Properties

Label 3174.2.a.ba.1.5
Level $3174$
Weight $2$
Character 3174.1
Self dual yes
Analytic conductor $25.345$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3174,2,Mod(1,3174)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3174, 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("3174.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.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: \(25.3445176016\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.91899\) of defining polynomial
Character \(\chi\) \(=\) 3174.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.82306 q^{5} -1.00000 q^{6} +2.89389 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.82306 q^{5} -1.00000 q^{6} +2.89389 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.82306 q^{10} -1.58065 q^{11} -1.00000 q^{12} +4.70981 q^{13} +2.89389 q^{14} -3.82306 q^{15} +1.00000 q^{16} -4.73780 q^{17} +1.00000 q^{18} +2.14832 q^{19} +3.82306 q^{20} -2.89389 q^{21} -1.58065 q^{22} -1.00000 q^{24} +9.61578 q^{25} +4.70981 q^{26} -1.00000 q^{27} +2.89389 q^{28} -0.588417 q^{29} -3.82306 q^{30} +0.170752 q^{31} +1.00000 q^{32} +1.58065 q^{33} -4.73780 q^{34} +11.0635 q^{35} +1.00000 q^{36} +4.81881 q^{37} +2.14832 q^{38} -4.70981 q^{39} +3.82306 q^{40} -10.3972 q^{41} -2.89389 q^{42} +12.2681 q^{43} -1.58065 q^{44} +3.82306 q^{45} +5.45520 q^{47} -1.00000 q^{48} +1.37462 q^{49} +9.61578 q^{50} +4.73780 q^{51} +4.70981 q^{52} -10.3078 q^{53} -1.00000 q^{54} -6.04290 q^{55} +2.89389 q^{56} -2.14832 q^{57} -0.588417 q^{58} -1.90272 q^{59} -3.82306 q^{60} -12.6545 q^{61} +0.170752 q^{62} +2.89389 q^{63} +1.00000 q^{64} +18.0059 q^{65} +1.58065 q^{66} +5.87608 q^{67} -4.73780 q^{68} +11.0635 q^{70} -11.8216 q^{71} +1.00000 q^{72} -13.2135 q^{73} +4.81881 q^{74} -9.61578 q^{75} +2.14832 q^{76} -4.57422 q^{77} -4.70981 q^{78} +11.6845 q^{79} +3.82306 q^{80} +1.00000 q^{81} -10.3972 q^{82} -0.113248 q^{83} -2.89389 q^{84} -18.1129 q^{85} +12.2681 q^{86} +0.588417 q^{87} -1.58065 q^{88} -12.2386 q^{89} +3.82306 q^{90} +13.6297 q^{91} -0.170752 q^{93} +5.45520 q^{94} +8.21316 q^{95} -1.00000 q^{96} +0.873537 q^{97} +1.37462 q^{98} -1.58065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9} - q^{10} + 11 q^{11} - 5 q^{12} + 12 q^{13} + 11 q^{14} + q^{15} + 5 q^{16} - q^{17} + 5 q^{18} + 15 q^{19} - q^{20} - 11 q^{21} + 11 q^{22} - 5 q^{24} + 6 q^{25} + 12 q^{26} - 5 q^{27} + 11 q^{28} + q^{29} + q^{30} - 18 q^{31} + 5 q^{32} - 11 q^{33} - q^{34} - 11 q^{35} + 5 q^{36} + 10 q^{37} + 15 q^{38} - 12 q^{39} - q^{40} - 16 q^{41} - 11 q^{42} + 18 q^{43} + 11 q^{44} - q^{45} - 4 q^{47} - 5 q^{48} + 20 q^{49} + 6 q^{50} + q^{51} + 12 q^{52} - q^{53} - 5 q^{54} - 22 q^{55} + 11 q^{56} - 15 q^{57} + q^{58} + 2 q^{59} + q^{60} - q^{61} - 18 q^{62} + 11 q^{63} + 5 q^{64} + 24 q^{65} - 11 q^{66} + 29 q^{67} - q^{68} - 11 q^{70} - 11 q^{71} + 5 q^{72} - 8 q^{73} + 10 q^{74} - 6 q^{75} + 15 q^{76} + 11 q^{77} - 12 q^{78} + 40 q^{79} - q^{80} + 5 q^{81} - 16 q^{82} + 8 q^{83} - 11 q^{84} - 13 q^{85} + 18 q^{86} - q^{87} + 11 q^{88} + 2 q^{89} - q^{90} + 33 q^{91} + 18 q^{93} - 4 q^{94} - 3 q^{95} - 5 q^{96} + 17 q^{97} + 20 q^{98} + 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.82306 1.70972 0.854862 0.518856i \(-0.173642\pi\)
0.854862 + 0.518856i \(0.173642\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.89389 1.09379 0.546895 0.837201i \(-0.315810\pi\)
0.546895 + 0.837201i \(0.315810\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.82306 1.20896
\(11\) −1.58065 −0.476583 −0.238291 0.971194i \(-0.576587\pi\)
−0.238291 + 0.971194i \(0.576587\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.70981 1.30627 0.653133 0.757243i \(-0.273454\pi\)
0.653133 + 0.757243i \(0.273454\pi\)
\(14\) 2.89389 0.773426
\(15\) −3.82306 −0.987109
\(16\) 1.00000 0.250000
\(17\) −4.73780 −1.14909 −0.574543 0.818475i \(-0.694820\pi\)
−0.574543 + 0.818475i \(0.694820\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.14832 0.492859 0.246430 0.969161i \(-0.420743\pi\)
0.246430 + 0.969161i \(0.420743\pi\)
\(20\) 3.82306 0.854862
\(21\) −2.89389 −0.631499
\(22\) −1.58065 −0.336995
\(23\) 0 0
\(24\) −1.00000 −0.204124
\(25\) 9.61578 1.92316
\(26\) 4.70981 0.923670
\(27\) −1.00000 −0.192450
\(28\) 2.89389 0.546895
\(29\) −0.588417 −0.109266 −0.0546332 0.998506i \(-0.517399\pi\)
−0.0546332 + 0.998506i \(0.517399\pi\)
\(30\) −3.82306 −0.697992
\(31\) 0.170752 0.0306680 0.0153340 0.999882i \(-0.495119\pi\)
0.0153340 + 0.999882i \(0.495119\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.58065 0.275155
\(34\) −4.73780 −0.812526
\(35\) 11.0635 1.87008
\(36\) 1.00000 0.166667
\(37\) 4.81881 0.792208 0.396104 0.918206i \(-0.370362\pi\)
0.396104 + 0.918206i \(0.370362\pi\)
\(38\) 2.14832 0.348504
\(39\) −4.70981 −0.754173
\(40\) 3.82306 0.604479
\(41\) −10.3972 −1.62378 −0.811889 0.583812i \(-0.801560\pi\)
−0.811889 + 0.583812i \(0.801560\pi\)
\(42\) −2.89389 −0.446538
\(43\) 12.2681 1.87087 0.935433 0.353505i \(-0.115010\pi\)
0.935433 + 0.353505i \(0.115010\pi\)
\(44\) −1.58065 −0.238291
\(45\) 3.82306 0.569908
\(46\) 0 0
\(47\) 5.45520 0.795723 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.37462 0.196375
\(50\) 9.61578 1.35988
\(51\) 4.73780 0.663425
\(52\) 4.70981 0.653133
\(53\) −10.3078 −1.41588 −0.707941 0.706272i \(-0.750376\pi\)
−0.707941 + 0.706272i \(0.750376\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.04290 −0.814825
\(56\) 2.89389 0.386713
\(57\) −2.14832 −0.284552
\(58\) −0.588417 −0.0772630
\(59\) −1.90272 −0.247714 −0.123857 0.992300i \(-0.539526\pi\)
−0.123857 + 0.992300i \(0.539526\pi\)
\(60\) −3.82306 −0.493555
\(61\) −12.6545 −1.62024 −0.810119 0.586265i \(-0.800598\pi\)
−0.810119 + 0.586265i \(0.800598\pi\)
\(62\) 0.170752 0.0216856
\(63\) 2.89389 0.364596
\(64\) 1.00000 0.125000
\(65\) 18.0059 2.23335
\(66\) 1.58065 0.194564
\(67\) 5.87608 0.717878 0.358939 0.933361i \(-0.383139\pi\)
0.358939 + 0.933361i \(0.383139\pi\)
\(68\) −4.73780 −0.574543
\(69\) 0 0
\(70\) 11.0635 1.32234
\(71\) −11.8216 −1.40297 −0.701483 0.712686i \(-0.747478\pi\)
−0.701483 + 0.712686i \(0.747478\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.2135 −1.54653 −0.773264 0.634084i \(-0.781377\pi\)
−0.773264 + 0.634084i \(0.781377\pi\)
\(74\) 4.81881 0.560176
\(75\) −9.61578 −1.11033
\(76\) 2.14832 0.246430
\(77\) −4.57422 −0.521281
\(78\) −4.70981 −0.533281
\(79\) 11.6845 1.31461 0.657307 0.753623i \(-0.271696\pi\)
0.657307 + 0.753623i \(0.271696\pi\)
\(80\) 3.82306 0.427431
\(81\) 1.00000 0.111111
\(82\) −10.3972 −1.14818
\(83\) −0.113248 −0.0124306 −0.00621528 0.999981i \(-0.501978\pi\)
−0.00621528 + 0.999981i \(0.501978\pi\)
\(84\) −2.89389 −0.315750
\(85\) −18.1129 −1.96462
\(86\) 12.2681 1.32290
\(87\) 0.588417 0.0630850
\(88\) −1.58065 −0.168497
\(89\) −12.2386 −1.29729 −0.648646 0.761090i \(-0.724664\pi\)
−0.648646 + 0.761090i \(0.724664\pi\)
\(90\) 3.82306 0.402986
\(91\) 13.6297 1.42878
\(92\) 0 0
\(93\) −0.170752 −0.0177062
\(94\) 5.45520 0.562661
\(95\) 8.21316 0.842653
\(96\) −1.00000 −0.102062
\(97\) 0.873537 0.0886943 0.0443471 0.999016i \(-0.485879\pi\)
0.0443471 + 0.999016i \(0.485879\pi\)
\(98\) 1.37462 0.138858
\(99\) −1.58065 −0.158861
\(100\) 9.61578 0.961578
\(101\) −10.9103 −1.08561 −0.542807 0.839858i \(-0.682638\pi\)
−0.542807 + 0.839858i \(0.682638\pi\)
\(102\) 4.73780 0.469112
\(103\) 3.69279 0.363862 0.181931 0.983311i \(-0.441765\pi\)
0.181931 + 0.983311i \(0.441765\pi\)
\(104\) 4.70981 0.461835
\(105\) −11.0635 −1.07969
\(106\) −10.3078 −1.00118
\(107\) 10.7665 1.04083 0.520416 0.853913i \(-0.325777\pi\)
0.520416 + 0.853913i \(0.325777\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.31357 −0.796296 −0.398148 0.917321i \(-0.630347\pi\)
−0.398148 + 0.917321i \(0.630347\pi\)
\(110\) −6.04290 −0.576168
\(111\) −4.81881 −0.457382
\(112\) 2.89389 0.273447
\(113\) 7.51278 0.706743 0.353372 0.935483i \(-0.385035\pi\)
0.353372 + 0.935483i \(0.385035\pi\)
\(114\) −2.14832 −0.201209
\(115\) 0 0
\(116\) −0.588417 −0.0546332
\(117\) 4.70981 0.435422
\(118\) −1.90272 −0.175160
\(119\) −13.7107 −1.25686
\(120\) −3.82306 −0.348996
\(121\) −8.50156 −0.772869
\(122\) −12.6545 −1.14568
\(123\) 10.3972 0.937488
\(124\) 0.170752 0.0153340
\(125\) 17.6464 1.57834
\(126\) 2.89389 0.257809
\(127\) −7.58758 −0.673289 −0.336645 0.941632i \(-0.609292\pi\)
−0.336645 + 0.941632i \(0.609292\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.2681 −1.08014
\(130\) 18.0059 1.57922
\(131\) 17.8085 1.55594 0.777970 0.628302i \(-0.216250\pi\)
0.777970 + 0.628302i \(0.216250\pi\)
\(132\) 1.58065 0.137578
\(133\) 6.21702 0.539084
\(134\) 5.87608 0.507616
\(135\) −3.82306 −0.329036
\(136\) −4.73780 −0.406263
\(137\) −0.762968 −0.0651847 −0.0325924 0.999469i \(-0.510376\pi\)
−0.0325924 + 0.999469i \(0.510376\pi\)
\(138\) 0 0
\(139\) 9.22612 0.782549 0.391275 0.920274i \(-0.372034\pi\)
0.391275 + 0.920274i \(0.372034\pi\)
\(140\) 11.0635 0.935039
\(141\) −5.45520 −0.459411
\(142\) −11.8216 −0.992047
\(143\) −7.44454 −0.622544
\(144\) 1.00000 0.0833333
\(145\) −2.24955 −0.186815
\(146\) −13.2135 −1.09356
\(147\) −1.37462 −0.113377
\(148\) 4.81881 0.396104
\(149\) 18.4911 1.51485 0.757424 0.652923i \(-0.226458\pi\)
0.757424 + 0.652923i \(0.226458\pi\)
\(150\) −9.61578 −0.785125
\(151\) −0.991887 −0.0807186 −0.0403593 0.999185i \(-0.512850\pi\)
−0.0403593 + 0.999185i \(0.512850\pi\)
\(152\) 2.14832 0.174252
\(153\) −4.73780 −0.383028
\(154\) −4.57422 −0.368601
\(155\) 0.652796 0.0524339
\(156\) −4.70981 −0.377087
\(157\) 8.10972 0.647226 0.323613 0.946189i \(-0.395102\pi\)
0.323613 + 0.946189i \(0.395102\pi\)
\(158\) 11.6845 0.929572
\(159\) 10.3078 0.817460
\(160\) 3.82306 0.302239
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −23.0759 −1.80745 −0.903724 0.428116i \(-0.859177\pi\)
−0.903724 + 0.428116i \(0.859177\pi\)
\(164\) −10.3972 −0.811889
\(165\) 6.04290 0.470439
\(166\) −0.113248 −0.00878973
\(167\) −5.28482 −0.408952 −0.204476 0.978872i \(-0.565549\pi\)
−0.204476 + 0.978872i \(0.565549\pi\)
\(168\) −2.89389 −0.223269
\(169\) 9.18232 0.706332
\(170\) −18.1129 −1.38919
\(171\) 2.14832 0.164286
\(172\) 12.2681 0.935433
\(173\) 14.3906 1.09410 0.547049 0.837101i \(-0.315751\pi\)
0.547049 + 0.837101i \(0.315751\pi\)
\(174\) 0.588417 0.0446078
\(175\) 27.8270 2.10353
\(176\) −1.58065 −0.119146
\(177\) 1.90272 0.143018
\(178\) −12.2386 −0.917324
\(179\) 11.2001 0.837132 0.418566 0.908186i \(-0.362533\pi\)
0.418566 + 0.908186i \(0.362533\pi\)
\(180\) 3.82306 0.284954
\(181\) 8.79467 0.653703 0.326852 0.945076i \(-0.394012\pi\)
0.326852 + 0.945076i \(0.394012\pi\)
\(182\) 13.6297 1.01030
\(183\) 12.6545 0.935445
\(184\) 0 0
\(185\) 18.4226 1.35446
\(186\) −0.170752 −0.0125202
\(187\) 7.48878 0.547634
\(188\) 5.45520 0.397862
\(189\) −2.89389 −0.210500
\(190\) 8.21316 0.595846
\(191\) −6.33200 −0.458168 −0.229084 0.973407i \(-0.573573\pi\)
−0.229084 + 0.973407i \(0.573573\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −3.07385 −0.221261 −0.110630 0.993862i \(-0.535287\pi\)
−0.110630 + 0.993862i \(0.535287\pi\)
\(194\) 0.873537 0.0627163
\(195\) −18.0059 −1.28943
\(196\) 1.37462 0.0981874
\(197\) −0.963415 −0.0686405 −0.0343202 0.999411i \(-0.510927\pi\)
−0.0343202 + 0.999411i \(0.510927\pi\)
\(198\) −1.58065 −0.112332
\(199\) 3.27328 0.232037 0.116018 0.993247i \(-0.462987\pi\)
0.116018 + 0.993247i \(0.462987\pi\)
\(200\) 9.61578 0.679938
\(201\) −5.87608 −0.414467
\(202\) −10.9103 −0.767645
\(203\) −1.70282 −0.119514
\(204\) 4.73780 0.331712
\(205\) −39.7493 −2.77621
\(206\) 3.69279 0.257289
\(207\) 0 0
\(208\) 4.70981 0.326567
\(209\) −3.39574 −0.234888
\(210\) −11.0635 −0.763456
\(211\) −24.4174 −1.68097 −0.840483 0.541838i \(-0.817728\pi\)
−0.840483 + 0.541838i \(0.817728\pi\)
\(212\) −10.3078 −0.707941
\(213\) 11.8216 0.810003
\(214\) 10.7665 0.735979
\(215\) 46.9016 3.19866
\(216\) −1.00000 −0.0680414
\(217\) 0.494139 0.0335444
\(218\) −8.31357 −0.563066
\(219\) 13.2135 0.892888
\(220\) −6.04290 −0.407412
\(221\) −22.3141 −1.50101
\(222\) −4.81881 −0.323418
\(223\) 15.1991 1.01781 0.508904 0.860823i \(-0.330051\pi\)
0.508904 + 0.860823i \(0.330051\pi\)
\(224\) 2.89389 0.193356
\(225\) 9.61578 0.641052
\(226\) 7.51278 0.499743
\(227\) 12.9325 0.858358 0.429179 0.903220i \(-0.358803\pi\)
0.429179 + 0.903220i \(0.358803\pi\)
\(228\) −2.14832 −0.142276
\(229\) −0.948502 −0.0626788 −0.0313394 0.999509i \(-0.509977\pi\)
−0.0313394 + 0.999509i \(0.509977\pi\)
\(230\) 0 0
\(231\) 4.57422 0.300962
\(232\) −0.588417 −0.0386315
\(233\) 13.9482 0.913778 0.456889 0.889524i \(-0.348964\pi\)
0.456889 + 0.889524i \(0.348964\pi\)
\(234\) 4.70981 0.307890
\(235\) 20.8556 1.36047
\(236\) −1.90272 −0.123857
\(237\) −11.6845 −0.758992
\(238\) −13.7107 −0.888732
\(239\) −2.44329 −0.158043 −0.0790217 0.996873i \(-0.525180\pi\)
−0.0790217 + 0.996873i \(0.525180\pi\)
\(240\) −3.82306 −0.246777
\(241\) 10.8636 0.699784 0.349892 0.936790i \(-0.386218\pi\)
0.349892 + 0.936790i \(0.386218\pi\)
\(242\) −8.50156 −0.546501
\(243\) −1.00000 −0.0641500
\(244\) −12.6545 −0.810119
\(245\) 5.25527 0.335747
\(246\) 10.3972 0.662904
\(247\) 10.1182 0.643805
\(248\) 0.170752 0.0108428
\(249\) 0.113248 0.00717678
\(250\) 17.6464 1.11606
\(251\) −14.3211 −0.903940 −0.451970 0.892033i \(-0.649279\pi\)
−0.451970 + 0.892033i \(0.649279\pi\)
\(252\) 2.89389 0.182298
\(253\) 0 0
\(254\) −7.58758 −0.476087
\(255\) 18.1129 1.13427
\(256\) 1.00000 0.0625000
\(257\) −4.24430 −0.264752 −0.132376 0.991200i \(-0.542261\pi\)
−0.132376 + 0.991200i \(0.542261\pi\)
\(258\) −12.2681 −0.763777
\(259\) 13.9451 0.866509
\(260\) 18.0059 1.11668
\(261\) −0.588417 −0.0364221
\(262\) 17.8085 1.10022
\(263\) −7.10375 −0.438036 −0.219018 0.975721i \(-0.570285\pi\)
−0.219018 + 0.975721i \(0.570285\pi\)
\(264\) 1.58065 0.0972820
\(265\) −39.4072 −2.42077
\(266\) 6.21702 0.381190
\(267\) 12.2386 0.748992
\(268\) 5.87608 0.358939
\(269\) −7.73235 −0.471450 −0.235725 0.971820i \(-0.575746\pi\)
−0.235725 + 0.971820i \(0.575746\pi\)
\(270\) −3.82306 −0.232664
\(271\) −5.08058 −0.308623 −0.154312 0.988022i \(-0.549316\pi\)
−0.154312 + 0.988022i \(0.549316\pi\)
\(272\) −4.73780 −0.287271
\(273\) −13.6297 −0.824907
\(274\) −0.762968 −0.0460926
\(275\) −15.1991 −0.916543
\(276\) 0 0
\(277\) −15.3909 −0.924751 −0.462376 0.886684i \(-0.653003\pi\)
−0.462376 + 0.886684i \(0.653003\pi\)
\(278\) 9.22612 0.553346
\(279\) 0.170752 0.0102227
\(280\) 11.0635 0.661172
\(281\) −3.29958 −0.196836 −0.0984181 0.995145i \(-0.531378\pi\)
−0.0984181 + 0.995145i \(0.531378\pi\)
\(282\) −5.45520 −0.324853
\(283\) −25.8826 −1.53856 −0.769280 0.638911i \(-0.779385\pi\)
−0.769280 + 0.638911i \(0.779385\pi\)
\(284\) −11.8216 −0.701483
\(285\) −8.21316 −0.486506
\(286\) −7.44454 −0.440205
\(287\) −30.0885 −1.77607
\(288\) 1.00000 0.0589256
\(289\) 5.44675 0.320397
\(290\) −2.24955 −0.132098
\(291\) −0.873537 −0.0512077
\(292\) −13.2135 −0.773264
\(293\) 6.52568 0.381234 0.190617 0.981664i \(-0.438951\pi\)
0.190617 + 0.981664i \(0.438951\pi\)
\(294\) −1.37462 −0.0801697
\(295\) −7.27423 −0.423522
\(296\) 4.81881 0.280088
\(297\) 1.58065 0.0917184
\(298\) 18.4911 1.07116
\(299\) 0 0
\(300\) −9.61578 −0.555167
\(301\) 35.5025 2.04633
\(302\) −0.991887 −0.0570767
\(303\) 10.9103 0.626779
\(304\) 2.14832 0.123215
\(305\) −48.3788 −2.77016
\(306\) −4.73780 −0.270842
\(307\) 4.61726 0.263521 0.131761 0.991282i \(-0.457937\pi\)
0.131761 + 0.991282i \(0.457937\pi\)
\(308\) −4.57422 −0.260641
\(309\) −3.69279 −0.210076
\(310\) 0.652796 0.0370763
\(311\) −10.7400 −0.609011 −0.304506 0.952511i \(-0.598491\pi\)
−0.304506 + 0.952511i \(0.598491\pi\)
\(312\) −4.70981 −0.266641
\(313\) 26.6899 1.50860 0.754301 0.656529i \(-0.227976\pi\)
0.754301 + 0.656529i \(0.227976\pi\)
\(314\) 8.10972 0.457658
\(315\) 11.0635 0.623359
\(316\) 11.6845 0.657307
\(317\) 19.1814 1.07733 0.538667 0.842518i \(-0.318928\pi\)
0.538667 + 0.842518i \(0.318928\pi\)
\(318\) 10.3078 0.578031
\(319\) 0.930080 0.0520745
\(320\) 3.82306 0.213715
\(321\) −10.7665 −0.600925
\(322\) 0 0
\(323\) −10.1783 −0.566337
\(324\) 1.00000 0.0555556
\(325\) 45.2885 2.51215
\(326\) −23.0759 −1.27806
\(327\) 8.31357 0.459742
\(328\) −10.3972 −0.574092
\(329\) 15.7868 0.870353
\(330\) 6.04290 0.332651
\(331\) 8.74066 0.480430 0.240215 0.970720i \(-0.422782\pi\)
0.240215 + 0.970720i \(0.422782\pi\)
\(332\) −0.113248 −0.00621528
\(333\) 4.81881 0.264069
\(334\) −5.28482 −0.289173
\(335\) 22.4646 1.22737
\(336\) −2.89389 −0.157875
\(337\) −3.25394 −0.177253 −0.0886267 0.996065i \(-0.528248\pi\)
−0.0886267 + 0.996065i \(0.528248\pi\)
\(338\) 9.18232 0.499452
\(339\) −7.51278 −0.408038
\(340\) −18.1129 −0.982309
\(341\) −0.269899 −0.0146159
\(342\) 2.14832 0.116168
\(343\) −16.2792 −0.878997
\(344\) 12.2681 0.661451
\(345\) 0 0
\(346\) 14.3906 0.773644
\(347\) −5.21179 −0.279783 −0.139892 0.990167i \(-0.544675\pi\)
−0.139892 + 0.990167i \(0.544675\pi\)
\(348\) 0.588417 0.0315425
\(349\) 15.4731 0.828256 0.414128 0.910219i \(-0.364087\pi\)
0.414128 + 0.910219i \(0.364087\pi\)
\(350\) 27.8270 1.48742
\(351\) −4.70981 −0.251391
\(352\) −1.58065 −0.0842487
\(353\) −1.61663 −0.0860448 −0.0430224 0.999074i \(-0.513699\pi\)
−0.0430224 + 0.999074i \(0.513699\pi\)
\(354\) 1.90272 0.101129
\(355\) −45.1947 −2.39868
\(356\) −12.2386 −0.648646
\(357\) 13.7107 0.725647
\(358\) 11.2001 0.591942
\(359\) 13.5579 0.715561 0.357780 0.933806i \(-0.383534\pi\)
0.357780 + 0.933806i \(0.383534\pi\)
\(360\) 3.82306 0.201493
\(361\) −14.3847 −0.757090
\(362\) 8.79467 0.462238
\(363\) 8.50156 0.446216
\(364\) 13.6297 0.714390
\(365\) −50.5161 −2.64414
\(366\) 12.6545 0.661460
\(367\) 16.2735 0.849467 0.424734 0.905318i \(-0.360368\pi\)
0.424734 + 0.905318i \(0.360368\pi\)
\(368\) 0 0
\(369\) −10.3972 −0.541259
\(370\) 18.4226 0.957746
\(371\) −29.8296 −1.54868
\(372\) −0.170752 −0.00885310
\(373\) 6.97489 0.361146 0.180573 0.983562i \(-0.442205\pi\)
0.180573 + 0.983562i \(0.442205\pi\)
\(374\) 7.48878 0.387236
\(375\) −17.6464 −0.911255
\(376\) 5.45520 0.281331
\(377\) −2.77133 −0.142731
\(378\) −2.89389 −0.148846
\(379\) 16.5116 0.848143 0.424072 0.905629i \(-0.360600\pi\)
0.424072 + 0.905629i \(0.360600\pi\)
\(380\) 8.21316 0.421326
\(381\) 7.58758 0.388724
\(382\) −6.33200 −0.323973
\(383\) −22.3879 −1.14397 −0.571984 0.820265i \(-0.693826\pi\)
−0.571984 + 0.820265i \(0.693826\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −17.4875 −0.891247
\(386\) −3.07385 −0.156455
\(387\) 12.2681 0.623622
\(388\) 0.873537 0.0443471
\(389\) −7.54227 −0.382408 −0.191204 0.981550i \(-0.561239\pi\)
−0.191204 + 0.981550i \(0.561239\pi\)
\(390\) −18.0059 −0.911763
\(391\) 0 0
\(392\) 1.37462 0.0694290
\(393\) −17.8085 −0.898322
\(394\) −0.963415 −0.0485361
\(395\) 44.6707 2.24763
\(396\) −1.58065 −0.0794305
\(397\) 28.7582 1.44333 0.721665 0.692242i \(-0.243377\pi\)
0.721665 + 0.692242i \(0.243377\pi\)
\(398\) 3.27328 0.164075
\(399\) −6.21702 −0.311240
\(400\) 9.61578 0.480789
\(401\) 6.81566 0.340358 0.170179 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(402\) −5.87608 −0.293072
\(403\) 0.804211 0.0400606
\(404\) −10.9103 −0.542807
\(405\) 3.82306 0.189969
\(406\) −1.70282 −0.0845094
\(407\) −7.61684 −0.377553
\(408\) 4.73780 0.234556
\(409\) −27.7947 −1.37436 −0.687179 0.726488i \(-0.741151\pi\)
−0.687179 + 0.726488i \(0.741151\pi\)
\(410\) −39.7493 −1.96308
\(411\) 0.762968 0.0376344
\(412\) 3.69279 0.181931
\(413\) −5.50628 −0.270947
\(414\) 0 0
\(415\) −0.432953 −0.0212528
\(416\) 4.70981 0.230917
\(417\) −9.22612 −0.451805
\(418\) −3.39574 −0.166091
\(419\) −15.1417 −0.739718 −0.369859 0.929088i \(-0.620594\pi\)
−0.369859 + 0.929088i \(0.620594\pi\)
\(420\) −11.0635 −0.539845
\(421\) 38.1445 1.85905 0.929524 0.368762i \(-0.120218\pi\)
0.929524 + 0.368762i \(0.120218\pi\)
\(422\) −24.4174 −1.18862
\(423\) 5.45520 0.265241
\(424\) −10.3078 −0.500590
\(425\) −45.5576 −2.20987
\(426\) 11.8216 0.572759
\(427\) −36.6207 −1.77220
\(428\) 10.7665 0.520416
\(429\) 7.44454 0.359426
\(430\) 46.9016 2.26180
\(431\) 10.2138 0.491981 0.245991 0.969272i \(-0.420887\pi\)
0.245991 + 0.969272i \(0.420887\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −9.82271 −0.472049 −0.236025 0.971747i \(-0.575845\pi\)
−0.236025 + 0.971747i \(0.575845\pi\)
\(434\) 0.494139 0.0237194
\(435\) 2.24955 0.107858
\(436\) −8.31357 −0.398148
\(437\) 0 0
\(438\) 13.2135 0.631367
\(439\) −7.79601 −0.372083 −0.186042 0.982542i \(-0.559566\pi\)
−0.186042 + 0.982542i \(0.559566\pi\)
\(440\) −6.04290 −0.288084
\(441\) 1.37462 0.0654583
\(442\) −22.3141 −1.06138
\(443\) −28.2208 −1.34081 −0.670406 0.741994i \(-0.733880\pi\)
−0.670406 + 0.741994i \(0.733880\pi\)
\(444\) −4.81881 −0.228691
\(445\) −46.7890 −2.21801
\(446\) 15.1991 0.719699
\(447\) −18.4911 −0.874598
\(448\) 2.89389 0.136724
\(449\) −12.7574 −0.602061 −0.301030 0.953615i \(-0.597331\pi\)
−0.301030 + 0.953615i \(0.597331\pi\)
\(450\) 9.61578 0.453292
\(451\) 16.4344 0.773864
\(452\) 7.51278 0.353372
\(453\) 0.991887 0.0466029
\(454\) 12.9325 0.606950
\(455\) 52.1071 2.44282
\(456\) −2.14832 −0.100604
\(457\) 9.37038 0.438328 0.219164 0.975688i \(-0.429667\pi\)
0.219164 + 0.975688i \(0.429667\pi\)
\(458\) −0.948502 −0.0443206
\(459\) 4.73780 0.221142
\(460\) 0 0
\(461\) −11.6569 −0.542916 −0.271458 0.962450i \(-0.587506\pi\)
−0.271458 + 0.962450i \(0.587506\pi\)
\(462\) 4.57422 0.212812
\(463\) 13.6279 0.633341 0.316670 0.948536i \(-0.397435\pi\)
0.316670 + 0.948536i \(0.397435\pi\)
\(464\) −0.588417 −0.0273166
\(465\) −0.652796 −0.0302727
\(466\) 13.9482 0.646139
\(467\) 6.57700 0.304347 0.152174 0.988354i \(-0.451373\pi\)
0.152174 + 0.988354i \(0.451373\pi\)
\(468\) 4.70981 0.217711
\(469\) 17.0048 0.785207
\(470\) 20.8556 0.961995
\(471\) −8.10972 −0.373676
\(472\) −1.90272 −0.0875800
\(473\) −19.3915 −0.891622
\(474\) −11.6845 −0.536689
\(475\) 20.6578 0.947845
\(476\) −13.7107 −0.628428
\(477\) −10.3078 −0.471961
\(478\) −2.44329 −0.111754
\(479\) 9.24047 0.422208 0.211104 0.977464i \(-0.432294\pi\)
0.211104 + 0.977464i \(0.432294\pi\)
\(480\) −3.82306 −0.174498
\(481\) 22.6957 1.03483
\(482\) 10.8636 0.494822
\(483\) 0 0
\(484\) −8.50156 −0.386434
\(485\) 3.33958 0.151643
\(486\) −1.00000 −0.0453609
\(487\) −38.0246 −1.72306 −0.861529 0.507708i \(-0.830493\pi\)
−0.861529 + 0.507708i \(0.830493\pi\)
\(488\) −12.6545 −0.572841
\(489\) 23.0759 1.04353
\(490\) 5.25527 0.237409
\(491\) −4.28664 −0.193453 −0.0967267 0.995311i \(-0.530837\pi\)
−0.0967267 + 0.995311i \(0.530837\pi\)
\(492\) 10.3972 0.468744
\(493\) 2.78780 0.125556
\(494\) 10.1182 0.455239
\(495\) −6.04290 −0.271608
\(496\) 0.170752 0.00766701
\(497\) −34.2105 −1.53455
\(498\) 0.113248 0.00507475
\(499\) −30.2794 −1.35549 −0.677746 0.735297i \(-0.737043\pi\)
−0.677746 + 0.735297i \(0.737043\pi\)
\(500\) 17.6464 0.789170
\(501\) 5.28482 0.236109
\(502\) −14.3211 −0.639182
\(503\) 11.6641 0.520077 0.260038 0.965598i \(-0.416265\pi\)
0.260038 + 0.965598i \(0.416265\pi\)
\(504\) 2.89389 0.128904
\(505\) −41.7107 −1.85610
\(506\) 0 0
\(507\) −9.18232 −0.407801
\(508\) −7.58758 −0.336645
\(509\) −15.3035 −0.678317 −0.339159 0.940729i \(-0.610142\pi\)
−0.339159 + 0.940729i \(0.610142\pi\)
\(510\) 18.1129 0.802052
\(511\) −38.2386 −1.69158
\(512\) 1.00000 0.0441942
\(513\) −2.14832 −0.0948508
\(514\) −4.24430 −0.187208
\(515\) 14.1178 0.622103
\(516\) −12.2681 −0.540072
\(517\) −8.62274 −0.379228
\(518\) 13.9451 0.612714
\(519\) −14.3906 −0.631678
\(520\) 18.0059 0.789610
\(521\) 1.93070 0.0845857 0.0422928 0.999105i \(-0.486534\pi\)
0.0422928 + 0.999105i \(0.486534\pi\)
\(522\) −0.588417 −0.0257543
\(523\) −18.2743 −0.799081 −0.399540 0.916716i \(-0.630830\pi\)
−0.399540 + 0.916716i \(0.630830\pi\)
\(524\) 17.8085 0.777970
\(525\) −27.8270 −1.21447
\(526\) −7.10375 −0.309738
\(527\) −0.808991 −0.0352402
\(528\) 1.58065 0.0687888
\(529\) 0 0
\(530\) −39.4072 −1.71174
\(531\) −1.90272 −0.0825712
\(532\) 6.21702 0.269542
\(533\) −48.9691 −2.12109
\(534\) 12.2386 0.529617
\(535\) 41.1608 1.77954
\(536\) 5.87608 0.253808
\(537\) −11.2001 −0.483319
\(538\) −7.73235 −0.333365
\(539\) −2.17279 −0.0935888
\(540\) −3.82306 −0.164518
\(541\) 44.4499 1.91105 0.955525 0.294910i \(-0.0952898\pi\)
0.955525 + 0.294910i \(0.0952898\pi\)
\(542\) −5.08058 −0.218230
\(543\) −8.79467 −0.377416
\(544\) −4.73780 −0.203131
\(545\) −31.7833 −1.36145
\(546\) −13.6297 −0.583297
\(547\) 19.1076 0.816981 0.408490 0.912763i \(-0.366055\pi\)
0.408490 + 0.912763i \(0.366055\pi\)
\(548\) −0.762968 −0.0325924
\(549\) −12.6545 −0.540079
\(550\) −15.1991 −0.648093
\(551\) −1.26411 −0.0538529
\(552\) 0 0
\(553\) 33.8138 1.43791
\(554\) −15.3909 −0.653898
\(555\) −18.4226 −0.781996
\(556\) 9.22612 0.391275
\(557\) −44.0806 −1.86776 −0.933878 0.357591i \(-0.883598\pi\)
−0.933878 + 0.357591i \(0.883598\pi\)
\(558\) 0.170752 0.00722852
\(559\) 57.7804 2.44385
\(560\) 11.0635 0.467519
\(561\) −7.48878 −0.316177
\(562\) −3.29958 −0.139184
\(563\) 39.4399 1.66219 0.831096 0.556129i \(-0.187714\pi\)
0.831096 + 0.556129i \(0.187714\pi\)
\(564\) −5.45520 −0.229705
\(565\) 28.7218 1.20834
\(566\) −25.8826 −1.08793
\(567\) 2.89389 0.121532
\(568\) −11.8216 −0.496024
\(569\) −38.3267 −1.60674 −0.803370 0.595480i \(-0.796962\pi\)
−0.803370 + 0.595480i \(0.796962\pi\)
\(570\) −8.21316 −0.344012
\(571\) −16.9708 −0.710204 −0.355102 0.934828i \(-0.615554\pi\)
−0.355102 + 0.934828i \(0.615554\pi\)
\(572\) −7.44454 −0.311272
\(573\) 6.33200 0.264523
\(574\) −30.0885 −1.25587
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −29.1794 −1.21475 −0.607377 0.794414i \(-0.707778\pi\)
−0.607377 + 0.794414i \(0.707778\pi\)
\(578\) 5.44675 0.226555
\(579\) 3.07385 0.127745
\(580\) −2.24955 −0.0934077
\(581\) −0.327727 −0.0135964
\(582\) −0.873537 −0.0362093
\(583\) 16.2929 0.674785
\(584\) −13.2135 −0.546780
\(585\) 18.0059 0.744452
\(586\) 6.52568 0.269573
\(587\) −15.1816 −0.626613 −0.313306 0.949652i \(-0.601437\pi\)
−0.313306 + 0.949652i \(0.601437\pi\)
\(588\) −1.37462 −0.0566885
\(589\) 0.366831 0.0151150
\(590\) −7.27423 −0.299475
\(591\) 0.963415 0.0396296
\(592\) 4.81881 0.198052
\(593\) −31.2609 −1.28373 −0.641865 0.766817i \(-0.721839\pi\)
−0.641865 + 0.766817i \(0.721839\pi\)
\(594\) 1.58065 0.0648547
\(595\) −52.4168 −2.14888
\(596\) 18.4911 0.757424
\(597\) −3.27328 −0.133967
\(598\) 0 0
\(599\) −30.8576 −1.26081 −0.630405 0.776267i \(-0.717111\pi\)
−0.630405 + 0.776267i \(0.717111\pi\)
\(600\) −9.61578 −0.392562
\(601\) 6.62774 0.270351 0.135176 0.990822i \(-0.456840\pi\)
0.135176 + 0.990822i \(0.456840\pi\)
\(602\) 35.5025 1.44698
\(603\) 5.87608 0.239293
\(604\) −0.991887 −0.0403593
\(605\) −32.5020 −1.32139
\(606\) 10.9103 0.443200
\(607\) −23.5777 −0.956991 −0.478495 0.878090i \(-0.658818\pi\)
−0.478495 + 0.878090i \(0.658818\pi\)
\(608\) 2.14832 0.0871260
\(609\) 1.70282 0.0690017
\(610\) −48.3788 −1.95880
\(611\) 25.6930 1.03943
\(612\) −4.73780 −0.191514
\(613\) −32.5703 −1.31550 −0.657750 0.753236i \(-0.728492\pi\)
−0.657750 + 0.753236i \(0.728492\pi\)
\(614\) 4.61726 0.186338
\(615\) 39.7493 1.60285
\(616\) −4.57422 −0.184301
\(617\) −6.00021 −0.241559 −0.120780 0.992679i \(-0.538539\pi\)
−0.120780 + 0.992679i \(0.538539\pi\)
\(618\) −3.69279 −0.148546
\(619\) −12.8869 −0.517968 −0.258984 0.965882i \(-0.583388\pi\)
−0.258984 + 0.965882i \(0.583388\pi\)
\(620\) 0.652796 0.0262169
\(621\) 0 0
\(622\) −10.7400 −0.430636
\(623\) −35.4173 −1.41896
\(624\) −4.70981 −0.188543
\(625\) 19.3843 0.775371
\(626\) 26.6899 1.06674
\(627\) 3.39574 0.135613
\(628\) 8.10972 0.323613
\(629\) −22.8306 −0.910315
\(630\) 11.0635 0.440781
\(631\) 21.4418 0.853585 0.426792 0.904350i \(-0.359644\pi\)
0.426792 + 0.904350i \(0.359644\pi\)
\(632\) 11.6845 0.464786
\(633\) 24.4174 0.970506
\(634\) 19.1814 0.761791
\(635\) −29.0078 −1.15114
\(636\) 10.3078 0.408730
\(637\) 6.47422 0.256518
\(638\) 0.930080 0.0368222
\(639\) −11.8216 −0.467655
\(640\) 3.82306 0.151120
\(641\) 30.7587 1.21489 0.607447 0.794360i \(-0.292194\pi\)
0.607447 + 0.794360i \(0.292194\pi\)
\(642\) −10.7665 −0.424918
\(643\) 28.7980 1.13568 0.567842 0.823138i \(-0.307778\pi\)
0.567842 + 0.823138i \(0.307778\pi\)
\(644\) 0 0
\(645\) −46.9016 −1.84675
\(646\) −10.1783 −0.400461
\(647\) −20.8594 −0.820069 −0.410035 0.912070i \(-0.634483\pi\)
−0.410035 + 0.912070i \(0.634483\pi\)
\(648\) 1.00000 0.0392837
\(649\) 3.00753 0.118056
\(650\) 45.2885 1.77636
\(651\) −0.494139 −0.0193668
\(652\) −23.0759 −0.903724
\(653\) 26.5838 1.04031 0.520153 0.854073i \(-0.325875\pi\)
0.520153 + 0.854073i \(0.325875\pi\)
\(654\) 8.31357 0.325086
\(655\) 68.0831 2.66023
\(656\) −10.3972 −0.405944
\(657\) −13.2135 −0.515509
\(658\) 15.7868 0.615433
\(659\) −27.4167 −1.06800 −0.534002 0.845483i \(-0.679313\pi\)
−0.534002 + 0.845483i \(0.679313\pi\)
\(660\) 6.04290 0.235220
\(661\) 8.13934 0.316583 0.158292 0.987392i \(-0.449401\pi\)
0.158292 + 0.987392i \(0.449401\pi\)
\(662\) 8.74066 0.339715
\(663\) 22.3141 0.866609
\(664\) −0.113248 −0.00439486
\(665\) 23.7680 0.921685
\(666\) 4.81881 0.186725
\(667\) 0 0
\(668\) −5.28482 −0.204476
\(669\) −15.1991 −0.587632
\(670\) 22.4646 0.867884
\(671\) 20.0022 0.772178
\(672\) −2.89389 −0.111634
\(673\) 31.0127 1.19545 0.597726 0.801700i \(-0.296071\pi\)
0.597726 + 0.801700i \(0.296071\pi\)
\(674\) −3.25394 −0.125337
\(675\) −9.61578 −0.370111
\(676\) 9.18232 0.353166
\(677\) −20.4045 −0.784208 −0.392104 0.919921i \(-0.628253\pi\)
−0.392104 + 0.919921i \(0.628253\pi\)
\(678\) −7.51278 −0.288527
\(679\) 2.52792 0.0970128
\(680\) −18.1129 −0.694597
\(681\) −12.9325 −0.495573
\(682\) −0.269899 −0.0103350
\(683\) 20.7857 0.795342 0.397671 0.917528i \(-0.369819\pi\)
0.397671 + 0.917528i \(0.369819\pi\)
\(684\) 2.14832 0.0821432
\(685\) −2.91687 −0.111448
\(686\) −16.2792 −0.621544
\(687\) 0.948502 0.0361876
\(688\) 12.2681 0.467716
\(689\) −48.5477 −1.84952
\(690\) 0 0
\(691\) −13.6695 −0.520012 −0.260006 0.965607i \(-0.583725\pi\)
−0.260006 + 0.965607i \(0.583725\pi\)
\(692\) 14.3906 0.547049
\(693\) −4.57422 −0.173760
\(694\) −5.21179 −0.197837
\(695\) 35.2720 1.33794
\(696\) 0.588417 0.0223039
\(697\) 49.2601 1.86586
\(698\) 15.4731 0.585665
\(699\) −13.9482 −0.527570
\(700\) 27.8270 1.05176
\(701\) −2.20983 −0.0834641 −0.0417320 0.999129i \(-0.513288\pi\)
−0.0417320 + 0.999129i \(0.513288\pi\)
\(702\) −4.70981 −0.177760
\(703\) 10.3524 0.390447
\(704\) −1.58065 −0.0595728
\(705\) −20.8556 −0.785466
\(706\) −1.61663 −0.0608428
\(707\) −31.5732 −1.18743
\(708\) 1.90272 0.0715088
\(709\) −18.7677 −0.704837 −0.352418 0.935843i \(-0.614641\pi\)
−0.352418 + 0.935843i \(0.614641\pi\)
\(710\) −45.1947 −1.69613
\(711\) 11.6845 0.438204
\(712\) −12.2386 −0.458662
\(713\) 0 0
\(714\) 13.7107 0.513110
\(715\) −28.4609 −1.06438
\(716\) 11.2001 0.418566
\(717\) 2.44329 0.0912465
\(718\) 13.5579 0.505978
\(719\) 20.6700 0.770862 0.385431 0.922737i \(-0.374053\pi\)
0.385431 + 0.922737i \(0.374053\pi\)
\(720\) 3.82306 0.142477
\(721\) 10.6865 0.397988
\(722\) −14.3847 −0.535343
\(723\) −10.8636 −0.404021
\(724\) 8.79467 0.326852
\(725\) −5.65809 −0.210136
\(726\) 8.50156 0.315522
\(727\) −31.5454 −1.16996 −0.584978 0.811049i \(-0.698897\pi\)
−0.584978 + 0.811049i \(0.698897\pi\)
\(728\) 13.6297 0.505150
\(729\) 1.00000 0.0370370
\(730\) −50.5161 −1.86969
\(731\) −58.1237 −2.14978
\(732\) 12.6545 0.467723
\(733\) −24.4270 −0.902232 −0.451116 0.892465i \(-0.648974\pi\)
−0.451116 + 0.892465i \(0.648974\pi\)
\(734\) 16.2735 0.600664
\(735\) −5.25527 −0.193843
\(736\) 0 0
\(737\) −9.28801 −0.342128
\(738\) −10.3972 −0.382728
\(739\) −42.1528 −1.55062 −0.775308 0.631583i \(-0.782405\pi\)
−0.775308 + 0.631583i \(0.782405\pi\)
\(740\) 18.4226 0.677229
\(741\) −10.1182 −0.371701
\(742\) −29.8296 −1.09508
\(743\) 2.98913 0.109661 0.0548303 0.998496i \(-0.482538\pi\)
0.0548303 + 0.998496i \(0.482538\pi\)
\(744\) −0.170752 −0.00626009
\(745\) 70.6925 2.58997
\(746\) 6.97489 0.255369
\(747\) −0.113248 −0.00414352
\(748\) 7.48878 0.273817
\(749\) 31.1570 1.13845
\(750\) −17.6464 −0.644355
\(751\) 39.2254 1.43135 0.715677 0.698431i \(-0.246118\pi\)
0.715677 + 0.698431i \(0.246118\pi\)
\(752\) 5.45520 0.198931
\(753\) 14.3211 0.521890
\(754\) −2.77133 −0.100926
\(755\) −3.79204 −0.138007
\(756\) −2.89389 −0.105250
\(757\) −47.4506 −1.72462 −0.862311 0.506379i \(-0.830984\pi\)
−0.862311 + 0.506379i \(0.830984\pi\)
\(758\) 16.5116 0.599728
\(759\) 0 0
\(760\) 8.21316 0.297923
\(761\) −30.7455 −1.11452 −0.557261 0.830337i \(-0.688148\pi\)
−0.557261 + 0.830337i \(0.688148\pi\)
\(762\) 7.58758 0.274869
\(763\) −24.0586 −0.870980
\(764\) −6.33200 −0.229084
\(765\) −18.1129 −0.654873
\(766\) −22.3879 −0.808907
\(767\) −8.96147 −0.323580
\(768\) −1.00000 −0.0360844
\(769\) −32.4747 −1.17107 −0.585533 0.810648i \(-0.699115\pi\)
−0.585533 + 0.810648i \(0.699115\pi\)
\(770\) −17.4875 −0.630206
\(771\) 4.24430 0.152855
\(772\) −3.07385 −0.110630
\(773\) 12.4335 0.447202 0.223601 0.974681i \(-0.428219\pi\)
0.223601 + 0.974681i \(0.428219\pi\)
\(774\) 12.2681 0.440967
\(775\) 1.64192 0.0589794
\(776\) 0.873537 0.0313582
\(777\) −13.9451 −0.500279
\(778\) −7.54227 −0.270403
\(779\) −22.3366 −0.800293
\(780\) −18.0059 −0.644714
\(781\) 18.6858 0.668629
\(782\) 0 0
\(783\) 0.588417 0.0210283
\(784\) 1.37462 0.0490937
\(785\) 31.0039 1.10658
\(786\) −17.8085 −0.635210
\(787\) −4.10539 −0.146341 −0.0731707 0.997319i \(-0.523312\pi\)
−0.0731707 + 0.997319i \(0.523312\pi\)
\(788\) −0.963415 −0.0343202
\(789\) 7.10375 0.252900
\(790\) 44.6707 1.58931
\(791\) 21.7412 0.773028
\(792\) −1.58065 −0.0561658
\(793\) −59.6001 −2.11646
\(794\) 28.7582 1.02059
\(795\) 39.4072 1.39763
\(796\) 3.27328 0.116018
\(797\) −38.6182 −1.36793 −0.683963 0.729517i \(-0.739745\pi\)
−0.683963 + 0.729517i \(0.739745\pi\)
\(798\) −6.21702 −0.220080
\(799\) −25.8457 −0.914354
\(800\) 9.61578 0.339969
\(801\) −12.2386 −0.432431
\(802\) 6.81566 0.240669
\(803\) 20.8859 0.737048
\(804\) −5.87608 −0.207233
\(805\) 0 0
\(806\) 0.804211 0.0283271
\(807\) 7.73235 0.272191
\(808\) −10.9103 −0.383822
\(809\) 36.8332 1.29499 0.647494 0.762071i \(-0.275817\pi\)
0.647494 + 0.762071i \(0.275817\pi\)
\(810\) 3.82306 0.134329
\(811\) −44.7013 −1.56968 −0.784838 0.619701i \(-0.787254\pi\)
−0.784838 + 0.619701i \(0.787254\pi\)
\(812\) −1.70282 −0.0597572
\(813\) 5.08058 0.178184
\(814\) −7.61684 −0.266970
\(815\) −88.2207 −3.09024
\(816\) 4.73780 0.165856
\(817\) 26.3558 0.922073
\(818\) −27.7947 −0.971818
\(819\) 13.6297 0.476260
\(820\) −39.7493 −1.38811
\(821\) 37.2008 1.29832 0.649159 0.760653i \(-0.275121\pi\)
0.649159 + 0.760653i \(0.275121\pi\)
\(822\) 0.762968 0.0266116
\(823\) 41.0413 1.43061 0.715304 0.698813i \(-0.246288\pi\)
0.715304 + 0.698813i \(0.246288\pi\)
\(824\) 3.69279 0.128645
\(825\) 15.1991 0.529166
\(826\) −5.50628 −0.191588
\(827\) 6.42848 0.223540 0.111770 0.993734i \(-0.464348\pi\)
0.111770 + 0.993734i \(0.464348\pi\)
\(828\) 0 0
\(829\) 15.8721 0.551262 0.275631 0.961264i \(-0.411113\pi\)
0.275631 + 0.961264i \(0.411113\pi\)
\(830\) −0.432953 −0.0150280
\(831\) 15.3909 0.533905
\(832\) 4.70981 0.163283
\(833\) −6.51269 −0.225651
\(834\) −9.22612 −0.319474
\(835\) −20.2042 −0.699195
\(836\) −3.39574 −0.117444
\(837\) −0.170752 −0.00590207
\(838\) −15.1417 −0.523060
\(839\) 10.4265 0.359963 0.179981 0.983670i \(-0.442396\pi\)
0.179981 + 0.983670i \(0.442396\pi\)
\(840\) −11.0635 −0.381728
\(841\) −28.6538 −0.988061
\(842\) 38.1445 1.31455
\(843\) 3.29958 0.113643
\(844\) −24.4174 −0.840483
\(845\) 35.1045 1.20763
\(846\) 5.45520 0.187554
\(847\) −24.6026 −0.845356
\(848\) −10.3078 −0.353971
\(849\) 25.8826 0.888289
\(850\) −45.5576 −1.56261
\(851\) 0 0
\(852\) 11.8216 0.405001
\(853\) 26.2935 0.900271 0.450135 0.892960i \(-0.351376\pi\)
0.450135 + 0.892960i \(0.351376\pi\)
\(854\) −36.6207 −1.25313
\(855\) 8.21316 0.280884
\(856\) 10.7665 0.367990
\(857\) 32.4336 1.10791 0.553955 0.832547i \(-0.313118\pi\)
0.553955 + 0.832547i \(0.313118\pi\)
\(858\) 7.44454 0.254153
\(859\) −55.1643 −1.88218 −0.941091 0.338154i \(-0.890198\pi\)
−0.941091 + 0.338154i \(0.890198\pi\)
\(860\) 46.9016 1.59933
\(861\) 30.0885 1.02541
\(862\) 10.2138 0.347883
\(863\) 18.2143 0.620023 0.310012 0.950733i \(-0.399667\pi\)
0.310012 + 0.950733i \(0.399667\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 55.0161 1.87060
\(866\) −9.82271 −0.333789
\(867\) −5.44675 −0.184981
\(868\) 0.494139 0.0167722
\(869\) −18.4691 −0.626522
\(870\) 2.24955 0.0762670
\(871\) 27.6752 0.937740
\(872\) −8.31357 −0.281533
\(873\) 0.873537 0.0295648
\(874\) 0 0
\(875\) 51.0668 1.72637
\(876\) 13.2135 0.446444
\(877\) −28.7970 −0.972407 −0.486203 0.873846i \(-0.661619\pi\)
−0.486203 + 0.873846i \(0.661619\pi\)
\(878\) −7.79601 −0.263103
\(879\) −6.52568 −0.220106
\(880\) −6.04290 −0.203706
\(881\) −21.0631 −0.709633 −0.354816 0.934936i \(-0.615457\pi\)
−0.354816 + 0.934936i \(0.615457\pi\)
\(882\) 1.37462 0.0462860
\(883\) −14.1593 −0.476498 −0.238249 0.971204i \(-0.576573\pi\)
−0.238249 + 0.971204i \(0.576573\pi\)
\(884\) −22.3141 −0.750506
\(885\) 7.27423 0.244521
\(886\) −28.2208 −0.948098
\(887\) 57.5302 1.93167 0.965837 0.259150i \(-0.0834424\pi\)
0.965837 + 0.259150i \(0.0834424\pi\)
\(888\) −4.81881 −0.161709
\(889\) −21.9577 −0.736436
\(890\) −46.7890 −1.56837
\(891\) −1.58065 −0.0529536
\(892\) 15.1991 0.508904
\(893\) 11.7195 0.392179
\(894\) −18.4911 −0.618434
\(895\) 42.8185 1.43127
\(896\) 2.89389 0.0966782
\(897\) 0 0
\(898\) −12.7574 −0.425721
\(899\) −0.100474 −0.00335098
\(900\) 9.61578 0.320526
\(901\) 48.8362 1.62697
\(902\) 16.4344 0.547205
\(903\) −35.5025 −1.18145
\(904\) 7.51278 0.249871
\(905\) 33.6226 1.11765
\(906\) 0.991887 0.0329532
\(907\) 45.1699 1.49984 0.749921 0.661528i \(-0.230092\pi\)
0.749921 + 0.661528i \(0.230092\pi\)
\(908\) 12.9325 0.429179
\(909\) −10.9103 −0.361871
\(910\) 52.1071 1.72733
\(911\) 24.4377 0.809656 0.404828 0.914393i \(-0.367331\pi\)
0.404828 + 0.914393i \(0.367331\pi\)
\(912\) −2.14832 −0.0711381
\(913\) 0.179005 0.00592419
\(914\) 9.37038 0.309945
\(915\) 48.3788 1.59935
\(916\) −0.948502 −0.0313394
\(917\) 51.5360 1.70187
\(918\) 4.73780 0.156371
\(919\) −1.40194 −0.0462457 −0.0231229 0.999733i \(-0.507361\pi\)
−0.0231229 + 0.999733i \(0.507361\pi\)
\(920\) 0 0
\(921\) −4.61726 −0.152144
\(922\) −11.6569 −0.383899
\(923\) −55.6775 −1.83265
\(924\) 4.57422 0.150481
\(925\) 46.3366 1.52354
\(926\) 13.6279 0.447840
\(927\) 3.69279 0.121287
\(928\) −0.588417 −0.0193157
\(929\) −8.73879 −0.286710 −0.143355 0.989671i \(-0.545789\pi\)
−0.143355 + 0.989671i \(0.545789\pi\)
\(930\) −0.652796 −0.0214060
\(931\) 2.95313 0.0967851
\(932\) 13.9482 0.456889
\(933\) 10.7400 0.351613
\(934\) 6.57700 0.215206
\(935\) 28.6301 0.936303
\(936\) 4.70981 0.153945
\(937\) 7.95252 0.259798 0.129899 0.991527i \(-0.458535\pi\)
0.129899 + 0.991527i \(0.458535\pi\)
\(938\) 17.0048 0.555225
\(939\) −26.6899 −0.870992
\(940\) 20.8556 0.680233
\(941\) 20.7252 0.675622 0.337811 0.941214i \(-0.390314\pi\)
0.337811 + 0.941214i \(0.390314\pi\)
\(942\) −8.10972 −0.264229
\(943\) 0 0
\(944\) −1.90272 −0.0619284
\(945\) −11.0635 −0.359897
\(946\) −19.3915 −0.630472
\(947\) 54.9184 1.78461 0.892304 0.451435i \(-0.149088\pi\)
0.892304 + 0.451435i \(0.149088\pi\)
\(948\) −11.6845 −0.379496
\(949\) −62.2333 −2.02018
\(950\) 20.6578 0.670227
\(951\) −19.1814 −0.622000
\(952\) −13.7107 −0.444366
\(953\) 8.68947 0.281480 0.140740 0.990047i \(-0.455052\pi\)
0.140740 + 0.990047i \(0.455052\pi\)
\(954\) −10.3078 −0.333727
\(955\) −24.2076 −0.783340
\(956\) −2.44329 −0.0790217
\(957\) −0.930080 −0.0300652
\(958\) 9.24047 0.298546
\(959\) −2.20795 −0.0712984
\(960\) −3.82306 −0.123389
\(961\) −30.9708 −0.999059
\(962\) 22.6957 0.731739
\(963\) 10.7665 0.346944
\(964\) 10.8636 0.349892
\(965\) −11.7515 −0.378295
\(966\) 0 0
\(967\) 39.3485 1.26536 0.632681 0.774412i \(-0.281954\pi\)
0.632681 + 0.774412i \(0.281954\pi\)
\(968\) −8.50156 −0.273250
\(969\) 10.1783 0.326975
\(970\) 3.33958 0.107228
\(971\) −16.1185 −0.517268 −0.258634 0.965975i \(-0.583272\pi\)
−0.258634 + 0.965975i \(0.583272\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 26.6994 0.855944
\(974\) −38.0246 −1.21839
\(975\) −45.2885 −1.45039
\(976\) −12.6545 −0.405060
\(977\) −35.5575 −1.13758 −0.568792 0.822482i \(-0.692589\pi\)
−0.568792 + 0.822482i \(0.692589\pi\)
\(978\) 23.0759 0.737887
\(979\) 19.3449 0.618267
\(980\) 5.25527 0.167873
\(981\) −8.31357 −0.265432
\(982\) −4.28664 −0.136792
\(983\) −54.6156 −1.74197 −0.870984 0.491311i \(-0.836518\pi\)
−0.870984 + 0.491311i \(0.836518\pi\)
\(984\) 10.3972 0.331452
\(985\) −3.68319 −0.117356
\(986\) 2.78780 0.0887818
\(987\) −15.7868 −0.502499
\(988\) 10.1182 0.321903
\(989\) 0 0
\(990\) −6.04290 −0.192056
\(991\) 26.8734 0.853663 0.426832 0.904331i \(-0.359630\pi\)
0.426832 + 0.904331i \(0.359630\pi\)
\(992\) 0.170752 0.00542139
\(993\) −8.74066 −0.277376
\(994\) −34.2105 −1.08509
\(995\) 12.5140 0.396719
\(996\) 0.113248 0.00358839
\(997\) −26.9729 −0.854239 −0.427119 0.904195i \(-0.640472\pi\)
−0.427119 + 0.904195i \(0.640472\pi\)
\(998\) −30.2794 −0.958477
\(999\) −4.81881 −0.152461
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 3174.2.a.ba.1.5 5
3.2 odd 2 9522.2.a.bs.1.1 5
23.3 even 11 138.2.e.b.55.1 10
23.8 even 11 138.2.e.b.133.1 yes 10
23.22 odd 2 3174.2.a.bb.1.1 5
69.8 odd 22 414.2.i.e.271.1 10
69.26 odd 22 414.2.i.e.55.1 10
69.68 even 2 9522.2.a.br.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.b.55.1 10 23.3 even 11
138.2.e.b.133.1 yes 10 23.8 even 11
414.2.i.e.55.1 10 69.26 odd 22
414.2.i.e.271.1 10 69.8 odd 22
3174.2.a.ba.1.5 5 1.1 even 1 trivial
3174.2.a.bb.1.1 5 23.22 odd 2
9522.2.a.br.1.5 5 69.68 even 2
9522.2.a.bs.1.1 5 3.2 odd 2