Properties

Label 1075.2.a.r.1.2
Level $1075$
Weight $2$
Character 1075.1
Self dual yes
Analytic conductor $8.584$
Analytic rank $0$
Dimension $6$
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] = [1075,2,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, 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("1075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.282109865.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 34x^{2} - 12x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.66064\) of defining polynomial
Character \(\chi\) \(=\) 1075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.66064 q^{2} -3.28593 q^{3} +0.757722 q^{4} +5.45675 q^{6} -0.418362 q^{7} +2.06297 q^{8} +7.79735 q^{9} +O(q^{10})\) \(q-1.66064 q^{2} -3.28593 q^{3} +0.757722 q^{4} +5.45675 q^{6} -0.418362 q^{7} +2.06297 q^{8} +7.79735 q^{9} -1.03535 q^{11} -2.48982 q^{12} -1.83145 q^{13} +0.694748 q^{14} -4.94130 q^{16} +5.10985 q^{17} -12.9486 q^{18} +3.10985 q^{19} +1.37471 q^{21} +1.71934 q^{22} -8.80565 q^{23} -6.77879 q^{24} +3.04138 q^{26} -15.7637 q^{27} -0.317002 q^{28} -1.66368 q^{29} -3.86680 q^{31} +4.07977 q^{32} +3.40208 q^{33} -8.48561 q^{34} +5.90822 q^{36} +5.86984 q^{37} -5.16433 q^{38} +6.01803 q^{39} -10.8578 q^{41} -2.28289 q^{42} -1.00000 q^{43} -0.784506 q^{44} +14.6230 q^{46} -0.275355 q^{47} +16.2368 q^{48} -6.82497 q^{49} -16.7906 q^{51} -1.38773 q^{52} +7.35014 q^{53} +26.1779 q^{54} -0.863070 q^{56} -10.2187 q^{57} +2.76277 q^{58} +10.0095 q^{59} -3.84598 q^{61} +6.42136 q^{62} -3.26211 q^{63} +3.10758 q^{64} -5.64963 q^{66} +6.49513 q^{67} +3.87185 q^{68} +28.9348 q^{69} +6.25380 q^{71} +16.0857 q^{72} +2.78582 q^{73} -9.74768 q^{74} +2.35640 q^{76} +0.433149 q^{77} -9.99378 q^{78} -16.3754 q^{79} +28.4066 q^{81} +18.0309 q^{82} +6.13740 q^{83} +1.04165 q^{84} +1.66064 q^{86} +5.46673 q^{87} -2.13589 q^{88} +15.3732 q^{89} +0.766210 q^{91} -6.67224 q^{92} +12.7060 q^{93} +0.457265 q^{94} -13.4058 q^{96} +9.11314 q^{97} +11.3338 q^{98} -8.07296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 2 q^{3} + 11 q^{4} + 11 q^{6} + 2 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + 2 q^{3} + 11 q^{4} + 11 q^{6} + 2 q^{7} + 6 q^{8} + 12 q^{9} - 6 q^{11} - 4 q^{12} + 17 q^{14} + 5 q^{16} + 7 q^{17} - 15 q^{18} - 5 q^{19} + 19 q^{21} + 34 q^{22} - 3 q^{23} + 9 q^{24} + 14 q^{26} + 5 q^{27} - 21 q^{28} + 18 q^{29} - 12 q^{31} + 3 q^{32} - 10 q^{33} + q^{34} - 10 q^{36} + 7 q^{37} - q^{38} - 3 q^{39} - 9 q^{42} - 6 q^{43} + 9 q^{44} - 14 q^{46} - 6 q^{47} + 2 q^{48} - q^{51} + 3 q^{52} - 8 q^{53} + 12 q^{54} + 20 q^{56} - 5 q^{57} - 24 q^{58} + 21 q^{59} - 8 q^{61} + 47 q^{62} + 19 q^{63} + 6 q^{64} + q^{66} - 35 q^{68} + 37 q^{69} + 14 q^{71} + 44 q^{72} - q^{73} + 2 q^{74} - 57 q^{76} + 13 q^{77} - 49 q^{78} - 31 q^{79} + 62 q^{81} + 32 q^{82} - 13 q^{83} + 21 q^{84} - q^{86} + 22 q^{87} + 39 q^{88} + 40 q^{89} + 11 q^{91} - 32 q^{92} - 15 q^{93} + 35 q^{94} - 50 q^{96} + 11 q^{97} - 44 q^{98} - 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.66064 −1.17425 −0.587125 0.809497i \(-0.699740\pi\)
−0.587125 + 0.809497i \(0.699740\pi\)
\(3\) −3.28593 −1.89713 −0.948567 0.316577i \(-0.897466\pi\)
−0.948567 + 0.316577i \(0.897466\pi\)
\(4\) 0.757722 0.378861
\(5\) 0 0
\(6\) 5.45675 2.22771
\(7\) −0.418362 −0.158126 −0.0790629 0.996870i \(-0.525193\pi\)
−0.0790629 + 0.996870i \(0.525193\pi\)
\(8\) 2.06297 0.729372
\(9\) 7.79735 2.59912
\(10\) 0 0
\(11\) −1.03535 −0.312169 −0.156084 0.987744i \(-0.549887\pi\)
−0.156084 + 0.987744i \(0.549887\pi\)
\(12\) −2.48982 −0.718750
\(13\) −1.83145 −0.507954 −0.253977 0.967210i \(-0.581739\pi\)
−0.253977 + 0.967210i \(0.581739\pi\)
\(14\) 0.694748 0.185679
\(15\) 0 0
\(16\) −4.94130 −1.23533
\(17\) 5.10985 1.23932 0.619660 0.784870i \(-0.287270\pi\)
0.619660 + 0.784870i \(0.287270\pi\)
\(18\) −12.9486 −3.05201
\(19\) 3.10985 0.713448 0.356724 0.934210i \(-0.383894\pi\)
0.356724 + 0.934210i \(0.383894\pi\)
\(20\) 0 0
\(21\) 1.37471 0.299986
\(22\) 1.71934 0.366564
\(23\) −8.80565 −1.83611 −0.918053 0.396458i \(-0.870239\pi\)
−0.918053 + 0.396458i \(0.870239\pi\)
\(24\) −6.77879 −1.38372
\(25\) 0 0
\(26\) 3.04138 0.596465
\(27\) −15.7637 −3.03373
\(28\) −0.317002 −0.0599078
\(29\) −1.66368 −0.308937 −0.154469 0.987998i \(-0.549367\pi\)
−0.154469 + 0.987998i \(0.549367\pi\)
\(30\) 0 0
\(31\) −3.86680 −0.694498 −0.347249 0.937773i \(-0.612884\pi\)
−0.347249 + 0.937773i \(0.612884\pi\)
\(32\) 4.07977 0.721208
\(33\) 3.40208 0.592226
\(34\) −8.48561 −1.45527
\(35\) 0 0
\(36\) 5.90822 0.984704
\(37\) 5.86984 0.964996 0.482498 0.875897i \(-0.339730\pi\)
0.482498 + 0.875897i \(0.339730\pi\)
\(38\) −5.16433 −0.837766
\(39\) 6.01803 0.963657
\(40\) 0 0
\(41\) −10.8578 −1.69570 −0.847851 0.530234i \(-0.822104\pi\)
−0.847851 + 0.530234i \(0.822104\pi\)
\(42\) −2.28289 −0.352258
\(43\) −1.00000 −0.152499
\(44\) −0.784506 −0.118269
\(45\) 0 0
\(46\) 14.6230 2.15605
\(47\) −0.275355 −0.0401647 −0.0200823 0.999798i \(-0.506393\pi\)
−0.0200823 + 0.999798i \(0.506393\pi\)
\(48\) 16.2368 2.34358
\(49\) −6.82497 −0.974996
\(50\) 0 0
\(51\) −16.7906 −2.35116
\(52\) −1.38773 −0.192444
\(53\) 7.35014 1.00962 0.504810 0.863230i \(-0.331563\pi\)
0.504810 + 0.863230i \(0.331563\pi\)
\(54\) 26.1779 3.56236
\(55\) 0 0
\(56\) −0.863070 −0.115333
\(57\) −10.2187 −1.35351
\(58\) 2.76277 0.362769
\(59\) 10.0095 1.30313 0.651566 0.758592i \(-0.274112\pi\)
0.651566 + 0.758592i \(0.274112\pi\)
\(60\) 0 0
\(61\) −3.84598 −0.492427 −0.246214 0.969216i \(-0.579186\pi\)
−0.246214 + 0.969216i \(0.579186\pi\)
\(62\) 6.42136 0.815514
\(63\) −3.26211 −0.410987
\(64\) 3.10758 0.388447
\(65\) 0 0
\(66\) −5.64963 −0.695421
\(67\) 6.49513 0.793507 0.396753 0.917925i \(-0.370137\pi\)
0.396753 + 0.917925i \(0.370137\pi\)
\(68\) 3.87185 0.469530
\(69\) 28.9348 3.48334
\(70\) 0 0
\(71\) 6.25380 0.742190 0.371095 0.928595i \(-0.378982\pi\)
0.371095 + 0.928595i \(0.378982\pi\)
\(72\) 16.0857 1.89572
\(73\) 2.78582 0.326056 0.163028 0.986621i \(-0.447874\pi\)
0.163028 + 0.986621i \(0.447874\pi\)
\(74\) −9.74768 −1.13315
\(75\) 0 0
\(76\) 2.35640 0.270298
\(77\) 0.433149 0.0493620
\(78\) −9.99378 −1.13157
\(79\) −16.3754 −1.84238 −0.921190 0.389114i \(-0.872781\pi\)
−0.921190 + 0.389114i \(0.872781\pi\)
\(80\) 0 0
\(81\) 28.4066 3.15628
\(82\) 18.0309 1.99118
\(83\) 6.13740 0.673667 0.336834 0.941564i \(-0.390644\pi\)
0.336834 + 0.941564i \(0.390644\pi\)
\(84\) 1.04165 0.113653
\(85\) 0 0
\(86\) 1.66064 0.179071
\(87\) 5.46673 0.586095
\(88\) −2.13589 −0.227687
\(89\) 15.3732 1.62956 0.814780 0.579770i \(-0.196858\pi\)
0.814780 + 0.579770i \(0.196858\pi\)
\(90\) 0 0
\(91\) 0.766210 0.0803207
\(92\) −6.67224 −0.695629
\(93\) 12.7060 1.31756
\(94\) 0.457265 0.0471633
\(95\) 0 0
\(96\) −13.4058 −1.36823
\(97\) 9.11314 0.925299 0.462649 0.886541i \(-0.346899\pi\)
0.462649 + 0.886541i \(0.346899\pi\)
\(98\) 11.3338 1.14489
\(99\) −8.07296 −0.811363
\(100\) 0 0
\(101\) 13.3938 1.33273 0.666365 0.745626i \(-0.267849\pi\)
0.666365 + 0.745626i \(0.267849\pi\)
\(102\) 27.8831 2.76084
\(103\) −16.1555 −1.59185 −0.795925 0.605396i \(-0.793015\pi\)
−0.795925 + 0.605396i \(0.793015\pi\)
\(104\) −3.77824 −0.370487
\(105\) 0 0
\(106\) −12.2059 −1.18555
\(107\) 1.25504 0.121329 0.0606647 0.998158i \(-0.480678\pi\)
0.0606647 + 0.998158i \(0.480678\pi\)
\(108\) −11.9445 −1.14936
\(109\) −7.35341 −0.704329 −0.352164 0.935938i \(-0.614554\pi\)
−0.352164 + 0.935938i \(0.614554\pi\)
\(110\) 0 0
\(111\) −19.2879 −1.83073
\(112\) 2.06725 0.195337
\(113\) 9.56479 0.899780 0.449890 0.893084i \(-0.351463\pi\)
0.449890 + 0.893084i \(0.351463\pi\)
\(114\) 16.9696 1.58935
\(115\) 0 0
\(116\) −1.26061 −0.117044
\(117\) −14.2805 −1.32023
\(118\) −16.6222 −1.53020
\(119\) −2.13776 −0.195969
\(120\) 0 0
\(121\) −9.92806 −0.902551
\(122\) 6.38678 0.578232
\(123\) 35.6780 3.21697
\(124\) −2.92996 −0.263118
\(125\) 0 0
\(126\) 5.41719 0.482601
\(127\) −7.99151 −0.709132 −0.354566 0.935031i \(-0.615371\pi\)
−0.354566 + 0.935031i \(0.615371\pi\)
\(128\) −13.3201 −1.17734
\(129\) 3.28593 0.289310
\(130\) 0 0
\(131\) 5.86435 0.512371 0.256185 0.966628i \(-0.417534\pi\)
0.256185 + 0.966628i \(0.417534\pi\)
\(132\) 2.57783 0.224371
\(133\) −1.30104 −0.112815
\(134\) −10.7861 −0.931775
\(135\) 0 0
\(136\) 10.5415 0.903925
\(137\) −19.6872 −1.68199 −0.840995 0.541042i \(-0.818030\pi\)
−0.840995 + 0.541042i \(0.818030\pi\)
\(138\) −48.0502 −4.09031
\(139\) −12.1919 −1.03410 −0.517051 0.855955i \(-0.672970\pi\)
−0.517051 + 0.855955i \(0.672970\pi\)
\(140\) 0 0
\(141\) 0.904798 0.0761977
\(142\) −10.3853 −0.871516
\(143\) 1.89619 0.158567
\(144\) −38.5290 −3.21075
\(145\) 0 0
\(146\) −4.62624 −0.382870
\(147\) 22.4264 1.84970
\(148\) 4.44771 0.365599
\(149\) 21.2313 1.73933 0.869666 0.493641i \(-0.164334\pi\)
0.869666 + 0.493641i \(0.164334\pi\)
\(150\) 0 0
\(151\) 23.9612 1.94993 0.974966 0.222355i \(-0.0713743\pi\)
0.974966 + 0.222355i \(0.0713743\pi\)
\(152\) 6.41554 0.520369
\(153\) 39.8432 3.22114
\(154\) −0.719305 −0.0579632
\(155\) 0 0
\(156\) 4.56000 0.365092
\(157\) 4.60922 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(158\) 27.1937 2.16341
\(159\) −24.1521 −1.91538
\(160\) 0 0
\(161\) 3.68395 0.290336
\(162\) −47.1731 −3.70626
\(163\) −8.79982 −0.689255 −0.344628 0.938739i \(-0.611995\pi\)
−0.344628 + 0.938739i \(0.611995\pi\)
\(164\) −8.22719 −0.642436
\(165\) 0 0
\(166\) −10.1920 −0.791053
\(167\) 17.4290 1.34870 0.674348 0.738413i \(-0.264425\pi\)
0.674348 + 0.738413i \(0.264425\pi\)
\(168\) 2.83599 0.218801
\(169\) −9.64578 −0.741983
\(170\) 0 0
\(171\) 24.2486 1.85433
\(172\) −0.757722 −0.0577758
\(173\) 1.56459 0.118953 0.0594767 0.998230i \(-0.481057\pi\)
0.0594767 + 0.998230i \(0.481057\pi\)
\(174\) −9.07826 −0.688221
\(175\) 0 0
\(176\) 5.11596 0.385630
\(177\) −32.8907 −2.47221
\(178\) −25.5294 −1.91351
\(179\) −7.25355 −0.542156 −0.271078 0.962557i \(-0.587380\pi\)
−0.271078 + 0.962557i \(0.587380\pi\)
\(180\) 0 0
\(181\) 16.6732 1.23931 0.619656 0.784873i \(-0.287272\pi\)
0.619656 + 0.784873i \(0.287272\pi\)
\(182\) −1.27240 −0.0943165
\(183\) 12.6376 0.934200
\(184\) −18.1658 −1.33920
\(185\) 0 0
\(186\) −21.1002 −1.54714
\(187\) −5.29046 −0.386877
\(188\) −0.208643 −0.0152168
\(189\) 6.59495 0.479712
\(190\) 0 0
\(191\) 10.5312 0.762014 0.381007 0.924572i \(-0.375577\pi\)
0.381007 + 0.924572i \(0.375577\pi\)
\(192\) −10.2113 −0.736936
\(193\) −3.40684 −0.245229 −0.122615 0.992454i \(-0.539128\pi\)
−0.122615 + 0.992454i \(0.539128\pi\)
\(194\) −15.1336 −1.08653
\(195\) 0 0
\(196\) −5.17144 −0.369388
\(197\) 12.7570 0.908897 0.454449 0.890773i \(-0.349836\pi\)
0.454449 + 0.890773i \(0.349836\pi\)
\(198\) 13.4063 0.952742
\(199\) 17.0212 1.20660 0.603300 0.797515i \(-0.293852\pi\)
0.603300 + 0.797515i \(0.293852\pi\)
\(200\) 0 0
\(201\) −21.3426 −1.50539
\(202\) −22.2422 −1.56496
\(203\) 0.696019 0.0488509
\(204\) −12.7226 −0.890762
\(205\) 0 0
\(206\) 26.8285 1.86923
\(207\) −68.6607 −4.77225
\(208\) 9.04977 0.627488
\(209\) −3.21977 −0.222716
\(210\) 0 0
\(211\) 15.9014 1.09470 0.547348 0.836905i \(-0.315637\pi\)
0.547348 + 0.836905i \(0.315637\pi\)
\(212\) 5.56937 0.382506
\(213\) −20.5496 −1.40803
\(214\) −2.08417 −0.142471
\(215\) 0 0
\(216\) −32.5202 −2.21272
\(217\) 1.61772 0.109818
\(218\) 12.2114 0.827057
\(219\) −9.15401 −0.618571
\(220\) 0 0
\(221\) −9.35845 −0.629518
\(222\) 32.0302 2.14973
\(223\) 6.51336 0.436167 0.218083 0.975930i \(-0.430020\pi\)
0.218083 + 0.975930i \(0.430020\pi\)
\(224\) −1.70682 −0.114042
\(225\) 0 0
\(226\) −15.8837 −1.05657
\(227\) 18.2375 1.21047 0.605234 0.796047i \(-0.293079\pi\)
0.605234 + 0.796047i \(0.293079\pi\)
\(228\) −7.74297 −0.512791
\(229\) −0.680942 −0.0449979 −0.0224989 0.999747i \(-0.507162\pi\)
−0.0224989 + 0.999747i \(0.507162\pi\)
\(230\) 0 0
\(231\) −1.42330 −0.0936462
\(232\) −3.43212 −0.225330
\(233\) −19.9212 −1.30508 −0.652540 0.757754i \(-0.726297\pi\)
−0.652540 + 0.757754i \(0.726297\pi\)
\(234\) 23.7147 1.55028
\(235\) 0 0
\(236\) 7.58446 0.493706
\(237\) 53.8085 3.49524
\(238\) 3.55005 0.230116
\(239\) 14.9202 0.965110 0.482555 0.875866i \(-0.339709\pi\)
0.482555 + 0.875866i \(0.339709\pi\)
\(240\) 0 0
\(241\) −10.8540 −0.699169 −0.349584 0.936905i \(-0.613677\pi\)
−0.349584 + 0.936905i \(0.613677\pi\)
\(242\) 16.4869 1.05982
\(243\) −46.0508 −2.95416
\(244\) −2.91419 −0.186562
\(245\) 0 0
\(246\) −59.2482 −3.77753
\(247\) −5.69554 −0.362399
\(248\) −7.97711 −0.506547
\(249\) −20.1671 −1.27804
\(250\) 0 0
\(251\) −12.1535 −0.767124 −0.383562 0.923515i \(-0.625303\pi\)
−0.383562 + 0.923515i \(0.625303\pi\)
\(252\) −2.47177 −0.155707
\(253\) 9.11691 0.573175
\(254\) 13.2710 0.832698
\(255\) 0 0
\(256\) 15.9047 0.994046
\(257\) −21.4603 −1.33866 −0.669328 0.742967i \(-0.733418\pi\)
−0.669328 + 0.742967i \(0.733418\pi\)
\(258\) −5.45675 −0.339722
\(259\) −2.45572 −0.152591
\(260\) 0 0
\(261\) −12.9723 −0.802963
\(262\) −9.73857 −0.601651
\(263\) −1.04713 −0.0645691 −0.0322846 0.999479i \(-0.510278\pi\)
−0.0322846 + 0.999479i \(0.510278\pi\)
\(264\) 7.01840 0.431953
\(265\) 0 0
\(266\) 2.16056 0.132472
\(267\) −50.5154 −3.09149
\(268\) 4.92151 0.300629
\(269\) 24.6494 1.50290 0.751450 0.659790i \(-0.229355\pi\)
0.751450 + 0.659790i \(0.229355\pi\)
\(270\) 0 0
\(271\) 10.9212 0.663418 0.331709 0.943382i \(-0.392375\pi\)
0.331709 + 0.943382i \(0.392375\pi\)
\(272\) −25.2493 −1.53096
\(273\) −2.51771 −0.152379
\(274\) 32.6933 1.97508
\(275\) 0 0
\(276\) 21.9245 1.31970
\(277\) 17.2987 1.03938 0.519690 0.854355i \(-0.326047\pi\)
0.519690 + 0.854355i \(0.326047\pi\)
\(278\) 20.2463 1.21429
\(279\) −30.1508 −1.80508
\(280\) 0 0
\(281\) 14.0056 0.835504 0.417752 0.908561i \(-0.362818\pi\)
0.417752 + 0.908561i \(0.362818\pi\)
\(282\) −1.50254 −0.0894751
\(283\) −5.90394 −0.350953 −0.175476 0.984484i \(-0.556147\pi\)
−0.175476 + 0.984484i \(0.556147\pi\)
\(284\) 4.73865 0.281187
\(285\) 0 0
\(286\) −3.14889 −0.186198
\(287\) 4.54248 0.268134
\(288\) 31.8114 1.87450
\(289\) 9.11054 0.535914
\(290\) 0 0
\(291\) −29.9451 −1.75542
\(292\) 2.11088 0.123530
\(293\) 2.93453 0.171437 0.0857184 0.996319i \(-0.472681\pi\)
0.0857184 + 0.996319i \(0.472681\pi\)
\(294\) −37.2422 −2.17201
\(295\) 0 0
\(296\) 12.1093 0.703840
\(297\) 16.3209 0.947037
\(298\) −35.2575 −2.04241
\(299\) 16.1272 0.932657
\(300\) 0 0
\(301\) 0.418362 0.0241140
\(302\) −39.7908 −2.28971
\(303\) −44.0110 −2.52837
\(304\) −15.3667 −0.881340
\(305\) 0 0
\(306\) −66.1653 −3.78242
\(307\) 7.24809 0.413670 0.206835 0.978376i \(-0.433684\pi\)
0.206835 + 0.978376i \(0.433684\pi\)
\(308\) 0.328207 0.0187013
\(309\) 53.0859 3.01995
\(310\) 0 0
\(311\) 13.5821 0.770171 0.385085 0.922881i \(-0.374172\pi\)
0.385085 + 0.922881i \(0.374172\pi\)
\(312\) 12.4151 0.702864
\(313\) 2.63634 0.149015 0.0745073 0.997220i \(-0.476262\pi\)
0.0745073 + 0.997220i \(0.476262\pi\)
\(314\) −7.65425 −0.431954
\(315\) 0 0
\(316\) −12.4080 −0.698006
\(317\) 14.4272 0.810312 0.405156 0.914247i \(-0.367217\pi\)
0.405156 + 0.914247i \(0.367217\pi\)
\(318\) 40.1079 2.24914
\(319\) 1.72248 0.0964405
\(320\) 0 0
\(321\) −4.12398 −0.230178
\(322\) −6.11771 −0.340926
\(323\) 15.8908 0.884190
\(324\) 21.5243 1.19579
\(325\) 0 0
\(326\) 14.6133 0.809357
\(327\) 24.1628 1.33621
\(328\) −22.3994 −1.23680
\(329\) 0.115198 0.00635107
\(330\) 0 0
\(331\) 14.3003 0.786016 0.393008 0.919535i \(-0.371434\pi\)
0.393008 + 0.919535i \(0.371434\pi\)
\(332\) 4.65045 0.255226
\(333\) 45.7692 2.50813
\(334\) −28.9433 −1.58371
\(335\) 0 0
\(336\) −6.79285 −0.370580
\(337\) 23.0364 1.25487 0.627436 0.778669i \(-0.284105\pi\)
0.627436 + 0.778669i \(0.284105\pi\)
\(338\) 16.0182 0.871273
\(339\) −31.4293 −1.70700
\(340\) 0 0
\(341\) 4.00348 0.216801
\(342\) −40.2681 −2.17745
\(343\) 5.78384 0.312298
\(344\) −2.06297 −0.111228
\(345\) 0 0
\(346\) −2.59821 −0.139681
\(347\) −1.03484 −0.0555531 −0.0277766 0.999614i \(-0.508843\pi\)
−0.0277766 + 0.999614i \(0.508843\pi\)
\(348\) 4.14226 0.222049
\(349\) 15.0905 0.807777 0.403889 0.914808i \(-0.367658\pi\)
0.403889 + 0.914808i \(0.367658\pi\)
\(350\) 0 0
\(351\) 28.8706 1.54100
\(352\) −4.22398 −0.225139
\(353\) 32.6499 1.73778 0.868888 0.495008i \(-0.164835\pi\)
0.868888 + 0.495008i \(0.164835\pi\)
\(354\) 54.6196 2.90300
\(355\) 0 0
\(356\) 11.6486 0.617377
\(357\) 7.02455 0.371778
\(358\) 12.0455 0.636626
\(359\) −9.63690 −0.508616 −0.254308 0.967123i \(-0.581848\pi\)
−0.254308 + 0.967123i \(0.581848\pi\)
\(360\) 0 0
\(361\) −9.32885 −0.490992
\(362\) −27.6883 −1.45526
\(363\) 32.6229 1.71226
\(364\) 0.580575 0.0304304
\(365\) 0 0
\(366\) −20.9865 −1.09698
\(367\) −6.54305 −0.341544 −0.170772 0.985311i \(-0.554626\pi\)
−0.170772 + 0.985311i \(0.554626\pi\)
\(368\) 43.5114 2.26819
\(369\) −84.6620 −4.40733
\(370\) 0 0
\(371\) −3.07502 −0.159647
\(372\) 9.62765 0.499171
\(373\) 25.3361 1.31185 0.655927 0.754824i \(-0.272278\pi\)
0.655927 + 0.754824i \(0.272278\pi\)
\(374\) 8.78555 0.454290
\(375\) 0 0
\(376\) −0.568051 −0.0292950
\(377\) 3.04695 0.156926
\(378\) −10.9518 −0.563301
\(379\) −9.44773 −0.485297 −0.242649 0.970114i \(-0.578016\pi\)
−0.242649 + 0.970114i \(0.578016\pi\)
\(380\) 0 0
\(381\) 26.2596 1.34532
\(382\) −17.4886 −0.894795
\(383\) 27.7715 1.41906 0.709530 0.704675i \(-0.248907\pi\)
0.709530 + 0.704675i \(0.248907\pi\)
\(384\) 43.7690 2.23357
\(385\) 0 0
\(386\) 5.65753 0.287960
\(387\) −7.79735 −0.396361
\(388\) 6.90523 0.350560
\(389\) 15.8897 0.805638 0.402819 0.915280i \(-0.368030\pi\)
0.402819 + 0.915280i \(0.368030\pi\)
\(390\) 0 0
\(391\) −44.9955 −2.27552
\(392\) −14.0797 −0.711135
\(393\) −19.2699 −0.972036
\(394\) −21.1847 −1.06727
\(395\) 0 0
\(396\) −6.11706 −0.307394
\(397\) −7.27891 −0.365318 −0.182659 0.983176i \(-0.558470\pi\)
−0.182659 + 0.983176i \(0.558470\pi\)
\(398\) −28.2660 −1.41685
\(399\) 4.27513 0.214024
\(400\) 0 0
\(401\) −36.0688 −1.80119 −0.900595 0.434659i \(-0.856869\pi\)
−0.900595 + 0.434659i \(0.856869\pi\)
\(402\) 35.4423 1.76770
\(403\) 7.08187 0.352773
\(404\) 10.1488 0.504920
\(405\) 0 0
\(406\) −1.15584 −0.0573632
\(407\) −6.07732 −0.301242
\(408\) −34.6386 −1.71487
\(409\) −2.84173 −0.140515 −0.0702573 0.997529i \(-0.522382\pi\)
−0.0702573 + 0.997529i \(0.522382\pi\)
\(410\) 0 0
\(411\) 64.6908 3.19096
\(412\) −12.2414 −0.603090
\(413\) −4.18761 −0.206059
\(414\) 114.021 5.60381
\(415\) 0 0
\(416\) −7.47191 −0.366341
\(417\) 40.0617 1.96183
\(418\) 5.34688 0.261524
\(419\) −29.8465 −1.45810 −0.729049 0.684462i \(-0.760037\pi\)
−0.729049 + 0.684462i \(0.760037\pi\)
\(420\) 0 0
\(421\) 28.8518 1.40615 0.703075 0.711116i \(-0.251810\pi\)
0.703075 + 0.711116i \(0.251810\pi\)
\(422\) −26.4064 −1.28545
\(423\) −2.14704 −0.104393
\(424\) 15.1632 0.736388
\(425\) 0 0
\(426\) 34.1254 1.65338
\(427\) 1.60901 0.0778655
\(428\) 0.950973 0.0459670
\(429\) −6.23075 −0.300824
\(430\) 0 0
\(431\) 2.38096 0.114687 0.0573435 0.998355i \(-0.481737\pi\)
0.0573435 + 0.998355i \(0.481737\pi\)
\(432\) 77.8934 3.74765
\(433\) −20.6586 −0.992790 −0.496395 0.868097i \(-0.665343\pi\)
−0.496395 + 0.868097i \(0.665343\pi\)
\(434\) −2.68645 −0.128954
\(435\) 0 0
\(436\) −5.57184 −0.266843
\(437\) −27.3842 −1.30997
\(438\) 15.2015 0.726356
\(439\) 23.2249 1.10846 0.554231 0.832363i \(-0.313012\pi\)
0.554231 + 0.832363i \(0.313012\pi\)
\(440\) 0 0
\(441\) −53.2167 −2.53413
\(442\) 15.5410 0.739211
\(443\) −36.2317 −1.72142 −0.860710 0.509095i \(-0.829980\pi\)
−0.860710 + 0.509095i \(0.829980\pi\)
\(444\) −14.6149 −0.693591
\(445\) 0 0
\(446\) −10.8163 −0.512168
\(447\) −69.7644 −3.29974
\(448\) −1.30009 −0.0614235
\(449\) −31.7118 −1.49657 −0.748286 0.663377i \(-0.769123\pi\)
−0.748286 + 0.663377i \(0.769123\pi\)
\(450\) 0 0
\(451\) 11.2416 0.529345
\(452\) 7.24746 0.340892
\(453\) −78.7347 −3.69928
\(454\) −30.2860 −1.42139
\(455\) 0 0
\(456\) −21.0810 −0.987209
\(457\) 15.4209 0.721358 0.360679 0.932690i \(-0.382545\pi\)
0.360679 + 0.932690i \(0.382545\pi\)
\(458\) 1.13080 0.0528387
\(459\) −80.5504 −3.75977
\(460\) 0 0
\(461\) −2.98422 −0.138989 −0.0694945 0.997582i \(-0.522139\pi\)
−0.0694945 + 0.997582i \(0.522139\pi\)
\(462\) 2.36359 0.109964
\(463\) 23.3271 1.08410 0.542052 0.840345i \(-0.317648\pi\)
0.542052 + 0.840345i \(0.317648\pi\)
\(464\) 8.22073 0.381638
\(465\) 0 0
\(466\) 33.0819 1.53249
\(467\) 12.3447 0.571243 0.285621 0.958343i \(-0.407800\pi\)
0.285621 + 0.958343i \(0.407800\pi\)
\(468\) −10.8206 −0.500184
\(469\) −2.71731 −0.125474
\(470\) 0 0
\(471\) −15.1456 −0.697871
\(472\) 20.6494 0.950467
\(473\) 1.03535 0.0476053
\(474\) −89.3566 −4.10428
\(475\) 0 0
\(476\) −1.61983 −0.0742449
\(477\) 57.3116 2.62412
\(478\) −24.7771 −1.13328
\(479\) 15.9091 0.726903 0.363452 0.931613i \(-0.381598\pi\)
0.363452 + 0.931613i \(0.381598\pi\)
\(480\) 0 0
\(481\) −10.7503 −0.490173
\(482\) 18.0246 0.820999
\(483\) −12.1052 −0.550806
\(484\) −7.52271 −0.341941
\(485\) 0 0
\(486\) 76.4737 3.46892
\(487\) −8.15431 −0.369507 −0.184753 0.982785i \(-0.559149\pi\)
−0.184753 + 0.982785i \(0.559149\pi\)
\(488\) −7.93416 −0.359163
\(489\) 28.9156 1.30761
\(490\) 0 0
\(491\) 37.4997 1.69234 0.846169 0.532915i \(-0.178904\pi\)
0.846169 + 0.532915i \(0.178904\pi\)
\(492\) 27.0340 1.21879
\(493\) −8.50114 −0.382872
\(494\) 9.45824 0.425546
\(495\) 0 0
\(496\) 19.1070 0.857931
\(497\) −2.61635 −0.117359
\(498\) 33.4902 1.50073
\(499\) −16.3210 −0.730629 −0.365315 0.930884i \(-0.619039\pi\)
−0.365315 + 0.930884i \(0.619039\pi\)
\(500\) 0 0
\(501\) −57.2705 −2.55866
\(502\) 20.1826 0.900795
\(503\) 7.34030 0.327288 0.163644 0.986519i \(-0.447675\pi\)
0.163644 + 0.986519i \(0.447675\pi\)
\(504\) −6.72965 −0.299762
\(505\) 0 0
\(506\) −15.1399 −0.673050
\(507\) 31.6954 1.40764
\(508\) −6.05535 −0.268663
\(509\) 27.1292 1.20248 0.601240 0.799069i \(-0.294674\pi\)
0.601240 + 0.799069i \(0.294674\pi\)
\(510\) 0 0
\(511\) −1.16548 −0.0515578
\(512\) 0.228189 0.0100846
\(513\) −49.0229 −2.16441
\(514\) 35.6378 1.57192
\(515\) 0 0
\(516\) 2.48982 0.109608
\(517\) 0.285088 0.0125382
\(518\) 4.07806 0.179180
\(519\) −5.14113 −0.225670
\(520\) 0 0
\(521\) 37.6617 1.64999 0.824995 0.565140i \(-0.191178\pi\)
0.824995 + 0.565140i \(0.191178\pi\)
\(522\) 21.5423 0.942879
\(523\) −18.4289 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(524\) 4.44355 0.194117
\(525\) 0 0
\(526\) 1.73891 0.0758202
\(527\) −19.7588 −0.860705
\(528\) −16.8107 −0.731592
\(529\) 54.5395 2.37128
\(530\) 0 0
\(531\) 78.0479 3.38699
\(532\) −0.985828 −0.0427411
\(533\) 19.8856 0.861339
\(534\) 83.8879 3.63018
\(535\) 0 0
\(536\) 13.3993 0.578761
\(537\) 23.8347 1.02854
\(538\) −40.9338 −1.76478
\(539\) 7.06622 0.304363
\(540\) 0 0
\(541\) −5.53559 −0.237994 −0.118997 0.992895i \(-0.537968\pi\)
−0.118997 + 0.992895i \(0.537968\pi\)
\(542\) −18.1362 −0.779018
\(543\) −54.7872 −2.35114
\(544\) 20.8470 0.893808
\(545\) 0 0
\(546\) 4.18101 0.178931
\(547\) −26.4946 −1.13283 −0.566413 0.824121i \(-0.691669\pi\)
−0.566413 + 0.824121i \(0.691669\pi\)
\(548\) −14.9174 −0.637241
\(549\) −29.9884 −1.27988
\(550\) 0 0
\(551\) −5.17378 −0.220411
\(552\) 59.6917 2.54065
\(553\) 6.85085 0.291328
\(554\) −28.7270 −1.22049
\(555\) 0 0
\(556\) −9.23807 −0.391781
\(557\) −39.4417 −1.67120 −0.835599 0.549340i \(-0.814879\pi\)
−0.835599 + 0.549340i \(0.814879\pi\)
\(558\) 50.0696 2.11961
\(559\) 1.83145 0.0774623
\(560\) 0 0
\(561\) 17.3841 0.733957
\(562\) −23.2583 −0.981090
\(563\) 16.1786 0.681847 0.340923 0.940091i \(-0.389260\pi\)
0.340923 + 0.940091i \(0.389260\pi\)
\(564\) 0.685586 0.0288684
\(565\) 0 0
\(566\) 9.80431 0.412106
\(567\) −11.8842 −0.499090
\(568\) 12.9014 0.541332
\(569\) 24.1001 1.01033 0.505164 0.863023i \(-0.331432\pi\)
0.505164 + 0.863023i \(0.331432\pi\)
\(570\) 0 0
\(571\) 1.47727 0.0618217 0.0309109 0.999522i \(-0.490159\pi\)
0.0309109 + 0.999522i \(0.490159\pi\)
\(572\) 1.43679 0.0600750
\(573\) −34.6050 −1.44564
\(574\) −7.54343 −0.314857
\(575\) 0 0
\(576\) 24.2309 1.00962
\(577\) 28.0608 1.16819 0.584094 0.811686i \(-0.301450\pi\)
0.584094 + 0.811686i \(0.301450\pi\)
\(578\) −15.1293 −0.629297
\(579\) 11.1946 0.465233
\(580\) 0 0
\(581\) −2.56765 −0.106524
\(582\) 49.7281 2.06129
\(583\) −7.60995 −0.315172
\(584\) 5.74708 0.237816
\(585\) 0 0
\(586\) −4.87319 −0.201310
\(587\) −0.883090 −0.0364490 −0.0182245 0.999834i \(-0.505801\pi\)
−0.0182245 + 0.999834i \(0.505801\pi\)
\(588\) 16.9930 0.700779
\(589\) −12.0252 −0.495488
\(590\) 0 0
\(591\) −41.9186 −1.72430
\(592\) −29.0046 −1.19208
\(593\) 1.46230 0.0600494 0.0300247 0.999549i \(-0.490441\pi\)
0.0300247 + 0.999549i \(0.490441\pi\)
\(594\) −27.1032 −1.11206
\(595\) 0 0
\(596\) 16.0874 0.658965
\(597\) −55.9304 −2.28908
\(598\) −26.7814 −1.09517
\(599\) 39.1820 1.60093 0.800467 0.599376i \(-0.204585\pi\)
0.800467 + 0.599376i \(0.204585\pi\)
\(600\) 0 0
\(601\) −12.5220 −0.510782 −0.255391 0.966838i \(-0.582204\pi\)
−0.255391 + 0.966838i \(0.582204\pi\)
\(602\) −0.694748 −0.0283158
\(603\) 50.6448 2.06242
\(604\) 18.1559 0.738754
\(605\) 0 0
\(606\) 73.0864 2.96893
\(607\) 31.4913 1.27819 0.639097 0.769126i \(-0.279308\pi\)
0.639097 + 0.769126i \(0.279308\pi\)
\(608\) 12.6875 0.514544
\(609\) −2.28707 −0.0926767
\(610\) 0 0
\(611\) 0.504300 0.0204018
\(612\) 30.1901 1.22036
\(613\) 8.81861 0.356180 0.178090 0.984014i \(-0.443008\pi\)
0.178090 + 0.984014i \(0.443008\pi\)
\(614\) −12.0365 −0.485752
\(615\) 0 0
\(616\) 0.893576 0.0360032
\(617\) −0.0225984 −0.000909776 0 −0.000454888 1.00000i \(-0.500145\pi\)
−0.000454888 1.00000i \(0.500145\pi\)
\(618\) −88.1565 −3.54618
\(619\) −35.2045 −1.41499 −0.707495 0.706719i \(-0.750175\pi\)
−0.707495 + 0.706719i \(0.750175\pi\)
\(620\) 0 0
\(621\) 138.810 5.57026
\(622\) −22.5550 −0.904372
\(623\) −6.43157 −0.257676
\(624\) −29.7369 −1.19043
\(625\) 0 0
\(626\) −4.37801 −0.174980
\(627\) 10.5799 0.422522
\(628\) 3.49251 0.139366
\(629\) 29.9940 1.19594
\(630\) 0 0
\(631\) −34.6406 −1.37902 −0.689511 0.724275i \(-0.742174\pi\)
−0.689511 + 0.724275i \(0.742174\pi\)
\(632\) −33.7821 −1.34378
\(633\) −52.2508 −2.07678
\(634\) −23.9584 −0.951509
\(635\) 0 0
\(636\) −18.3006 −0.725665
\(637\) 12.4996 0.495253
\(638\) −2.86042 −0.113245
\(639\) 48.7631 1.92904
\(640\) 0 0
\(641\) 5.72453 0.226105 0.113053 0.993589i \(-0.463937\pi\)
0.113053 + 0.993589i \(0.463937\pi\)
\(642\) 6.84844 0.270287
\(643\) −5.48384 −0.216262 −0.108131 0.994137i \(-0.534487\pi\)
−0.108131 + 0.994137i \(0.534487\pi\)
\(644\) 2.79141 0.109997
\(645\) 0 0
\(646\) −26.3890 −1.03826
\(647\) 12.4219 0.488357 0.244178 0.969730i \(-0.421482\pi\)
0.244178 + 0.969730i \(0.421482\pi\)
\(648\) 58.6020 2.30210
\(649\) −10.3634 −0.406797
\(650\) 0 0
\(651\) −5.31572 −0.208340
\(652\) −6.66782 −0.261132
\(653\) −29.0925 −1.13848 −0.569238 0.822173i \(-0.692762\pi\)
−0.569238 + 0.822173i \(0.692762\pi\)
\(654\) −40.1257 −1.56904
\(655\) 0 0
\(656\) 53.6516 2.09474
\(657\) 21.7220 0.847456
\(658\) −0.191302 −0.00745774
\(659\) −8.60292 −0.335122 −0.167561 0.985862i \(-0.553589\pi\)
−0.167561 + 0.985862i \(0.553589\pi\)
\(660\) 0 0
\(661\) −0.394246 −0.0153344 −0.00766719 0.999971i \(-0.502441\pi\)
−0.00766719 + 0.999971i \(0.502441\pi\)
\(662\) −23.7476 −0.922978
\(663\) 30.7512 1.19428
\(664\) 12.6613 0.491354
\(665\) 0 0
\(666\) −76.0061 −2.94518
\(667\) 14.6498 0.567241
\(668\) 13.2063 0.510969
\(669\) −21.4024 −0.827466
\(670\) 0 0
\(671\) 3.98192 0.153720
\(672\) 5.60849 0.216352
\(673\) −18.8806 −0.727794 −0.363897 0.931439i \(-0.618554\pi\)
−0.363897 + 0.931439i \(0.618554\pi\)
\(674\) −38.2551 −1.47353
\(675\) 0 0
\(676\) −7.30882 −0.281108
\(677\) 18.7810 0.721814 0.360907 0.932602i \(-0.382467\pi\)
0.360907 + 0.932602i \(0.382467\pi\)
\(678\) 52.1927 2.00445
\(679\) −3.81259 −0.146314
\(680\) 0 0
\(681\) −59.9273 −2.29642
\(682\) −6.64834 −0.254578
\(683\) 28.7666 1.10072 0.550361 0.834927i \(-0.314490\pi\)
0.550361 + 0.834927i \(0.314490\pi\)
\(684\) 18.3737 0.702535
\(685\) 0 0
\(686\) −9.60487 −0.366716
\(687\) 2.23753 0.0853670
\(688\) 4.94130 0.188385
\(689\) −13.4615 −0.512841
\(690\) 0 0
\(691\) −18.9619 −0.721346 −0.360673 0.932692i \(-0.617453\pi\)
−0.360673 + 0.932692i \(0.617453\pi\)
\(692\) 1.18552 0.0450668
\(693\) 3.37742 0.128297
\(694\) 1.71850 0.0652332
\(695\) 0 0
\(696\) 11.2777 0.427481
\(697\) −55.4817 −2.10152
\(698\) −25.0599 −0.948532
\(699\) 65.4597 2.47591
\(700\) 0 0
\(701\) −17.4331 −0.658438 −0.329219 0.944254i \(-0.606785\pi\)
−0.329219 + 0.944254i \(0.606785\pi\)
\(702\) −47.9436 −1.80952
\(703\) 18.2543 0.688474
\(704\) −3.21742 −0.121261
\(705\) 0 0
\(706\) −54.2196 −2.04058
\(707\) −5.60344 −0.210739
\(708\) −24.9220 −0.936626
\(709\) −23.2515 −0.873228 −0.436614 0.899649i \(-0.643822\pi\)
−0.436614 + 0.899649i \(0.643822\pi\)
\(710\) 0 0
\(711\) −127.685 −4.78856
\(712\) 31.7146 1.18856
\(713\) 34.0497 1.27517
\(714\) −11.6652 −0.436561
\(715\) 0 0
\(716\) −5.49618 −0.205402
\(717\) −49.0269 −1.83094
\(718\) 16.0034 0.597242
\(719\) 19.8428 0.740013 0.370006 0.929029i \(-0.379355\pi\)
0.370006 + 0.929029i \(0.379355\pi\)
\(720\) 0 0
\(721\) 6.75885 0.251713
\(722\) 15.4919 0.576547
\(723\) 35.6656 1.32642
\(724\) 12.6337 0.469528
\(725\) 0 0
\(726\) −54.1749 −2.01062
\(727\) −9.98392 −0.370283 −0.185142 0.982712i \(-0.559274\pi\)
−0.185142 + 0.982712i \(0.559274\pi\)
\(728\) 1.58067 0.0585836
\(729\) 66.1000 2.44815
\(730\) 0 0
\(731\) −5.10985 −0.188995
\(732\) 9.57581 0.353932
\(733\) 5.80810 0.214527 0.107264 0.994231i \(-0.465791\pi\)
0.107264 + 0.994231i \(0.465791\pi\)
\(734\) 10.8656 0.401058
\(735\) 0 0
\(736\) −35.9250 −1.32421
\(737\) −6.72471 −0.247708
\(738\) 140.593 5.17530
\(739\) −24.2801 −0.893157 −0.446579 0.894744i \(-0.647358\pi\)
−0.446579 + 0.894744i \(0.647358\pi\)
\(740\) 0 0
\(741\) 18.7152 0.687519
\(742\) 5.10650 0.187465
\(743\) −42.5948 −1.56265 −0.781326 0.624123i \(-0.785457\pi\)
−0.781326 + 0.624123i \(0.785457\pi\)
\(744\) 26.2122 0.960988
\(745\) 0 0
\(746\) −42.0742 −1.54044
\(747\) 47.8554 1.75094
\(748\) −4.00870 −0.146573
\(749\) −0.525061 −0.0191853
\(750\) 0 0
\(751\) 10.0708 0.367487 0.183744 0.982974i \(-0.441178\pi\)
0.183744 + 0.982974i \(0.441178\pi\)
\(752\) 1.36061 0.0496164
\(753\) 39.9357 1.45534
\(754\) −5.05988 −0.184270
\(755\) 0 0
\(756\) 4.99714 0.181744
\(757\) −28.3498 −1.03039 −0.515195 0.857073i \(-0.672280\pi\)
−0.515195 + 0.857073i \(0.672280\pi\)
\(758\) 15.6893 0.569860
\(759\) −29.9575 −1.08739
\(760\) 0 0
\(761\) −25.1916 −0.913195 −0.456598 0.889673i \(-0.650932\pi\)
−0.456598 + 0.889673i \(0.650932\pi\)
\(762\) −43.6077 −1.57974
\(763\) 3.07638 0.111373
\(764\) 7.97976 0.288698
\(765\) 0 0
\(766\) −46.1185 −1.66633
\(767\) −18.3320 −0.661931
\(768\) −52.2619 −1.88584
\(769\) 48.9390 1.76479 0.882394 0.470512i \(-0.155931\pi\)
0.882394 + 0.470512i \(0.155931\pi\)
\(770\) 0 0
\(771\) 70.5170 2.53961
\(772\) −2.58144 −0.0929079
\(773\) −47.5716 −1.71103 −0.855517 0.517775i \(-0.826760\pi\)
−0.855517 + 0.517775i \(0.826760\pi\)
\(774\) 12.9486 0.465427
\(775\) 0 0
\(776\) 18.8002 0.674887
\(777\) 8.06931 0.289485
\(778\) −26.3870 −0.946020
\(779\) −33.7661 −1.20980
\(780\) 0 0
\(781\) −6.47486 −0.231689
\(782\) 74.7214 2.67203
\(783\) 26.2258 0.937233
\(784\) 33.7243 1.20444
\(785\) 0 0
\(786\) 32.0003 1.14141
\(787\) 43.9966 1.56831 0.784155 0.620564i \(-0.213096\pi\)
0.784155 + 0.620564i \(0.213096\pi\)
\(788\) 9.66625 0.344346
\(789\) 3.44081 0.122496
\(790\) 0 0
\(791\) −4.00154 −0.142278
\(792\) −16.6543 −0.591785
\(793\) 7.04374 0.250130
\(794\) 12.0876 0.428974
\(795\) 0 0
\(796\) 12.8973 0.457134
\(797\) 0.857030 0.0303576 0.0151788 0.999885i \(-0.495168\pi\)
0.0151788 + 0.999885i \(0.495168\pi\)
\(798\) −7.09945 −0.251318
\(799\) −1.40702 −0.0497769
\(800\) 0 0
\(801\) 119.870 4.23541
\(802\) 59.8973 2.11505
\(803\) −2.88429 −0.101784
\(804\) −16.1717 −0.570333
\(805\) 0 0
\(806\) −11.7604 −0.414243
\(807\) −80.9962 −2.85120
\(808\) 27.6310 0.972056
\(809\) −12.4204 −0.436677 −0.218338 0.975873i \(-0.570064\pi\)
−0.218338 + 0.975873i \(0.570064\pi\)
\(810\) 0 0
\(811\) −21.0981 −0.740855 −0.370427 0.928861i \(-0.620789\pi\)
−0.370427 + 0.928861i \(0.620789\pi\)
\(812\) 0.527389 0.0185077
\(813\) −35.8864 −1.25859
\(814\) 10.0922 0.353733
\(815\) 0 0
\(816\) 82.9675 2.90444
\(817\) −3.10985 −0.108800
\(818\) 4.71909 0.164999
\(819\) 5.97441 0.208763
\(820\) 0 0
\(821\) −39.1333 −1.36576 −0.682881 0.730529i \(-0.739273\pi\)
−0.682881 + 0.730529i \(0.739273\pi\)
\(822\) −107.428 −3.74698
\(823\) −17.1258 −0.596966 −0.298483 0.954415i \(-0.596481\pi\)
−0.298483 + 0.954415i \(0.596481\pi\)
\(824\) −33.3284 −1.16105
\(825\) 0 0
\(826\) 6.95411 0.241964
\(827\) 3.68695 0.128208 0.0641039 0.997943i \(-0.479581\pi\)
0.0641039 + 0.997943i \(0.479581\pi\)
\(828\) −52.0258 −1.80802
\(829\) 1.80157 0.0625711 0.0312855 0.999510i \(-0.490040\pi\)
0.0312855 + 0.999510i \(0.490040\pi\)
\(830\) 0 0
\(831\) −56.8425 −1.97184
\(832\) −5.69139 −0.197313
\(833\) −34.8746 −1.20833
\(834\) −66.5280 −2.30368
\(835\) 0 0
\(836\) −2.43969 −0.0843785
\(837\) 60.9553 2.10692
\(838\) 49.5643 1.71217
\(839\) 23.5253 0.812184 0.406092 0.913832i \(-0.366891\pi\)
0.406092 + 0.913832i \(0.366891\pi\)
\(840\) 0 0
\(841\) −26.2322 −0.904558
\(842\) −47.9124 −1.65117
\(843\) −46.0215 −1.58506
\(844\) 12.0488 0.414738
\(845\) 0 0
\(846\) 3.56546 0.122583
\(847\) 4.15352 0.142717
\(848\) −36.3193 −1.24721
\(849\) 19.3999 0.665804
\(850\) 0 0
\(851\) −51.6878 −1.77183
\(852\) −15.5709 −0.533449
\(853\) 34.8867 1.19450 0.597249 0.802056i \(-0.296260\pi\)
0.597249 + 0.802056i \(0.296260\pi\)
\(854\) −2.67199 −0.0914335
\(855\) 0 0
\(856\) 2.58912 0.0884943
\(857\) −3.91889 −0.133867 −0.0669333 0.997757i \(-0.521321\pi\)
−0.0669333 + 0.997757i \(0.521321\pi\)
\(858\) 10.3470 0.353242
\(859\) 39.2696 1.33986 0.669931 0.742424i \(-0.266324\pi\)
0.669931 + 0.742424i \(0.266324\pi\)
\(860\) 0 0
\(861\) −14.9263 −0.508687
\(862\) −3.95392 −0.134671
\(863\) 25.1270 0.855332 0.427666 0.903937i \(-0.359336\pi\)
0.427666 + 0.903937i \(0.359336\pi\)
\(864\) −64.3125 −2.18795
\(865\) 0 0
\(866\) 34.3065 1.16578
\(867\) −29.9366 −1.01670
\(868\) 1.22578 0.0416058
\(869\) 16.9542 0.575133
\(870\) 0 0
\(871\) −11.8955 −0.403065
\(872\) −15.1699 −0.513717
\(873\) 71.0583 2.40496
\(874\) 45.4753 1.53823
\(875\) 0 0
\(876\) −6.93620 −0.234353
\(877\) −27.5451 −0.930130 −0.465065 0.885276i \(-0.653969\pi\)
−0.465065 + 0.885276i \(0.653969\pi\)
\(878\) −38.5681 −1.30161
\(879\) −9.64265 −0.325239
\(880\) 0 0
\(881\) 17.6072 0.593203 0.296602 0.955001i \(-0.404147\pi\)
0.296602 + 0.955001i \(0.404147\pi\)
\(882\) 88.3737 2.97570
\(883\) −44.8242 −1.50845 −0.754227 0.656614i \(-0.771988\pi\)
−0.754227 + 0.656614i \(0.771988\pi\)
\(884\) −7.09111 −0.238500
\(885\) 0 0
\(886\) 60.1678 2.02138
\(887\) −7.46972 −0.250809 −0.125404 0.992106i \(-0.540023\pi\)
−0.125404 + 0.992106i \(0.540023\pi\)
\(888\) −39.7904 −1.33528
\(889\) 3.34334 0.112132
\(890\) 0 0
\(891\) −29.4106 −0.985294
\(892\) 4.93532 0.165247
\(893\) −0.856312 −0.0286554
\(894\) 115.854 3.87472
\(895\) 0 0
\(896\) 5.57262 0.186168
\(897\) −52.9927 −1.76938
\(898\) 52.6618 1.75735
\(899\) 6.43311 0.214556
\(900\) 0 0
\(901\) 37.5581 1.25124
\(902\) −18.6682 −0.621584
\(903\) −1.37471 −0.0457474
\(904\) 19.7319 0.656274
\(905\) 0 0
\(906\) 130.750 4.34388
\(907\) 43.4335 1.44219 0.721093 0.692838i \(-0.243640\pi\)
0.721093 + 0.692838i \(0.243640\pi\)
\(908\) 13.8190 0.458600
\(909\) 104.436 3.46392
\(910\) 0 0
\(911\) 17.1269 0.567440 0.283720 0.958907i \(-0.408431\pi\)
0.283720 + 0.958907i \(0.408431\pi\)
\(912\) 50.4939 1.67202
\(913\) −6.35434 −0.210298
\(914\) −25.6085 −0.847054
\(915\) 0 0
\(916\) −0.515965 −0.0170480
\(917\) −2.45342 −0.0810191
\(918\) 133.765 4.41491
\(919\) −17.0359 −0.561964 −0.280982 0.959713i \(-0.590660\pi\)
−0.280982 + 0.959713i \(0.590660\pi\)
\(920\) 0 0
\(921\) −23.8167 −0.784787
\(922\) 4.95571 0.163208
\(923\) −11.4536 −0.376998
\(924\) −1.07847 −0.0354789
\(925\) 0 0
\(926\) −38.7379 −1.27301
\(927\) −125.970 −4.13740
\(928\) −6.78742 −0.222808
\(929\) −6.23233 −0.204476 −0.102238 0.994760i \(-0.532600\pi\)
−0.102238 + 0.994760i \(0.532600\pi\)
\(930\) 0 0
\(931\) −21.2246 −0.695609
\(932\) −15.0947 −0.494445
\(933\) −44.6299 −1.46112
\(934\) −20.5000 −0.670781
\(935\) 0 0
\(936\) −29.4603 −0.962939
\(937\) 3.88579 0.126943 0.0634717 0.997984i \(-0.479783\pi\)
0.0634717 + 0.997984i \(0.479783\pi\)
\(938\) 4.51248 0.147338
\(939\) −8.66283 −0.282701
\(940\) 0 0
\(941\) 16.7775 0.546932 0.273466 0.961882i \(-0.411830\pi\)
0.273466 + 0.961882i \(0.411830\pi\)
\(942\) 25.1513 0.819475
\(943\) 95.6100 3.11349
\(944\) −49.4602 −1.60979
\(945\) 0 0
\(946\) −1.71934 −0.0559005
\(947\) 3.90586 0.126923 0.0634617 0.997984i \(-0.479786\pi\)
0.0634617 + 0.997984i \(0.479786\pi\)
\(948\) 40.7719 1.32421
\(949\) −5.10210 −0.165621
\(950\) 0 0
\(951\) −47.4068 −1.53727
\(952\) −4.41015 −0.142934
\(953\) −30.0787 −0.974345 −0.487173 0.873306i \(-0.661972\pi\)
−0.487173 + 0.873306i \(0.661972\pi\)
\(954\) −95.1739 −3.08137
\(955\) 0 0
\(956\) 11.3054 0.365643
\(957\) −5.65996 −0.182961
\(958\) −26.4192 −0.853566
\(959\) 8.23637 0.265966
\(960\) 0 0
\(961\) −16.0478 −0.517673
\(962\) 17.8524 0.575586
\(963\) 9.78599 0.315349
\(964\) −8.22434 −0.264888
\(965\) 0 0
\(966\) 20.1024 0.646783
\(967\) 50.4756 1.62319 0.811593 0.584223i \(-0.198601\pi\)
0.811593 + 0.584223i \(0.198601\pi\)
\(968\) −20.4813 −0.658295
\(969\) −52.2162 −1.67743
\(970\) 0 0
\(971\) −51.3772 −1.64877 −0.824387 0.566026i \(-0.808480\pi\)
−0.824387 + 0.566026i \(0.808480\pi\)
\(972\) −34.8937 −1.11922
\(973\) 5.10062 0.163518
\(974\) 13.5414 0.433893
\(975\) 0 0
\(976\) 19.0041 0.608308
\(977\) −7.93563 −0.253883 −0.126942 0.991910i \(-0.540516\pi\)
−0.126942 + 0.991910i \(0.540516\pi\)
\(978\) −48.0184 −1.53546
\(979\) −15.9166 −0.508698
\(980\) 0 0
\(981\) −57.3371 −1.83063
\(982\) −62.2734 −1.98723
\(983\) 9.69159 0.309114 0.154557 0.987984i \(-0.450605\pi\)
0.154557 + 0.987984i \(0.450605\pi\)
\(984\) 73.6027 2.34637
\(985\) 0 0
\(986\) 14.1173 0.449587
\(987\) −0.378533 −0.0120488
\(988\) −4.31564 −0.137299
\(989\) 8.80565 0.280003
\(990\) 0 0
\(991\) −28.9030 −0.918136 −0.459068 0.888401i \(-0.651816\pi\)
−0.459068 + 0.888401i \(0.651816\pi\)
\(992\) −15.7757 −0.500878
\(993\) −46.9898 −1.49118
\(994\) 4.34482 0.137809
\(995\) 0 0
\(996\) −15.2810 −0.484198
\(997\) −12.1891 −0.386034 −0.193017 0.981195i \(-0.561827\pi\)
−0.193017 + 0.981195i \(0.561827\pi\)
\(998\) 27.1033 0.857941
\(999\) −92.5307 −2.92754
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 1075.2.a.r.1.2 yes 6
3.2 odd 2 9675.2.a.cj.1.5 6
5.2 odd 4 1075.2.b.j.474.4 12
5.3 odd 4 1075.2.b.j.474.9 12
5.4 even 2 1075.2.a.q.1.5 6
15.14 odd 2 9675.2.a.ck.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1075.2.a.q.1.5 6 5.4 even 2
1075.2.a.r.1.2 yes 6 1.1 even 1 trivial
1075.2.b.j.474.4 12 5.2 odd 4
1075.2.b.j.474.9 12 5.3 odd 4
9675.2.a.cj.1.5 6 3.2 odd 2
9675.2.a.ck.1.2 6 15.14 odd 2