Properties

Label 931.2.a.n.1.2
Level $931$
Weight $2$
Character 931.1
Self dual yes
Analytic conductor $7.434$
Analytic rank $0$
Dimension $7$
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] = [931,2,Mod(1,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, 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("931.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 4x^{3} - 12x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 133)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.862998\) of defining polynomial
Character \(\chi\) \(=\) 931.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.40650 q^{2} +3.19504 q^{3} -0.0217491 q^{4} +0.295752 q^{5} -4.49383 q^{6} +2.84360 q^{8} +7.20828 q^{9} +O(q^{10})\) \(q-1.40650 q^{2} +3.19504 q^{3} -0.0217491 q^{4} +0.295752 q^{5} -4.49383 q^{6} +2.84360 q^{8} +7.20828 q^{9} -0.415977 q^{10} -0.372561 q^{11} -0.0694893 q^{12} +3.46904 q^{13} +0.944941 q^{15} -3.95603 q^{16} +3.29575 q^{17} -10.1385 q^{18} +1.00000 q^{19} -0.00643235 q^{20} +0.524008 q^{22} -7.41280 q^{23} +9.08541 q^{24} -4.91253 q^{25} -4.87922 q^{26} +13.4456 q^{27} +3.21894 q^{29} -1.32906 q^{30} -3.96455 q^{31} -0.123026 q^{32} -1.19035 q^{33} -4.63549 q^{34} -0.156774 q^{36} +9.83661 q^{37} -1.40650 q^{38} +11.0837 q^{39} +0.841001 q^{40} +2.81301 q^{41} -6.08278 q^{43} +0.00810286 q^{44} +2.13187 q^{45} +10.4261 q^{46} +6.82129 q^{47} -12.6397 q^{48} +6.90949 q^{50} +10.5301 q^{51} -0.0754486 q^{52} +10.4357 q^{53} -18.9113 q^{54} -0.110186 q^{55} +3.19504 q^{57} -4.52745 q^{58} -4.68799 q^{59} -0.0205516 q^{60} +7.09816 q^{61} +5.57615 q^{62} +8.08509 q^{64} +1.02598 q^{65} +1.67423 q^{66} -7.66601 q^{67} -0.0716797 q^{68} -23.6842 q^{69} +1.88170 q^{71} +20.4974 q^{72} -1.01791 q^{73} -13.8352 q^{74} -15.6957 q^{75} -0.0217491 q^{76} -15.5893 q^{78} -10.6056 q^{79} -1.17001 q^{80} +21.3345 q^{81} -3.95650 q^{82} -12.3958 q^{83} +0.974727 q^{85} +8.55545 q^{86} +10.2847 q^{87} -1.05941 q^{88} -6.18557 q^{89} -2.99848 q^{90} +0.161222 q^{92} -12.6669 q^{93} -9.59417 q^{94} +0.295752 q^{95} -0.393073 q^{96} -0.235136 q^{97} -2.68552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} - 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 2 q^{2} - 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 12 q^{8} + 15 q^{9} + 7 q^{11} - 22 q^{12} + 6 q^{13} + 2 q^{15} + 24 q^{16} + 19 q^{17} - 12 q^{18} + 7 q^{19} - 8 q^{20} - 6 q^{22} - q^{23} + 20 q^{24} - 3 q^{25} + 12 q^{26} - 14 q^{27} + 24 q^{29} - 20 q^{30} + 26 q^{32} - 14 q^{33} + 6 q^{34} + 46 q^{36} + 8 q^{37} + 2 q^{38} + 16 q^{39} - 10 q^{40} - 4 q^{41} + 4 q^{43} + 26 q^{44} + 14 q^{45} - 16 q^{46} + 5 q^{47} - 28 q^{48} + 16 q^{50} - 4 q^{51} + 42 q^{52} + 20 q^{53} - 24 q^{54} - 30 q^{55} - 2 q^{57} - 16 q^{59} - 44 q^{60} + 5 q^{61} + 24 q^{62} + 32 q^{64} + 26 q^{65} + 68 q^{66} - 4 q^{67} + 22 q^{68} - 36 q^{69} + 12 q^{71} + 3 q^{73} - 4 q^{74} + 18 q^{75} + 10 q^{76} - 14 q^{78} - 20 q^{79} - 4 q^{80} + 27 q^{81} - 48 q^{82} - 11 q^{83} + 26 q^{85} + 36 q^{86} - 16 q^{87} - 32 q^{88} - 10 q^{89} + 32 q^{90} - 30 q^{92} - 4 q^{93} + 16 q^{94} - 2 q^{95} - 12 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40650 −0.994548 −0.497274 0.867594i \(-0.665666\pi\)
−0.497274 + 0.867594i \(0.665666\pi\)
\(3\) 3.19504 1.84466 0.922329 0.386406i \(-0.126284\pi\)
0.922329 + 0.386406i \(0.126284\pi\)
\(4\) −0.0217491 −0.0108746
\(5\) 0.295752 0.132265 0.0661323 0.997811i \(-0.478934\pi\)
0.0661323 + 0.997811i \(0.478934\pi\)
\(6\) −4.49383 −1.83460
\(7\) 0 0
\(8\) 2.84360 1.00536
\(9\) 7.20828 2.40276
\(10\) −0.415977 −0.131543
\(11\) −0.372561 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(12\) −0.0694893 −0.0200598
\(13\) 3.46904 0.962140 0.481070 0.876682i \(-0.340248\pi\)
0.481070 + 0.876682i \(0.340248\pi\)
\(14\) 0 0
\(15\) 0.944941 0.243983
\(16\) −3.95603 −0.989007
\(17\) 3.29575 0.799337 0.399669 0.916660i \(-0.369125\pi\)
0.399669 + 0.916660i \(0.369125\pi\)
\(18\) −10.1385 −2.38966
\(19\) 1.00000 0.229416
\(20\) −0.00643235 −0.00143832
\(21\) 0 0
\(22\) 0.524008 0.111719
\(23\) −7.41280 −1.54568 −0.772838 0.634604i \(-0.781163\pi\)
−0.772838 + 0.634604i \(0.781163\pi\)
\(24\) 9.08541 1.85455
\(25\) −4.91253 −0.982506
\(26\) −4.87922 −0.956894
\(27\) 13.4456 2.58761
\(28\) 0 0
\(29\) 3.21894 0.597743 0.298871 0.954293i \(-0.403390\pi\)
0.298871 + 0.954293i \(0.403390\pi\)
\(30\) −1.32906 −0.242652
\(31\) −3.96455 −0.712054 −0.356027 0.934476i \(-0.615869\pi\)
−0.356027 + 0.934476i \(0.615869\pi\)
\(32\) −0.123026 −0.0217482
\(33\) −1.19035 −0.207213
\(34\) −4.63549 −0.794979
\(35\) 0 0
\(36\) −0.156774 −0.0261290
\(37\) 9.83661 1.61713 0.808564 0.588408i \(-0.200245\pi\)
0.808564 + 0.588408i \(0.200245\pi\)
\(38\) −1.40650 −0.228165
\(39\) 11.0837 1.77482
\(40\) 0.841001 0.132974
\(41\) 2.81301 0.439318 0.219659 0.975577i \(-0.429506\pi\)
0.219659 + 0.975577i \(0.429506\pi\)
\(42\) 0 0
\(43\) −6.08278 −0.927615 −0.463808 0.885936i \(-0.653517\pi\)
−0.463808 + 0.885936i \(0.653517\pi\)
\(44\) 0.00810286 0.00122155
\(45\) 2.13187 0.317800
\(46\) 10.4261 1.53725
\(47\) 6.82129 0.994988 0.497494 0.867468i \(-0.334254\pi\)
0.497494 + 0.867468i \(0.334254\pi\)
\(48\) −12.6397 −1.82438
\(49\) 0 0
\(50\) 6.90949 0.977149
\(51\) 10.5301 1.47450
\(52\) −0.0754486 −0.0104628
\(53\) 10.4357 1.43345 0.716724 0.697356i \(-0.245641\pi\)
0.716724 + 0.697356i \(0.245641\pi\)
\(54\) −18.9113 −2.57351
\(55\) −0.110186 −0.0148574
\(56\) 0 0
\(57\) 3.19504 0.423193
\(58\) −4.52745 −0.594484
\(59\) −4.68799 −0.610324 −0.305162 0.952300i \(-0.598711\pi\)
−0.305162 + 0.952300i \(0.598711\pi\)
\(60\) −0.0205516 −0.00265320
\(61\) 7.09816 0.908827 0.454413 0.890791i \(-0.349849\pi\)
0.454413 + 0.890791i \(0.349849\pi\)
\(62\) 5.57615 0.708172
\(63\) 0 0
\(64\) 8.08509 1.01064
\(65\) 1.02598 0.127257
\(66\) 1.67423 0.206083
\(67\) −7.66601 −0.936552 −0.468276 0.883582i \(-0.655125\pi\)
−0.468276 + 0.883582i \(0.655125\pi\)
\(68\) −0.0716797 −0.00869244
\(69\) −23.6842 −2.85124
\(70\) 0 0
\(71\) 1.88170 0.223317 0.111658 0.993747i \(-0.464384\pi\)
0.111658 + 0.993747i \(0.464384\pi\)
\(72\) 20.4974 2.41565
\(73\) −1.01791 −0.119138 −0.0595688 0.998224i \(-0.518973\pi\)
−0.0595688 + 0.998224i \(0.518973\pi\)
\(74\) −13.8352 −1.60831
\(75\) −15.6957 −1.81239
\(76\) −0.0217491 −0.00249480
\(77\) 0 0
\(78\) −15.5893 −1.76514
\(79\) −10.6056 −1.19323 −0.596614 0.802529i \(-0.703487\pi\)
−0.596614 + 0.802529i \(0.703487\pi\)
\(80\) −1.17001 −0.130811
\(81\) 21.3345 2.37050
\(82\) −3.95650 −0.436923
\(83\) −12.3958 −1.36061 −0.680306 0.732928i \(-0.738153\pi\)
−0.680306 + 0.732928i \(0.738153\pi\)
\(84\) 0 0
\(85\) 0.974727 0.105724
\(86\) 8.55545 0.922558
\(87\) 10.2847 1.10263
\(88\) −1.05941 −0.112934
\(89\) −6.18557 −0.655669 −0.327834 0.944735i \(-0.606319\pi\)
−0.327834 + 0.944735i \(0.606319\pi\)
\(90\) −2.99848 −0.316067
\(91\) 0 0
\(92\) 0.161222 0.0168085
\(93\) −12.6669 −1.31350
\(94\) −9.59417 −0.989563
\(95\) 0.295752 0.0303436
\(96\) −0.393073 −0.0401179
\(97\) −0.235136 −0.0238745 −0.0119372 0.999929i \(-0.503800\pi\)
−0.0119372 + 0.999929i \(0.503800\pi\)
\(98\) 0 0
\(99\) −2.68552 −0.269905
\(100\) 0.106843 0.0106843
\(101\) 0.771429 0.0767601 0.0383800 0.999263i \(-0.487780\pi\)
0.0383800 + 0.999263i \(0.487780\pi\)
\(102\) −14.8106 −1.46646
\(103\) 10.6191 1.04633 0.523167 0.852230i \(-0.324750\pi\)
0.523167 + 0.852230i \(0.324750\pi\)
\(104\) 9.86456 0.967300
\(105\) 0 0
\(106\) −14.6778 −1.42563
\(107\) 8.14803 0.787700 0.393850 0.919175i \(-0.371143\pi\)
0.393850 + 0.919175i \(0.371143\pi\)
\(108\) −0.292431 −0.0281392
\(109\) 6.70289 0.642020 0.321010 0.947076i \(-0.395978\pi\)
0.321010 + 0.947076i \(0.395978\pi\)
\(110\) 0.154977 0.0147764
\(111\) 31.4284 2.98305
\(112\) 0 0
\(113\) 15.6066 1.46814 0.734072 0.679072i \(-0.237617\pi\)
0.734072 + 0.679072i \(0.237617\pi\)
\(114\) −4.49383 −0.420886
\(115\) −2.19235 −0.204438
\(116\) −0.0700092 −0.00650019
\(117\) 25.0058 2.31179
\(118\) 6.59367 0.606996
\(119\) 0 0
\(120\) 2.68703 0.245291
\(121\) −10.8612 −0.987382
\(122\) −9.98359 −0.903872
\(123\) 8.98767 0.810391
\(124\) 0.0862254 0.00774327
\(125\) −2.93165 −0.262215
\(126\) 0 0
\(127\) −3.34348 −0.296686 −0.148343 0.988936i \(-0.547394\pi\)
−0.148343 + 0.988936i \(0.547394\pi\)
\(128\) −11.1257 −0.983378
\(129\) −19.4347 −1.71113
\(130\) −1.44304 −0.126563
\(131\) −15.7131 −1.37286 −0.686430 0.727196i \(-0.740823\pi\)
−0.686430 + 0.727196i \(0.740823\pi\)
\(132\) 0.0258890 0.00225335
\(133\) 0 0
\(134\) 10.7823 0.931445
\(135\) 3.97658 0.342249
\(136\) 9.37179 0.803624
\(137\) 13.5984 1.16179 0.580893 0.813980i \(-0.302703\pi\)
0.580893 + 0.813980i \(0.302703\pi\)
\(138\) 33.3119 2.83570
\(139\) −21.3319 −1.80935 −0.904675 0.426102i \(-0.859886\pi\)
−0.904675 + 0.426102i \(0.859886\pi\)
\(140\) 0 0
\(141\) 21.7943 1.83541
\(142\) −2.64662 −0.222099
\(143\) −1.29243 −0.108078
\(144\) −28.5162 −2.37635
\(145\) 0.952011 0.0790602
\(146\) 1.43170 0.118488
\(147\) 0 0
\(148\) −0.213938 −0.0175856
\(149\) 10.2208 0.837320 0.418660 0.908143i \(-0.362500\pi\)
0.418660 + 0.908143i \(0.362500\pi\)
\(150\) 22.0761 1.80251
\(151\) −2.58011 −0.209966 −0.104983 0.994474i \(-0.533479\pi\)
−0.104983 + 0.994474i \(0.533479\pi\)
\(152\) 2.84360 0.230646
\(153\) 23.7567 1.92062
\(154\) 0 0
\(155\) −1.17252 −0.0941795
\(156\) −0.241061 −0.0193004
\(157\) −21.6002 −1.72388 −0.861941 0.507009i \(-0.830751\pi\)
−0.861941 + 0.507009i \(0.830751\pi\)
\(158\) 14.9169 1.18672
\(159\) 33.3424 2.64422
\(160\) −0.0363853 −0.00287651
\(161\) 0 0
\(162\) −30.0070 −2.35758
\(163\) −15.4704 −1.21174 −0.605868 0.795565i \(-0.707174\pi\)
−0.605868 + 0.795565i \(0.707174\pi\)
\(164\) −0.0611804 −0.00477739
\(165\) −0.352048 −0.0274069
\(166\) 17.4347 1.35319
\(167\) 7.96646 0.616463 0.308231 0.951311i \(-0.400263\pi\)
0.308231 + 0.951311i \(0.400263\pi\)
\(168\) 0 0
\(169\) −0.965736 −0.0742874
\(170\) −1.37096 −0.105148
\(171\) 7.20828 0.551231
\(172\) 0.132295 0.0100874
\(173\) −5.61526 −0.426920 −0.213460 0.976952i \(-0.568473\pi\)
−0.213460 + 0.976952i \(0.568473\pi\)
\(174\) −14.4654 −1.09662
\(175\) 0 0
\(176\) 1.47386 0.111096
\(177\) −14.9783 −1.12584
\(178\) 8.70002 0.652094
\(179\) 7.43709 0.555874 0.277937 0.960599i \(-0.410349\pi\)
0.277937 + 0.960599i \(0.410349\pi\)
\(180\) −0.0463662 −0.00345594
\(181\) −8.99507 −0.668598 −0.334299 0.942467i \(-0.608500\pi\)
−0.334299 + 0.942467i \(0.608500\pi\)
\(182\) 0 0
\(183\) 22.6789 1.67647
\(184\) −21.0790 −1.55396
\(185\) 2.90920 0.213889
\(186\) 17.8160 1.30633
\(187\) −1.22787 −0.0897906
\(188\) −0.148357 −0.0108201
\(189\) 0 0
\(190\) −0.415977 −0.0301781
\(191\) 22.1575 1.60326 0.801629 0.597822i \(-0.203967\pi\)
0.801629 + 0.597822i \(0.203967\pi\)
\(192\) 25.8322 1.86428
\(193\) 0.407408 0.0293259 0.0146629 0.999892i \(-0.495332\pi\)
0.0146629 + 0.999892i \(0.495332\pi\)
\(194\) 0.330720 0.0237443
\(195\) 3.27804 0.234745
\(196\) 0 0
\(197\) −20.5203 −1.46201 −0.731006 0.682371i \(-0.760949\pi\)
−0.731006 + 0.682371i \(0.760949\pi\)
\(198\) 3.77720 0.268434
\(199\) −7.01563 −0.497325 −0.248662 0.968590i \(-0.579991\pi\)
−0.248662 + 0.968590i \(0.579991\pi\)
\(200\) −13.9693 −0.987775
\(201\) −24.4932 −1.72762
\(202\) −1.08502 −0.0763416
\(203\) 0 0
\(204\) −0.229020 −0.0160346
\(205\) 0.831953 0.0581061
\(206\) −14.9358 −1.04063
\(207\) −53.4335 −3.71389
\(208\) −13.7236 −0.951563
\(209\) −0.372561 −0.0257706
\(210\) 0 0
\(211\) −15.2256 −1.04817 −0.524087 0.851665i \(-0.675593\pi\)
−0.524087 + 0.851665i \(0.675593\pi\)
\(212\) −0.226967 −0.0155881
\(213\) 6.01211 0.411943
\(214\) −11.4602 −0.783405
\(215\) −1.79900 −0.122691
\(216\) 38.2340 2.60149
\(217\) 0 0
\(218\) −9.42763 −0.638520
\(219\) −3.25227 −0.219768
\(220\) 0.00239644 0.000161568 0
\(221\) 11.4331 0.769074
\(222\) −44.2041 −2.96678
\(223\) −0.0430042 −0.00287977 −0.00143989 0.999999i \(-0.500458\pi\)
−0.00143989 + 0.999999i \(0.500458\pi\)
\(224\) 0 0
\(225\) −35.4109 −2.36073
\(226\) −21.9507 −1.46014
\(227\) 15.7366 1.04448 0.522238 0.852800i \(-0.325097\pi\)
0.522238 + 0.852800i \(0.325097\pi\)
\(228\) −0.0694893 −0.00460204
\(229\) −10.5762 −0.698898 −0.349449 0.936955i \(-0.613631\pi\)
−0.349449 + 0.936955i \(0.613631\pi\)
\(230\) 3.08355 0.203323
\(231\) 0 0
\(232\) 9.15338 0.600949
\(233\) −0.526501 −0.0344922 −0.0172461 0.999851i \(-0.505490\pi\)
−0.0172461 + 0.999851i \(0.505490\pi\)
\(234\) −35.1708 −2.29919
\(235\) 2.01741 0.131602
\(236\) 0.101960 0.00663701
\(237\) −33.8854 −2.20110
\(238\) 0 0
\(239\) −4.56967 −0.295588 −0.147794 0.989018i \(-0.547217\pi\)
−0.147794 + 0.989018i \(0.547217\pi\)
\(240\) −3.73821 −0.241301
\(241\) −22.8596 −1.47252 −0.736258 0.676701i \(-0.763409\pi\)
−0.736258 + 0.676701i \(0.763409\pi\)
\(242\) 15.2763 0.981998
\(243\) 27.8277 1.78515
\(244\) −0.154379 −0.00988309
\(245\) 0 0
\(246\) −12.6412 −0.805972
\(247\) 3.46904 0.220730
\(248\) −11.2736 −0.715873
\(249\) −39.6050 −2.50986
\(250\) 4.12338 0.260786
\(251\) −8.49789 −0.536382 −0.268191 0.963366i \(-0.586426\pi\)
−0.268191 + 0.963366i \(0.586426\pi\)
\(252\) 0 0
\(253\) 2.76172 0.173628
\(254\) 4.70261 0.295068
\(255\) 3.11429 0.195024
\(256\) −0.521917 −0.0326198
\(257\) −11.3461 −0.707749 −0.353875 0.935293i \(-0.615136\pi\)
−0.353875 + 0.935293i \(0.615136\pi\)
\(258\) 27.3350 1.70180
\(259\) 0 0
\(260\) −0.0223141 −0.00138386
\(261\) 23.2031 1.43623
\(262\) 22.1005 1.36537
\(263\) −4.91687 −0.303187 −0.151594 0.988443i \(-0.548441\pi\)
−0.151594 + 0.988443i \(0.548441\pi\)
\(264\) −3.38486 −0.208324
\(265\) 3.08637 0.189594
\(266\) 0 0
\(267\) −19.7631 −1.20948
\(268\) 0.166729 0.0101846
\(269\) −28.8234 −1.75739 −0.878696 0.477381i \(-0.841586\pi\)
−0.878696 + 0.477381i \(0.841586\pi\)
\(270\) −5.59307 −0.340383
\(271\) −2.65446 −0.161247 −0.0806234 0.996745i \(-0.525691\pi\)
−0.0806234 + 0.996745i \(0.525691\pi\)
\(272\) −13.0381 −0.790550
\(273\) 0 0
\(274\) −19.1261 −1.15545
\(275\) 1.83022 0.110366
\(276\) 0.515110 0.0310060
\(277\) −3.59970 −0.216285 −0.108143 0.994135i \(-0.534490\pi\)
−0.108143 + 0.994135i \(0.534490\pi\)
\(278\) 30.0034 1.79949
\(279\) −28.5776 −1.71090
\(280\) 0 0
\(281\) 1.35683 0.0809415 0.0404708 0.999181i \(-0.487114\pi\)
0.0404708 + 0.999181i \(0.487114\pi\)
\(282\) −30.6538 −1.82540
\(283\) −6.05006 −0.359639 −0.179819 0.983700i \(-0.557551\pi\)
−0.179819 + 0.983700i \(0.557551\pi\)
\(284\) −0.0409253 −0.00242847
\(285\) 0.944941 0.0559735
\(286\) 1.81781 0.107489
\(287\) 0 0
\(288\) −0.886807 −0.0522556
\(289\) −6.13802 −0.361060
\(290\) −1.33901 −0.0786291
\(291\) −0.751270 −0.0440402
\(292\) 0.0221387 0.00129557
\(293\) −3.98610 −0.232871 −0.116435 0.993198i \(-0.537147\pi\)
−0.116435 + 0.993198i \(0.537147\pi\)
\(294\) 0 0
\(295\) −1.38648 −0.0807242
\(296\) 27.9713 1.62580
\(297\) −5.00931 −0.290670
\(298\) −14.3756 −0.832755
\(299\) −25.7153 −1.48716
\(300\) 0.341368 0.0197089
\(301\) 0 0
\(302\) 3.62893 0.208822
\(303\) 2.46475 0.141596
\(304\) −3.95603 −0.226894
\(305\) 2.09930 0.120206
\(306\) −33.4139 −1.91015
\(307\) 17.9160 1.02252 0.511259 0.859427i \(-0.329179\pi\)
0.511259 + 0.859427i \(0.329179\pi\)
\(308\) 0 0
\(309\) 33.9286 1.93013
\(310\) 1.64916 0.0936660
\(311\) 18.8468 1.06870 0.534352 0.845262i \(-0.320556\pi\)
0.534352 + 0.845262i \(0.320556\pi\)
\(312\) 31.5177 1.78434
\(313\) −18.8133 −1.06339 −0.531694 0.846936i \(-0.678444\pi\)
−0.531694 + 0.846936i \(0.678444\pi\)
\(314\) 30.3807 1.71448
\(315\) 0 0
\(316\) 0.230663 0.0129758
\(317\) 14.0847 0.791078 0.395539 0.918449i \(-0.370558\pi\)
0.395539 + 0.918449i \(0.370558\pi\)
\(318\) −46.8962 −2.62981
\(319\) −1.19925 −0.0671452
\(320\) 2.39119 0.133671
\(321\) 26.0333 1.45304
\(322\) 0 0
\(323\) 3.29575 0.183381
\(324\) −0.464006 −0.0257781
\(325\) −17.0418 −0.945308
\(326\) 21.7592 1.20513
\(327\) 21.4160 1.18431
\(328\) 7.99905 0.441674
\(329\) 0 0
\(330\) 0.495156 0.0272575
\(331\) 7.13182 0.392000 0.196000 0.980604i \(-0.437205\pi\)
0.196000 + 0.980604i \(0.437205\pi\)
\(332\) 0.269597 0.0147961
\(333\) 70.9051 3.88557
\(334\) −11.2048 −0.613102
\(335\) −2.26724 −0.123873
\(336\) 0 0
\(337\) −6.45355 −0.351547 −0.175774 0.984431i \(-0.556243\pi\)
−0.175774 + 0.984431i \(0.556243\pi\)
\(338\) 1.35831 0.0738824
\(339\) 49.8637 2.70822
\(340\) −0.0211994 −0.00114970
\(341\) 1.47703 0.0799859
\(342\) −10.1385 −0.548226
\(343\) 0 0
\(344\) −17.2970 −0.932590
\(345\) −7.00466 −0.377118
\(346\) 7.89788 0.424593
\(347\) 4.87230 0.261559 0.130779 0.991411i \(-0.458252\pi\)
0.130779 + 0.991411i \(0.458252\pi\)
\(348\) −0.223682 −0.0119906
\(349\) 11.2362 0.601459 0.300729 0.953709i \(-0.402770\pi\)
0.300729 + 0.953709i \(0.402770\pi\)
\(350\) 0 0
\(351\) 46.6435 2.48965
\(352\) 0.0458347 0.00244300
\(353\) 6.20110 0.330051 0.165026 0.986289i \(-0.447229\pi\)
0.165026 + 0.986289i \(0.447229\pi\)
\(354\) 21.0670 1.11970
\(355\) 0.556517 0.0295369
\(356\) 0.134531 0.00713011
\(357\) 0 0
\(358\) −10.4603 −0.552844
\(359\) 19.4926 1.02878 0.514390 0.857557i \(-0.328018\pi\)
0.514390 + 0.857557i \(0.328018\pi\)
\(360\) 6.06217 0.319504
\(361\) 1.00000 0.0526316
\(362\) 12.6516 0.664953
\(363\) −34.7020 −1.82138
\(364\) 0 0
\(365\) −0.301050 −0.0157577
\(366\) −31.8980 −1.66733
\(367\) 23.3201 1.21730 0.608649 0.793440i \(-0.291712\pi\)
0.608649 + 0.793440i \(0.291712\pi\)
\(368\) 29.3252 1.52868
\(369\) 20.2769 1.05558
\(370\) −4.09180 −0.212723
\(371\) 0 0
\(372\) 0.275494 0.0142837
\(373\) −0.251690 −0.0130320 −0.00651601 0.999979i \(-0.502074\pi\)
−0.00651601 + 0.999979i \(0.502074\pi\)
\(374\) 1.72700 0.0893010
\(375\) −9.36676 −0.483697
\(376\) 19.3970 1.00032
\(377\) 11.1667 0.575112
\(378\) 0 0
\(379\) −6.23458 −0.320249 −0.160125 0.987097i \(-0.551190\pi\)
−0.160125 + 0.987097i \(0.551190\pi\)
\(380\) −0.00643235 −0.000329973 0
\(381\) −10.6826 −0.547284
\(382\) −31.1645 −1.59452
\(383\) −11.5020 −0.587723 −0.293862 0.955848i \(-0.594940\pi\)
−0.293862 + 0.955848i \(0.594940\pi\)
\(384\) −35.5469 −1.81400
\(385\) 0 0
\(386\) −0.573021 −0.0291660
\(387\) −43.8464 −2.22884
\(388\) 0.00511401 0.000259624 0
\(389\) −34.8083 −1.76485 −0.882426 0.470452i \(-0.844091\pi\)
−0.882426 + 0.470452i \(0.844091\pi\)
\(390\) −4.61058 −0.233466
\(391\) −24.4307 −1.23552
\(392\) 0 0
\(393\) −50.2040 −2.53246
\(394\) 28.8619 1.45404
\(395\) −3.13664 −0.157822
\(396\) 0.0584077 0.00293510
\(397\) 29.1524 1.46311 0.731557 0.681780i \(-0.238794\pi\)
0.731557 + 0.681780i \(0.238794\pi\)
\(398\) 9.86750 0.494613
\(399\) 0 0
\(400\) 19.4341 0.971706
\(401\) −14.5746 −0.727820 −0.363910 0.931434i \(-0.618558\pi\)
−0.363910 + 0.931434i \(0.618558\pi\)
\(402\) 34.4498 1.71820
\(403\) −13.7532 −0.685095
\(404\) −0.0167779 −0.000834732 0
\(405\) 6.30973 0.313533
\(406\) 0 0
\(407\) −3.66473 −0.181654
\(408\) 29.9432 1.48241
\(409\) −9.60370 −0.474872 −0.237436 0.971403i \(-0.576307\pi\)
−0.237436 + 0.971403i \(0.576307\pi\)
\(410\) −1.17015 −0.0577893
\(411\) 43.4473 2.14310
\(412\) −0.230957 −0.0113784
\(413\) 0 0
\(414\) 75.1544 3.69364
\(415\) −3.66608 −0.179961
\(416\) −0.426783 −0.0209248
\(417\) −68.1564 −3.33763
\(418\) 0.524008 0.0256301
\(419\) −17.2401 −0.842234 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(420\) 0 0
\(421\) 15.3528 0.748251 0.374126 0.927378i \(-0.377943\pi\)
0.374126 + 0.927378i \(0.377943\pi\)
\(422\) 21.4149 1.04246
\(423\) 49.1698 2.39072
\(424\) 29.6748 1.44114
\(425\) −16.1905 −0.785354
\(426\) −8.45605 −0.409697
\(427\) 0 0
\(428\) −0.177212 −0.00856589
\(429\) −4.12936 −0.199368
\(430\) 2.53029 0.122022
\(431\) −5.98348 −0.288214 −0.144107 0.989562i \(-0.546031\pi\)
−0.144107 + 0.989562i \(0.546031\pi\)
\(432\) −53.1913 −2.55917
\(433\) −15.2443 −0.732596 −0.366298 0.930498i \(-0.619375\pi\)
−0.366298 + 0.930498i \(0.619375\pi\)
\(434\) 0 0
\(435\) 3.04171 0.145839
\(436\) −0.145782 −0.00698169
\(437\) −7.41280 −0.354602
\(438\) 4.57433 0.218570
\(439\) −31.9500 −1.52489 −0.762445 0.647053i \(-0.776001\pi\)
−0.762445 + 0.647053i \(0.776001\pi\)
\(440\) −0.313324 −0.0149371
\(441\) 0 0
\(442\) −16.0807 −0.764881
\(443\) 33.7439 1.60322 0.801610 0.597848i \(-0.203977\pi\)
0.801610 + 0.597848i \(0.203977\pi\)
\(444\) −0.683539 −0.0324393
\(445\) −1.82940 −0.0867217
\(446\) 0.0604855 0.00286407
\(447\) 32.6559 1.54457
\(448\) 0 0
\(449\) 21.1734 0.999234 0.499617 0.866246i \(-0.333474\pi\)
0.499617 + 0.866246i \(0.333474\pi\)
\(450\) 49.8056 2.34786
\(451\) −1.04802 −0.0493491
\(452\) −0.339430 −0.0159654
\(453\) −8.24356 −0.387316
\(454\) −22.1336 −1.03878
\(455\) 0 0
\(456\) 9.08541 0.425463
\(457\) −8.06009 −0.377035 −0.188518 0.982070i \(-0.560368\pi\)
−0.188518 + 0.982070i \(0.560368\pi\)
\(458\) 14.8755 0.695088
\(459\) 44.3135 2.06838
\(460\) 0.0476817 0.00222317
\(461\) 12.4647 0.580537 0.290268 0.956945i \(-0.406255\pi\)
0.290268 + 0.956945i \(0.406255\pi\)
\(462\) 0 0
\(463\) 14.8402 0.689685 0.344842 0.938661i \(-0.387932\pi\)
0.344842 + 0.938661i \(0.387932\pi\)
\(464\) −12.7342 −0.591172
\(465\) −3.74626 −0.173729
\(466\) 0.740525 0.0343042
\(467\) −2.49691 −0.115543 −0.0577715 0.998330i \(-0.518399\pi\)
−0.0577715 + 0.998330i \(0.518399\pi\)
\(468\) −0.543855 −0.0251397
\(469\) 0 0
\(470\) −2.83750 −0.130884
\(471\) −69.0134 −3.17997
\(472\) −13.3307 −0.613597
\(473\) 2.26620 0.104200
\(474\) 47.6600 2.18909
\(475\) −4.91253 −0.225402
\(476\) 0 0
\(477\) 75.2232 3.44424
\(478\) 6.42726 0.293976
\(479\) −15.7163 −0.718097 −0.359049 0.933319i \(-0.616899\pi\)
−0.359049 + 0.933319i \(0.616899\pi\)
\(480\) −0.116252 −0.00530617
\(481\) 34.1236 1.55590
\(482\) 32.1521 1.46449
\(483\) 0 0
\(484\) 0.236222 0.0107373
\(485\) −0.0695421 −0.00315775
\(486\) −39.1397 −1.77541
\(487\) 14.8609 0.673412 0.336706 0.941610i \(-0.390687\pi\)
0.336706 + 0.941610i \(0.390687\pi\)
\(488\) 20.1843 0.913701
\(489\) −49.4286 −2.23524
\(490\) 0 0
\(491\) −2.79345 −0.126067 −0.0630334 0.998011i \(-0.520077\pi\)
−0.0630334 + 0.998011i \(0.520077\pi\)
\(492\) −0.195474 −0.00881264
\(493\) 10.6088 0.477798
\(494\) −4.87922 −0.219527
\(495\) −0.794250 −0.0356989
\(496\) 15.6839 0.704226
\(497\) 0 0
\(498\) 55.7045 2.49618
\(499\) 31.0687 1.39082 0.695412 0.718611i \(-0.255222\pi\)
0.695412 + 0.718611i \(0.255222\pi\)
\(500\) 0.0637609 0.00285147
\(501\) 25.4532 1.13716
\(502\) 11.9523 0.533458
\(503\) −9.93140 −0.442819 −0.221410 0.975181i \(-0.571066\pi\)
−0.221410 + 0.975181i \(0.571066\pi\)
\(504\) 0 0
\(505\) 0.228152 0.0101526
\(506\) −3.88436 −0.172681
\(507\) −3.08557 −0.137035
\(508\) 0.0727177 0.00322633
\(509\) 14.3464 0.635894 0.317947 0.948108i \(-0.397007\pi\)
0.317947 + 0.948108i \(0.397007\pi\)
\(510\) −4.38026 −0.193961
\(511\) 0 0
\(512\) 22.9854 1.01582
\(513\) 13.4456 0.593639
\(514\) 15.9583 0.703891
\(515\) 3.14063 0.138393
\(516\) 0.422688 0.0186078
\(517\) −2.54134 −0.111768
\(518\) 0 0
\(519\) −17.9410 −0.787522
\(520\) 2.91747 0.127939
\(521\) 35.4447 1.55286 0.776431 0.630203i \(-0.217028\pi\)
0.776431 + 0.630203i \(0.217028\pi\)
\(522\) −32.6352 −1.42840
\(523\) −33.8791 −1.48143 −0.740714 0.671820i \(-0.765513\pi\)
−0.740714 + 0.671820i \(0.765513\pi\)
\(524\) 0.341746 0.0149292
\(525\) 0 0
\(526\) 6.91559 0.301534
\(527\) −13.0662 −0.569171
\(528\) 4.70904 0.204935
\(529\) 31.9496 1.38911
\(530\) −4.34099 −0.188561
\(531\) −33.7923 −1.46646
\(532\) 0 0
\(533\) 9.75844 0.422685
\(534\) 27.7969 1.20289
\(535\) 2.40980 0.104185
\(536\) −21.7990 −0.941574
\(537\) 23.7618 1.02540
\(538\) 40.5402 1.74781
\(539\) 0 0
\(540\) −0.0864871 −0.00372181
\(541\) −31.4110 −1.35046 −0.675232 0.737606i \(-0.735956\pi\)
−0.675232 + 0.737606i \(0.735956\pi\)
\(542\) 3.73350 0.160368
\(543\) −28.7396 −1.23333
\(544\) −0.405464 −0.0173841
\(545\) 1.98240 0.0849165
\(546\) 0 0
\(547\) −39.3436 −1.68221 −0.841106 0.540870i \(-0.818095\pi\)
−0.841106 + 0.540870i \(0.818095\pi\)
\(548\) −0.295752 −0.0126339
\(549\) 51.1656 2.18369
\(550\) −2.57420 −0.109764
\(551\) 3.21894 0.137132
\(552\) −67.3483 −2.86653
\(553\) 0 0
\(554\) 5.06300 0.215106
\(555\) 9.29501 0.394551
\(556\) 0.463951 0.0196759
\(557\) 22.2814 0.944091 0.472046 0.881574i \(-0.343516\pi\)
0.472046 + 0.881574i \(0.343516\pi\)
\(558\) 40.1945 1.70157
\(559\) −21.1014 −0.892495
\(560\) 0 0
\(561\) −3.92309 −0.165633
\(562\) −1.90838 −0.0805002
\(563\) −22.9600 −0.967649 −0.483825 0.875165i \(-0.660753\pi\)
−0.483825 + 0.875165i \(0.660753\pi\)
\(564\) −0.474007 −0.0199593
\(565\) 4.61569 0.194183
\(566\) 8.50943 0.357678
\(567\) 0 0
\(568\) 5.35080 0.224514
\(569\) −5.71545 −0.239604 −0.119802 0.992798i \(-0.538226\pi\)
−0.119802 + 0.992798i \(0.538226\pi\)
\(570\) −1.32906 −0.0556683
\(571\) 20.7502 0.868370 0.434185 0.900824i \(-0.357036\pi\)
0.434185 + 0.900824i \(0.357036\pi\)
\(572\) 0.0281092 0.00117530
\(573\) 70.7940 2.95746
\(574\) 0 0
\(575\) 36.4156 1.51864
\(576\) 58.2796 2.42832
\(577\) 3.83652 0.159717 0.0798583 0.996806i \(-0.474553\pi\)
0.0798583 + 0.996806i \(0.474553\pi\)
\(578\) 8.63314 0.359091
\(579\) 1.30169 0.0540962
\(580\) −0.0207054 −0.000859745 0
\(581\) 0 0
\(582\) 1.05666 0.0438001
\(583\) −3.88792 −0.161021
\(584\) −2.89453 −0.119777
\(585\) 7.39554 0.305768
\(586\) 5.60647 0.231601
\(587\) 36.8951 1.52282 0.761412 0.648269i \(-0.224507\pi\)
0.761412 + 0.648269i \(0.224507\pi\)
\(588\) 0 0
\(589\) −3.96455 −0.163356
\(590\) 1.95009 0.0802841
\(591\) −65.5632 −2.69691
\(592\) −38.9139 −1.59935
\(593\) −37.4036 −1.53598 −0.767990 0.640462i \(-0.778743\pi\)
−0.767990 + 0.640462i \(0.778743\pi\)
\(594\) 7.04561 0.289085
\(595\) 0 0
\(596\) −0.222293 −0.00910549
\(597\) −22.4152 −0.917394
\(598\) 36.1687 1.47905
\(599\) −14.1382 −0.577672 −0.288836 0.957379i \(-0.593268\pi\)
−0.288836 + 0.957379i \(0.593268\pi\)
\(600\) −44.6323 −1.82211
\(601\) 42.8102 1.74627 0.873133 0.487482i \(-0.162084\pi\)
0.873133 + 0.487482i \(0.162084\pi\)
\(602\) 0 0
\(603\) −55.2587 −2.25031
\(604\) 0.0561151 0.00228329
\(605\) −3.21223 −0.130596
\(606\) −3.46667 −0.140824
\(607\) 13.3315 0.541107 0.270554 0.962705i \(-0.412793\pi\)
0.270554 + 0.962705i \(0.412793\pi\)
\(608\) −0.123026 −0.00498937
\(609\) 0 0
\(610\) −2.95267 −0.119550
\(611\) 23.6634 0.957317
\(612\) −0.516688 −0.0208859
\(613\) 3.39651 0.137184 0.0685919 0.997645i \(-0.478149\pi\)
0.0685919 + 0.997645i \(0.478149\pi\)
\(614\) −25.1989 −1.01694
\(615\) 2.65812 0.107186
\(616\) 0 0
\(617\) −26.1522 −1.05285 −0.526424 0.850222i \(-0.676467\pi\)
−0.526424 + 0.850222i \(0.676467\pi\)
\(618\) −47.7206 −1.91961
\(619\) 46.8265 1.88211 0.941057 0.338247i \(-0.109834\pi\)
0.941057 + 0.338247i \(0.109834\pi\)
\(620\) 0.0255014 0.00102416
\(621\) −99.6698 −3.99961
\(622\) −26.5081 −1.06288
\(623\) 0 0
\(624\) −43.8476 −1.75531
\(625\) 23.6956 0.947824
\(626\) 26.4609 1.05759
\(627\) −1.19035 −0.0475378
\(628\) 0.469785 0.0187465
\(629\) 32.4190 1.29263
\(630\) 0 0
\(631\) 12.1061 0.481935 0.240968 0.970533i \(-0.422535\pi\)
0.240968 + 0.970533i \(0.422535\pi\)
\(632\) −30.1581 −1.19963
\(633\) −48.6464 −1.93352
\(634\) −19.8102 −0.786765
\(635\) −0.988842 −0.0392410
\(636\) −0.725167 −0.0287548
\(637\) 0 0
\(638\) 1.68675 0.0667791
\(639\) 13.5638 0.536577
\(640\) −3.29044 −0.130066
\(641\) 17.4856 0.690638 0.345319 0.938485i \(-0.387771\pi\)
0.345319 + 0.938485i \(0.387771\pi\)
\(642\) −36.6159 −1.44511
\(643\) 0.553783 0.0218391 0.0109195 0.999940i \(-0.496524\pi\)
0.0109195 + 0.999940i \(0.496524\pi\)
\(644\) 0 0
\(645\) −5.74787 −0.226322
\(646\) −4.63549 −0.182381
\(647\) 13.1248 0.515991 0.257995 0.966146i \(-0.416938\pi\)
0.257995 + 0.966146i \(0.416938\pi\)
\(648\) 60.6667 2.38321
\(649\) 1.74656 0.0685585
\(650\) 23.9693 0.940154
\(651\) 0 0
\(652\) 0.336468 0.0131771
\(653\) 3.42592 0.134067 0.0670333 0.997751i \(-0.478647\pi\)
0.0670333 + 0.997751i \(0.478647\pi\)
\(654\) −30.1217 −1.17785
\(655\) −4.64719 −0.181581
\(656\) −11.1283 −0.434488
\(657\) −7.33740 −0.286259
\(658\) 0 0
\(659\) 3.34771 0.130408 0.0652041 0.997872i \(-0.479230\pi\)
0.0652041 + 0.997872i \(0.479230\pi\)
\(660\) 0.00765673 0.000298038 0
\(661\) 25.2423 0.981810 0.490905 0.871213i \(-0.336666\pi\)
0.490905 + 0.871213i \(0.336666\pi\)
\(662\) −10.0309 −0.389863
\(663\) 36.5292 1.41868
\(664\) −35.2486 −1.36791
\(665\) 0 0
\(666\) −99.7282 −3.86439
\(667\) −23.8614 −0.923916
\(668\) −0.173263 −0.00670376
\(669\) −0.137400 −0.00531219
\(670\) 3.18888 0.123197
\(671\) −2.64450 −0.102090
\(672\) 0 0
\(673\) −5.61847 −0.216576 −0.108288 0.994120i \(-0.534537\pi\)
−0.108288 + 0.994120i \(0.534537\pi\)
\(674\) 9.07693 0.349630
\(675\) −66.0521 −2.54235
\(676\) 0.0210039 0.000807843 0
\(677\) −51.6090 −1.98349 −0.991747 0.128209i \(-0.959077\pi\)
−0.991747 + 0.128209i \(0.959077\pi\)
\(678\) −70.1334 −2.69346
\(679\) 0 0
\(680\) 2.77173 0.106291
\(681\) 50.2792 1.92670
\(682\) −2.07745 −0.0795498
\(683\) 16.4961 0.631207 0.315603 0.948891i \(-0.397793\pi\)
0.315603 + 0.948891i \(0.397793\pi\)
\(684\) −0.156774 −0.00599440
\(685\) 4.02175 0.153663
\(686\) 0 0
\(687\) −33.7915 −1.28923
\(688\) 24.0637 0.917418
\(689\) 36.2018 1.37918
\(690\) 9.85207 0.375062
\(691\) 11.7493 0.446965 0.223482 0.974708i \(-0.428257\pi\)
0.223482 + 0.974708i \(0.428257\pi\)
\(692\) 0.122127 0.00464257
\(693\) 0 0
\(694\) −6.85290 −0.260133
\(695\) −6.30897 −0.239313
\(696\) 29.2454 1.10854
\(697\) 9.27097 0.351163
\(698\) −15.8037 −0.598180
\(699\) −1.68219 −0.0636263
\(700\) 0 0
\(701\) −28.7953 −1.08758 −0.543792 0.839220i \(-0.683012\pi\)
−0.543792 + 0.839220i \(0.683012\pi\)
\(702\) −65.6042 −2.47607
\(703\) 9.83661 0.370995
\(704\) −3.01219 −0.113526
\(705\) 6.44572 0.242760
\(706\) −8.72187 −0.328252
\(707\) 0 0
\(708\) 0.325765 0.0122430
\(709\) 18.1688 0.682345 0.341172 0.940001i \(-0.389176\pi\)
0.341172 + 0.940001i \(0.389176\pi\)
\(710\) −0.782743 −0.0293758
\(711\) −76.4484 −2.86704
\(712\) −17.5893 −0.659185
\(713\) 29.3884 1.10060
\(714\) 0 0
\(715\) −0.382239 −0.0142949
\(716\) −0.161750 −0.00604489
\(717\) −14.6003 −0.545258
\(718\) −27.4164 −1.02317
\(719\) 34.2823 1.27851 0.639256 0.768994i \(-0.279242\pi\)
0.639256 + 0.768994i \(0.279242\pi\)
\(720\) −8.43373 −0.314306
\(721\) 0 0
\(722\) −1.40650 −0.0523446
\(723\) −73.0373 −2.71629
\(724\) 0.195635 0.00727071
\(725\) −15.8132 −0.587286
\(726\) 48.8084 1.81145
\(727\) −13.1214 −0.486646 −0.243323 0.969945i \(-0.578237\pi\)
−0.243323 + 0.969945i \(0.578237\pi\)
\(728\) 0 0
\(729\) 24.9070 0.922483
\(730\) 0.423428 0.0156718
\(731\) −20.0473 −0.741478
\(732\) −0.493247 −0.0182309
\(733\) −6.06773 −0.224117 −0.112058 0.993702i \(-0.535744\pi\)
−0.112058 + 0.993702i \(0.535744\pi\)
\(734\) −32.7997 −1.21066
\(735\) 0 0
\(736\) 0.911968 0.0336156
\(737\) 2.85605 0.105204
\(738\) −28.5196 −1.04982
\(739\) 23.7399 0.873288 0.436644 0.899634i \(-0.356167\pi\)
0.436644 + 0.899634i \(0.356167\pi\)
\(740\) −0.0632726 −0.00232595
\(741\) 11.0837 0.407171
\(742\) 0 0
\(743\) −33.1525 −1.21625 −0.608124 0.793842i \(-0.708078\pi\)
−0.608124 + 0.793842i \(0.708078\pi\)
\(744\) −36.0195 −1.32054
\(745\) 3.02283 0.110748
\(746\) 0.354003 0.0129610
\(747\) −89.3522 −3.26923
\(748\) 0.0267050 0.000976433 0
\(749\) 0 0
\(750\) 13.1744 0.481060
\(751\) 37.5315 1.36954 0.684772 0.728757i \(-0.259902\pi\)
0.684772 + 0.728757i \(0.259902\pi\)
\(752\) −26.9852 −0.984050
\(753\) −27.1511 −0.989441
\(754\) −15.7059 −0.571977
\(755\) −0.763074 −0.0277711
\(756\) 0 0
\(757\) 14.2165 0.516708 0.258354 0.966050i \(-0.416820\pi\)
0.258354 + 0.966050i \(0.416820\pi\)
\(758\) 8.76896 0.318503
\(759\) 8.82379 0.320283
\(760\) 0.841001 0.0305063
\(761\) 51.4365 1.86457 0.932285 0.361725i \(-0.117812\pi\)
0.932285 + 0.361725i \(0.117812\pi\)
\(762\) 15.0250 0.544300
\(763\) 0 0
\(764\) −0.481905 −0.0174347
\(765\) 7.02611 0.254029
\(766\) 16.1776 0.584519
\(767\) −16.2628 −0.587217
\(768\) −1.66755 −0.0601724
\(769\) 29.5161 1.06438 0.532188 0.846626i \(-0.321370\pi\)
0.532188 + 0.846626i \(0.321370\pi\)
\(770\) 0 0
\(771\) −36.2512 −1.30556
\(772\) −0.00886077 −0.000318906 0
\(773\) −47.3158 −1.70183 −0.850916 0.525301i \(-0.823953\pi\)
−0.850916 + 0.525301i \(0.823953\pi\)
\(774\) 61.6701 2.21669
\(775\) 19.4760 0.699597
\(776\) −0.668633 −0.0240025
\(777\) 0 0
\(778\) 48.9580 1.75523
\(779\) 2.81301 0.100786
\(780\) −0.0712945 −0.00255275
\(781\) −0.701047 −0.0250854
\(782\) 34.3619 1.22878
\(783\) 43.2807 1.54673
\(784\) 0 0
\(785\) −6.38830 −0.228008
\(786\) 70.6120 2.51865
\(787\) −20.6740 −0.736948 −0.368474 0.929638i \(-0.620120\pi\)
−0.368474 + 0.929638i \(0.620120\pi\)
\(788\) 0.446299 0.0158987
\(789\) −15.7096 −0.559277
\(790\) 4.41170 0.156961
\(791\) 0 0
\(792\) −7.63654 −0.271353
\(793\) 24.6238 0.874418
\(794\) −41.0029 −1.45514
\(795\) 9.86109 0.349737
\(796\) 0.152584 0.00540819
\(797\) −19.0955 −0.676398 −0.338199 0.941075i \(-0.609818\pi\)
−0.338199 + 0.941075i \(0.609818\pi\)
\(798\) 0 0
\(799\) 22.4813 0.795331
\(800\) 0.604370 0.0213677
\(801\) −44.5873 −1.57542
\(802\) 20.4992 0.723851
\(803\) 0.379234 0.0133829
\(804\) 0.532705 0.0187871
\(805\) 0 0
\(806\) 19.3439 0.681360
\(807\) −92.0919 −3.24179
\(808\) 2.19363 0.0771717
\(809\) 7.56887 0.266107 0.133054 0.991109i \(-0.457522\pi\)
0.133054 + 0.991109i \(0.457522\pi\)
\(810\) −8.87465 −0.311824
\(811\) 10.5995 0.372200 0.186100 0.982531i \(-0.440415\pi\)
0.186100 + 0.982531i \(0.440415\pi\)
\(812\) 0 0
\(813\) −8.48110 −0.297445
\(814\) 5.15446 0.180664
\(815\) −4.57541 −0.160270
\(816\) −41.6572 −1.45829
\(817\) −6.08278 −0.212810
\(818\) 13.5076 0.472283
\(819\) 0 0
\(820\) −0.0180943 −0.000631879 0
\(821\) 5.17484 0.180603 0.0903016 0.995914i \(-0.471217\pi\)
0.0903016 + 0.995914i \(0.471217\pi\)
\(822\) −61.1088 −2.13141
\(823\) −11.4852 −0.400348 −0.200174 0.979760i \(-0.564151\pi\)
−0.200174 + 0.979760i \(0.564151\pi\)
\(824\) 30.1965 1.05195
\(825\) 5.84761 0.203588
\(826\) 0 0
\(827\) −4.29977 −0.149518 −0.0747588 0.997202i \(-0.523819\pi\)
−0.0747588 + 0.997202i \(0.523819\pi\)
\(828\) 1.16213 0.0403869
\(829\) 56.5964 1.96567 0.982837 0.184474i \(-0.0590582\pi\)
0.982837 + 0.184474i \(0.0590582\pi\)
\(830\) 5.15635 0.178980
\(831\) −11.5012 −0.398972
\(832\) 28.0475 0.972374
\(833\) 0 0
\(834\) 95.8621 3.31943
\(835\) 2.35610 0.0815362
\(836\) 0.00810286 0.000280243 0
\(837\) −53.3059 −1.84252
\(838\) 24.2483 0.837642
\(839\) −8.94673 −0.308875 −0.154438 0.988003i \(-0.549357\pi\)
−0.154438 + 0.988003i \(0.549357\pi\)
\(840\) 0 0
\(841\) −18.6384 −0.642703
\(842\) −21.5938 −0.744172
\(843\) 4.33512 0.149309
\(844\) 0.331143 0.0113984
\(845\) −0.285619 −0.00982558
\(846\) −69.1575 −2.37768
\(847\) 0 0
\(848\) −41.2838 −1.41769
\(849\) −19.3302 −0.663410
\(850\) 22.7720 0.781072
\(851\) −72.9168 −2.49956
\(852\) −0.130758 −0.00447970
\(853\) −21.1366 −0.723702 −0.361851 0.932236i \(-0.617855\pi\)
−0.361851 + 0.932236i \(0.617855\pi\)
\(854\) 0 0
\(855\) 2.13187 0.0729083
\(856\) 23.1697 0.791924
\(857\) 57.6877 1.97058 0.985288 0.170903i \(-0.0546683\pi\)
0.985288 + 0.170903i \(0.0546683\pi\)
\(858\) 5.80796 0.198281
\(859\) 28.3309 0.966638 0.483319 0.875444i \(-0.339431\pi\)
0.483319 + 0.875444i \(0.339431\pi\)
\(860\) 0.0391266 0.00133421
\(861\) 0 0
\(862\) 8.41578 0.286643
\(863\) 8.74556 0.297702 0.148851 0.988860i \(-0.452442\pi\)
0.148851 + 0.988860i \(0.452442\pi\)
\(864\) −1.65416 −0.0562758
\(865\) −1.66073 −0.0564664
\(866\) 21.4412 0.728602
\(867\) −19.6112 −0.666032
\(868\) 0 0
\(869\) 3.95124 0.134037
\(870\) −4.27818 −0.145044
\(871\) −26.5937 −0.901093
\(872\) 19.0603 0.645463
\(873\) −1.69493 −0.0573646
\(874\) 10.4261 0.352669
\(875\) 0 0
\(876\) 0.0707340 0.00238988
\(877\) 33.7988 1.14131 0.570653 0.821191i \(-0.306690\pi\)
0.570653 + 0.821191i \(0.306690\pi\)
\(878\) 44.9378 1.51658
\(879\) −12.7358 −0.429566
\(880\) 0.435898 0.0146941
\(881\) −7.64434 −0.257544 −0.128772 0.991674i \(-0.541104\pi\)
−0.128772 + 0.991674i \(0.541104\pi\)
\(882\) 0 0
\(883\) 35.1172 1.18179 0.590894 0.806749i \(-0.298775\pi\)
0.590894 + 0.806749i \(0.298775\pi\)
\(884\) −0.248660 −0.00836334
\(885\) −4.42987 −0.148909
\(886\) −47.4608 −1.59448
\(887\) 38.9233 1.30692 0.653459 0.756962i \(-0.273317\pi\)
0.653459 + 0.756962i \(0.273317\pi\)
\(888\) 89.3696 2.99905
\(889\) 0 0
\(890\) 2.57305 0.0862489
\(891\) −7.94839 −0.266281
\(892\) 0.000935303 0 3.13163e−5 0
\(893\) 6.82129 0.228266
\(894\) −45.9306 −1.53615
\(895\) 2.19954 0.0735225
\(896\) 0 0
\(897\) −82.1615 −2.74329
\(898\) −29.7804 −0.993786
\(899\) −12.7617 −0.425625
\(900\) 0.770156 0.0256719
\(901\) 34.3934 1.14581
\(902\) 1.47404 0.0490801
\(903\) 0 0
\(904\) 44.3788 1.47602
\(905\) −2.66031 −0.0884318
\(906\) 11.5946 0.385204
\(907\) −34.8962 −1.15871 −0.579355 0.815075i \(-0.696696\pi\)
−0.579355 + 0.815075i \(0.696696\pi\)
\(908\) −0.342258 −0.0113582
\(909\) 5.56068 0.184436
\(910\) 0 0
\(911\) 15.0789 0.499588 0.249794 0.968299i \(-0.419637\pi\)
0.249794 + 0.968299i \(0.419637\pi\)
\(912\) −12.6397 −0.418541
\(913\) 4.61818 0.152839
\(914\) 11.3365 0.374980
\(915\) 6.70735 0.221738
\(916\) 0.230024 0.00760021
\(917\) 0 0
\(918\) −62.3270 −2.05710
\(919\) −39.1404 −1.29112 −0.645562 0.763708i \(-0.723377\pi\)
−0.645562 + 0.763708i \(0.723377\pi\)
\(920\) −6.23417 −0.205534
\(921\) 57.2422 1.88620
\(922\) −17.5316 −0.577372
\(923\) 6.52770 0.214862
\(924\) 0 0
\(925\) −48.3226 −1.58884
\(926\) −20.8729 −0.685925
\(927\) 76.5457 2.51409
\(928\) −0.396014 −0.0129998
\(929\) 45.5626 1.49486 0.747430 0.664341i \(-0.231288\pi\)
0.747430 + 0.664341i \(0.231288\pi\)
\(930\) 5.26913 0.172782
\(931\) 0 0
\(932\) 0.0114509 0.000375088 0
\(933\) 60.2163 1.97139
\(934\) 3.51191 0.114913
\(935\) −0.363145 −0.0118761
\(936\) 71.1065 2.32419
\(937\) −34.5829 −1.12977 −0.564887 0.825168i \(-0.691080\pi\)
−0.564887 + 0.825168i \(0.691080\pi\)
\(938\) 0 0
\(939\) −60.1091 −1.96159
\(940\) −0.0438770 −0.00143111
\(941\) 37.4816 1.22186 0.610932 0.791683i \(-0.290795\pi\)
0.610932 + 0.791683i \(0.290795\pi\)
\(942\) 97.0676 3.16263
\(943\) −20.8522 −0.679042
\(944\) 18.5458 0.603615
\(945\) 0 0
\(946\) −3.18742 −0.103632
\(947\) −31.6815 −1.02951 −0.514755 0.857337i \(-0.672117\pi\)
−0.514755 + 0.857337i \(0.672117\pi\)
\(948\) 0.736978 0.0239359
\(949\) −3.53118 −0.114627
\(950\) 6.90949 0.224173
\(951\) 45.0013 1.45927
\(952\) 0 0
\(953\) 48.2237 1.56212 0.781060 0.624456i \(-0.214679\pi\)
0.781060 + 0.624456i \(0.214679\pi\)
\(954\) −105.802 −3.42546
\(955\) 6.55312 0.212054
\(956\) 0.0993864 0.00321439
\(957\) −3.83166 −0.123860
\(958\) 22.1051 0.714182
\(959\) 0 0
\(960\) 7.63994 0.246578
\(961\) −15.2824 −0.492979
\(962\) −47.9950 −1.54742
\(963\) 58.7333 1.89265
\(964\) 0.497176 0.0160130
\(965\) 0.120492 0.00387877
\(966\) 0 0
\(967\) 33.3518 1.07252 0.536260 0.844053i \(-0.319836\pi\)
0.536260 + 0.844053i \(0.319836\pi\)
\(968\) −30.8849 −0.992677
\(969\) 10.5301 0.338274
\(970\) 0.0978112 0.00314053
\(971\) −52.4761 −1.68404 −0.842019 0.539448i \(-0.818633\pi\)
−0.842019 + 0.539448i \(0.818633\pi\)
\(972\) −0.605227 −0.0194127
\(973\) 0 0
\(974\) −20.9019 −0.669740
\(975\) −54.4492 −1.74377
\(976\) −28.0805 −0.898836
\(977\) 21.5232 0.688588 0.344294 0.938862i \(-0.388118\pi\)
0.344294 + 0.938862i \(0.388118\pi\)
\(978\) 69.5215 2.22305
\(979\) 2.30450 0.0736521
\(980\) 0 0
\(981\) 48.3163 1.54262
\(982\) 3.92900 0.125380
\(983\) −9.88684 −0.315341 −0.157671 0.987492i \(-0.550398\pi\)
−0.157671 + 0.987492i \(0.550398\pi\)
\(984\) 25.5573 0.814737
\(985\) −6.06893 −0.193372
\(986\) −14.9214 −0.475193
\(987\) 0 0
\(988\) −0.0754486 −0.00240034
\(989\) 45.0904 1.43379
\(990\) 1.11711 0.0355042
\(991\) 25.5753 0.812426 0.406213 0.913778i \(-0.366849\pi\)
0.406213 + 0.913778i \(0.366849\pi\)
\(992\) 0.487743 0.0154859
\(993\) 22.7865 0.723107
\(994\) 0 0
\(995\) −2.07489 −0.0657784
\(996\) 0.861374 0.0272937
\(997\) −5.06941 −0.160550 −0.0802750 0.996773i \(-0.525580\pi\)
−0.0802750 + 0.996773i \(0.525580\pi\)
\(998\) −43.6982 −1.38324
\(999\) 132.259 4.18450
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 931.2.a.n.1.2 7
3.2 odd 2 8379.2.a.cl.1.6 7
7.2 even 3 133.2.f.d.39.6 14
7.3 odd 6 931.2.f.p.324.6 14
7.4 even 3 133.2.f.d.58.6 yes 14
7.5 odd 6 931.2.f.p.704.6 14
7.6 odd 2 931.2.a.o.1.2 7
21.2 odd 6 1197.2.j.l.172.2 14
21.11 odd 6 1197.2.j.l.856.2 14
21.20 even 2 8379.2.a.ck.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
133.2.f.d.39.6 14 7.2 even 3
133.2.f.d.58.6 yes 14 7.4 even 3
931.2.a.n.1.2 7 1.1 even 1 trivial
931.2.a.o.1.2 7 7.6 odd 2
931.2.f.p.324.6 14 7.3 odd 6
931.2.f.p.704.6 14 7.5 odd 6
1197.2.j.l.172.2 14 21.2 odd 6
1197.2.j.l.856.2 14 21.11 odd 6
8379.2.a.ck.1.6 7 21.20 even 2
8379.2.a.cl.1.6 7 3.2 odd 2