Properties

Label 1725.2.a.y.1.2
Level $1725$
Weight $2$
Character 1725.1
Self dual yes
Analytic conductor $13.774$
Analytic rank $0$
Dimension $2$
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] = [1725,2,Mod(1,1725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1725, 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("1725.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1725.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: \(13.7741943487\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1725.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} -2.44949 q^{6} +1.00000 q^{7} +4.89898 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} -2.44949 q^{6} +1.00000 q^{7} +4.89898 q^{8} +1.00000 q^{9} +2.44949 q^{11} -4.00000 q^{12} -4.44949 q^{13} +2.44949 q^{14} +4.00000 q^{16} +5.44949 q^{17} +2.44949 q^{18} +4.44949 q^{19} -1.00000 q^{21} +6.00000 q^{22} +1.00000 q^{23} -4.89898 q^{24} -10.8990 q^{26} -1.00000 q^{27} +4.00000 q^{28} +10.3485 q^{29} +0.101021 q^{31} -2.44949 q^{33} +13.3485 q^{34} +4.00000 q^{36} -3.89898 q^{37} +10.8990 q^{38} +4.44949 q^{39} +5.44949 q^{41} -2.44949 q^{42} -2.00000 q^{43} +9.79796 q^{44} +2.44949 q^{46} -8.44949 q^{47} -4.00000 q^{48} -6.00000 q^{49} -5.44949 q^{51} -17.7980 q^{52} -0.550510 q^{53} -2.44949 q^{54} +4.89898 q^{56} -4.44949 q^{57} +25.3485 q^{58} +10.3485 q^{59} +0.651531 q^{61} +0.247449 q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{66} +7.00000 q^{67} +21.7980 q^{68} -1.00000 q^{69} -4.34847 q^{71} +4.89898 q^{72} +5.34847 q^{73} -9.55051 q^{74} +17.7980 q^{76} +2.44949 q^{77} +10.8990 q^{78} -4.00000 q^{79} +1.00000 q^{81} +13.3485 q^{82} -15.2474 q^{83} -4.00000 q^{84} -4.89898 q^{86} -10.3485 q^{87} +12.0000 q^{88} -16.8990 q^{89} -4.44949 q^{91} +4.00000 q^{92} -0.101021 q^{93} -20.6969 q^{94} -3.10102 q^{97} -14.6969 q^{98} +2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9} - 8 q^{12} - 4 q^{13} + 8 q^{16} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 12 q^{22} + 2 q^{23} - 12 q^{26} - 2 q^{27} + 8 q^{28} + 6 q^{29} + 10 q^{31} + 12 q^{34} + 8 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{39} + 6 q^{41} - 4 q^{43} - 12 q^{47} - 8 q^{48} - 12 q^{49} - 6 q^{51} - 16 q^{52} - 6 q^{53} - 4 q^{57} + 36 q^{58} + 6 q^{59} + 16 q^{61} - 24 q^{62} + 2 q^{63} - 16 q^{64} - 12 q^{66} + 14 q^{67} + 24 q^{68} - 2 q^{69} + 6 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} + 12 q^{78} - 8 q^{79} + 2 q^{81} + 12 q^{82} - 6 q^{83} - 8 q^{84} - 6 q^{87} + 24 q^{88} - 24 q^{89} - 4 q^{91} + 8 q^{92} - 10 q^{93} - 12 q^{94} - 16 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.44949 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.00000 2.00000
\(5\) 0 0
\(6\) −2.44949 −1.00000
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 4.89898 1.73205
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) −4.00000 −1.15470
\(13\) −4.44949 −1.23407 −0.617033 0.786937i \(-0.711666\pi\)
−0.617033 + 0.786937i \(0.711666\pi\)
\(14\) 2.44949 0.654654
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 5.44949 1.32170 0.660848 0.750520i \(-0.270197\pi\)
0.660848 + 0.750520i \(0.270197\pi\)
\(18\) 2.44949 0.577350
\(19\) 4.44949 1.02078 0.510391 0.859942i \(-0.329501\pi\)
0.510391 + 0.859942i \(0.329501\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 6.00000 1.27920
\(23\) 1.00000 0.208514
\(24\) −4.89898 −1.00000
\(25\) 0 0
\(26\) −10.8990 −2.13747
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 10.3485 1.92166 0.960831 0.277134i \(-0.0893846\pi\)
0.960831 + 0.277134i \(0.0893846\pi\)
\(30\) 0 0
\(31\) 0.101021 0.0181438 0.00907191 0.999959i \(-0.497112\pi\)
0.00907191 + 0.999959i \(0.497112\pi\)
\(32\) 0 0
\(33\) −2.44949 −0.426401
\(34\) 13.3485 2.28924
\(35\) 0 0
\(36\) 4.00000 0.666667
\(37\) −3.89898 −0.640988 −0.320494 0.947250i \(-0.603849\pi\)
−0.320494 + 0.947250i \(0.603849\pi\)
\(38\) 10.8990 1.76805
\(39\) 4.44949 0.712489
\(40\) 0 0
\(41\) 5.44949 0.851067 0.425534 0.904943i \(-0.360086\pi\)
0.425534 + 0.904943i \(0.360086\pi\)
\(42\) −2.44949 −0.377964
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 9.79796 1.47710
\(45\) 0 0
\(46\) 2.44949 0.361158
\(47\) −8.44949 −1.23248 −0.616242 0.787557i \(-0.711346\pi\)
−0.616242 + 0.787557i \(0.711346\pi\)
\(48\) −4.00000 −0.577350
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −5.44949 −0.763081
\(52\) −17.7980 −2.46813
\(53\) −0.550510 −0.0756184 −0.0378092 0.999285i \(-0.512038\pi\)
−0.0378092 + 0.999285i \(0.512038\pi\)
\(54\) −2.44949 −0.333333
\(55\) 0 0
\(56\) 4.89898 0.654654
\(57\) −4.44949 −0.589349
\(58\) 25.3485 3.32842
\(59\) 10.3485 1.34726 0.673628 0.739071i \(-0.264735\pi\)
0.673628 + 0.739071i \(0.264735\pi\)
\(60\) 0 0
\(61\) 0.651531 0.0834200 0.0417100 0.999130i \(-0.486719\pi\)
0.0417100 + 0.999130i \(0.486719\pi\)
\(62\) 0.247449 0.0314260
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 21.7980 2.64339
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −4.34847 −0.516068 −0.258034 0.966136i \(-0.583075\pi\)
−0.258034 + 0.966136i \(0.583075\pi\)
\(72\) 4.89898 0.577350
\(73\) 5.34847 0.625991 0.312995 0.949755i \(-0.398668\pi\)
0.312995 + 0.949755i \(0.398668\pi\)
\(74\) −9.55051 −1.11022
\(75\) 0 0
\(76\) 17.7980 2.04157
\(77\) 2.44949 0.279145
\(78\) 10.8990 1.23407
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 13.3485 1.47409
\(83\) −15.2474 −1.67362 −0.836812 0.547490i \(-0.815584\pi\)
−0.836812 + 0.547490i \(0.815584\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −4.89898 −0.528271
\(87\) −10.3485 −1.10947
\(88\) 12.0000 1.27920
\(89\) −16.8990 −1.79129 −0.895644 0.444771i \(-0.853285\pi\)
−0.895644 + 0.444771i \(0.853285\pi\)
\(90\) 0 0
\(91\) −4.44949 −0.466433
\(92\) 4.00000 0.417029
\(93\) −0.101021 −0.0104753
\(94\) −20.6969 −2.13473
\(95\) 0 0
\(96\) 0 0
\(97\) −3.10102 −0.314861 −0.157430 0.987530i \(-0.550321\pi\)
−0.157430 + 0.987530i \(0.550321\pi\)
\(98\) −14.6969 −1.48461
\(99\) 2.44949 0.246183
\(100\) 0 0
\(101\) −12.5505 −1.24882 −0.624411 0.781096i \(-0.714661\pi\)
−0.624411 + 0.781096i \(0.714661\pi\)
\(102\) −13.3485 −1.32170
\(103\) 12.6969 1.25107 0.625533 0.780198i \(-0.284881\pi\)
0.625533 + 0.780198i \(0.284881\pi\)
\(104\) −21.7980 −2.13747
\(105\) 0 0
\(106\) −1.34847 −0.130975
\(107\) 5.44949 0.526822 0.263411 0.964684i \(-0.415152\pi\)
0.263411 + 0.964684i \(0.415152\pi\)
\(108\) −4.00000 −0.384900
\(109\) 11.5505 1.10634 0.553169 0.833069i \(-0.313418\pi\)
0.553169 + 0.833069i \(0.313418\pi\)
\(110\) 0 0
\(111\) 3.89898 0.370075
\(112\) 4.00000 0.377964
\(113\) 1.65153 0.155363 0.0776815 0.996978i \(-0.475248\pi\)
0.0776815 + 0.996978i \(0.475248\pi\)
\(114\) −10.8990 −1.02078
\(115\) 0 0
\(116\) 41.3939 3.84332
\(117\) −4.44949 −0.411355
\(118\) 25.3485 2.33352
\(119\) 5.44949 0.499554
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 1.59592 0.144488
\(123\) −5.44949 −0.491364
\(124\) 0.404082 0.0362876
\(125\) 0 0
\(126\) 2.44949 0.218218
\(127\) 1.55051 0.137586 0.0687928 0.997631i \(-0.478085\pi\)
0.0687928 + 0.997631i \(0.478085\pi\)
\(128\) −19.5959 −1.73205
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −19.5959 −1.71210 −0.856052 0.516890i \(-0.827090\pi\)
−0.856052 + 0.516890i \(0.827090\pi\)
\(132\) −9.79796 −0.852803
\(133\) 4.44949 0.385820
\(134\) 17.1464 1.48123
\(135\) 0 0
\(136\) 26.6969 2.28924
\(137\) 16.8990 1.44378 0.721889 0.692009i \(-0.243274\pi\)
0.721889 + 0.692009i \(0.243274\pi\)
\(138\) −2.44949 −0.208514
\(139\) −4.79796 −0.406958 −0.203479 0.979079i \(-0.565225\pi\)
−0.203479 + 0.979079i \(0.565225\pi\)
\(140\) 0 0
\(141\) 8.44949 0.711575
\(142\) −10.6515 −0.893857
\(143\) −10.8990 −0.911418
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 13.1010 1.08425
\(147\) 6.00000 0.494872
\(148\) −15.5959 −1.28198
\(149\) −1.34847 −0.110471 −0.0552355 0.998473i \(-0.517591\pi\)
−0.0552355 + 0.998473i \(0.517591\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 21.7980 1.76805
\(153\) 5.44949 0.440565
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 17.7980 1.42498
\(157\) −18.5959 −1.48412 −0.742058 0.670336i \(-0.766150\pi\)
−0.742058 + 0.670336i \(0.766150\pi\)
\(158\) −9.79796 −0.779484
\(159\) 0.550510 0.0436583
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 2.44949 0.192450
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 21.7980 1.70213
\(165\) 0 0
\(166\) −37.3485 −2.89880
\(167\) −13.3485 −1.03294 −0.516468 0.856307i \(-0.672753\pi\)
−0.516468 + 0.856307i \(0.672753\pi\)
\(168\) −4.89898 −0.377964
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 4.44949 0.340261
\(172\) −8.00000 −0.609994
\(173\) −19.5959 −1.48985 −0.744925 0.667148i \(-0.767515\pi\)
−0.744925 + 0.667148i \(0.767515\pi\)
\(174\) −25.3485 −1.92166
\(175\) 0 0
\(176\) 9.79796 0.738549
\(177\) −10.3485 −0.777839
\(178\) −41.3939 −3.10260
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −8.89898 −0.661456 −0.330728 0.943726i \(-0.607294\pi\)
−0.330728 + 0.943726i \(0.607294\pi\)
\(182\) −10.8990 −0.807886
\(183\) −0.651531 −0.0481625
\(184\) 4.89898 0.361158
\(185\) 0 0
\(186\) −0.247449 −0.0181438
\(187\) 13.3485 0.976137
\(188\) −33.7980 −2.46497
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.2474 −1.32034 −0.660170 0.751117i \(-0.729516\pi\)
−0.660170 + 0.751117i \(0.729516\pi\)
\(192\) 8.00000 0.577350
\(193\) 6.69694 0.482056 0.241028 0.970518i \(-0.422515\pi\)
0.241028 + 0.970518i \(0.422515\pi\)
\(194\) −7.59592 −0.545355
\(195\) 0 0
\(196\) −24.0000 −1.71429
\(197\) 22.8990 1.63148 0.815742 0.578415i \(-0.196329\pi\)
0.815742 + 0.578415i \(0.196329\pi\)
\(198\) 6.00000 0.426401
\(199\) −2.89898 −0.205503 −0.102752 0.994707i \(-0.532765\pi\)
−0.102752 + 0.994707i \(0.532765\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) −30.7423 −2.16302
\(203\) 10.3485 0.726320
\(204\) −21.7980 −1.52616
\(205\) 0 0
\(206\) 31.1010 2.16691
\(207\) 1.00000 0.0695048
\(208\) −17.7980 −1.23407
\(209\) 10.8990 0.753898
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) −2.20204 −0.151237
\(213\) 4.34847 0.297952
\(214\) 13.3485 0.912483
\(215\) 0 0
\(216\) −4.89898 −0.333333
\(217\) 0.101021 0.00685772
\(218\) 28.2929 1.91623
\(219\) −5.34847 −0.361416
\(220\) 0 0
\(221\) −24.2474 −1.63106
\(222\) 9.55051 0.640988
\(223\) −21.5959 −1.44617 −0.723085 0.690759i \(-0.757276\pi\)
−0.723085 + 0.690759i \(0.757276\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4.04541 0.269097
\(227\) −21.7980 −1.44678 −0.723391 0.690439i \(-0.757417\pi\)
−0.723391 + 0.690439i \(0.757417\pi\)
\(228\) −17.7980 −1.17870
\(229\) −28.4949 −1.88300 −0.941498 0.337019i \(-0.890581\pi\)
−0.941498 + 0.337019i \(0.890581\pi\)
\(230\) 0 0
\(231\) −2.44949 −0.161165
\(232\) 50.6969 3.32842
\(233\) −21.7980 −1.42803 −0.714016 0.700129i \(-0.753126\pi\)
−0.714016 + 0.700129i \(0.753126\pi\)
\(234\) −10.8990 −0.712489
\(235\) 0 0
\(236\) 41.3939 2.69451
\(237\) 4.00000 0.259828
\(238\) 13.3485 0.865253
\(239\) 6.55051 0.423717 0.211859 0.977300i \(-0.432048\pi\)
0.211859 + 0.977300i \(0.432048\pi\)
\(240\) 0 0
\(241\) −29.3485 −1.89050 −0.945251 0.326346i \(-0.894183\pi\)
−0.945251 + 0.326346i \(0.894183\pi\)
\(242\) −12.2474 −0.787296
\(243\) −1.00000 −0.0641500
\(244\) 2.60612 0.166840
\(245\) 0 0
\(246\) −13.3485 −0.851067
\(247\) −19.7980 −1.25971
\(248\) 0.494897 0.0314260
\(249\) 15.2474 0.966268
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 4.00000 0.251976
\(253\) 2.44949 0.153998
\(254\) 3.79796 0.238305
\(255\) 0 0
\(256\) −32.0000 −2.00000
\(257\) 25.3485 1.58119 0.790597 0.612337i \(-0.209770\pi\)
0.790597 + 0.612337i \(0.209770\pi\)
\(258\) 4.89898 0.304997
\(259\) −3.89898 −0.242271
\(260\) 0 0
\(261\) 10.3485 0.640554
\(262\) −48.0000 −2.96545
\(263\) −27.2474 −1.68015 −0.840075 0.542471i \(-0.817489\pi\)
−0.840075 + 0.542471i \(0.817489\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 10.8990 0.668259
\(267\) 16.8990 1.03420
\(268\) 28.0000 1.71037
\(269\) −5.44949 −0.332261 −0.166131 0.986104i \(-0.553127\pi\)
−0.166131 + 0.986104i \(0.553127\pi\)
\(270\) 0 0
\(271\) 31.6969 1.92545 0.962726 0.270479i \(-0.0871820\pi\)
0.962726 + 0.270479i \(0.0871820\pi\)
\(272\) 21.7980 1.32170
\(273\) 4.44949 0.269295
\(274\) 41.3939 2.50070
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −11.7526 −0.704871
\(279\) 0.101021 0.00604794
\(280\) 0 0
\(281\) −19.3485 −1.15423 −0.577116 0.816662i \(-0.695822\pi\)
−0.577116 + 0.816662i \(0.695822\pi\)
\(282\) 20.6969 1.23248
\(283\) 2.10102 0.124893 0.0624464 0.998048i \(-0.480110\pi\)
0.0624464 + 0.998048i \(0.480110\pi\)
\(284\) −17.3939 −1.03214
\(285\) 0 0
\(286\) −26.6969 −1.57862
\(287\) 5.44949 0.321673
\(288\) 0 0
\(289\) 12.6969 0.746879
\(290\) 0 0
\(291\) 3.10102 0.181785
\(292\) 21.3939 1.25198
\(293\) 21.2474 1.24129 0.620645 0.784092i \(-0.286871\pi\)
0.620645 + 0.784092i \(0.286871\pi\)
\(294\) 14.6969 0.857143
\(295\) 0 0
\(296\) −19.1010 −1.11022
\(297\) −2.44949 −0.142134
\(298\) −3.30306 −0.191341
\(299\) −4.44949 −0.257321
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 34.2929 1.97333
\(303\) 12.5505 0.721008
\(304\) 17.7980 1.02078
\(305\) 0 0
\(306\) 13.3485 0.763081
\(307\) 2.65153 0.151331 0.0756654 0.997133i \(-0.475892\pi\)
0.0756654 + 0.997133i \(0.475892\pi\)
\(308\) 9.79796 0.558291
\(309\) −12.6969 −0.722304
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 21.7980 1.23407
\(313\) −19.6969 −1.11334 −0.556668 0.830735i \(-0.687921\pi\)
−0.556668 + 0.830735i \(0.687921\pi\)
\(314\) −45.5505 −2.57056
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −6.24745 −0.350892 −0.175446 0.984489i \(-0.556137\pi\)
−0.175446 + 0.984489i \(0.556137\pi\)
\(318\) 1.34847 0.0756184
\(319\) 25.3485 1.41924
\(320\) 0 0
\(321\) −5.44949 −0.304161
\(322\) 2.44949 0.136505
\(323\) 24.2474 1.34916
\(324\) 4.00000 0.222222
\(325\) 0 0
\(326\) 24.4949 1.35665
\(327\) −11.5505 −0.638745
\(328\) 26.6969 1.47409
\(329\) −8.44949 −0.465835
\(330\) 0 0
\(331\) 24.5959 1.35191 0.675957 0.736941i \(-0.263731\pi\)
0.675957 + 0.736941i \(0.263731\pi\)
\(332\) −60.9898 −3.34725
\(333\) −3.89898 −0.213663
\(334\) −32.6969 −1.78910
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 19.7980 1.07846 0.539232 0.842157i \(-0.318715\pi\)
0.539232 + 0.842157i \(0.318715\pi\)
\(338\) 16.6515 0.905724
\(339\) −1.65153 −0.0896988
\(340\) 0 0
\(341\) 0.247449 0.0134001
\(342\) 10.8990 0.589349
\(343\) −13.0000 −0.701934
\(344\) −9.79796 −0.528271
\(345\) 0 0
\(346\) −48.0000 −2.58050
\(347\) 28.8990 1.55138 0.775689 0.631115i \(-0.217402\pi\)
0.775689 + 0.631115i \(0.217402\pi\)
\(348\) −41.3939 −2.21894
\(349\) 23.4949 1.25765 0.628827 0.777546i \(-0.283536\pi\)
0.628827 + 0.777546i \(0.283536\pi\)
\(350\) 0 0
\(351\) 4.44949 0.237496
\(352\) 0 0
\(353\) 15.5505 0.827670 0.413835 0.910352i \(-0.364189\pi\)
0.413835 + 0.910352i \(0.364189\pi\)
\(354\) −25.3485 −1.34726
\(355\) 0 0
\(356\) −67.5959 −3.58258
\(357\) −5.44949 −0.288418
\(358\) −14.6969 −0.776757
\(359\) −3.55051 −0.187389 −0.0936944 0.995601i \(-0.529868\pi\)
−0.0936944 + 0.995601i \(0.529868\pi\)
\(360\) 0 0
\(361\) 0.797959 0.0419978
\(362\) −21.7980 −1.14568
\(363\) 5.00000 0.262432
\(364\) −17.7980 −0.932867
\(365\) 0 0
\(366\) −1.59592 −0.0834200
\(367\) −13.6969 −0.714974 −0.357487 0.933918i \(-0.616366\pi\)
−0.357487 + 0.933918i \(0.616366\pi\)
\(368\) 4.00000 0.208514
\(369\) 5.44949 0.283689
\(370\) 0 0
\(371\) −0.550510 −0.0285811
\(372\) −0.404082 −0.0209507
\(373\) −27.5959 −1.42886 −0.714431 0.699706i \(-0.753314\pi\)
−0.714431 + 0.699706i \(0.753314\pi\)
\(374\) 32.6969 1.69072
\(375\) 0 0
\(376\) −41.3939 −2.13473
\(377\) −46.0454 −2.37146
\(378\) −2.44949 −0.125988
\(379\) 25.3939 1.30440 0.652198 0.758049i \(-0.273847\pi\)
0.652198 + 0.758049i \(0.273847\pi\)
\(380\) 0 0
\(381\) −1.55051 −0.0794350
\(382\) −44.6969 −2.28689
\(383\) −1.65153 −0.0843893 −0.0421946 0.999109i \(-0.513435\pi\)
−0.0421946 + 0.999109i \(0.513435\pi\)
\(384\) 19.5959 1.00000
\(385\) 0 0
\(386\) 16.4041 0.834946
\(387\) −2.00000 −0.101666
\(388\) −12.4041 −0.629722
\(389\) −3.30306 −0.167472 −0.0837359 0.996488i \(-0.526685\pi\)
−0.0837359 + 0.996488i \(0.526685\pi\)
\(390\) 0 0
\(391\) 5.44949 0.275593
\(392\) −29.3939 −1.48461
\(393\) 19.5959 0.988483
\(394\) 56.0908 2.82581
\(395\) 0 0
\(396\) 9.79796 0.492366
\(397\) 13.7980 0.692500 0.346250 0.938142i \(-0.387455\pi\)
0.346250 + 0.938142i \(0.387455\pi\)
\(398\) −7.10102 −0.355942
\(399\) −4.44949 −0.222753
\(400\) 0 0
\(401\) −13.1010 −0.654234 −0.327117 0.944984i \(-0.606077\pi\)
−0.327117 + 0.944984i \(0.606077\pi\)
\(402\) −17.1464 −0.855186
\(403\) −0.449490 −0.0223907
\(404\) −50.2020 −2.49764
\(405\) 0 0
\(406\) 25.3485 1.25802
\(407\) −9.55051 −0.473401
\(408\) −26.6969 −1.32170
\(409\) 21.8990 1.08283 0.541417 0.840754i \(-0.317888\pi\)
0.541417 + 0.840754i \(0.317888\pi\)
\(410\) 0 0
\(411\) −16.8990 −0.833565
\(412\) 50.7878 2.50213
\(413\) 10.3485 0.509215
\(414\) 2.44949 0.120386
\(415\) 0 0
\(416\) 0 0
\(417\) 4.79796 0.234957
\(418\) 26.6969 1.30579
\(419\) 33.5505 1.63905 0.819525 0.573044i \(-0.194237\pi\)
0.819525 + 0.573044i \(0.194237\pi\)
\(420\) 0 0
\(421\) −22.2474 −1.08427 −0.542137 0.840290i \(-0.682385\pi\)
−0.542137 + 0.840290i \(0.682385\pi\)
\(422\) 26.9444 1.31163
\(423\) −8.44949 −0.410828
\(424\) −2.69694 −0.130975
\(425\) 0 0
\(426\) 10.6515 0.516068
\(427\) 0.651531 0.0315298
\(428\) 21.7980 1.05364
\(429\) 10.8990 0.526208
\(430\) 0 0
\(431\) −24.4949 −1.17988 −0.589939 0.807448i \(-0.700848\pi\)
−0.589939 + 0.807448i \(0.700848\pi\)
\(432\) −4.00000 −0.192450
\(433\) −2.79796 −0.134461 −0.0672307 0.997737i \(-0.521416\pi\)
−0.0672307 + 0.997737i \(0.521416\pi\)
\(434\) 0.247449 0.0118779
\(435\) 0 0
\(436\) 46.2020 2.21268
\(437\) 4.44949 0.212848
\(438\) −13.1010 −0.625991
\(439\) −36.6969 −1.75145 −0.875725 0.482811i \(-0.839616\pi\)
−0.875725 + 0.482811i \(0.839616\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −59.3939 −2.82508
\(443\) 20.4495 0.971585 0.485792 0.874074i \(-0.338531\pi\)
0.485792 + 0.874074i \(0.338531\pi\)
\(444\) 15.5959 0.740150
\(445\) 0 0
\(446\) −52.8990 −2.50484
\(447\) 1.34847 0.0637804
\(448\) −8.00000 −0.377964
\(449\) −21.2474 −1.00273 −0.501365 0.865236i \(-0.667168\pi\)
−0.501365 + 0.865236i \(0.667168\pi\)
\(450\) 0 0
\(451\) 13.3485 0.628555
\(452\) 6.60612 0.310726
\(453\) −14.0000 −0.657777
\(454\) −53.3939 −2.50590
\(455\) 0 0
\(456\) −21.7980 −1.02078
\(457\) −0.101021 −0.00472554 −0.00236277 0.999997i \(-0.500752\pi\)
−0.00236277 + 0.999997i \(0.500752\pi\)
\(458\) −69.7980 −3.26144
\(459\) −5.44949 −0.254360
\(460\) 0 0
\(461\) −18.4949 −0.861393 −0.430697 0.902497i \(-0.641732\pi\)
−0.430697 + 0.902497i \(0.641732\pi\)
\(462\) −6.00000 −0.279145
\(463\) 24.9444 1.15926 0.579632 0.814878i \(-0.303196\pi\)
0.579632 + 0.814878i \(0.303196\pi\)
\(464\) 41.3939 1.92166
\(465\) 0 0
\(466\) −53.3939 −2.47342
\(467\) 34.8434 1.61236 0.806179 0.591671i \(-0.201532\pi\)
0.806179 + 0.591671i \(0.201532\pi\)
\(468\) −17.7980 −0.822711
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) 18.5959 0.856855
\(472\) 50.6969 2.33352
\(473\) −4.89898 −0.225255
\(474\) 9.79796 0.450035
\(475\) 0 0
\(476\) 21.7980 0.999108
\(477\) −0.550510 −0.0252061
\(478\) 16.0454 0.733900
\(479\) 32.9444 1.50527 0.752634 0.658439i \(-0.228783\pi\)
0.752634 + 0.658439i \(0.228783\pi\)
\(480\) 0 0
\(481\) 17.3485 0.791022
\(482\) −71.8888 −3.27444
\(483\) −1.00000 −0.0455016
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) −2.44949 −0.111111
\(487\) 18.9444 0.858452 0.429226 0.903197i \(-0.358786\pi\)
0.429226 + 0.903197i \(0.358786\pi\)
\(488\) 3.19184 0.144488
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 12.5505 0.566397 0.283198 0.959061i \(-0.408605\pi\)
0.283198 + 0.959061i \(0.408605\pi\)
\(492\) −21.7980 −0.982728
\(493\) 56.3939 2.53985
\(494\) −48.4949 −2.18189
\(495\) 0 0
\(496\) 0.404082 0.0181438
\(497\) −4.34847 −0.195056
\(498\) 37.3485 1.67362
\(499\) −1.00000 −0.0447661 −0.0223831 0.999749i \(-0.507125\pi\)
−0.0223831 + 0.999749i \(0.507125\pi\)
\(500\) 0 0
\(501\) 13.3485 0.596366
\(502\) 44.0908 1.96787
\(503\) −7.04541 −0.314139 −0.157070 0.987588i \(-0.550205\pi\)
−0.157070 + 0.987588i \(0.550205\pi\)
\(504\) 4.89898 0.218218
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) −6.79796 −0.301908
\(508\) 6.20204 0.275171
\(509\) 28.8990 1.28092 0.640462 0.767990i \(-0.278743\pi\)
0.640462 + 0.767990i \(0.278743\pi\)
\(510\) 0 0
\(511\) 5.34847 0.236602
\(512\) −39.1918 −1.73205
\(513\) −4.44949 −0.196450
\(514\) 62.0908 2.73871
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −20.6969 −0.910250
\(518\) −9.55051 −0.419625
\(519\) 19.5959 0.860165
\(520\) 0 0
\(521\) −43.8434 −1.92081 −0.960406 0.278603i \(-0.910129\pi\)
−0.960406 + 0.278603i \(0.910129\pi\)
\(522\) 25.3485 1.10947
\(523\) 33.3939 1.46021 0.730106 0.683334i \(-0.239471\pi\)
0.730106 + 0.683334i \(0.239471\pi\)
\(524\) −78.3837 −3.42421
\(525\) 0 0
\(526\) −66.7423 −2.91010
\(527\) 0.550510 0.0239806
\(528\) −9.79796 −0.426401
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.3485 0.449085
\(532\) 17.7980 0.771639
\(533\) −24.2474 −1.05027
\(534\) 41.3939 1.79129
\(535\) 0 0
\(536\) 34.2929 1.48123
\(537\) 6.00000 0.258919
\(538\) −13.3485 −0.575493
\(539\) −14.6969 −0.633042
\(540\) 0 0
\(541\) 12.8990 0.554570 0.277285 0.960788i \(-0.410565\pi\)
0.277285 + 0.960788i \(0.410565\pi\)
\(542\) 77.6413 3.33498
\(543\) 8.89898 0.381892
\(544\) 0 0
\(545\) 0 0
\(546\) 10.8990 0.466433
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 67.5959 2.88755
\(549\) 0.651531 0.0278067
\(550\) 0 0
\(551\) 46.0454 1.96160
\(552\) −4.89898 −0.208514
\(553\) −4.00000 −0.170097
\(554\) 24.4949 1.04069
\(555\) 0 0
\(556\) −19.1918 −0.813915
\(557\) 19.0454 0.806980 0.403490 0.914984i \(-0.367797\pi\)
0.403490 + 0.914984i \(0.367797\pi\)
\(558\) 0.247449 0.0104753
\(559\) 8.89898 0.376387
\(560\) 0 0
\(561\) −13.3485 −0.563573
\(562\) −47.3939 −1.99919
\(563\) −5.44949 −0.229669 −0.114834 0.993385i \(-0.536634\pi\)
−0.114834 + 0.993385i \(0.536634\pi\)
\(564\) 33.7980 1.42315
\(565\) 0 0
\(566\) 5.14643 0.216321
\(567\) 1.00000 0.0419961
\(568\) −21.3031 −0.893857
\(569\) −28.2929 −1.18610 −0.593049 0.805166i \(-0.702076\pi\)
−0.593049 + 0.805166i \(0.702076\pi\)
\(570\) 0 0
\(571\) 13.1464 0.550161 0.275080 0.961421i \(-0.411296\pi\)
0.275080 + 0.961421i \(0.411296\pi\)
\(572\) −43.5959 −1.82284
\(573\) 18.2474 0.762298
\(574\) 13.3485 0.557154
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 31.1010 1.29363
\(579\) −6.69694 −0.278315
\(580\) 0 0
\(581\) −15.2474 −0.632571
\(582\) 7.59592 0.314861
\(583\) −1.34847 −0.0558479
\(584\) 26.2020 1.08425
\(585\) 0 0
\(586\) 52.0454 2.14998
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 24.0000 0.989743
\(589\) 0.449490 0.0185209
\(590\) 0 0
\(591\) −22.8990 −0.941938
\(592\) −15.5959 −0.640988
\(593\) −27.5505 −1.13136 −0.565682 0.824624i \(-0.691387\pi\)
−0.565682 + 0.824624i \(0.691387\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −5.39388 −0.220942
\(597\) 2.89898 0.118647
\(598\) −10.8990 −0.445692
\(599\) −2.69694 −0.110194 −0.0550970 0.998481i \(-0.517547\pi\)
−0.0550970 + 0.998481i \(0.517547\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) −4.89898 −0.199667
\(603\) 7.00000 0.285062
\(604\) 56.0000 2.27861
\(605\) 0 0
\(606\) 30.7423 1.24882
\(607\) 16.2474 0.659464 0.329732 0.944075i \(-0.393042\pi\)
0.329732 + 0.944075i \(0.393042\pi\)
\(608\) 0 0
\(609\) −10.3485 −0.419341
\(610\) 0 0
\(611\) 37.5959 1.52097
\(612\) 21.7980 0.881130
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 6.49490 0.262113
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) −12.5505 −0.505265 −0.252632 0.967562i \(-0.581296\pi\)
−0.252632 + 0.967562i \(0.581296\pi\)
\(618\) −31.1010 −1.25107
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 29.3939 1.17859
\(623\) −16.8990 −0.677043
\(624\) 17.7980 0.712489
\(625\) 0 0
\(626\) −48.2474 −1.92836
\(627\) −10.8990 −0.435263
\(628\) −74.3837 −2.96823
\(629\) −21.2474 −0.847191
\(630\) 0 0
\(631\) 29.5505 1.17639 0.588194 0.808720i \(-0.299839\pi\)
0.588194 + 0.808720i \(0.299839\pi\)
\(632\) −19.5959 −0.779484
\(633\) −11.0000 −0.437211
\(634\) −15.3031 −0.607762
\(635\) 0 0
\(636\) 2.20204 0.0873166
\(637\) 26.6969 1.05777
\(638\) 62.0908 2.45820
\(639\) −4.34847 −0.172023
\(640\) 0 0
\(641\) 26.9444 1.06424 0.532120 0.846669i \(-0.321396\pi\)
0.532120 + 0.846669i \(0.321396\pi\)
\(642\) −13.3485 −0.526822
\(643\) 9.69694 0.382410 0.191205 0.981550i \(-0.438760\pi\)
0.191205 + 0.981550i \(0.438760\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 59.3939 2.33682
\(647\) −1.95459 −0.0768430 −0.0384215 0.999262i \(-0.512233\pi\)
−0.0384215 + 0.999262i \(0.512233\pi\)
\(648\) 4.89898 0.192450
\(649\) 25.3485 0.995014
\(650\) 0 0
\(651\) −0.101021 −0.00395931
\(652\) 40.0000 1.56652
\(653\) −43.8434 −1.71572 −0.857862 0.513881i \(-0.828207\pi\)
−0.857862 + 0.513881i \(0.828207\pi\)
\(654\) −28.2929 −1.10634
\(655\) 0 0
\(656\) 21.7980 0.851067
\(657\) 5.34847 0.208664
\(658\) −20.6969 −0.806851
\(659\) −5.14643 −0.200476 −0.100238 0.994963i \(-0.531960\pi\)
−0.100238 + 0.994963i \(0.531960\pi\)
\(660\) 0 0
\(661\) 46.0908 1.79272 0.896362 0.443322i \(-0.146200\pi\)
0.896362 + 0.443322i \(0.146200\pi\)
\(662\) 60.2474 2.34158
\(663\) 24.2474 0.941693
\(664\) −74.6969 −2.89880
\(665\) 0 0
\(666\) −9.55051 −0.370075
\(667\) 10.3485 0.400694
\(668\) −53.3939 −2.06587
\(669\) 21.5959 0.834946
\(670\) 0 0
\(671\) 1.59592 0.0616097
\(672\) 0 0
\(673\) −22.4495 −0.865364 −0.432682 0.901547i \(-0.642433\pi\)
−0.432682 + 0.901547i \(0.642433\pi\)
\(674\) 48.4949 1.86795
\(675\) 0 0
\(676\) 27.1918 1.04584
\(677\) 12.5505 0.482355 0.241178 0.970481i \(-0.422466\pi\)
0.241178 + 0.970481i \(0.422466\pi\)
\(678\) −4.04541 −0.155363
\(679\) −3.10102 −0.119006
\(680\) 0 0
\(681\) 21.7980 0.835300
\(682\) 0.606123 0.0232097
\(683\) −18.2474 −0.698219 −0.349110 0.937082i \(-0.613516\pi\)
−0.349110 + 0.937082i \(0.613516\pi\)
\(684\) 17.7980 0.680522
\(685\) 0 0
\(686\) −31.8434 −1.21579
\(687\) 28.4949 1.08715
\(688\) −8.00000 −0.304997
\(689\) 2.44949 0.0933181
\(690\) 0 0
\(691\) 16.2020 0.616355 0.308177 0.951329i \(-0.400281\pi\)
0.308177 + 0.951329i \(0.400281\pi\)
\(692\) −78.3837 −2.97970
\(693\) 2.44949 0.0930484
\(694\) 70.7878 2.68707
\(695\) 0 0
\(696\) −50.6969 −1.92166
\(697\) 29.6969 1.12485
\(698\) 57.5505 2.17832
\(699\) 21.7980 0.824475
\(700\) 0 0
\(701\) −52.0454 −1.96573 −0.982864 0.184332i \(-0.940988\pi\)
−0.982864 + 0.184332i \(0.940988\pi\)
\(702\) 10.8990 0.411355
\(703\) −17.3485 −0.654310
\(704\) −19.5959 −0.738549
\(705\) 0 0
\(706\) 38.0908 1.43357
\(707\) −12.5505 −0.472011
\(708\) −41.3939 −1.55568
\(709\) −28.7423 −1.07944 −0.539721 0.841844i \(-0.681470\pi\)
−0.539721 + 0.841844i \(0.681470\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) −82.7878 −3.10260
\(713\) 0.101021 0.00378325
\(714\) −13.3485 −0.499554
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) −6.55051 −0.244633
\(718\) −8.69694 −0.324567
\(719\) −11.9444 −0.445450 −0.222725 0.974881i \(-0.571495\pi\)
−0.222725 + 0.974881i \(0.571495\pi\)
\(720\) 0 0
\(721\) 12.6969 0.472859
\(722\) 1.95459 0.0727424
\(723\) 29.3485 1.09148
\(724\) −35.5959 −1.32291
\(725\) 0 0
\(726\) 12.2474 0.454545
\(727\) 16.3031 0.604647 0.302324 0.953205i \(-0.402238\pi\)
0.302324 + 0.953205i \(0.402238\pi\)
\(728\) −21.7980 −0.807886
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.8990 −0.403113
\(732\) −2.60612 −0.0963251
\(733\) 8.59592 0.317497 0.158749 0.987319i \(-0.449254\pi\)
0.158749 + 0.987319i \(0.449254\pi\)
\(734\) −33.5505 −1.23837
\(735\) 0 0
\(736\) 0 0
\(737\) 17.1464 0.631597
\(738\) 13.3485 0.491364
\(739\) −9.20204 −0.338503 −0.169251 0.985573i \(-0.554135\pi\)
−0.169251 + 0.985573i \(0.554135\pi\)
\(740\) 0 0
\(741\) 19.7980 0.727296
\(742\) −1.34847 −0.0495039
\(743\) 21.7980 0.799690 0.399845 0.916583i \(-0.369064\pi\)
0.399845 + 0.916583i \(0.369064\pi\)
\(744\) −0.494897 −0.0181438
\(745\) 0 0
\(746\) −67.5959 −2.47486
\(747\) −15.2474 −0.557875
\(748\) 53.3939 1.95227
\(749\) 5.44949 0.199120
\(750\) 0 0
\(751\) 30.6515 1.11849 0.559245 0.829002i \(-0.311091\pi\)
0.559245 + 0.829002i \(0.311091\pi\)
\(752\) −33.7980 −1.23248
\(753\) −18.0000 −0.655956
\(754\) −112.788 −4.10749
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 3.69694 0.134368 0.0671838 0.997741i \(-0.478599\pi\)
0.0671838 + 0.997741i \(0.478599\pi\)
\(758\) 62.2020 2.25928
\(759\) −2.44949 −0.0889108
\(760\) 0 0
\(761\) 14.1464 0.512808 0.256404 0.966570i \(-0.417462\pi\)
0.256404 + 0.966570i \(0.417462\pi\)
\(762\) −3.79796 −0.137586
\(763\) 11.5505 0.418157
\(764\) −72.9898 −2.64068
\(765\) 0 0
\(766\) −4.04541 −0.146167
\(767\) −46.0454 −1.66260
\(768\) 32.0000 1.15470
\(769\) 10.4495 0.376818 0.188409 0.982091i \(-0.439667\pi\)
0.188409 + 0.982091i \(0.439667\pi\)
\(770\) 0 0
\(771\) −25.3485 −0.912903
\(772\) 26.7878 0.964112
\(773\) 16.2929 0.586013 0.293007 0.956110i \(-0.405344\pi\)
0.293007 + 0.956110i \(0.405344\pi\)
\(774\) −4.89898 −0.176090
\(775\) 0 0
\(776\) −15.1918 −0.545355
\(777\) 3.89898 0.139875
\(778\) −8.09082 −0.290070
\(779\) 24.2474 0.868755
\(780\) 0 0
\(781\) −10.6515 −0.381142
\(782\) 13.3485 0.477340
\(783\) −10.3485 −0.369824
\(784\) −24.0000 −0.857143
\(785\) 0 0
\(786\) 48.0000 1.71210
\(787\) 21.6969 0.773412 0.386706 0.922203i \(-0.373613\pi\)
0.386706 + 0.922203i \(0.373613\pi\)
\(788\) 91.5959 3.26297
\(789\) 27.2474 0.970035
\(790\) 0 0
\(791\) 1.65153 0.0587217
\(792\) 12.0000 0.426401
\(793\) −2.89898 −0.102946
\(794\) 33.7980 1.19944
\(795\) 0 0
\(796\) −11.5959 −0.411006
\(797\) −9.24745 −0.327561 −0.163781 0.986497i \(-0.552369\pi\)
−0.163781 + 0.986497i \(0.552369\pi\)
\(798\) −10.8990 −0.385820
\(799\) −46.0454 −1.62897
\(800\) 0 0
\(801\) −16.8990 −0.597096
\(802\) −32.0908 −1.13317
\(803\) 13.1010 0.462325
\(804\) −28.0000 −0.987484
\(805\) 0 0
\(806\) −1.10102 −0.0387818
\(807\) 5.44949 0.191831
\(808\) −61.4847 −2.16302
\(809\) 6.55051 0.230304 0.115152 0.993348i \(-0.463265\pi\)
0.115152 + 0.993348i \(0.463265\pi\)
\(810\) 0 0
\(811\) −42.3939 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(812\) 41.3939 1.45264
\(813\) −31.6969 −1.11166
\(814\) −23.3939 −0.819955
\(815\) 0 0
\(816\) −21.7980 −0.763081
\(817\) −8.89898 −0.311336
\(818\) 53.6413 1.87552
\(819\) −4.44949 −0.155478
\(820\) 0 0
\(821\) 21.3031 0.743482 0.371741 0.928336i \(-0.378761\pi\)
0.371741 + 0.928336i \(0.378761\pi\)
\(822\) −41.3939 −1.44378
\(823\) −3.59592 −0.125346 −0.0626729 0.998034i \(-0.519962\pi\)
−0.0626729 + 0.998034i \(0.519962\pi\)
\(824\) 62.2020 2.16691
\(825\) 0 0
\(826\) 25.3485 0.881986
\(827\) −32.1464 −1.11784 −0.558920 0.829221i \(-0.688784\pi\)
−0.558920 + 0.829221i \(0.688784\pi\)
\(828\) 4.00000 0.139010
\(829\) −27.6969 −0.961954 −0.480977 0.876733i \(-0.659718\pi\)
−0.480977 + 0.876733i \(0.659718\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 35.5959 1.23407
\(833\) −32.6969 −1.13288
\(834\) 11.7526 0.406958
\(835\) 0 0
\(836\) 43.5959 1.50780
\(837\) −0.101021 −0.00349178
\(838\) 82.1816 2.83892
\(839\) −45.1918 −1.56020 −0.780098 0.625658i \(-0.784831\pi\)
−0.780098 + 0.625658i \(0.784831\pi\)
\(840\) 0 0
\(841\) 78.0908 2.69279
\(842\) −54.4949 −1.87802
\(843\) 19.3485 0.666397
\(844\) 44.0000 1.51454
\(845\) 0 0
\(846\) −20.6969 −0.711575
\(847\) −5.00000 −0.171802
\(848\) −2.20204 −0.0756184
\(849\) −2.10102 −0.0721068
\(850\) 0 0
\(851\) −3.89898 −0.133655
\(852\) 17.3939 0.595904
\(853\) −46.0908 −1.57812 −0.789060 0.614316i \(-0.789432\pi\)
−0.789060 + 0.614316i \(0.789432\pi\)
\(854\) 1.59592 0.0546112
\(855\) 0 0
\(856\) 26.6969 0.912483
\(857\) 13.5959 0.464428 0.232214 0.972665i \(-0.425403\pi\)
0.232214 + 0.972665i \(0.425403\pi\)
\(858\) 26.6969 0.911418
\(859\) 38.1918 1.30309 0.651544 0.758611i \(-0.274121\pi\)
0.651544 + 0.758611i \(0.274121\pi\)
\(860\) 0 0
\(861\) −5.44949 −0.185718
\(862\) −60.0000 −2.04361
\(863\) −6.49490 −0.221089 −0.110544 0.993871i \(-0.535259\pi\)
−0.110544 + 0.993871i \(0.535259\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −6.85357 −0.232894
\(867\) −12.6969 −0.431211
\(868\) 0.404082 0.0137154
\(869\) −9.79796 −0.332373
\(870\) 0 0
\(871\) −31.1464 −1.05536
\(872\) 56.5857 1.91623
\(873\) −3.10102 −0.104954
\(874\) 10.8990 0.368663
\(875\) 0 0
\(876\) −21.3939 −0.722832
\(877\) 10.4949 0.354388 0.177194 0.984176i \(-0.443298\pi\)
0.177194 + 0.984176i \(0.443298\pi\)
\(878\) −89.8888 −3.03360
\(879\) −21.2474 −0.716659
\(880\) 0 0
\(881\) 8.44949 0.284671 0.142335 0.989819i \(-0.454539\pi\)
0.142335 + 0.989819i \(0.454539\pi\)
\(882\) −14.6969 −0.494872
\(883\) 37.5505 1.26368 0.631838 0.775101i \(-0.282301\pi\)
0.631838 + 0.775101i \(0.282301\pi\)
\(884\) −96.9898 −3.26212
\(885\) 0 0
\(886\) 50.0908 1.68283
\(887\) 6.49490 0.218077 0.109039 0.994038i \(-0.465223\pi\)
0.109039 + 0.994038i \(0.465223\pi\)
\(888\) 19.1010 0.640988
\(889\) 1.55051 0.0520024
\(890\) 0 0
\(891\) 2.44949 0.0820610
\(892\) −86.3837 −2.89234
\(893\) −37.5959 −1.25810
\(894\) 3.30306 0.110471
\(895\) 0 0
\(896\) −19.5959 −0.654654
\(897\) 4.44949 0.148564
\(898\) −52.0454 −1.73678
\(899\) 1.04541 0.0348663
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 32.6969 1.08869
\(903\) 2.00000 0.0665558
\(904\) 8.09082 0.269097
\(905\) 0 0
\(906\) −34.2929 −1.13930
\(907\) −38.7980 −1.28827 −0.644133 0.764914i \(-0.722781\pi\)
−0.644133 + 0.764914i \(0.722781\pi\)
\(908\) −87.1918 −2.89356
\(909\) −12.5505 −0.416274
\(910\) 0 0
\(911\) −14.2020 −0.470535 −0.235267 0.971931i \(-0.575597\pi\)
−0.235267 + 0.971931i \(0.575597\pi\)
\(912\) −17.7980 −0.589349
\(913\) −37.3485 −1.23605
\(914\) −0.247449 −0.00818488
\(915\) 0 0
\(916\) −113.980 −3.76599
\(917\) −19.5959 −0.647114
\(918\) −13.3485 −0.440565
\(919\) −45.3939 −1.49741 −0.748703 0.662906i \(-0.769323\pi\)
−0.748703 + 0.662906i \(0.769323\pi\)
\(920\) 0 0
\(921\) −2.65153 −0.0873709
\(922\) −45.3031 −1.49198
\(923\) 19.3485 0.636863
\(924\) −9.79796 −0.322329
\(925\) 0 0
\(926\) 61.1010 2.00790
\(927\) 12.6969 0.417022
\(928\) 0 0
\(929\) 22.8434 0.749467 0.374733 0.927133i \(-0.377734\pi\)
0.374733 + 0.927133i \(0.377734\pi\)
\(930\) 0 0
\(931\) −26.6969 −0.874957
\(932\) −87.1918 −2.85606
\(933\) −12.0000 −0.392862
\(934\) 85.3485 2.79269
\(935\) 0 0
\(936\) −21.7980 −0.712489
\(937\) 8.89898 0.290717 0.145358 0.989379i \(-0.453566\pi\)
0.145358 + 0.989379i \(0.453566\pi\)
\(938\) 17.1464 0.559851
\(939\) 19.6969 0.642785
\(940\) 0 0
\(941\) −24.8536 −0.810203 −0.405102 0.914272i \(-0.632764\pi\)
−0.405102 + 0.914272i \(0.632764\pi\)
\(942\) 45.5505 1.48412
\(943\) 5.44949 0.177460
\(944\) 41.3939 1.34726
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 0.494897 0.0160820 0.00804100 0.999968i \(-0.497440\pi\)
0.00804100 + 0.999968i \(0.497440\pi\)
\(948\) 16.0000 0.519656
\(949\) −23.7980 −0.772514
\(950\) 0 0
\(951\) 6.24745 0.202587
\(952\) 26.6969 0.865253
\(953\) −10.4041 −0.337021 −0.168511 0.985700i \(-0.553896\pi\)
−0.168511 + 0.985700i \(0.553896\pi\)
\(954\) −1.34847 −0.0436583
\(955\) 0 0
\(956\) 26.2020 0.847435
\(957\) −25.3485 −0.819400
\(958\) 80.6969 2.60720
\(959\) 16.8990 0.545697
\(960\) 0 0
\(961\) −30.9898 −0.999671
\(962\) 42.4949 1.37009
\(963\) 5.44949 0.175607
\(964\) −117.394 −3.78100
\(965\) 0 0
\(966\) −2.44949 −0.0788110
\(967\) −54.0454 −1.73798 −0.868992 0.494827i \(-0.835231\pi\)
−0.868992 + 0.494827i \(0.835231\pi\)
\(968\) −24.4949 −0.787296
\(969\) −24.2474 −0.778940
\(970\) 0 0
\(971\) 22.2929 0.715412 0.357706 0.933834i \(-0.383559\pi\)
0.357706 + 0.933834i \(0.383559\pi\)
\(972\) −4.00000 −0.128300
\(973\) −4.79796 −0.153816
\(974\) 46.4041 1.48688
\(975\) 0 0
\(976\) 2.60612 0.0834200
\(977\) −19.6515 −0.628708 −0.314354 0.949306i \(-0.601788\pi\)
−0.314354 + 0.949306i \(0.601788\pi\)
\(978\) −24.4949 −0.783260
\(979\) −41.3939 −1.32295
\(980\) 0 0
\(981\) 11.5505 0.368779
\(982\) 30.7423 0.981028
\(983\) 33.7423 1.07621 0.538107 0.842877i \(-0.319140\pi\)
0.538107 + 0.842877i \(0.319140\pi\)
\(984\) −26.6969 −0.851067
\(985\) 0 0
\(986\) 138.136 4.39915
\(987\) 8.44949 0.268950
\(988\) −79.1918 −2.51943
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 57.7878 1.83569 0.917844 0.396941i \(-0.129928\pi\)
0.917844 + 0.396941i \(0.129928\pi\)
\(992\) 0 0
\(993\) −24.5959 −0.780528
\(994\) −10.6515 −0.337846
\(995\) 0 0
\(996\) 60.9898 1.93254
\(997\) −40.0908 −1.26969 −0.634844 0.772640i \(-0.718936\pi\)
−0.634844 + 0.772640i \(0.718936\pi\)
\(998\) −2.44949 −0.0775372
\(999\) 3.89898 0.123358
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 1725.2.a.y.1.2 2
3.2 odd 2 5175.2.a.bl.1.1 2
5.2 odd 4 1725.2.b.m.1174.4 4
5.3 odd 4 1725.2.b.m.1174.1 4
5.4 even 2 345.2.a.i.1.1 2
15.14 odd 2 1035.2.a.k.1.2 2
20.19 odd 2 5520.2.a.bi.1.1 2
115.114 odd 2 7935.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.i.1.1 2 5.4 even 2
1035.2.a.k.1.2 2 15.14 odd 2
1725.2.a.y.1.2 2 1.1 even 1 trivial
1725.2.b.m.1174.1 4 5.3 odd 4
1725.2.b.m.1174.4 4 5.2 odd 4
5175.2.a.bl.1.1 2 3.2 odd 2
5520.2.a.bi.1.1 2 20.19 odd 2
7935.2.a.t.1.1 2 115.114 odd 2