) 31.0420 1.52933
\(413\) 29.3401 1.44373
\(414\) −0.864095 −0.0424679
\(415\) 0 0
\(416\) 0 0
\(417\) 31.3401 1.53473
\(418\) −15.6327 −0.764620
\(419\) −5.67004 −0.277000 −0.138500 0.990362i \(-0.544228\pi\)
−0.138500 + 0.990362i \(0.544228\pi\)
\(420\) 0 0
\(421\) 26.3009 1.28183 0.640913 0.767613i \(-0.278556\pi\)
0.640913 + 0.767613i \(0.278556\pi\)
\(422\) 29.6700 1.44432
\(423\) 2.34916 0.114220
\(424\) −68.0950 −3.30698
\(425\) 0 0
\(426\) 8.34916 0.404518
\(427\) −12.3118 −0.595810
\(428\) 41.3538 1.99891
\(429\) 0 0
\(430\) 0 0
\(431\) −5.09029 −0.245190 −0.122595 0.992457i \(-0.539122\pi\)
−0.122595 + 0.992457i \(0.539122\pi\)
\(432\) 33.8397 1.62811
\(433\) −24.0565 −1.15608 −0.578042 0.816007i \(-0.696183\pi\)
−0.578042 + 0.816007i \(0.696183\pi\)
\(434\) −31.7831 −1.52564
\(435\) 0 0
\(436\) 46.2262 2.21383
\(437\) −1.06562 −0.0509755
\(438\) −48.4815 −2.31653
\(439\) −7.21606 −0.344404 −0.172202 0.985062i \(-0.555088\pi\)
−0.172202 + 0.985062i \(0.555088\pi\)
\(440\) 0 0
\(441\) −2.84956 −0.135693
\(442\) 0 0
\(443\) −27.3829 −1.30100 −0.650500 0.759506i \(-0.725441\pi\)
−0.650500 + 0.759506i \(0.725441\pi\)
\(444\) −60.9811 −2.89403
\(445\) 0 0
\(446\) −50.3310 −2.38324
\(447\) 0.971726 0.0459611
\(448\) −10.9344 −0.516601
\(449\) −1.42571 −0.0672833 −0.0336416 0.999434i \(-0.510710\pi\)
−0.0336416 + 0.999434i \(0.510710\pi\)
\(450\) 0 0
\(451\) 4.57429 0.215395
\(452\) 13.9435 0.655845
\(453\) 25.1896 1.18351
\(454\) −56.4732 −2.65042
\(455\) 0 0
\(456\) 19.3774 0.907431
\(457\) −35.3219 −1.65229 −0.826145 0.563457i \(-0.809471\pi\)
−0.826145 + 0.563457i \(0.809471\pi\)
\(458\) 26.0675 1.21805
\(459\) 37.2835 1.74025
\(460\) 0 0
\(461\) −29.1979 −1.35988 −0.679941 0.733267i \(-0.737995\pi\)
−0.679941 + 0.733267i \(0.737995\pi\)
\(462\) −35.8397 −1.66741
\(463\) −3.32088 −0.154335 −0.0771673 0.997018i \(-0.524588\pi\)
−0.0771673 + 0.997018i \(0.524588\pi\)
\(464\) −20.0192 −0.929368
\(465\) 0 0
\(466\) −28.5561 −1.32284
\(467\) −21.5525 −0.997333 −0.498666 0.866794i \(-0.666177\pi\)
−0.498666 + 0.866794i \(0.666177\pi\)
\(468\) 0 0
\(469\) 14.2553 0.658247
\(470\) 0 0
\(471\) 15.5279 0.715487
\(472\) 51.5525 2.37290
\(473\) 2.47318 0.113717
\(474\) −2.22779 −0.102326
\(475\) 0 0
\(476\) 95.3038 4.36824
\(477\) 8.25526 0.377983
\(478\) −70.6939 −3.23346
\(479\) −5.42024 −0.247657 −0.123829 0.992304i \(-0.539517\pi\)
−0.123829 + 0.992304i \(0.539517\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 37.9253 1.72745
\(483\) −2.44305 −0.111163
\(484\) −12.8013 −0.581877
\(485\) 0 0
\(486\) −18.0620 −0.819309
\(487\) 17.4521 0.790831 0.395416 0.918502i \(-0.370601\pi\)
0.395416 + 0.918502i \(0.370601\pi\)
\(488\) −21.6327 −0.979266
\(489\) 2.08562 0.0943150
\(490\) 0 0
\(491\) 24.6236 1.11125 0.555624 0.831433i \(-0.312479\pi\)
0.555624 + 0.831433i \(0.312479\pi\)
\(492\) −10.5561 −0.475908
\(493\) −22.0565 −0.993377
\(494\) 0 0
\(495\) 0 0
\(496\) −22.9481 −1.03040
\(497\) −7.28354 −0.326711
\(498\) −29.3401 −1.31476
\(499\) −32.3629 −1.44876 −0.724381 0.689400i \(-0.757874\pi\)
−0.724381 + 0.689400i \(0.757874\pi\)
\(500\) 0 0
\(501\) −26.6127 −1.18897
\(502\) 2.91518 0.130111
\(503\) 26.7411 1.19233 0.596164 0.802863i \(-0.296691\pi\)
0.596164 + 0.802863i \(0.296691\pi\)
\(504\) −13.7074 −0.610576
\(505\) 0 0
\(506\) 3.46305 0.153951
\(507\) 0 0
\(508\) −25.4960 −1.13120
\(509\) −9.74474 −0.431928 −0.215964 0.976401i \(-0.569289\pi\)
−0.215964 + 0.976401i \(0.569289\pi\)
\(510\) 0 0
\(511\) 42.2937 1.87096
\(512\) −49.3365 −2.18038
\(513\) −12.3118 −0.543580
\(514\) 2.08562 0.0919928
\(515\) 0 0
\(516\) −5.70739 −0.251254
\(517\) −9.41478 −0.414061
\(518\) 77.8215 3.41928
\(519\) 9.55695 0.419503
\(520\) 0 0
\(521\) −11.4340 −0.500932 −0.250466 0.968125i \(-0.580584\pi\)
−0.250466 + 0.968125i \(0.580584\pi\)
\(522\) 5.90611 0.258503
\(523\) −33.6272 −1.47042 −0.735208 0.677841i \(-0.762916\pi\)
−0.735208 + 0.677841i \(0.762916\pi\)
\(524\) 14.7549 0.644569
\(525\) 0 0
\(526\) 64.4304 2.80930
\(527\) −25.2835 −1.10137
\(528\) −25.8770 −1.12615
\(529\) −22.7639 −0.989736
\(530\) 0 0
\(531\) −6.24980 −0.271218
\(532\) −31.4713 −1.36446
\(533\) 0 0
\(534\) 12.9992 0.562530
\(535\) 0 0
\(536\) 25.0475 1.08189
\(537\) 13.7557 0.593601
\(538\) 12.6418 0.545025
\(539\) 11.4202 0.491905
\(540\) 0 0
\(541\) −26.4431 −1.13688 −0.568438 0.822726i \(-0.692452\pi\)
−0.568438 + 0.822726i \(0.692452\pi\)
\(542\) 47.4960 2.04013
\(543\) −18.1987 −0.780982
\(544\) 23.1523 0.992647
\(545\) 0 0
\(546\) 0 0
\(547\) 21.9681 0.939289 0.469644 0.882856i \(-0.344382\pi\)
0.469644 + 0.882856i \(0.344382\pi\)
\(548\) −65.1798 −2.78434
\(549\) 2.62257 0.111928
\(550\) 0 0
\(551\) 7.28354 0.310289
\(552\) −4.29261 −0.182706
\(553\) 1.94345 0.0826440
\(554\) −59.7941 −2.54041
\(555\) 0 0
\(556\) −89.4350 −3.79289
\(557\) −39.3027 −1.66531 −0.832655 0.553792i \(-0.813180\pi\)
−0.832655 + 0.553792i \(0.813180\pi\)
\(558\) 6.77020 0.286605
\(559\) 0 0
\(560\) 0 0
\(561\) −28.5105 −1.20372
\(562\) −79.4350 −3.35076
\(563\) 23.0420 0.971105 0.485552 0.874208i \(-0.338618\pi\)
0.485552 + 0.874208i \(0.338618\pi\)
\(564\) 21.7266 0.914854
\(565\) 0 0
\(566\) −15.9764 −0.671538
\(567\) −21.1787 −0.889422
\(568\) −12.7977 −0.536979
\(569\) −3.63270 −0.152291 −0.0761453 0.997097i \(-0.524261\pi\)
−0.0761453 + 0.997097i \(0.524261\pi\)
\(570\) 0 0
\(571\) −43.3785 −1.81533 −0.907667 0.419692i \(-0.862138\pi\)
−0.907667 + 0.419692i \(0.862138\pi\)
\(572\) 0 0
\(573\) −20.1131 −0.840237
\(574\) 13.4713 0.562282
\(575\) 0 0
\(576\) 2.32916 0.0970482
\(577\) 8.84876 0.368379 0.184189 0.982891i \(-0.441034\pi\)
0.184189 + 0.982891i \(0.441034\pi\)
\(578\) 68.1660 2.83533
\(579\) −24.7839 −1.02999
\(580\) 0 0
\(581\) 25.5953 1.06187
\(582\) −2.44305 −0.101268
\(583\) −33.0848 −1.37023
\(584\) 74.3129 3.07509
\(585\) 0 0
\(586\) −20.2070 −0.834743
\(587\) −15.3209 −0.632361 −0.316180 0.948699i \(-0.602400\pi\)
−0.316180 + 0.948699i \(0.602400\pi\)
\(588\) −26.3546 −1.08685
\(589\) 8.34916 0.344021
\(590\) 0 0
\(591\) −22.5561 −0.927836
\(592\) 56.1888 2.30935
\(593\) −4.52867 −0.185970 −0.0929852 0.995667i \(-0.529641\pi\)
−0.0929852 + 0.995667i \(0.529641\pi\)
\(594\) 40.0109 1.64167
\(595\) 0 0
\(596\) −2.77301 −0.113587
\(597\) 24.6983 1.01083
\(598\) 0 0
\(599\) 23.0283 0.940910 0.470455 0.882424i \(-0.344090\pi\)
0.470455 + 0.882424i \(0.344090\pi\)
\(600\) 0 0
\(601\) −12.0565 −0.491797 −0.245898 0.969296i \(-0.579083\pi\)
−0.245898 + 0.969296i \(0.579083\pi\)
\(602\) 7.28354 0.296855
\(603\) −3.03655 −0.123658
\(604\) −71.8836 −2.92490
\(605\) 0 0
\(606\) −26.5396 −1.07810
\(607\) 8.99639 0.365152 0.182576 0.983192i \(-0.441556\pi\)
0.182576 + 0.983192i \(0.441556\pi\)
\(608\) −7.64538 −0.310061
\(609\) 16.6983 0.676650
\(610\) 0 0
\(611\) 0 0
\(612\) −20.3009 −0.820615
\(613\) −14.1131 −0.570023 −0.285011 0.958524i \(-0.591997\pi\)
−0.285011 + 0.958524i \(0.591997\pi\)
\(614\) −39.7749 −1.60518
\(615\) 0 0
\(616\) 54.9354 2.21341
\(617\) −2.46120 −0.0990841 −0.0495420 0.998772i \(-0.515776\pi\)
−0.0495420 + 0.998772i \(0.515776\pi\)
\(618\) −27.3484 −1.10011
\(619\) −5.73205 −0.230391 −0.115195 0.993343i \(-0.536749\pi\)
−0.115195 + 0.993343i \(0.536749\pi\)
\(620\) 0 0
\(621\) 2.72739 0.109446
\(622\) 61.1232 2.45082
\(623\) −11.3401 −0.454331
\(624\) 0 0
\(625\) 0 0
\(626\) 28.2262 1.12815
\(627\) 9.41478 0.375990
\(628\) −44.3118 −1.76823
\(629\) 61.9072 2.46840
\(630\) 0 0
\(631\) 5.27807 0.210117 0.105058 0.994466i \(-0.466497\pi\)
0.105058 + 0.994466i \(0.466497\pi\)
\(632\) 3.41478 0.135833
\(633\) −17.8688 −0.710219
\(634\) −40.6044 −1.61261
\(635\) 0 0
\(636\) 76.3502 3.02748
\(637\) 0 0
\(638\) −23.6700 −0.937106
\(639\) 1.55148 0.0613757
\(640\) 0 0
\(641\) −42.9992 −1.69837 −0.849183 0.528098i \(-0.822905\pi\)
−0.849183 + 0.528098i \(0.822905\pi\)
\(642\) −36.4332 −1.43790
\(643\) 43.3593 1.70992 0.854962 0.518691i \(-0.173581\pi\)
0.854962 + 0.518691i \(0.173581\pi\)
\(644\) 6.97173 0.274724
\(645\) 0 0
\(646\) −36.6236 −1.44094
\(647\) 19.6454 0.772339 0.386170 0.922428i \(-0.373798\pi\)
0.386170 + 0.922428i \(0.373798\pi\)
\(648\) −37.2125 −1.46184
\(649\) 25.0475 0.983199
\(650\) 0 0
\(651\) 19.1414 0.750209
\(652\) −5.95173 −0.233088
\(653\) −9.93252 −0.388690 −0.194345 0.980933i \(-0.562258\pi\)
−0.194345 + 0.980933i \(0.562258\pi\)
\(654\) −40.7258 −1.59250
\(655\) 0 0
\(656\) 9.72659 0.379760
\(657\) −9.00907 −0.351477
\(658\) −27.7266 −1.08090
\(659\) 10.1806 0.396579 0.198289 0.980144i \(-0.436461\pi\)
0.198289 + 0.980144i \(0.436461\pi\)
\(660\) 0 0
\(661\) 32.9427 1.28132 0.640660 0.767824i \(-0.278661\pi\)
0.640660 + 0.767824i \(0.278661\pi\)
\(662\) −75.5800 −2.93750
\(663\) 0 0
\(664\) 44.9728 1.74528
\(665\) 0 0
\(666\) −16.5769 −0.642344
\(667\) −1.61350 −0.0624748
\(668\) 75.9445 2.93838
\(669\) 30.3118 1.17192
\(670\) 0 0
\(671\) −10.5105 −0.405754
\(672\) −17.5279 −0.676152
\(673\) 11.2270 0.432769 0.216384 0.976308i \(-0.430574\pi\)
0.216384 + 0.976308i \(0.430574\pi\)
\(674\) 43.9253 1.69194
\(675\) 0 0
\(676\) 0 0
\(677\) 15.7447 0.605119 0.302560 0.953130i \(-0.402159\pi\)
0.302560 + 0.953130i \(0.402159\pi\)
\(678\) −12.2843 −0.471777
\(679\) 2.13124 0.0817895
\(680\) 0 0
\(681\) 34.0109 1.30330
\(682\) −27.1331 −1.03898
\(683\) 50.3492 1.92656 0.963279 0.268504i \(-0.0865293\pi\)
0.963279 + 0.268504i \(0.0865293\pi\)
\(684\) 6.70378 0.256325
\(685\) 0 0
\(686\) −24.8114 −0.947304
\(687\) −15.6991 −0.598959
\(688\) 5.25887 0.200493
\(689\) 0 0
\(690\) 0 0
\(691\) −8.19325 −0.311686 −0.155843 0.987782i \(-0.549809\pi\)
−0.155843 + 0.987782i \(0.549809\pi\)
\(692\) −27.2726 −1.03675
\(693\) −6.65991 −0.252989
\(694\) −83.5717 −3.17234
\(695\) 0 0
\(696\) 29.3401 1.11213
\(697\) 10.7165 0.405915
\(698\) 52.5380 1.98859
\(699\) 17.1979 0.650485
\(700\) 0 0
\(701\) −8.25526 −0.311797 −0.155899 0.987773i \(-0.549827\pi\)
−0.155899 + 0.987773i \(0.549827\pi\)
\(702\) 0 0
\(703\) −20.4431 −0.771024
\(704\) −9.33462 −0.351812
\(705\) 0 0
\(706\) 31.5471 1.18729
\(707\) 23.1523 0.870732
\(708\) −57.8023 −2.17234
\(709\) −37.7831 −1.41898 −0.709488 0.704718i \(-0.751074\pi\)
−0.709488 + 0.704718i \(0.751074\pi\)
\(710\) 0 0
\(711\) −0.413979 −0.0155254
\(712\) −19.9253 −0.746732
\(713\) −1.84956 −0.0692665
\(714\) −83.9637 −3.14226
\(715\) 0 0
\(716\) −39.2545 −1.46701
\(717\) 42.5753 1.59001
\(718\) 7.48506 0.279340
\(719\) −22.2443 −0.829574 −0.414787 0.909919i \(-0.636144\pi\)
−0.414787 + 0.909919i \(0.636144\pi\)
\(720\) 0 0
\(721\) 23.8578 0.888512
\(722\) −35.6747 −1.32768
\(723\) −22.8405 −0.849447
\(724\) 51.9336 1.93010
\(725\) 0 0
\(726\) 11.2781 0.418569
\(727\) 30.0812 1.11565 0.557825 0.829958i \(-0.311636\pi\)
0.557825 + 0.829958i \(0.311636\pi\)
\(728\) 0 0
\(729\) 30.0101 1.11149
\(730\) 0 0
\(731\) 5.79407 0.214301
\(732\) 24.2553 0.896500
\(733\) −35.0101 −1.29313 −0.646564 0.762859i \(-0.723795\pi\)
−0.646564 + 0.762859i \(0.723795\pi\)
\(734\) 76.9300 2.83954
\(735\) 0 0
\(736\) 1.69365 0.0624288
\(737\) 12.1696 0.448275
\(738\) −2.86956 −0.105630
\(739\) −42.6856 −1.57022 −0.785108 0.619359i \(-0.787393\pi\)
−0.785108 + 0.619359i \(0.787393\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −97.4350 −3.57695
\(743\) 13.6892 0.502210 0.251105 0.967960i \(-0.419206\pi\)
0.251105 + 0.967960i \(0.419206\pi\)
\(744\) 33.6327 1.23303
\(745\) 0 0
\(746\) −45.8962 −1.68038
\(747\) −5.45213 −0.199483
\(748\) 81.3603 2.97483
\(749\) 31.7831 1.16133
\(750\) 0 0
\(751\) 17.7831 0.648916 0.324458 0.945900i \(-0.394818\pi\)
0.324458 + 0.945900i \(0.394818\pi\)
\(752\) −20.0192 −0.730025
\(753\) −1.75566 −0.0639799
\(754\) 0 0
\(755\) 0 0
\(756\) 80.5489 2.92954
\(757\) 26.9717 0.980304 0.490152 0.871637i \(-0.336941\pi\)
0.490152 + 0.871637i \(0.336941\pi\)
\(758\) −40.2125 −1.46058
\(759\) −2.08562 −0.0757032
\(760\) 0 0
\(761\) 35.1523 1.27427 0.637135 0.770752i \(-0.280119\pi\)
0.637135 + 0.770752i \(0.280119\pi\)
\(762\) 22.4623 0.813722
\(763\) 35.5279 1.28620
\(764\) 57.3966 2.07654
\(765\) 0 0
\(766\) −5.43398 −0.196338
\(767\) 0 0
\(768\) 48.0812 1.73498
\(769\) −23.1523 −0.834893 −0.417447 0.908701i \(-0.637075\pi\)
−0.417447 + 0.908701i \(0.637075\pi\)
\(770\) 0 0
\(771\) −1.25606 −0.0452360
\(772\) 70.7258 2.54548
\(773\) −41.6218 −1.49703 −0.748515 0.663117i \(-0.769233\pi\)
−0.748515 + 0.663117i \(0.769233\pi\)
\(774\) −1.55148 −0.0557669
\(775\) 0 0
\(776\) 3.74474 0.134428
\(777\) −46.8680 −1.68138
\(778\) −1.14137 −0.0409201
\(779\) −3.53880 −0.126791
\(780\) 0 0
\(781\) −6.21792 −0.222495
\(782\) 8.11310 0.290124
\(783\) −18.6418 −0.666202
\(784\) 24.2835 0.867269
\(785\) 0 0
\(786\) −12.9992 −0.463666
\(787\) 43.2353 1.54117 0.770585 0.637337i \(-0.219964\pi\)
0.770585 + 0.637337i \(0.219964\pi\)
\(788\) 64.3684 2.29303
\(789\) −38.8031 −1.38143
\(790\) 0 0
\(791\) 10.7165 0.381034
\(792\) −11.7019 −0.415810
\(793\) 0 0
\(794\) −73.5289 −2.60944
\(795\) 0 0
\(796\) −70.4815 −2.49815
\(797\) 14.4431 0.511599 0.255800 0.966730i \(-0.417661\pi\)
0.255800 + 0.966730i \(0.417661\pi\)
\(798\) 27.7266 0.981511
\(799\) −22.0565 −0.780305
\(800\) 0 0
\(801\) 2.41558 0.0853503
\(802\) −38.2827 −1.35181
\(803\) 36.1059 1.27415
\(804\) −28.0840 −0.990447
\(805\) 0 0
\(806\) 0 0
\(807\) −7.61350 −0.268008
\(808\) 40.6802 1.43112
\(809\) 17.8880 0.628907 0.314454 0.949273i \(-0.398179\pi\)
0.314454 + 0.949273i \(0.398179\pi\)
\(810\) 0 0
\(811\) 8.69285 0.305247 0.152624 0.988284i \(-0.451228\pi\)
0.152624 + 0.988284i \(0.451228\pi\)
\(812\) −47.6519 −1.67225
\(813\) −28.6044 −1.00320
\(814\) 66.4358 2.32857
\(815\) 0 0
\(816\) −60.6236 −2.12225
\(817\) −1.91332 −0.0669387
\(818\) 59.5097 2.08071
\(819\) 0 0
\(820\) 0 0
\(821\) 9.41478 0.328578 0.164289 0.986412i \(-0.447467\pi\)
0.164289 + 0.986412i \(0.447467\pi\)
\(822\) 57.4241 2.00290
\(823\) 30.3365 1.05746 0.528732 0.848789i \(-0.322668\pi\)
0.528732 + 0.848789i \(0.322668\pi\)
\(824\) 41.9198 1.46035
\(825\) 0 0
\(826\) 73.7650 2.56661
\(827\) −18.2361 −0.634130 −0.317065 0.948404i \(-0.602697\pi\)
−0.317065 + 0.948404i \(0.602697\pi\)
\(828\) −1.48506 −0.0516095
\(829\) 29.8023 1.03508 0.517539 0.855660i \(-0.326848\pi\)
0.517539 + 0.855660i \(0.326848\pi\)
\(830\) 0 0
\(831\) 36.0109 1.24921
\(832\) 0 0
\(833\) 26.7549 0.927001
\(834\) 78.7933 2.72839
\(835\) 0 0
\(836\) −26.8669 −0.929211
\(837\) −21.3692 −0.738626
\(838\) −14.2553 −0.492440
\(839\) 46.1004 1.59156 0.795782 0.605584i \(-0.207060\pi\)
0.795782 + 0.605584i \(0.207060\pi\)
\(840\) 0 0
\(841\) −17.9717 −0.619715
\(842\) 66.1240 2.27878
\(843\) 47.8397 1.64769
\(844\) 50.9920 1.75522
\(845\) 0 0
\(846\) 5.90611 0.203056
\(847\) −9.83863 −0.338059
\(848\) −70.3502 −2.41584
\(849\) 9.62177 0.330218
\(850\) 0 0
\(851\) 4.52867 0.155241
\(852\) 14.3492 0.491594
\(853\) −19.3774 −0.663471 −0.331735 0.943373i \(-0.607634\pi\)
−0.331735 + 0.943373i \(0.607634\pi\)
\(854\) −30.9536 −1.05921
\(855\) 0 0
\(856\) 55.8452 1.90875
\(857\) 43.4713 1.48495 0.742476 0.669873i \(-0.233651\pi\)
0.742476 + 0.669873i \(0.233651\pi\)
\(858\) 0 0
\(859\) −27.7375 −0.946392 −0.473196 0.880957i \(-0.656900\pi\)
−0.473196 + 0.880957i \(0.656900\pi\)
\(860\) 0 0
\(861\) −8.11310 −0.276494
\(862\) −12.7977 −0.435891
\(863\) 3.69646 0.125829 0.0629145 0.998019i \(-0.479960\pi\)
0.0629145 + 0.998019i \(0.479960\pi\)
\(864\) 19.5679 0.665713
\(865\) 0 0
\(866\) −60.4815 −2.05524
\(867\) −41.0529 −1.39423
\(868\) −54.6236 −1.85405
\(869\) 1.65911 0.0562816
\(870\) 0 0
\(871\) 0 0
\(872\) 62.4249 2.11397
\(873\) −0.453981 −0.0153649
\(874\) −2.67912 −0.0906224
\(875\) 0 0
\(876\) −83.3219 −2.81519
\(877\) −20.5671 −0.694501 −0.347250 0.937772i \(-0.612885\pi\)
−0.347250 + 0.937772i \(0.612885\pi\)
\(878\) −18.1422 −0.612269
\(879\) 12.1696 0.410472
\(880\) 0 0
\(881\) −14.5369 −0.489762 −0.244881 0.969553i \(-0.578749\pi\)
−0.244881 + 0.969553i \(0.578749\pi\)
\(882\) −7.16418 −0.241230
\(883\) 36.2871 1.22116 0.610580 0.791955i \(-0.290936\pi\)
0.610580 + 0.791955i \(0.290936\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −68.8444 −2.31287
\(887\) 45.2407 1.51903 0.759517 0.650487i \(-0.225435\pi\)
0.759517 + 0.650487i \(0.225435\pi\)
\(888\) −82.3502 −2.76349
\(889\) −19.5953 −0.657207
\(890\) 0 0
\(891\) −18.0802 −0.605708
\(892\) −86.5007 −2.89626
\(893\) 7.28354 0.243734
\(894\) 2.44305 0.0817079
\(895\) 0 0
\(896\) −50.6428 −1.69186
\(897\) 0 0
\(898\) −3.58442 −0.119614
\(899\) 12.6418 0.421627
\(900\) 0 0
\(901\) −77.5097 −2.58222
\(902\) 11.5004 0.382921
\(903\) −4.38650 −0.145974
\(904\) 18.8296 0.626262
\(905\) 0 0
\(906\) 63.3302 2.10401
\(907\) 11.2973 0.375120 0.187560 0.982253i \(-0.439942\pi\)
0.187560 + 0.982253i \(0.439942\pi\)
\(908\) −97.0568 −3.22094
\(909\) −4.93172 −0.163575
\(910\) 0 0
\(911\) −41.1979 −1.36495 −0.682474 0.730910i \(-0.739096\pi\)
−0.682474 + 0.730910i \(0.739096\pi\)
\(912\) 20.0192 0.662902
\(913\) 21.8506 0.723150
\(914\) −88.8042 −2.93738
\(915\) 0 0
\(916\) 44.8005 1.48025
\(917\) 11.3401 0.374483
\(918\) 93.7359 3.09375
\(919\) 1.74474 0.0575535 0.0287768 0.999586i \(-0.490839\pi\)
0.0287768 + 0.999586i \(0.490839\pi\)
\(920\) 0 0
\(921\) 23.9544 0.789324
\(922\) −73.4076 −2.41755
\(923\) 0 0
\(924\) −61.5953 −2.02634
\(925\) 0 0
\(926\) −8.34916 −0.274370
\(927\) −5.08201 −0.166915
\(928\) −11.5761 −0.380006
\(929\) −15.5569 −0.510407 −0.255203 0.966887i \(-0.582142\pi\)
−0.255203 + 0.966887i \(0.582142\pi\)
\(930\) 0 0
\(931\) −8.83502 −0.289556
\(932\) −49.0776 −1.60759
\(933\) −36.8114 −1.20515
\(934\) −54.1860 −1.77302
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0384 −0.850638 −0.425319 0.905044i \(-0.639838\pi\)
−0.425319 + 0.905044i \(0.639838\pi\)
\(938\) 35.8397 1.17021
\(939\) −16.9992 −0.554748
\(940\) 0 0
\(941\) −50.8789 −1.65860 −0.829302 0.558800i \(-0.811262\pi\)
−0.829302 + 0.558800i \(0.811262\pi\)
\(942\) 39.0392 1.27197
\(943\) 0.783938 0.0255285
\(944\) 53.2599 1.73346
\(945\) 0 0
\(946\) 6.21792 0.202162
\(947\) 37.8770 1.23084 0.615419 0.788200i \(-0.288987\pi\)
0.615419 + 0.788200i \(0.288987\pi\)
\(948\) −3.82876 −0.124352
\(949\) 0 0
\(950\) 0 0
\(951\) 24.4540 0.792975
\(952\) 128.700 4.17120
\(953\) 32.0950 1.03966 0.519829 0.854271i \(-0.325996\pi\)
0.519829 + 0.854271i \(0.325996\pi\)
\(954\) 20.7549 0.671964
\(955\) 0 0
\(956\) −121.497 −3.92950
\(957\) 14.2553 0.460807
\(958\) −13.6272 −0.440276
\(959\) −50.0950 −1.61765
\(960\) 0 0
\(961\) −16.5087 −0.532538
\(962\) 0 0
\(963\) −6.77020 −0.218167
\(964\) 65.1798 2.09930
\(965\) 0 0
\(966\) −6.14217 −0.197621
\(967\) −24.4057 −0.784835 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(968\) −17.2871 −0.555630
\(969\) 22.0565 0.708558
\(970\) 0 0
\(971\) −36.9354 −1.18531 −0.592657 0.805455i \(-0.701921\pi\)
−0.592657 + 0.805455i \(0.701921\pi\)
\(972\) −31.0420 −0.995673
\(973\) −68.7367 −2.20360
\(974\) 43.8770 1.40591
\(975\) 0 0
\(976\) −22.3492 −0.715379
\(977\) −25.1715 −0.805308 −0.402654 0.915352i \(-0.631912\pi\)
−0.402654 + 0.915352i \(0.631912\pi\)
\(978\) 5.24354 0.167670
\(979\) −9.68097 −0.309405
\(980\) 0 0
\(981\) −7.56788 −0.241624
\(982\) 61.9072 1.97554
\(983\) 13.1896 0.420684 0.210342 0.977628i \(-0.432542\pi\)
0.210342 + 0.977628i \(0.432542\pi\)
\(984\) −14.2553 −0.454441
\(985\) 0 0
\(986\) −55.4532 −1.76599
\(987\) 16.6983 0.531513
\(988\) 0 0
\(989\) 0.423851 0.0134777
\(990\) 0 0
\(991\) 23.0667 0.732737 0.366369 0.930470i \(-0.380601\pi\)
0.366369 + 0.930470i \(0.380601\pi\)
\(992\) −13.2698 −0.421317
\(993\) 45.5180 1.44447
\(994\) −18.3118 −0.580815
\(995\) 0 0
\(996\) −50.4249 −1.59777
\(997\) 0.630841 0.0199789 0.00998947 0.999950i \(-0.496820\pi\)
0.00998947 + 0.999950i \(0.496820\pi\)
\(998\) −81.3648 −2.57556
\(999\) 52.3227 1.65542
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.be.1.3 3
5.4 even 2 845.2.a.i.1.1 3
13.5 odd 4 325.2.c.g.51.1 6
13.8 odd 4 325.2.c.g.51.6 6
13.12 even 2 4225.2.a.bc.1.1 3
15.14 odd 2 7605.2.a.cc.1.3 3
65.4 even 6 845.2.e.i.146.1 6
65.8 even 4 325.2.d.f.324.5 6
65.9 even 6 845.2.e.k.146.3 6
65.18 even 4 325.2.d.e.324.1 6
65.19 odd 12 845.2.m.h.361.6 12
65.24 odd 12 845.2.m.h.316.6 12
65.29 even 6 845.2.e.k.191.3 6
65.34 odd 4 65.2.c.a.51.1 6
65.44 odd 4 65.2.c.a.51.6 yes 6
65.47 even 4 325.2.d.e.324.2 6
65.49 even 6 845.2.e.i.191.1 6
65.54 odd 12 845.2.m.h.316.1 12
65.57 even 4 325.2.d.f.324.6 6
65.59 odd 12 845.2.m.h.361.1 12
65.64 even 2 845.2.a.k.1.3 3
195.44 even 4 585.2.b.g.181.1 6
195.164 even 4 585.2.b.g.181.6 6
195.194 odd 2 7605.2.a.bs.1.1 3
260.99 even 4 1040.2.k.d.961.2 6
260.239 even 4 1040.2.k.d.961.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.c.a.51.1 6 65.34 odd 4
65.2.c.a.51.6 yes 6 65.44 odd 4
325.2.c.g.51.1 6 13.5 odd 4
325.2.c.g.51.6 6 13.8 odd 4
325.2.d.e.324.1 6 65.18 even 4
325.2.d.e.324.2 6 65.47 even 4
325.2.d.f.324.5 6 65.8 even 4
325.2.d.f.324.6 6 65.57 even 4
585.2.b.g.181.1 6 195.44 even 4
585.2.b.g.181.6 6 195.164 even 4
845.2.a.i.1.1 3 5.4 even 2
845.2.a.k.1.3 3 65.64 even 2
845.2.e.i.146.1 6 65.4 even 6
845.2.e.i.191.1 6 65.49 even 6
845.2.e.k.146.3 6 65.9 even 6
845.2.e.k.191.3 6 65.29 even 6
845.2.m.h.316.1 12 65.54 odd 12
845.2.m.h.316.6 12 65.24 odd 12
845.2.m.h.361.1 12 65.59 odd 12
845.2.m.h.361.6 12 65.19 odd 12
1040.2.k.d.961.1 6 260.239 even 4
1040.2.k.d.961.2 6 260.99 even 4
4225.2.a.bc.1.1 3 13.12 even 2
4225.2.a.be.1.3 3 1.1 even 1 trivial
7605.2.a.bs.1.1 3 195.194 odd 2
7605.2.a.cc.1.3 3 15.14 odd 2