Properties

Label 931.2.a.g.1.2
Level $931$
Weight $2$
Character 931.1
Self dual yes
Analytic conductor $7.434$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) 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)
 
\(f(q)\) \(=\) \(q+1.30278 q^{2} +3.30278 q^{3} -0.302776 q^{4} +3.00000 q^{5} +4.30278 q^{6} -3.00000 q^{8} +7.90833 q^{9} +O(q^{10})\) \(q+1.30278 q^{2} +3.30278 q^{3} -0.302776 q^{4} +3.00000 q^{5} +4.30278 q^{6} -3.00000 q^{8} +7.90833 q^{9} +3.90833 q^{10} -4.30278 q^{11} -1.00000 q^{12} -1.60555 q^{13} +9.90833 q^{15} -3.30278 q^{16} +1.69722 q^{17} +10.3028 q^{18} -1.00000 q^{19} -0.908327 q^{20} -5.60555 q^{22} -3.00000 q^{23} -9.90833 q^{24} +4.00000 q^{25} -2.09167 q^{26} +16.2111 q^{27} -0.908327 q^{29} +12.9083 q^{30} +2.30278 q^{31} +1.69722 q^{32} -14.2111 q^{33} +2.21110 q^{34} -2.39445 q^{36} -3.60555 q^{37} -1.30278 q^{38} -5.30278 q^{39} -9.00000 q^{40} -4.30278 q^{41} -10.0000 q^{43} +1.30278 q^{44} +23.7250 q^{45} -3.90833 q^{46} +8.21110 q^{47} -10.9083 q^{48} +5.21110 q^{50} +5.60555 q^{51} +0.486122 q^{52} +3.90833 q^{53} +21.1194 q^{54} -12.9083 q^{55} -3.30278 q^{57} -1.18335 q^{58} -8.21110 q^{59} -3.00000 q^{60} -10.2111 q^{61} +3.00000 q^{62} +8.81665 q^{64} -4.81665 q^{65} -18.5139 q^{66} +8.90833 q^{67} -0.513878 q^{68} -9.90833 q^{69} -2.21110 q^{71} -23.7250 q^{72} +16.5139 q^{73} -4.69722 q^{74} +13.2111 q^{75} +0.302776 q^{76} -6.90833 q^{78} -3.21110 q^{79} -9.90833 q^{80} +29.8167 q^{81} -5.60555 q^{82} +2.09167 q^{83} +5.09167 q^{85} -13.0278 q^{86} -3.00000 q^{87} +12.9083 q^{88} +3.39445 q^{89} +30.9083 q^{90} +0.908327 q^{92} +7.60555 q^{93} +10.6972 q^{94} -3.00000 q^{95} +5.60555 q^{96} -2.39445 q^{97} -34.0278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{3} + 3 q^{4} + 6 q^{5} + 5 q^{6} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{3} + 3 q^{4} + 6 q^{5} + 5 q^{6} - 6 q^{8} + 5 q^{9} - 3 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} + 9 q^{15} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{19} + 9 q^{20} - 4 q^{22} - 6 q^{23} - 9 q^{24} + 8 q^{25} - 15 q^{26} + 18 q^{27} + 9 q^{29} + 15 q^{30} + q^{31} + 7 q^{32} - 14 q^{33} - 10 q^{34} - 12 q^{36} + q^{38} - 7 q^{39} - 18 q^{40} - 5 q^{41} - 20 q^{43} - q^{44} + 15 q^{45} + 3 q^{46} + 2 q^{47} - 11 q^{48} - 4 q^{50} + 4 q^{51} + 19 q^{52} - 3 q^{53} + 17 q^{54} - 15 q^{55} - 3 q^{57} - 24 q^{58} - 2 q^{59} - 6 q^{60} - 6 q^{61} + 6 q^{62} - 4 q^{64} + 12 q^{65} - 19 q^{66} + 7 q^{67} + 17 q^{68} - 9 q^{69} + 10 q^{71} - 15 q^{72} + 15 q^{73} - 13 q^{74} + 12 q^{75} - 3 q^{76} - 3 q^{78} + 8 q^{79} - 9 q^{80} + 38 q^{81} - 4 q^{82} + 15 q^{83} + 21 q^{85} + 10 q^{86} - 6 q^{87} + 15 q^{88} + 14 q^{89} + 51 q^{90} - 9 q^{92} + 8 q^{93} + 25 q^{94} - 6 q^{95} + 4 q^{96} - 12 q^{97} - 32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30278 0.921201 0.460601 0.887607i \(-0.347634\pi\)
0.460601 + 0.887607i \(0.347634\pi\)
\(3\) 3.30278 1.90686 0.953429 0.301617i \(-0.0975264\pi\)
0.953429 + 0.301617i \(0.0975264\pi\)
\(4\) −0.302776 −0.151388
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 4.30278 1.75660
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) 7.90833 2.63611
\(10\) 3.90833 1.23592
\(11\) −4.30278 −1.29734 −0.648668 0.761072i \(-0.724674\pi\)
−0.648668 + 0.761072i \(0.724674\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.60555 −0.445300 −0.222650 0.974898i \(-0.571471\pi\)
−0.222650 + 0.974898i \(0.571471\pi\)
\(14\) 0 0
\(15\) 9.90833 2.55832
\(16\) −3.30278 −0.825694
\(17\) 1.69722 0.411637 0.205819 0.978590i \(-0.434014\pi\)
0.205819 + 0.978590i \(0.434014\pi\)
\(18\) 10.3028 2.42839
\(19\) −1.00000 −0.229416
\(20\) −0.908327 −0.203108
\(21\) 0 0
\(22\) −5.60555 −1.19511
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −9.90833 −2.02253
\(25\) 4.00000 0.800000
\(26\) −2.09167 −0.410211
\(27\) 16.2111 3.11983
\(28\) 0 0
\(29\) −0.908327 −0.168672 −0.0843360 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 12.9083 2.35673
\(31\) 2.30278 0.413591 0.206795 0.978384i \(-0.433697\pi\)
0.206795 + 0.978384i \(0.433697\pi\)
\(32\) 1.69722 0.300030
\(33\) −14.2111 −2.47384
\(34\) 2.21110 0.379201
\(35\) 0 0
\(36\) −2.39445 −0.399075
\(37\) −3.60555 −0.592749 −0.296374 0.955072i \(-0.595778\pi\)
−0.296374 + 0.955072i \(0.595778\pi\)
\(38\) −1.30278 −0.211338
\(39\) −5.30278 −0.849124
\(40\) −9.00000 −1.42302
\(41\) −4.30278 −0.671981 −0.335990 0.941865i \(-0.609071\pi\)
−0.335990 + 0.941865i \(0.609071\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 1.30278 0.196401
\(45\) 23.7250 3.53671
\(46\) −3.90833 −0.576251
\(47\) 8.21110 1.19771 0.598856 0.800857i \(-0.295622\pi\)
0.598856 + 0.800857i \(0.295622\pi\)
\(48\) −10.9083 −1.57448
\(49\) 0 0
\(50\) 5.21110 0.736961
\(51\) 5.60555 0.784934
\(52\) 0.486122 0.0674130
\(53\) 3.90833 0.536850 0.268425 0.963301i \(-0.413497\pi\)
0.268425 + 0.963301i \(0.413497\pi\)
\(54\) 21.1194 2.87399
\(55\) −12.9083 −1.74056
\(56\) 0 0
\(57\) −3.30278 −0.437463
\(58\) −1.18335 −0.155381
\(59\) −8.21110 −1.06899 −0.534497 0.845170i \(-0.679499\pi\)
−0.534497 + 0.845170i \(0.679499\pi\)
\(60\) −3.00000 −0.387298
\(61\) −10.2111 −1.30740 −0.653699 0.756755i \(-0.726784\pi\)
−0.653699 + 0.756755i \(0.726784\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 8.81665 1.10208
\(65\) −4.81665 −0.597432
\(66\) −18.5139 −2.27890
\(67\) 8.90833 1.08833 0.544163 0.838980i \(-0.316847\pi\)
0.544163 + 0.838980i \(0.316847\pi\)
\(68\) −0.513878 −0.0623169
\(69\) −9.90833 −1.19282
\(70\) 0 0
\(71\) −2.21110 −0.262410 −0.131205 0.991355i \(-0.541885\pi\)
−0.131205 + 0.991355i \(0.541885\pi\)
\(72\) −23.7250 −2.79602
\(73\) 16.5139 1.93280 0.966402 0.257037i \(-0.0827461\pi\)
0.966402 + 0.257037i \(0.0827461\pi\)
\(74\) −4.69722 −0.546041
\(75\) 13.2111 1.52549
\(76\) 0.302776 0.0347307
\(77\) 0 0
\(78\) −6.90833 −0.782214
\(79\) −3.21110 −0.361277 −0.180639 0.983550i \(-0.557816\pi\)
−0.180639 + 0.983550i \(0.557816\pi\)
\(80\) −9.90833 −1.10778
\(81\) 29.8167 3.31296
\(82\) −5.60555 −0.619030
\(83\) 2.09167 0.229591 0.114795 0.993389i \(-0.463379\pi\)
0.114795 + 0.993389i \(0.463379\pi\)
\(84\) 0 0
\(85\) 5.09167 0.552269
\(86\) −13.0278 −1.40482
\(87\) −3.00000 −0.321634
\(88\) 12.9083 1.37603
\(89\) 3.39445 0.359811 0.179905 0.983684i \(-0.442421\pi\)
0.179905 + 0.983684i \(0.442421\pi\)
\(90\) 30.9083 3.25802
\(91\) 0 0
\(92\) 0.908327 0.0946996
\(93\) 7.60555 0.788659
\(94\) 10.6972 1.10333
\(95\) −3.00000 −0.307794
\(96\) 5.60555 0.572114
\(97\) −2.39445 −0.243119 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(98\) 0 0
\(99\) −34.0278 −3.41992
\(100\) −1.21110 −0.121110
\(101\) 11.6056 1.15480 0.577398 0.816463i \(-0.304068\pi\)
0.577398 + 0.816463i \(0.304068\pi\)
\(102\) 7.30278 0.723083
\(103\) 1.78890 0.176265 0.0881327 0.996109i \(-0.471910\pi\)
0.0881327 + 0.996109i \(0.471910\pi\)
\(104\) 4.81665 0.472312
\(105\) 0 0
\(106\) 5.09167 0.494547
\(107\) −20.2111 −1.95388 −0.976941 0.213512i \(-0.931510\pi\)
−0.976941 + 0.213512i \(0.931510\pi\)
\(108\) −4.90833 −0.472304
\(109\) 4.21110 0.403350 0.201675 0.979452i \(-0.435361\pi\)
0.201675 + 0.979452i \(0.435361\pi\)
\(110\) −16.8167 −1.60341
\(111\) −11.9083 −1.13029
\(112\) 0 0
\(113\) 13.3028 1.25142 0.625710 0.780056i \(-0.284809\pi\)
0.625710 + 0.780056i \(0.284809\pi\)
\(114\) −4.30278 −0.402992
\(115\) −9.00000 −0.839254
\(116\) 0.275019 0.0255349
\(117\) −12.6972 −1.17386
\(118\) −10.6972 −0.984759
\(119\) 0 0
\(120\) −29.7250 −2.71351
\(121\) 7.51388 0.683080
\(122\) −13.3028 −1.20438
\(123\) −14.2111 −1.28137
\(124\) −0.697224 −0.0626126
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 9.81665 0.871087 0.435544 0.900168i \(-0.356556\pi\)
0.435544 + 0.900168i \(0.356556\pi\)
\(128\) 8.09167 0.715210
\(129\) −33.0278 −2.90793
\(130\) −6.27502 −0.550356
\(131\) 2.48612 0.217213 0.108607 0.994085i \(-0.465361\pi\)
0.108607 + 0.994085i \(0.465361\pi\)
\(132\) 4.30278 0.374509
\(133\) 0 0
\(134\) 11.6056 1.00257
\(135\) 48.6333 4.18569
\(136\) −5.09167 −0.436607
\(137\) −21.6333 −1.84826 −0.924129 0.382080i \(-0.875208\pi\)
−0.924129 + 0.382080i \(0.875208\pi\)
\(138\) −12.9083 −1.09883
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 27.1194 2.28387
\(142\) −2.88057 −0.241732
\(143\) 6.90833 0.577703
\(144\) −26.1194 −2.17662
\(145\) −2.72498 −0.226297
\(146\) 21.5139 1.78050
\(147\) 0 0
\(148\) 1.09167 0.0897350
\(149\) 8.21110 0.672680 0.336340 0.941741i \(-0.390811\pi\)
0.336340 + 0.941741i \(0.390811\pi\)
\(150\) 17.2111 1.40528
\(151\) −4.90833 −0.399434 −0.199717 0.979854i \(-0.564002\pi\)
−0.199717 + 0.979854i \(0.564002\pi\)
\(152\) 3.00000 0.243332
\(153\) 13.4222 1.08512
\(154\) 0 0
\(155\) 6.90833 0.554890
\(156\) 1.60555 0.128547
\(157\) 13.5139 1.07852 0.539262 0.842138i \(-0.318703\pi\)
0.539262 + 0.842138i \(0.318703\pi\)
\(158\) −4.18335 −0.332809
\(159\) 12.9083 1.02370
\(160\) 5.09167 0.402532
\(161\) 0 0
\(162\) 38.8444 3.05191
\(163\) 9.30278 0.728650 0.364325 0.931272i \(-0.381300\pi\)
0.364325 + 0.931272i \(0.381300\pi\)
\(164\) 1.30278 0.101730
\(165\) −42.6333 −3.31900
\(166\) 2.72498 0.211500
\(167\) −11.6056 −0.898065 −0.449032 0.893516i \(-0.648231\pi\)
−0.449032 + 0.893516i \(0.648231\pi\)
\(168\) 0 0
\(169\) −10.4222 −0.801708
\(170\) 6.63331 0.508751
\(171\) −7.90833 −0.604765
\(172\) 3.02776 0.230864
\(173\) 13.8167 1.05046 0.525230 0.850960i \(-0.323979\pi\)
0.525230 + 0.850960i \(0.323979\pi\)
\(174\) −3.90833 −0.296289
\(175\) 0 0
\(176\) 14.2111 1.07120
\(177\) −27.1194 −2.03842
\(178\) 4.42221 0.331458
\(179\) 20.7250 1.54906 0.774529 0.632539i \(-0.217987\pi\)
0.774529 + 0.632539i \(0.217987\pi\)
\(180\) −7.18335 −0.535415
\(181\) 14.3028 1.06312 0.531558 0.847022i \(-0.321607\pi\)
0.531558 + 0.847022i \(0.321607\pi\)
\(182\) 0 0
\(183\) −33.7250 −2.49302
\(184\) 9.00000 0.663489
\(185\) −10.8167 −0.795256
\(186\) 9.90833 0.726514
\(187\) −7.30278 −0.534032
\(188\) −2.48612 −0.181319
\(189\) 0 0
\(190\) −3.90833 −0.283540
\(191\) −14.3305 −1.03692 −0.518460 0.855102i \(-0.673495\pi\)
−0.518460 + 0.855102i \(0.673495\pi\)
\(192\) 29.1194 2.10151
\(193\) 2.11943 0.152560 0.0762799 0.997086i \(-0.475696\pi\)
0.0762799 + 0.997086i \(0.475696\pi\)
\(194\) −3.11943 −0.223962
\(195\) −15.9083 −1.13922
\(196\) 0 0
\(197\) 3.90833 0.278457 0.139228 0.990260i \(-0.455538\pi\)
0.139228 + 0.990260i \(0.455538\pi\)
\(198\) −44.3305 −3.15043
\(199\) −15.4222 −1.09325 −0.546626 0.837377i \(-0.684088\pi\)
−0.546626 + 0.837377i \(0.684088\pi\)
\(200\) −12.0000 −0.848528
\(201\) 29.4222 2.07528
\(202\) 15.1194 1.06380
\(203\) 0 0
\(204\) −1.69722 −0.118829
\(205\) −12.9083 −0.901557
\(206\) 2.33053 0.162376
\(207\) −23.7250 −1.64900
\(208\) 5.30278 0.367681
\(209\) 4.30278 0.297629
\(210\) 0 0
\(211\) −11.3028 −0.778115 −0.389058 0.921213i \(-0.627199\pi\)
−0.389058 + 0.921213i \(0.627199\pi\)
\(212\) −1.18335 −0.0812725
\(213\) −7.30278 −0.500378
\(214\) −26.3305 −1.79992
\(215\) −30.0000 −2.04598
\(216\) −48.6333 −3.30908
\(217\) 0 0
\(218\) 5.48612 0.371567
\(219\) 54.5416 3.68558
\(220\) 3.90833 0.263499
\(221\) −2.72498 −0.183302
\(222\) −15.5139 −1.04122
\(223\) 8.81665 0.590407 0.295203 0.955434i \(-0.404613\pi\)
0.295203 + 0.955434i \(0.404613\pi\)
\(224\) 0 0
\(225\) 31.6333 2.10889
\(226\) 17.3305 1.15281
\(227\) −0.119429 −0.00792681 −0.00396341 0.999992i \(-0.501262\pi\)
−0.00396341 + 0.999992i \(0.501262\pi\)
\(228\) 1.00000 0.0662266
\(229\) 15.2111 1.00518 0.502589 0.864525i \(-0.332381\pi\)
0.502589 + 0.864525i \(0.332381\pi\)
\(230\) −11.7250 −0.773122
\(231\) 0 0
\(232\) 2.72498 0.178904
\(233\) −12.9083 −0.845653 −0.422826 0.906211i \(-0.638962\pi\)
−0.422826 + 0.906211i \(0.638962\pi\)
\(234\) −16.5416 −1.08136
\(235\) 24.6333 1.60690
\(236\) 2.48612 0.161833
\(237\) −10.6056 −0.688905
\(238\) 0 0
\(239\) 1.42221 0.0919948 0.0459974 0.998942i \(-0.485353\pi\)
0.0459974 + 0.998942i \(0.485353\pi\)
\(240\) −32.7250 −2.11239
\(241\) 13.3944 0.862812 0.431406 0.902158i \(-0.358018\pi\)
0.431406 + 0.902158i \(0.358018\pi\)
\(242\) 9.78890 0.629254
\(243\) 49.8444 3.19752
\(244\) 3.09167 0.197924
\(245\) 0 0
\(246\) −18.5139 −1.18040
\(247\) 1.60555 0.102159
\(248\) −6.90833 −0.438679
\(249\) 6.90833 0.437797
\(250\) −3.90833 −0.247184
\(251\) 5.09167 0.321384 0.160692 0.987005i \(-0.448627\pi\)
0.160692 + 0.987005i \(0.448627\pi\)
\(252\) 0 0
\(253\) 12.9083 0.811540
\(254\) 12.7889 0.802447
\(255\) 16.8167 1.05310
\(256\) −7.09167 −0.443230
\(257\) −9.90833 −0.618064 −0.309032 0.951052i \(-0.600005\pi\)
−0.309032 + 0.951052i \(0.600005\pi\)
\(258\) −43.0278 −2.67879
\(259\) 0 0
\(260\) 1.45837 0.0904440
\(261\) −7.18335 −0.444638
\(262\) 3.23886 0.200097
\(263\) −13.3028 −0.820284 −0.410142 0.912022i \(-0.634521\pi\)
−0.410142 + 0.912022i \(0.634521\pi\)
\(264\) 42.6333 2.62390
\(265\) 11.7250 0.720260
\(266\) 0 0
\(267\) 11.2111 0.686108
\(268\) −2.69722 −0.164759
\(269\) 11.7250 0.714885 0.357442 0.933935i \(-0.383649\pi\)
0.357442 + 0.933935i \(0.383649\pi\)
\(270\) 63.3583 3.85586
\(271\) −7.09167 −0.430788 −0.215394 0.976527i \(-0.569104\pi\)
−0.215394 + 0.976527i \(0.569104\pi\)
\(272\) −5.60555 −0.339886
\(273\) 0 0
\(274\) −28.1833 −1.70262
\(275\) −17.2111 −1.03787
\(276\) 3.00000 0.180579
\(277\) −6.60555 −0.396889 −0.198445 0.980112i \(-0.563589\pi\)
−0.198445 + 0.980112i \(0.563589\pi\)
\(278\) −6.51388 −0.390676
\(279\) 18.2111 1.09027
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 35.3305 2.10390
\(283\) 12.3305 0.732974 0.366487 0.930423i \(-0.380560\pi\)
0.366487 + 0.930423i \(0.380560\pi\)
\(284\) 0.669468 0.0397256
\(285\) −9.90833 −0.586919
\(286\) 9.00000 0.532181
\(287\) 0 0
\(288\) 13.4222 0.790911
\(289\) −14.1194 −0.830555
\(290\) −3.55004 −0.208465
\(291\) −7.90833 −0.463594
\(292\) −5.00000 −0.292603
\(293\) −30.6333 −1.78962 −0.894808 0.446450i \(-0.852688\pi\)
−0.894808 + 0.446450i \(0.852688\pi\)
\(294\) 0 0
\(295\) −24.6333 −1.43421
\(296\) 10.8167 0.628705
\(297\) −69.7527 −4.04746
\(298\) 10.6972 0.619674
\(299\) 4.81665 0.278554
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −6.39445 −0.367959
\(303\) 38.3305 2.20203
\(304\) 3.30278 0.189427
\(305\) −30.6333 −1.75406
\(306\) 17.4861 0.999615
\(307\) −18.3028 −1.04459 −0.522297 0.852763i \(-0.674925\pi\)
−0.522297 + 0.852763i \(0.674925\pi\)
\(308\) 0 0
\(309\) 5.90833 0.336113
\(310\) 9.00000 0.511166
\(311\) −15.5139 −0.879711 −0.439856 0.898068i \(-0.644970\pi\)
−0.439856 + 0.898068i \(0.644970\pi\)
\(312\) 15.9083 0.900632
\(313\) −9.02776 −0.510279 −0.255139 0.966904i \(-0.582121\pi\)
−0.255139 + 0.966904i \(0.582121\pi\)
\(314\) 17.6056 0.993539
\(315\) 0 0
\(316\) 0.972244 0.0546930
\(317\) 23.2111 1.30367 0.651833 0.758363i \(-0.274000\pi\)
0.651833 + 0.758363i \(0.274000\pi\)
\(318\) 16.8167 0.943031
\(319\) 3.90833 0.218824
\(320\) 26.4500 1.47860
\(321\) −66.7527 −3.72577
\(322\) 0 0
\(323\) −1.69722 −0.0944361
\(324\) −9.02776 −0.501542
\(325\) −6.42221 −0.356240
\(326\) 12.1194 0.671233
\(327\) 13.9083 0.769132
\(328\) 12.9083 0.712743
\(329\) 0 0
\(330\) −55.5416 −3.05747
\(331\) 15.3028 0.841117 0.420558 0.907266i \(-0.361834\pi\)
0.420558 + 0.907266i \(0.361834\pi\)
\(332\) −0.633308 −0.0347573
\(333\) −28.5139 −1.56255
\(334\) −15.1194 −0.827298
\(335\) 26.7250 1.46014
\(336\) 0 0
\(337\) −24.7250 −1.34686 −0.673428 0.739253i \(-0.735179\pi\)
−0.673428 + 0.739253i \(0.735179\pi\)
\(338\) −13.5778 −0.738535
\(339\) 43.9361 2.38628
\(340\) −1.54163 −0.0836069
\(341\) −9.90833 −0.536566
\(342\) −10.3028 −0.557110
\(343\) 0 0
\(344\) 30.0000 1.61749
\(345\) −29.7250 −1.60034
\(346\) 18.0000 0.967686
\(347\) 25.5416 1.37115 0.685573 0.728004i \(-0.259552\pi\)
0.685573 + 0.728004i \(0.259552\pi\)
\(348\) 0.908327 0.0486914
\(349\) 10.5139 0.562795 0.281397 0.959591i \(-0.409202\pi\)
0.281397 + 0.959591i \(0.409202\pi\)
\(350\) 0 0
\(351\) −26.0278 −1.38926
\(352\) −7.30278 −0.389239
\(353\) 0.908327 0.0483454 0.0241727 0.999708i \(-0.492305\pi\)
0.0241727 + 0.999708i \(0.492305\pi\)
\(354\) −35.3305 −1.87780
\(355\) −6.63331 −0.352059
\(356\) −1.02776 −0.0544710
\(357\) 0 0
\(358\) 27.0000 1.42699
\(359\) 28.5416 1.50637 0.753185 0.657809i \(-0.228517\pi\)
0.753185 + 0.657809i \(0.228517\pi\)
\(360\) −71.1749 −3.75125
\(361\) 1.00000 0.0526316
\(362\) 18.6333 0.979345
\(363\) 24.8167 1.30254
\(364\) 0 0
\(365\) 49.5416 2.59313
\(366\) −43.9361 −2.29658
\(367\) −36.0278 −1.88063 −0.940317 0.340300i \(-0.889471\pi\)
−0.940317 + 0.340300i \(0.889471\pi\)
\(368\) 9.90833 0.516507
\(369\) −34.0278 −1.77141
\(370\) −14.0917 −0.732591
\(371\) 0 0
\(372\) −2.30278 −0.119393
\(373\) −1.11943 −0.0579619 −0.0289809 0.999580i \(-0.509226\pi\)
−0.0289809 + 0.999580i \(0.509226\pi\)
\(374\) −9.51388 −0.491951
\(375\) −9.90833 −0.511664
\(376\) −24.6333 −1.27037
\(377\) 1.45837 0.0751096
\(378\) 0 0
\(379\) −14.8167 −0.761080 −0.380540 0.924764i \(-0.624262\pi\)
−0.380540 + 0.924764i \(0.624262\pi\)
\(380\) 0.908327 0.0465962
\(381\) 32.4222 1.66104
\(382\) −18.6695 −0.955213
\(383\) 36.6333 1.87187 0.935937 0.352167i \(-0.114555\pi\)
0.935937 + 0.352167i \(0.114555\pi\)
\(384\) 26.7250 1.36380
\(385\) 0 0
\(386\) 2.76114 0.140538
\(387\) −79.0833 −4.02003
\(388\) 0.724981 0.0368053
\(389\) 37.1472 1.88344 0.941719 0.336402i \(-0.109210\pi\)
0.941719 + 0.336402i \(0.109210\pi\)
\(390\) −20.7250 −1.04945
\(391\) −5.09167 −0.257497
\(392\) 0 0
\(393\) 8.21110 0.414195
\(394\) 5.09167 0.256515
\(395\) −9.63331 −0.484704
\(396\) 10.3028 0.517734
\(397\) −0.972244 −0.0487955 −0.0243978 0.999702i \(-0.507767\pi\)
−0.0243978 + 0.999702i \(0.507767\pi\)
\(398\) −20.0917 −1.00710
\(399\) 0 0
\(400\) −13.2111 −0.660555
\(401\) −21.5139 −1.07435 −0.537176 0.843470i \(-0.680509\pi\)
−0.537176 + 0.843470i \(0.680509\pi\)
\(402\) 38.3305 1.91175
\(403\) −3.69722 −0.184172
\(404\) −3.51388 −0.174822
\(405\) 89.4500 4.44480
\(406\) 0 0
\(407\) 15.5139 0.768994
\(408\) −16.8167 −0.832548
\(409\) −10.0917 −0.499001 −0.249501 0.968375i \(-0.580266\pi\)
−0.249501 + 0.968375i \(0.580266\pi\)
\(410\) −16.8167 −0.830515
\(411\) −71.4500 −3.52437
\(412\) −0.541635 −0.0266844
\(413\) 0 0
\(414\) −30.9083 −1.51906
\(415\) 6.27502 0.308029
\(416\) −2.72498 −0.133603
\(417\) −16.5139 −0.808688
\(418\) 5.60555 0.274176
\(419\) −15.0000 −0.732798 −0.366399 0.930458i \(-0.619409\pi\)
−0.366399 + 0.930458i \(0.619409\pi\)
\(420\) 0 0
\(421\) −22.3944 −1.09144 −0.545719 0.837968i \(-0.683744\pi\)
−0.545719 + 0.837968i \(0.683744\pi\)
\(422\) −14.7250 −0.716801
\(423\) 64.9361 3.15730
\(424\) −11.7250 −0.569415
\(425\) 6.78890 0.329310
\(426\) −9.51388 −0.460949
\(427\) 0 0
\(428\) 6.11943 0.295794
\(429\) 22.8167 1.10160
\(430\) −39.0833 −1.88476
\(431\) 11.2111 0.540020 0.270010 0.962858i \(-0.412973\pi\)
0.270010 + 0.962858i \(0.412973\pi\)
\(432\) −53.5416 −2.57602
\(433\) 29.4222 1.41394 0.706970 0.707243i \(-0.250061\pi\)
0.706970 + 0.707243i \(0.250061\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) −1.27502 −0.0610623
\(437\) 3.00000 0.143509
\(438\) 71.0555 3.39516
\(439\) −37.2111 −1.77599 −0.887995 0.459854i \(-0.847902\pi\)
−0.887995 + 0.459854i \(0.847902\pi\)
\(440\) 38.7250 1.84614
\(441\) 0 0
\(442\) −3.55004 −0.168858
\(443\) 9.11943 0.433277 0.216639 0.976252i \(-0.430491\pi\)
0.216639 + 0.976252i \(0.430491\pi\)
\(444\) 3.60555 0.171112
\(445\) 10.1833 0.482737
\(446\) 11.4861 0.543884
\(447\) 27.1194 1.28270
\(448\) 0 0
\(449\) −18.5139 −0.873724 −0.436862 0.899529i \(-0.643910\pi\)
−0.436862 + 0.899529i \(0.643910\pi\)
\(450\) 41.2111 1.94271
\(451\) 18.5139 0.871784
\(452\) −4.02776 −0.189450
\(453\) −16.2111 −0.761664
\(454\) −0.155590 −0.00730219
\(455\) 0 0
\(456\) 9.90833 0.464000
\(457\) 18.6972 0.874619 0.437310 0.899311i \(-0.355931\pi\)
0.437310 + 0.899311i \(0.355931\pi\)
\(458\) 19.8167 0.925971
\(459\) 27.5139 1.28424
\(460\) 2.72498 0.127053
\(461\) 9.90833 0.461477 0.230738 0.973016i \(-0.425886\pi\)
0.230738 + 0.973016i \(0.425886\pi\)
\(462\) 0 0
\(463\) 28.4500 1.32218 0.661091 0.750306i \(-0.270094\pi\)
0.661091 + 0.750306i \(0.270094\pi\)
\(464\) 3.00000 0.139272
\(465\) 22.8167 1.05810
\(466\) −16.8167 −0.779016
\(467\) 37.5416 1.73722 0.868610 0.495497i \(-0.165014\pi\)
0.868610 + 0.495497i \(0.165014\pi\)
\(468\) 3.84441 0.177708
\(469\) 0 0
\(470\) 32.0917 1.48028
\(471\) 44.6333 2.05659
\(472\) 24.6333 1.13384
\(473\) 43.0278 1.97842
\(474\) −13.8167 −0.634620
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 30.9083 1.41520
\(478\) 1.85281 0.0847457
\(479\) 4.30278 0.196599 0.0982994 0.995157i \(-0.468660\pi\)
0.0982994 + 0.995157i \(0.468660\pi\)
\(480\) 16.8167 0.767572
\(481\) 5.78890 0.263951
\(482\) 17.4500 0.794824
\(483\) 0 0
\(484\) −2.27502 −0.103410
\(485\) −7.18335 −0.326179
\(486\) 64.9361 2.94556
\(487\) 33.4222 1.51450 0.757252 0.653122i \(-0.226541\pi\)
0.757252 + 0.653122i \(0.226541\pi\)
\(488\) 30.6333 1.38670
\(489\) 30.7250 1.38943
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 4.30278 0.193984
\(493\) −1.54163 −0.0694317
\(494\) 2.09167 0.0941088
\(495\) −102.083 −4.58830
\(496\) −7.60555 −0.341499
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) −26.9361 −1.20582 −0.602912 0.797807i \(-0.705993\pi\)
−0.602912 + 0.797807i \(0.705993\pi\)
\(500\) 0.908327 0.0406216
\(501\) −38.3305 −1.71248
\(502\) 6.63331 0.296059
\(503\) 22.4222 0.999757 0.499878 0.866096i \(-0.333378\pi\)
0.499878 + 0.866096i \(0.333378\pi\)
\(504\) 0 0
\(505\) 34.8167 1.54932
\(506\) 16.8167 0.747591
\(507\) −34.4222 −1.52874
\(508\) −2.97224 −0.131872
\(509\) 9.63331 0.426989 0.213494 0.976944i \(-0.431515\pi\)
0.213494 + 0.976944i \(0.431515\pi\)
\(510\) 21.9083 0.970117
\(511\) 0 0
\(512\) −25.4222 −1.12351
\(513\) −16.2111 −0.715738
\(514\) −12.9083 −0.569362
\(515\) 5.36669 0.236485
\(516\) 10.0000 0.440225
\(517\) −35.3305 −1.55384
\(518\) 0 0
\(519\) 45.6333 2.00308
\(520\) 14.4500 0.633673
\(521\) 6.63331 0.290610 0.145305 0.989387i \(-0.453584\pi\)
0.145305 + 0.989387i \(0.453584\pi\)
\(522\) −9.35829 −0.409601
\(523\) 41.6611 1.82171 0.910856 0.412725i \(-0.135423\pi\)
0.910856 + 0.412725i \(0.135423\pi\)
\(524\) −0.752737 −0.0328835
\(525\) 0 0
\(526\) −17.3305 −0.755647
\(527\) 3.90833 0.170249
\(528\) 46.9361 2.04263
\(529\) −14.0000 −0.608696
\(530\) 15.2750 0.663504
\(531\) −64.9361 −2.81799
\(532\) 0 0
\(533\) 6.90833 0.299233
\(534\) 14.6056 0.632044
\(535\) −60.6333 −2.62141
\(536\) −26.7250 −1.15434
\(537\) 68.4500 2.95383
\(538\) 15.2750 0.658553
\(539\) 0 0
\(540\) −14.7250 −0.633662
\(541\) 31.2111 1.34187 0.670935 0.741516i \(-0.265893\pi\)
0.670935 + 0.741516i \(0.265893\pi\)
\(542\) −9.23886 −0.396843
\(543\) 47.2389 2.02721
\(544\) 2.88057 0.123503
\(545\) 12.6333 0.541151
\(546\) 0 0
\(547\) −28.5139 −1.21917 −0.609583 0.792722i \(-0.708663\pi\)
−0.609583 + 0.792722i \(0.708663\pi\)
\(548\) 6.55004 0.279804
\(549\) −80.7527 −3.44644
\(550\) −22.4222 −0.956086
\(551\) 0.908327 0.0386960
\(552\) 29.7250 1.26518
\(553\) 0 0
\(554\) −8.60555 −0.365615
\(555\) −35.7250 −1.51644
\(556\) 1.51388 0.0642027
\(557\) 24.1194 1.02197 0.510987 0.859589i \(-0.329280\pi\)
0.510987 + 0.859589i \(0.329280\pi\)
\(558\) 23.7250 1.00436
\(559\) 16.0555 0.679076
\(560\) 0 0
\(561\) −24.1194 −1.01832
\(562\) 3.90833 0.164863
\(563\) 9.78890 0.412553 0.206276 0.978494i \(-0.433865\pi\)
0.206276 + 0.978494i \(0.433865\pi\)
\(564\) −8.21110 −0.345750
\(565\) 39.9083 1.67896
\(566\) 16.0639 0.675217
\(567\) 0 0
\(568\) 6.63331 0.278327
\(569\) 28.8167 1.20806 0.604028 0.796963i \(-0.293561\pi\)
0.604028 + 0.796963i \(0.293561\pi\)
\(570\) −12.9083 −0.540670
\(571\) −9.60555 −0.401980 −0.200990 0.979593i \(-0.564416\pi\)
−0.200990 + 0.979593i \(0.564416\pi\)
\(572\) −2.09167 −0.0874572
\(573\) −47.3305 −1.97726
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 69.7250 2.90521
\(577\) −1.48612 −0.0618681 −0.0309340 0.999521i \(-0.509848\pi\)
−0.0309340 + 0.999521i \(0.509848\pi\)
\(578\) −18.3944 −0.765108
\(579\) 7.00000 0.290910
\(580\) 0.825058 0.0342587
\(581\) 0 0
\(582\) −10.3028 −0.427064
\(583\) −16.8167 −0.696475
\(584\) −49.5416 −2.05005
\(585\) −38.0917 −1.57490
\(586\) −39.9083 −1.64860
\(587\) 28.4222 1.17311 0.586555 0.809909i \(-0.300484\pi\)
0.586555 + 0.809909i \(0.300484\pi\)
\(588\) 0 0
\(589\) −2.30278 −0.0948842
\(590\) −32.0917 −1.32119
\(591\) 12.9083 0.530978
\(592\) 11.9083 0.489429
\(593\) −16.8167 −0.690577 −0.345289 0.938497i \(-0.612219\pi\)
−0.345289 + 0.938497i \(0.612219\pi\)
\(594\) −90.8722 −3.72853
\(595\) 0 0
\(596\) −2.48612 −0.101836
\(597\) −50.9361 −2.08468
\(598\) 6.27502 0.256605
\(599\) 9.90833 0.404843 0.202422 0.979298i \(-0.435119\pi\)
0.202422 + 0.979298i \(0.435119\pi\)
\(600\) −39.6333 −1.61802
\(601\) 9.48612 0.386947 0.193473 0.981106i \(-0.438025\pi\)
0.193473 + 0.981106i \(0.438025\pi\)
\(602\) 0 0
\(603\) 70.4500 2.86894
\(604\) 1.48612 0.0604694
\(605\) 22.5416 0.916448
\(606\) 49.9361 2.02851
\(607\) 13.0000 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) −1.69722 −0.0688315
\(609\) 0 0
\(610\) −39.9083 −1.61584
\(611\) −13.1833 −0.533341
\(612\) −4.06392 −0.164274
\(613\) 31.7250 1.28136 0.640680 0.767808i \(-0.278653\pi\)
0.640680 + 0.767808i \(0.278653\pi\)
\(614\) −23.8444 −0.962282
\(615\) −42.6333 −1.71914
\(616\) 0 0
\(617\) −36.1194 −1.45411 −0.727057 0.686577i \(-0.759112\pi\)
−0.727057 + 0.686577i \(0.759112\pi\)
\(618\) 7.69722 0.309628
\(619\) 1.66947 0.0671016 0.0335508 0.999437i \(-0.489318\pi\)
0.0335508 + 0.999437i \(0.489318\pi\)
\(620\) −2.09167 −0.0840036
\(621\) −48.6333 −1.95159
\(622\) −20.2111 −0.810391
\(623\) 0 0
\(624\) 17.5139 0.701116
\(625\) −29.0000 −1.16000
\(626\) −11.7611 −0.470070
\(627\) 14.2111 0.567537
\(628\) −4.09167 −0.163276
\(629\) −6.11943 −0.243998
\(630\) 0 0
\(631\) −37.2389 −1.48246 −0.741228 0.671254i \(-0.765756\pi\)
−0.741228 + 0.671254i \(0.765756\pi\)
\(632\) 9.63331 0.383192
\(633\) −37.3305 −1.48376
\(634\) 30.2389 1.20094
\(635\) 29.4500 1.16869
\(636\) −3.90833 −0.154975
\(637\) 0 0
\(638\) 5.09167 0.201581
\(639\) −17.4861 −0.691740
\(640\) 24.2750 0.959554
\(641\) −0.908327 −0.0358768 −0.0179384 0.999839i \(-0.505710\pi\)
−0.0179384 + 0.999839i \(0.505710\pi\)
\(642\) −86.9638 −3.43219
\(643\) −37.6056 −1.48302 −0.741509 0.670943i \(-0.765890\pi\)
−0.741509 + 0.670943i \(0.765890\pi\)
\(644\) 0 0
\(645\) −99.0833 −3.90140
\(646\) −2.21110 −0.0869947
\(647\) 13.4222 0.527681 0.263841 0.964566i \(-0.415011\pi\)
0.263841 + 0.964566i \(0.415011\pi\)
\(648\) −89.4500 −3.51393
\(649\) 35.3305 1.38684
\(650\) −8.36669 −0.328169
\(651\) 0 0
\(652\) −2.81665 −0.110309
\(653\) 4.81665 0.188490 0.0942451 0.995549i \(-0.469956\pi\)
0.0942451 + 0.995549i \(0.469956\pi\)
\(654\) 18.1194 0.708526
\(655\) 7.45837 0.291422
\(656\) 14.2111 0.554850
\(657\) 130.597 5.09508
\(658\) 0 0
\(659\) −23.3305 −0.908828 −0.454414 0.890790i \(-0.650151\pi\)
−0.454414 + 0.890790i \(0.650151\pi\)
\(660\) 12.9083 0.502456
\(661\) 15.2111 0.591643 0.295822 0.955243i \(-0.404407\pi\)
0.295822 + 0.955243i \(0.404407\pi\)
\(662\) 19.9361 0.774838
\(663\) −9.00000 −0.349531
\(664\) −6.27502 −0.243518
\(665\) 0 0
\(666\) −37.1472 −1.43942
\(667\) 2.72498 0.105512
\(668\) 3.51388 0.135956
\(669\) 29.1194 1.12582
\(670\) 34.8167 1.34508
\(671\) 43.9361 1.69613
\(672\) 0 0
\(673\) 4.09167 0.157722 0.0788612 0.996886i \(-0.474872\pi\)
0.0788612 + 0.996886i \(0.474872\pi\)
\(674\) −32.2111 −1.24073
\(675\) 64.8444 2.49586
\(676\) 3.15559 0.121369
\(677\) 28.9361 1.11210 0.556052 0.831147i \(-0.312316\pi\)
0.556052 + 0.831147i \(0.312316\pi\)
\(678\) 57.2389 2.19825
\(679\) 0 0
\(680\) −15.2750 −0.585770
\(681\) −0.394449 −0.0151153
\(682\) −12.9083 −0.494285
\(683\) −2.21110 −0.0846055 −0.0423027 0.999105i \(-0.513469\pi\)
−0.0423027 + 0.999105i \(0.513469\pi\)
\(684\) 2.39445 0.0915540
\(685\) −64.8999 −2.47970
\(686\) 0 0
\(687\) 50.2389 1.91673
\(688\) 33.0278 1.25917
\(689\) −6.27502 −0.239059
\(690\) −38.7250 −1.47423
\(691\) 20.1833 0.767811 0.383905 0.923372i \(-0.374579\pi\)
0.383905 + 0.923372i \(0.374579\pi\)
\(692\) −4.18335 −0.159027
\(693\) 0 0
\(694\) 33.2750 1.26310
\(695\) −15.0000 −0.568982
\(696\) 9.00000 0.341144
\(697\) −7.30278 −0.276612
\(698\) 13.6972 0.518448
\(699\) −42.6333 −1.61254
\(700\) 0 0
\(701\) 6.78890 0.256413 0.128207 0.991747i \(-0.459078\pi\)
0.128207 + 0.991747i \(0.459078\pi\)
\(702\) −33.9083 −1.27979
\(703\) 3.60555 0.135986
\(704\) −37.9361 −1.42977
\(705\) 81.3583 3.06413
\(706\) 1.18335 0.0445358
\(707\) 0 0
\(708\) 8.21110 0.308592
\(709\) −4.39445 −0.165037 −0.0825185 0.996590i \(-0.526296\pi\)
−0.0825185 + 0.996590i \(0.526296\pi\)
\(710\) −8.64171 −0.324318
\(711\) −25.3944 −0.952366
\(712\) −10.1833 −0.381637
\(713\) −6.90833 −0.258719
\(714\) 0 0
\(715\) 20.7250 0.775070
\(716\) −6.27502 −0.234508
\(717\) 4.69722 0.175421
\(718\) 37.1833 1.38767
\(719\) −15.6333 −0.583024 −0.291512 0.956567i \(-0.594158\pi\)
−0.291512 + 0.956567i \(0.594158\pi\)
\(720\) −78.3583 −2.92024
\(721\) 0 0
\(722\) 1.30278 0.0484843
\(723\) 44.2389 1.64526
\(724\) −4.33053 −0.160943
\(725\) −3.63331 −0.134938
\(726\) 32.3305 1.19990
\(727\) −51.6611 −1.91600 −0.958001 0.286764i \(-0.907421\pi\)
−0.958001 + 0.286764i \(0.907421\pi\)
\(728\) 0 0
\(729\) 75.1749 2.78426
\(730\) 64.5416 2.38879
\(731\) −16.9722 −0.627741
\(732\) 10.2111 0.377413
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) −46.9361 −1.73244
\(735\) 0 0
\(736\) −5.09167 −0.187682
\(737\) −38.3305 −1.41192
\(738\) −44.3305 −1.63183
\(739\) −47.4222 −1.74445 −0.872227 0.489101i \(-0.837325\pi\)
−0.872227 + 0.489101i \(0.837325\pi\)
\(740\) 3.27502 0.120392
\(741\) 5.30278 0.194802
\(742\) 0 0
\(743\) 30.6333 1.12383 0.561914 0.827196i \(-0.310065\pi\)
0.561914 + 0.827196i \(0.310065\pi\)
\(744\) −22.8167 −0.836499
\(745\) 24.6333 0.902495
\(746\) −1.45837 −0.0533946
\(747\) 16.5416 0.605227
\(748\) 2.21110 0.0808459
\(749\) 0 0
\(750\) −12.9083 −0.471345
\(751\) −49.3583 −1.80111 −0.900555 0.434743i \(-0.856839\pi\)
−0.900555 + 0.434743i \(0.856839\pi\)
\(752\) −27.1194 −0.988944
\(753\) 16.8167 0.612833
\(754\) 1.89992 0.0691911
\(755\) −14.7250 −0.535897
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −19.3028 −0.701108
\(759\) 42.6333 1.54749
\(760\) 9.00000 0.326464
\(761\) 9.78890 0.354847 0.177424 0.984135i \(-0.443224\pi\)
0.177424 + 0.984135i \(0.443224\pi\)
\(762\) 42.2389 1.53015
\(763\) 0 0
\(764\) 4.33894 0.156977
\(765\) 40.2666 1.45584
\(766\) 47.7250 1.72437
\(767\) 13.1833 0.476023
\(768\) −23.4222 −0.845176
\(769\) 10.7889 0.389058 0.194529 0.980897i \(-0.437682\pi\)
0.194529 + 0.980897i \(0.437682\pi\)
\(770\) 0 0
\(771\) −32.7250 −1.17856
\(772\) −0.641712 −0.0230957
\(773\) 43.8167 1.57598 0.787988 0.615691i \(-0.211123\pi\)
0.787988 + 0.615691i \(0.211123\pi\)
\(774\) −103.028 −3.70326
\(775\) 9.21110 0.330873
\(776\) 7.18335 0.257867
\(777\) 0 0
\(778\) 48.3944 1.73503
\(779\) 4.30278 0.154163
\(780\) 4.81665 0.172464
\(781\) 9.51388 0.340433
\(782\) −6.63331 −0.237207
\(783\) −14.7250 −0.526228
\(784\) 0 0
\(785\) 40.5416 1.44699
\(786\) 10.6972 0.381557
\(787\) −11.6333 −0.414683 −0.207341 0.978269i \(-0.566481\pi\)
−0.207341 + 0.978269i \(0.566481\pi\)
\(788\) −1.18335 −0.0421550
\(789\) −43.9361 −1.56417
\(790\) −12.5500 −0.446510
\(791\) 0 0
\(792\) 102.083 3.62737
\(793\) 16.3944 0.582184
\(794\) −1.26662 −0.0449505
\(795\) 38.7250 1.37343
\(796\) 4.66947 0.165505
\(797\) −36.9083 −1.30736 −0.653680 0.756771i \(-0.726776\pi\)
−0.653680 + 0.756771i \(0.726776\pi\)
\(798\) 0 0
\(799\) 13.9361 0.493023
\(800\) 6.78890 0.240024
\(801\) 26.8444 0.948501
\(802\) −28.0278 −0.989694
\(803\) −71.0555 −2.50749
\(804\) −8.90833 −0.314172
\(805\) 0 0
\(806\) −4.81665 −0.169659
\(807\) 38.7250 1.36318
\(808\) −34.8167 −1.22485
\(809\) 10.0278 0.352557 0.176279 0.984340i \(-0.443594\pi\)
0.176279 + 0.984340i \(0.443594\pi\)
\(810\) 116.533 4.09456
\(811\) −40.0555 −1.40654 −0.703270 0.710923i \(-0.748277\pi\)
−0.703270 + 0.710923i \(0.748277\pi\)
\(812\) 0 0
\(813\) −23.4222 −0.821453
\(814\) 20.2111 0.708399
\(815\) 27.9083 0.977586
\(816\) −18.5139 −0.648115
\(817\) 10.0000 0.349856
\(818\) −13.1472 −0.459681
\(819\) 0 0
\(820\) 3.90833 0.136485
\(821\) −8.36669 −0.292000 −0.146000 0.989285i \(-0.546640\pi\)
−0.146000 + 0.989285i \(0.546640\pi\)
\(822\) −93.0833 −3.24665
\(823\) −34.2389 −1.19349 −0.596746 0.802430i \(-0.703540\pi\)
−0.596746 + 0.802430i \(0.703540\pi\)
\(824\) −5.36669 −0.186958
\(825\) −56.8444 −1.97907
\(826\) 0 0
\(827\) −45.0000 −1.56480 −0.782402 0.622774i \(-0.786006\pi\)
−0.782402 + 0.622774i \(0.786006\pi\)
\(828\) 7.18335 0.249639
\(829\) 18.0555 0.627094 0.313547 0.949573i \(-0.398483\pi\)
0.313547 + 0.949573i \(0.398483\pi\)
\(830\) 8.17494 0.283756
\(831\) −21.8167 −0.756811
\(832\) −14.1556 −0.490757
\(833\) 0 0
\(834\) −21.5139 −0.744965
\(835\) −34.8167 −1.20488
\(836\) −1.30278 −0.0450574
\(837\) 37.3305 1.29033
\(838\) −19.5416 −0.675055
\(839\) 17.2111 0.594193 0.297097 0.954847i \(-0.403982\pi\)
0.297097 + 0.954847i \(0.403982\pi\)
\(840\) 0 0
\(841\) −28.1749 −0.971550
\(842\) −29.1749 −1.00543
\(843\) 9.90833 0.341261
\(844\) 3.42221 0.117797
\(845\) −31.2666 −1.07560
\(846\) 84.5971 2.90851
\(847\) 0 0
\(848\) −12.9083 −0.443274
\(849\) 40.7250 1.39768
\(850\) 8.84441 0.303361
\(851\) 10.8167 0.370790
\(852\) 2.21110 0.0757511
\(853\) 36.3305 1.24393 0.621967 0.783044i \(-0.286334\pi\)
0.621967 + 0.783044i \(0.286334\pi\)
\(854\) 0 0
\(855\) −23.7250 −0.811377
\(856\) 60.6333 2.07240
\(857\) −42.9083 −1.46572 −0.732860 0.680379i \(-0.761815\pi\)
−0.732860 + 0.680379i \(0.761815\pi\)
\(858\) 29.7250 1.01479
\(859\) 23.6972 0.808539 0.404269 0.914640i \(-0.367526\pi\)
0.404269 + 0.914640i \(0.367526\pi\)
\(860\) 9.08327 0.309737
\(861\) 0 0
\(862\) 14.6056 0.497467
\(863\) 37.5416 1.27793 0.638966 0.769235i \(-0.279362\pi\)
0.638966 + 0.769235i \(0.279362\pi\)
\(864\) 27.5139 0.936041
\(865\) 41.4500 1.40934
\(866\) 38.3305 1.30252
\(867\) −46.6333 −1.58375
\(868\) 0 0
\(869\) 13.8167 0.468698
\(870\) −11.7250 −0.397514
\(871\) −14.3028 −0.484631
\(872\) −12.6333 −0.427818
\(873\) −18.9361 −0.640889
\(874\) 3.90833 0.132201
\(875\) 0 0
\(876\) −16.5139 −0.557952
\(877\) −29.8167 −1.00684 −0.503418 0.864043i \(-0.667925\pi\)
−0.503418 + 0.864043i \(0.667925\pi\)
\(878\) −48.4777 −1.63604
\(879\) −101.175 −3.41255
\(880\) 42.6333 1.43717
\(881\) −48.7527 −1.64252 −0.821261 0.570553i \(-0.806729\pi\)
−0.821261 + 0.570553i \(0.806729\pi\)
\(882\) 0 0
\(883\) −31.7889 −1.06978 −0.534891 0.844921i \(-0.679647\pi\)
−0.534891 + 0.844921i \(0.679647\pi\)
\(884\) 0.825058 0.0277497
\(885\) −81.3583 −2.73483
\(886\) 11.8806 0.399136
\(887\) −40.8167 −1.37049 −0.685245 0.728313i \(-0.740305\pi\)
−0.685245 + 0.728313i \(0.740305\pi\)
\(888\) 35.7250 1.19885
\(889\) 0 0
\(890\) 13.2666 0.444698
\(891\) −128.294 −4.29802
\(892\) −2.66947 −0.0893804
\(893\) −8.21110 −0.274774
\(894\) 35.3305 1.18163
\(895\) 62.1749 2.07828
\(896\) 0 0
\(897\) 15.9083 0.531164
\(898\) −24.1194 −0.804876
\(899\) −2.09167 −0.0697612
\(900\) −9.57779 −0.319260
\(901\) 6.63331 0.220988
\(902\) 24.1194 0.803089
\(903\) 0 0
\(904\) −39.9083 −1.32733
\(905\) 42.9083 1.42632
\(906\) −21.1194 −0.701646
\(907\) −26.8167 −0.890432 −0.445216 0.895423i \(-0.646873\pi\)
−0.445216 + 0.895423i \(0.646873\pi\)
\(908\) 0.0361603 0.00120002
\(909\) 91.7805 3.04417
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 10.9083 0.361211
\(913\) −9.00000 −0.297857
\(914\) 24.3583 0.805701
\(915\) −101.175 −3.34474
\(916\) −4.60555 −0.152172
\(917\) 0 0
\(918\) 35.8444 1.18304
\(919\) −44.0278 −1.45234 −0.726171 0.687514i \(-0.758702\pi\)
−0.726171 + 0.687514i \(0.758702\pi\)
\(920\) 27.0000 0.890164
\(921\) −60.4500 −1.99189
\(922\) 12.9083 0.425113
\(923\) 3.55004 0.116851
\(924\) 0 0
\(925\) −14.4222 −0.474199
\(926\) 37.0639 1.21800
\(927\) 14.1472 0.464655
\(928\) −1.54163 −0.0506066
\(929\) 6.51388 0.213713 0.106857 0.994274i \(-0.465921\pi\)
0.106857 + 0.994274i \(0.465921\pi\)
\(930\) 29.7250 0.974721
\(931\) 0 0
\(932\) 3.90833 0.128022
\(933\) −51.2389 −1.67748
\(934\) 48.9083 1.60033
\(935\) −21.9083 −0.716479
\(936\) 38.0917 1.24507
\(937\) 4.11943 0.134576 0.0672879 0.997734i \(-0.478565\pi\)
0.0672879 + 0.997734i \(0.478565\pi\)
\(938\) 0 0
\(939\) −29.8167 −0.973030
\(940\) −7.45837 −0.243265
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 58.1472 1.89454
\(943\) 12.9083 0.420353
\(944\) 27.1194 0.882662
\(945\) 0 0
\(946\) 56.0555 1.82252
\(947\) 26.0917 0.847865 0.423933 0.905694i \(-0.360649\pi\)
0.423933 + 0.905694i \(0.360649\pi\)
\(948\) 3.21110 0.104292
\(949\) −26.5139 −0.860677
\(950\) −5.21110 −0.169070
\(951\) 76.6611 2.48591
\(952\) 0 0
\(953\) −35.3305 −1.14447 −0.572234 0.820090i \(-0.693923\pi\)
−0.572234 + 0.820090i \(0.693923\pi\)
\(954\) 40.2666 1.30368
\(955\) −42.9916 −1.39118
\(956\) −0.430609 −0.0139269
\(957\) 12.9083 0.417267
\(958\) 5.60555 0.181107
\(959\) 0 0
\(960\) 87.3583 2.81948
\(961\) −25.6972 −0.828943
\(962\) 7.54163 0.243152
\(963\) −159.836 −5.15064
\(964\) −4.05551 −0.130619
\(965\) 6.35829 0.204681
\(966\) 0 0
\(967\) 30.3028 0.974472 0.487236 0.873270i \(-0.338005\pi\)
0.487236 + 0.873270i \(0.338005\pi\)
\(968\) −22.5416 −0.724516
\(969\) −5.60555 −0.180076
\(970\) −9.35829 −0.300477
\(971\) −24.6333 −0.790520 −0.395260 0.918569i \(-0.629346\pi\)
−0.395260 + 0.918569i \(0.629346\pi\)
\(972\) −15.0917 −0.484066
\(973\) 0 0
\(974\) 43.5416 1.39516
\(975\) −21.2111 −0.679299
\(976\) 33.7250 1.07951
\(977\) 47.4500 1.51806 0.759029 0.651056i \(-0.225674\pi\)
0.759029 + 0.651056i \(0.225674\pi\)
\(978\) 40.0278 1.27995
\(979\) −14.6056 −0.466795
\(980\) 0 0
\(981\) 33.3028 1.06328
\(982\) −15.6333 −0.498879
\(983\) −23.0555 −0.735357 −0.367678 0.929953i \(-0.619847\pi\)
−0.367678 + 0.929953i \(0.619847\pi\)
\(984\) 42.6333 1.35910
\(985\) 11.7250 0.373589
\(986\) −2.00840 −0.0639606
\(987\) 0 0
\(988\) −0.486122 −0.0154656
\(989\) 30.0000 0.953945
\(990\) −132.992 −4.22675
\(991\) −53.0278 −1.68448 −0.842241 0.539101i \(-0.818764\pi\)
−0.842241 + 0.539101i \(0.818764\pi\)
\(992\) 3.90833 0.124090
\(993\) 50.5416 1.60389
\(994\) 0 0
\(995\) −46.2666 −1.46675
\(996\) −2.09167 −0.0662772
\(997\) −37.7250 −1.19476 −0.597381 0.801958i \(-0.703792\pi\)
−0.597381 + 0.801958i \(0.703792\pi\)
\(998\) −35.0917 −1.11081
\(999\) −58.4500 −1.84927
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.g.1.2 2
3.2 odd 2 8379.2.a.bf.1.1 2
7.2 even 3 931.2.f.g.704.1 4
7.3 odd 6 931.2.f.h.324.1 4
7.4 even 3 931.2.f.g.324.1 4
7.5 odd 6 931.2.f.h.704.1 4
7.6 odd 2 133.2.a.b.1.2 2
21.20 even 2 1197.2.a.h.1.1 2
28.27 even 2 2128.2.a.l.1.2 2
35.34 odd 2 3325.2.a.n.1.1 2
56.13 odd 2 8512.2.a.bh.1.2 2
56.27 even 2 8512.2.a.l.1.1 2
133.132 even 2 2527.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
133.2.a.b.1.2 2 7.6 odd 2
931.2.a.g.1.2 2 1.1 even 1 trivial
931.2.f.g.324.1 4 7.4 even 3
931.2.f.g.704.1 4 7.2 even 3
931.2.f.h.324.1 4 7.3 odd 6
931.2.f.h.704.1 4 7.5 odd 6
1197.2.a.h.1.1 2 21.20 even 2
2128.2.a.l.1.2 2 28.27 even 2
2527.2.a.d.1.1 2 133.132 even 2
3325.2.a.n.1.1 2 35.34 odd 2
8379.2.a.bf.1.1 2 3.2 odd 2
8512.2.a.l.1.1 2 56.27 even 2
8512.2.a.bh.1.2 2 56.13 odd 2