Properties

Label 1183.2.c.g.337.7
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11667456256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.7
Root \(2.22001i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.g.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.22001i q^{2} +0.549551 q^{3} -2.92843 q^{4} -4.22001i q^{5} +1.22001i q^{6} +1.00000i q^{7} -2.06113i q^{8} -2.69799 q^{9} +O(q^{10})\) \(q+2.22001i q^{2} +0.549551 q^{3} -2.92843 q^{4} -4.22001i q^{5} +1.22001i q^{6} +1.00000i q^{7} -2.06113i q^{8} -2.69799 q^{9} +9.36845 q^{10} +0.549551i q^{11} -1.60932 q^{12} -2.22001 q^{14} -2.31911i q^{15} -1.28114 q^{16} +2.37888 q^{17} -5.98957i q^{18} -3.61068i q^{19} +12.3580i q^{20} +0.549551i q^{21} -1.22001 q^{22} -5.81890 q^{23} -1.13270i q^{24} -12.8085 q^{25} -3.13134 q^{27} -2.92843i q^{28} -3.59889 q^{29} +5.14844 q^{30} -5.14844i q^{31} -6.96640i q^{32} +0.302006i q^{33} +5.28114i q^{34} +4.22001 q^{35} +7.90090 q^{36} +0.329543i q^{37} +8.01574 q^{38} -8.69799 q^{40} -6.29157i q^{41} -1.22001 q^{42} -3.22001 q^{43} -1.60932i q^{44} +11.3856i q^{45} -12.9180i q^{46} -8.20957i q^{47} -0.704052 q^{48} -1.00000 q^{49} -28.4349i q^{50} +1.30732 q^{51} +2.65866 q^{53} -6.95160i q^{54} +2.31911 q^{55} +2.06113 q^{56} -1.98426i q^{57} -7.98957i q^{58} +1.80753i q^{59} +6.79136i q^{60} +0.609325 q^{61} +11.4296 q^{62} -2.69799i q^{63} +12.9032 q^{64} -0.670457 q^{66} +10.3698i q^{67} -6.96640 q^{68} -3.19778 q^{69} +9.36845i q^{70} -11.1978i q^{71} +5.56092i q^{72} -4.90621i q^{73} -0.731589 q^{74} -7.03891 q^{75} +10.5736i q^{76} -0.549551 q^{77} -14.0171 q^{79} +5.40642i q^{80} +6.37315 q^{81} +13.9673 q^{82} -5.73159i q^{83} -1.60932i q^{84} -10.0389i q^{85} -7.14844i q^{86} -1.97777 q^{87} +1.13270 q^{88} +7.46755i q^{89} -25.2760 q^{90} +17.0403 q^{92} -2.82933i q^{93} +18.2253 q^{94} -15.2371 q^{95} -3.82840i q^{96} -6.84070i q^{97} -2.22001i q^{98} -1.48269i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9} + 22 q^{10} - 24 q^{12} + 2 q^{14} + 38 q^{16} + 8 q^{17} + 10 q^{22} + 4 q^{23} - 10 q^{25} - 52 q^{27} + 2 q^{29} + 8 q^{30} + 14 q^{35} + 68 q^{36} + 46 q^{38} - 34 q^{40} + 10 q^{42} - 6 q^{43} + 22 q^{48} - 8 q^{49} - 14 q^{51} + 4 q^{53} - 6 q^{55} - 12 q^{56} + 16 q^{61} + 10 q^{62} - 28 q^{64} + 12 q^{66} - 66 q^{68} + 36 q^{69} + 40 q^{74} + 14 q^{75} - 2 q^{77} - 52 q^{79} + 48 q^{81} + 28 q^{82} + 26 q^{87} - 6 q^{88} - 52 q^{90} + 24 q^{92} + 66 q^{94} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.22001i 1.56978i 0.619633 + 0.784891i \(0.287281\pi\)
−0.619633 + 0.784891i \(0.712719\pi\)
\(3\) 0.549551 0.317283 0.158642 0.987336i \(-0.449289\pi\)
0.158642 + 0.987336i \(0.449289\pi\)
\(4\) −2.92843 −1.46422
\(5\) − 4.22001i − 1.88724i −0.331024 0.943622i \(-0.607394\pi\)
0.331024 0.943622i \(-0.392606\pi\)
\(6\) 1.22001i 0.498066i
\(7\) 1.00000i 0.377964i
\(8\) − 2.06113i − 0.728720i
\(9\) −2.69799 −0.899331
\(10\) 9.36845 2.96256
\(11\) 0.549551i 0.165696i 0.996562 + 0.0828480i \(0.0264016\pi\)
−0.996562 + 0.0828480i \(0.973598\pi\)
\(12\) −1.60932 −0.464572
\(13\) 0 0
\(14\) −2.22001 −0.593322
\(15\) − 2.31911i − 0.598792i
\(16\) −1.28114 −0.320285
\(17\) 2.37888 0.576964 0.288482 0.957485i \(-0.406849\pi\)
0.288482 + 0.957485i \(0.406849\pi\)
\(18\) − 5.98957i − 1.41175i
\(19\) − 3.61068i − 0.828348i −0.910198 0.414174i \(-0.864071\pi\)
0.910198 0.414174i \(-0.135929\pi\)
\(20\) 12.3580i 2.76334i
\(21\) 0.549551i 0.119922i
\(22\) −1.22001 −0.260107
\(23\) −5.81890 −1.21332 −0.606662 0.794960i \(-0.707492\pi\)
−0.606662 + 0.794960i \(0.707492\pi\)
\(24\) − 1.13270i − 0.231211i
\(25\) −12.8085 −2.56169
\(26\) 0 0
\(27\) −3.13134 −0.602626
\(28\) − 2.92843i − 0.553422i
\(29\) −3.59889 −0.668297 −0.334149 0.942520i \(-0.608449\pi\)
−0.334149 + 0.942520i \(0.608449\pi\)
\(30\) 5.14844 0.939973
\(31\) − 5.14844i − 0.924688i −0.886701 0.462344i \(-0.847009\pi\)
0.886701 0.462344i \(-0.152991\pi\)
\(32\) − 6.96640i − 1.23150i
\(33\) 0.302006i 0.0525726i
\(34\) 5.28114i 0.905708i
\(35\) 4.22001 0.713312
\(36\) 7.90090 1.31682
\(37\) 0.329543i 0.0541766i 0.999633 + 0.0270883i \(0.00862353\pi\)
−0.999633 + 0.0270883i \(0.991376\pi\)
\(38\) 8.01574 1.30033
\(39\) 0 0
\(40\) −8.69799 −1.37527
\(41\) − 6.29157i − 0.982579i −0.870996 0.491289i \(-0.836526\pi\)
0.870996 0.491289i \(-0.163474\pi\)
\(42\) −1.22001 −0.188251
\(43\) −3.22001 −0.491047 −0.245523 0.969391i \(-0.578960\pi\)
−0.245523 + 0.969391i \(0.578960\pi\)
\(44\) − 1.60932i − 0.242615i
\(45\) 11.3856i 1.69726i
\(46\) − 12.9180i − 1.90466i
\(47\) − 8.20957i − 1.19749i −0.800940 0.598745i \(-0.795666\pi\)
0.800940 0.598745i \(-0.204334\pi\)
\(48\) −0.704052 −0.101621
\(49\) −1.00000 −0.142857
\(50\) − 28.4349i − 4.02130i
\(51\) 1.30732 0.183061
\(52\) 0 0
\(53\) 2.65866 0.365196 0.182598 0.983188i \(-0.441549\pi\)
0.182598 + 0.983188i \(0.441549\pi\)
\(54\) − 6.95160i − 0.945992i
\(55\) 2.31911 0.312709
\(56\) 2.06113 0.275430
\(57\) − 1.98426i − 0.262821i
\(58\) − 7.98957i − 1.04908i
\(59\) 1.80753i 0.235320i 0.993054 + 0.117660i \(0.0375393\pi\)
−0.993054 + 0.117660i \(0.962461\pi\)
\(60\) 6.79136i 0.876761i
\(61\) 0.609325 0.0780160 0.0390080 0.999239i \(-0.487580\pi\)
0.0390080 + 0.999239i \(0.487580\pi\)
\(62\) 11.4296 1.45156
\(63\) − 2.69799i − 0.339915i
\(64\) 12.9032 1.61290
\(65\) 0 0
\(66\) −0.670457 −0.0825275
\(67\) 10.3698i 1.26687i 0.773794 + 0.633437i \(0.218356\pi\)
−0.773794 + 0.633437i \(0.781644\pi\)
\(68\) −6.96640 −0.844801
\(69\) −3.19778 −0.384968
\(70\) 9.36845i 1.11974i
\(71\) − 11.1978i − 1.32893i −0.747318 0.664466i \(-0.768659\pi\)
0.747318 0.664466i \(-0.231341\pi\)
\(72\) 5.56092i 0.655361i
\(73\) − 4.90621i − 0.574228i −0.957896 0.287114i \(-0.907304\pi\)
0.957896 0.287114i \(-0.0926959\pi\)
\(74\) −0.731589 −0.0850455
\(75\) −7.03891 −0.812783
\(76\) 10.5736i 1.21288i
\(77\) −0.549551 −0.0626272
\(78\) 0 0
\(79\) −14.0171 −1.57705 −0.788524 0.615004i \(-0.789154\pi\)
−0.788524 + 0.615004i \(0.789154\pi\)
\(80\) 5.40642i 0.604456i
\(81\) 6.37315 0.708128
\(82\) 13.9673 1.54243
\(83\) − 5.73159i − 0.629124i −0.949237 0.314562i \(-0.898142\pi\)
0.949237 0.314562i \(-0.101858\pi\)
\(84\) − 1.60932i − 0.175592i
\(85\) − 10.0389i − 1.08887i
\(86\) − 7.14844i − 0.770836i
\(87\) −1.97777 −0.212040
\(88\) 1.13270 0.120746
\(89\) 7.46755i 0.791559i 0.918346 + 0.395779i \(0.129526\pi\)
−0.918346 + 0.395779i \(0.870474\pi\)
\(90\) −25.2760 −2.66433
\(91\) 0 0
\(92\) 17.0403 1.77657
\(93\) − 2.82933i − 0.293388i
\(94\) 18.2253 1.87980
\(95\) −15.2371 −1.56329
\(96\) − 3.82840i − 0.390734i
\(97\) − 6.84070i − 0.694568i −0.937760 0.347284i \(-0.887104\pi\)
0.937760 0.347284i \(-0.112896\pi\)
\(98\) − 2.22001i − 0.224255i
\(99\) − 1.48269i − 0.149015i
\(100\) 37.5088 3.75088
\(101\) −5.75913 −0.573054 −0.286527 0.958072i \(-0.592501\pi\)
−0.286527 + 0.958072i \(0.592501\pi\)
\(102\) 2.90226i 0.287366i
\(103\) −0.571776 −0.0563388 −0.0281694 0.999603i \(-0.508968\pi\)
−0.0281694 + 0.999603i \(0.508968\pi\)
\(104\) 0 0
\(105\) 2.31911 0.226322
\(106\) 5.90226i 0.573278i
\(107\) 4.07157 0.393613 0.196807 0.980442i \(-0.436943\pi\)
0.196807 + 0.980442i \(0.436943\pi\)
\(108\) 9.16992 0.882376
\(109\) 15.3087i 1.46631i 0.680064 + 0.733153i \(0.261952\pi\)
−0.680064 + 0.733153i \(0.738048\pi\)
\(110\) 5.14844i 0.490885i
\(111\) 0.181101i 0.0171893i
\(112\) − 1.28114i − 0.121056i
\(113\) 12.1769 1.14551 0.572754 0.819727i \(-0.305875\pi\)
0.572754 + 0.819727i \(0.305875\pi\)
\(114\) 4.40506 0.412572
\(115\) 24.5558i 2.28984i
\(116\) 10.5391 0.978533
\(117\) 0 0
\(118\) −4.01273 −0.369402
\(119\) 2.37888i 0.218072i
\(120\) −4.77999 −0.436352
\(121\) 10.6980 0.972545
\(122\) 1.35271i 0.122468i
\(123\) − 3.45754i − 0.311756i
\(124\) 15.0769i 1.35394i
\(125\) 32.9518i 2.94730i
\(126\) 5.98957 0.533593
\(127\) −1.96067 −0.173981 −0.0869907 0.996209i \(-0.527725\pi\)
−0.0869907 + 0.996209i \(0.527725\pi\)
\(128\) 14.7124i 1.30040i
\(129\) −1.76956 −0.155801
\(130\) 0 0
\(131\) −6.50021 −0.567926 −0.283963 0.958835i \(-0.591649\pi\)
−0.283963 + 0.958835i \(0.591649\pi\)
\(132\) − 0.884406i − 0.0769777i
\(133\) 3.61068 0.313086
\(134\) −23.0211 −1.98872
\(135\) 13.2143i 1.13730i
\(136\) − 4.90319i − 0.420445i
\(137\) 15.2576i 1.30354i 0.758416 + 0.651770i \(0.225973\pi\)
−0.758416 + 0.651770i \(0.774027\pi\)
\(138\) − 7.09910i − 0.604316i
\(139\) −17.4960 −1.48399 −0.741997 0.670403i \(-0.766121\pi\)
−0.741997 + 0.670403i \(0.766121\pi\)
\(140\) −12.3580 −1.04444
\(141\) − 4.51158i − 0.379944i
\(142\) 24.8592 2.08613
\(143\) 0 0
\(144\) 3.45651 0.288042
\(145\) 15.1873i 1.26124i
\(146\) 10.8918 0.901414
\(147\) −0.549551 −0.0453262
\(148\) − 0.965046i − 0.0793263i
\(149\) 4.55486i 0.373149i 0.982441 + 0.186574i \(0.0597386\pi\)
−0.982441 + 0.186574i \(0.940261\pi\)
\(150\) − 15.6264i − 1.27589i
\(151\) 6.32912i 0.515057i 0.966271 + 0.257528i \(0.0829081\pi\)
−0.966271 + 0.257528i \(0.917092\pi\)
\(152\) −7.44210 −0.603634
\(153\) −6.41821 −0.518882
\(154\) − 1.22001i − 0.0983110i
\(155\) −21.7265 −1.74511
\(156\) 0 0
\(157\) 16.3100 1.30168 0.650841 0.759214i \(-0.274416\pi\)
0.650841 + 0.759214i \(0.274416\pi\)
\(158\) − 31.1181i − 2.47562i
\(159\) 1.46107 0.115871
\(160\) −29.3983 −2.32414
\(161\) − 5.81890i − 0.458593i
\(162\) 14.1484i 1.11161i
\(163\) − 23.5998i − 1.84848i −0.381811 0.924241i \(-0.624699\pi\)
0.381811 0.924241i \(-0.375301\pi\)
\(164\) 18.4245i 1.43871i
\(165\) 1.27447 0.0992173
\(166\) 12.7242 0.987587
\(167\) 17.8303i 1.37975i 0.723930 + 0.689874i \(0.242334\pi\)
−0.723930 + 0.689874i \(0.757666\pi\)
\(168\) 1.13270 0.0873895
\(169\) 0 0
\(170\) 22.2865 1.70929
\(171\) 9.74160i 0.744959i
\(172\) 9.42958 0.718999
\(173\) 7.57135 0.575639 0.287820 0.957685i \(-0.407070\pi\)
0.287820 + 0.957685i \(0.407070\pi\)
\(174\) − 4.39068i − 0.332856i
\(175\) − 12.8085i − 0.968229i
\(176\) − 0.704052i − 0.0530699i
\(177\) 0.993330i 0.0746632i
\(178\) −16.5780 −1.24258
\(179\) 22.8033 1.70440 0.852201 0.523215i \(-0.175267\pi\)
0.852201 + 0.523215i \(0.175267\pi\)
\(180\) − 33.3419i − 2.48515i
\(181\) −13.9294 −1.03536 −0.517681 0.855574i \(-0.673205\pi\)
−0.517681 + 0.855574i \(0.673205\pi\)
\(182\) 0 0
\(183\) 0.334855 0.0247532
\(184\) 11.9935i 0.884174i
\(185\) 1.39068 0.102244
\(186\) 6.28114 0.460556
\(187\) 1.30732i 0.0956006i
\(188\) 24.0412i 1.75338i
\(189\) − 3.13134i − 0.227771i
\(190\) − 33.8265i − 2.45403i
\(191\) −12.6718 −0.916900 −0.458450 0.888720i \(-0.651595\pi\)
−0.458450 + 0.888720i \(0.651595\pi\)
\(192\) 7.09096 0.511746
\(193\) 4.15492i 0.299078i 0.988756 + 0.149539i \(0.0477789\pi\)
−0.988756 + 0.149539i \(0.952221\pi\)
\(194\) 15.1864 1.09032
\(195\) 0 0
\(196\) 2.92843 0.209174
\(197\) − 6.85020i − 0.488056i −0.969768 0.244028i \(-0.921531\pi\)
0.969768 0.244028i \(-0.0784690\pi\)
\(198\) 3.29157 0.233922
\(199\) 0.813587 0.0576737 0.0288368 0.999584i \(-0.490820\pi\)
0.0288368 + 0.999584i \(0.490820\pi\)
\(200\) 26.3999i 1.86676i
\(201\) 5.69874i 0.401958i
\(202\) − 12.7853i − 0.899571i
\(203\) − 3.59889i − 0.252593i
\(204\) −3.82840 −0.268041
\(205\) −26.5505 −1.85437
\(206\) − 1.26935i − 0.0884397i
\(207\) 15.6994 1.09118
\(208\) 0 0
\(209\) 1.98426 0.137254
\(210\) 5.14844i 0.355276i
\(211\) −13.9734 −0.961969 −0.480984 0.876729i \(-0.659721\pi\)
−0.480984 + 0.876729i \(0.659721\pi\)
\(212\) −7.78573 −0.534726
\(213\) − 6.15375i − 0.421648i
\(214\) 9.03891i 0.617887i
\(215\) 13.5885i 0.926725i
\(216\) 6.45410i 0.439146i
\(217\) 5.14844 0.349499
\(218\) −33.9854 −2.30178
\(219\) − 2.69621i − 0.182193i
\(220\) −6.79136 −0.457874
\(221\) 0 0
\(222\) −0.402045 −0.0269835
\(223\) − 13.5340i − 0.906303i −0.891433 0.453152i \(-0.850300\pi\)
0.891433 0.453152i \(-0.149700\pi\)
\(224\) 6.96640 0.465462
\(225\) 34.5572 2.30381
\(226\) 27.0328i 1.79820i
\(227\) − 5.36751i − 0.356254i −0.984007 0.178127i \(-0.942996\pi\)
0.984007 0.178127i \(-0.0570038\pi\)
\(228\) 5.81076i 0.384827i
\(229\) − 3.09910i − 0.204794i −0.994744 0.102397i \(-0.967349\pi\)
0.994744 0.102397i \(-0.0326513\pi\)
\(230\) −54.5141 −3.59455
\(231\) −0.302006 −0.0198706
\(232\) 7.41779i 0.487002i
\(233\) 20.3712 1.33456 0.667280 0.744807i \(-0.267459\pi\)
0.667280 + 0.744807i \(0.267459\pi\)
\(234\) 0 0
\(235\) −34.6445 −2.25996
\(236\) − 5.29323i − 0.344560i
\(237\) −7.70312 −0.500371
\(238\) −5.28114 −0.342325
\(239\) − 1.29157i − 0.0835449i −0.999127 0.0417725i \(-0.986700\pi\)
0.999127 0.0417725i \(-0.0133005\pi\)
\(240\) 2.97110i 0.191784i
\(241\) − 2.13270i − 0.137379i −0.997638 0.0686896i \(-0.978118\pi\)
0.997638 0.0686896i \(-0.0218818\pi\)
\(242\) 23.7496i 1.52668i
\(243\) 12.8964 0.827304
\(244\) −1.78437 −0.114232
\(245\) 4.22001i 0.269606i
\(246\) 7.67577 0.489389
\(247\) 0 0
\(248\) −10.6116 −0.673839
\(249\) − 3.14980i − 0.199611i
\(250\) −73.1532 −4.62662
\(251\) 30.7711 1.94226 0.971128 0.238561i \(-0.0766757\pi\)
0.971128 + 0.238561i \(0.0766757\pi\)
\(252\) 7.90090i 0.497710i
\(253\) − 3.19778i − 0.201043i
\(254\) − 4.35271i − 0.273113i
\(255\) − 5.51689i − 0.345481i
\(256\) −6.85521 −0.428451
\(257\) 1.47361 0.0919213 0.0459607 0.998943i \(-0.485365\pi\)
0.0459607 + 0.998943i \(0.485365\pi\)
\(258\) − 3.92843i − 0.244574i
\(259\) −0.329543 −0.0204768
\(260\) 0 0
\(261\) 9.70979 0.601021
\(262\) − 14.4305i − 0.891520i
\(263\) 6.67694 0.411718 0.205859 0.978582i \(-0.434001\pi\)
0.205859 + 0.978582i \(0.434001\pi\)
\(264\) 0.622475 0.0383107
\(265\) − 11.2196i − 0.689214i
\(266\) 8.01574i 0.491477i
\(267\) 4.10380i 0.251149i
\(268\) − 30.3673i − 1.85498i
\(269\) 7.57573 0.461900 0.230950 0.972966i \(-0.425817\pi\)
0.230950 + 0.972966i \(0.425817\pi\)
\(270\) −29.3358 −1.78532
\(271\) − 20.5680i − 1.24942i −0.780858 0.624709i \(-0.785218\pi\)
0.780858 0.624709i \(-0.214782\pi\)
\(272\) −3.04768 −0.184793
\(273\) 0 0
\(274\) −33.8719 −2.04628
\(275\) − 7.03891i − 0.424462i
\(276\) 9.36450 0.563676
\(277\) −5.70541 −0.342805 −0.171402 0.985201i \(-0.554830\pi\)
−0.171402 + 0.985201i \(0.554830\pi\)
\(278\) − 38.8413i − 2.32955i
\(279\) 13.8905i 0.831600i
\(280\) − 8.69799i − 0.519805i
\(281\) 6.37315i 0.380190i 0.981766 + 0.190095i \(0.0608796\pi\)
−0.981766 + 0.190095i \(0.939120\pi\)
\(282\) 10.0157 0.596429
\(283\) −27.0194 −1.60614 −0.803068 0.595887i \(-0.796801\pi\)
−0.803068 + 0.595887i \(0.796801\pi\)
\(284\) 32.7920i 1.94585i
\(285\) −8.37357 −0.496008
\(286\) 0 0
\(287\) 6.29157 0.371380
\(288\) 18.7953i 1.10752i
\(289\) −11.3409 −0.667113
\(290\) −33.7160 −1.97987
\(291\) − 3.75932i − 0.220375i
\(292\) 14.3675i 0.840795i
\(293\) 4.87472i 0.284784i 0.989810 + 0.142392i \(0.0454794\pi\)
−0.989810 + 0.142392i \(0.954521\pi\)
\(294\) − 1.22001i − 0.0711523i
\(295\) 7.62779 0.444107
\(296\) 0.679232 0.0394796
\(297\) − 1.72083i − 0.0998527i
\(298\) −10.1118 −0.585763
\(299\) 0 0
\(300\) 20.6130 1.19009
\(301\) − 3.22001i − 0.185598i
\(302\) −14.0507 −0.808527
\(303\) −3.16493 −0.181821
\(304\) 4.62579i 0.265307i
\(305\) − 2.57135i − 0.147235i
\(306\) − 14.2485i − 0.814531i
\(307\) − 16.1760i − 0.923212i −0.887085 0.461606i \(-0.847273\pi\)
0.887085 0.461606i \(-0.152727\pi\)
\(308\) 1.60932 0.0916998
\(309\) −0.314220 −0.0178754
\(310\) − 48.2329i − 2.73945i
\(311\) 1.30806 0.0741735 0.0370868 0.999312i \(-0.488192\pi\)
0.0370868 + 0.999312i \(0.488192\pi\)
\(312\) 0 0
\(313\) 13.1978 0.745983 0.372991 0.927835i \(-0.378332\pi\)
0.372991 + 0.927835i \(0.378332\pi\)
\(314\) 36.2084i 2.04336i
\(315\) −11.3856 −0.641503
\(316\) 41.0482 2.30914
\(317\) − 8.07552i − 0.453566i −0.973945 0.226783i \(-0.927179\pi\)
0.973945 0.226783i \(-0.0728209\pi\)
\(318\) 3.24359i 0.181892i
\(319\) − 1.97777i − 0.110734i
\(320\) − 54.4516i − 3.04394i
\(321\) 2.23753 0.124887
\(322\) 12.9180 0.719892
\(323\) − 8.58939i − 0.477927i
\(324\) −18.6634 −1.03685
\(325\) 0 0
\(326\) 52.3918 2.90171
\(327\) 8.41290i 0.465234i
\(328\) −12.9678 −0.716025
\(329\) 8.20957 0.452608
\(330\) 2.82933i 0.155750i
\(331\) − 14.9451i − 0.821458i −0.911757 0.410729i \(-0.865274\pi\)
0.911757 0.410729i \(-0.134726\pi\)
\(332\) 16.7846i 0.921174i
\(333\) − 0.889106i − 0.0487227i
\(334\) −39.5833 −2.16590
\(335\) 43.7607 2.39090
\(336\) − 0.704052i − 0.0384092i
\(337\) 17.1695 0.935282 0.467641 0.883918i \(-0.345104\pi\)
0.467641 + 0.883918i \(0.345104\pi\)
\(338\) 0 0
\(339\) 6.69184 0.363451
\(340\) 29.3983i 1.59435i
\(341\) 2.82933 0.153217
\(342\) −21.6264 −1.16942
\(343\) − 1.00000i − 0.0539949i
\(344\) 6.63686i 0.357836i
\(345\) 13.4947i 0.726528i
\(346\) 16.8085i 0.903629i
\(347\) −3.93845 −0.211427 −0.105713 0.994397i \(-0.533713\pi\)
−0.105713 + 0.994397i \(0.533713\pi\)
\(348\) 5.79178 0.310472
\(349\) − 17.1777i − 0.919499i −0.888049 0.459750i \(-0.847939\pi\)
0.888049 0.459750i \(-0.152061\pi\)
\(350\) 28.4349 1.51991
\(351\) 0 0
\(352\) 3.82840 0.204054
\(353\) − 18.1964i − 0.968498i −0.874930 0.484249i \(-0.839093\pi\)
0.874930 0.484249i \(-0.160907\pi\)
\(354\) −2.20520 −0.117205
\(355\) −47.2547 −2.50802
\(356\) − 21.8682i − 1.15901i
\(357\) 1.30732i 0.0691906i
\(358\) 50.6236i 2.67554i
\(359\) − 16.3126i − 0.860948i −0.902603 0.430474i \(-0.858346\pi\)
0.902603 0.430474i \(-0.141654\pi\)
\(360\) 23.4671 1.23683
\(361\) 5.96297 0.313840
\(362\) − 30.9233i − 1.62529i
\(363\) 5.87909 0.308572
\(364\) 0 0
\(365\) −20.7042 −1.08371
\(366\) 0.743381i 0.0388571i
\(367\) −36.1963 −1.88943 −0.944716 0.327889i \(-0.893663\pi\)
−0.944716 + 0.327889i \(0.893663\pi\)
\(368\) 7.45482 0.388610
\(369\) 16.9746i 0.883664i
\(370\) 3.08731i 0.160502i
\(371\) 2.65866i 0.138031i
\(372\) 8.28551i 0.429584i
\(373\) 9.79784 0.507313 0.253657 0.967294i \(-0.418367\pi\)
0.253657 + 0.967294i \(0.418367\pi\)
\(374\) −2.90226 −0.150072
\(375\) 18.1087i 0.935129i
\(376\) −16.9210 −0.872635
\(377\) 0 0
\(378\) 6.95160 0.357552
\(379\) 13.0655i 0.671130i 0.942017 + 0.335565i \(0.108927\pi\)
−0.942017 + 0.335565i \(0.891073\pi\)
\(380\) 44.6209 2.28900
\(381\) −1.07749 −0.0552014
\(382\) − 28.1315i − 1.43933i
\(383\) 27.7929i 1.42015i 0.704125 + 0.710076i \(0.251339\pi\)
−0.704125 + 0.710076i \(0.748661\pi\)
\(384\) 8.08520i 0.412596i
\(385\) 2.31911i 0.118193i
\(386\) −9.22396 −0.469487
\(387\) 8.68756 0.441614
\(388\) 20.0325i 1.01700i
\(389\) −13.7047 −0.694854 −0.347427 0.937707i \(-0.612945\pi\)
−0.347427 + 0.937707i \(0.612945\pi\)
\(390\) 0 0
\(391\) −13.8425 −0.700044
\(392\) 2.06113i 0.104103i
\(393\) −3.57220 −0.180194
\(394\) 15.2075 0.766143
\(395\) 59.1523i 2.97627i
\(396\) 4.34195i 0.218191i
\(397\) − 7.91194i − 0.397089i −0.980092 0.198545i \(-0.936379\pi\)
0.980092 0.198545i \(-0.0636214\pi\)
\(398\) 1.80617i 0.0905351i
\(399\) 1.98426 0.0993370
\(400\) 16.4094 0.820472
\(401\) 16.5442i 0.826180i 0.910690 + 0.413090i \(0.135550\pi\)
−0.910690 + 0.413090i \(0.864450\pi\)
\(402\) −12.6512 −0.630987
\(403\) 0 0
\(404\) 16.8652 0.839076
\(405\) − 26.8947i − 1.33641i
\(406\) 7.98957 0.396516
\(407\) −0.181101 −0.00897684
\(408\) − 2.69456i − 0.133400i
\(409\) 25.7819i 1.27483i 0.770520 + 0.637416i \(0.219997\pi\)
−0.770520 + 0.637416i \(0.780003\pi\)
\(410\) − 58.9423i − 2.91095i
\(411\) 8.38481i 0.413592i
\(412\) 1.67441 0.0824922
\(413\) −1.80753 −0.0889427
\(414\) 34.8527i 1.71292i
\(415\) −24.1873 −1.18731
\(416\) 0 0
\(417\) −9.61496 −0.470847
\(418\) 4.40506i 0.215459i
\(419\) 23.6871 1.15719 0.578596 0.815614i \(-0.303601\pi\)
0.578596 + 0.815614i \(0.303601\pi\)
\(420\) −6.79136 −0.331385
\(421\) − 20.8246i − 1.01493i −0.861672 0.507465i \(-0.830583\pi\)
0.861672 0.507465i \(-0.169417\pi\)
\(422\) − 31.0211i − 1.51008i
\(423\) 22.1494i 1.07694i
\(424\) − 5.47986i − 0.266125i
\(425\) −30.4698 −1.47800
\(426\) 13.6614 0.661896
\(427\) 0.609325i 0.0294873i
\(428\) −11.9233 −0.576335
\(429\) 0 0
\(430\) −30.1665 −1.45476
\(431\) 19.9504i 0.960978i 0.877001 + 0.480489i \(0.159541\pi\)
−0.877001 + 0.480489i \(0.840459\pi\)
\(432\) 4.01168 0.193012
\(433\) −0.0166817 −0.000801670 0 −0.000400835 1.00000i \(-0.500128\pi\)
−0.000400835 1.00000i \(0.500128\pi\)
\(434\) 11.4296i 0.548638i
\(435\) 8.34623i 0.400171i
\(436\) − 44.8305i − 2.14699i
\(437\) 21.0102i 1.00505i
\(438\) 5.98561 0.286004
\(439\) −13.4960 −0.644130 −0.322065 0.946718i \(-0.604377\pi\)
−0.322065 + 0.946718i \(0.604377\pi\)
\(440\) − 4.77999i − 0.227877i
\(441\) 2.69799 0.128476
\(442\) 0 0
\(443\) −15.0110 −0.713196 −0.356598 0.934258i \(-0.616063\pi\)
−0.356598 + 0.934258i \(0.616063\pi\)
\(444\) − 0.530342i − 0.0251689i
\(445\) 31.5131 1.49387
\(446\) 30.0456 1.42270
\(447\) 2.50313i 0.118394i
\(448\) 12.9032i 0.609619i
\(449\) 23.7836i 1.12242i 0.827674 + 0.561210i \(0.189664\pi\)
−0.827674 + 0.561210i \(0.810336\pi\)
\(450\) 76.7172i 3.61648i
\(451\) 3.45754 0.162809
\(452\) −35.6593 −1.67727
\(453\) 3.47818i 0.163419i
\(454\) 11.9159 0.559242
\(455\) 0 0
\(456\) −4.08981 −0.191523
\(457\) 18.1313i 0.848148i 0.905627 + 0.424074i \(0.139400\pi\)
−0.905627 + 0.424074i \(0.860600\pi\)
\(458\) 6.88003 0.321483
\(459\) −7.44909 −0.347694
\(460\) − 71.9101i − 3.35282i
\(461\) − 6.07959i − 0.283155i −0.989927 0.141577i \(-0.954783\pi\)
0.989927 0.141577i \(-0.0452174\pi\)
\(462\) − 0.670457i − 0.0311925i
\(463\) − 5.19289i − 0.241334i −0.992693 0.120667i \(-0.961497\pi\)
0.992693 0.120667i \(-0.0385034\pi\)
\(464\) 4.61068 0.214046
\(465\) −11.9398 −0.553695
\(466\) 45.2242i 2.09497i
\(467\) 8.69968 0.402573 0.201287 0.979532i \(-0.435488\pi\)
0.201287 + 0.979532i \(0.435488\pi\)
\(468\) 0 0
\(469\) −10.3698 −0.478833
\(470\) − 76.9110i − 3.54764i
\(471\) 8.96320 0.413002
\(472\) 3.72556 0.171483
\(473\) − 1.76956i − 0.0813644i
\(474\) − 17.1010i − 0.785474i
\(475\) 46.2473i 2.12197i
\(476\) − 6.96640i − 0.319305i
\(477\) −7.17306 −0.328432
\(478\) 2.86730 0.131147
\(479\) − 24.2188i − 1.10659i −0.832986 0.553294i \(-0.813371\pi\)
0.832986 0.553294i \(-0.186629\pi\)
\(480\) −16.1559 −0.737411
\(481\) 0 0
\(482\) 4.73461 0.215655
\(483\) − 3.19778i − 0.145504i
\(484\) −31.3284 −1.42402
\(485\) −28.8678 −1.31082
\(486\) 28.6301i 1.29869i
\(487\) 1.77393i 0.0803846i 0.999192 + 0.0401923i \(0.0127971\pi\)
−0.999192 + 0.0401923i \(0.987203\pi\)
\(488\) − 1.25590i − 0.0568519i
\(489\) − 12.9693i − 0.586493i
\(490\) −9.36845 −0.423223
\(491\) 6.68967 0.301900 0.150950 0.988541i \(-0.451767\pi\)
0.150950 + 0.988541i \(0.451767\pi\)
\(492\) 10.1252i 0.456479i
\(493\) −8.56134 −0.385584
\(494\) 0 0
\(495\) −6.25694 −0.281229
\(496\) 6.59588i 0.296164i
\(497\) 11.1978 0.502289
\(498\) 6.99258 0.313345
\(499\) 24.6387i 1.10298i 0.834181 + 0.551491i \(0.185941\pi\)
−0.834181 + 0.551491i \(0.814059\pi\)
\(500\) − 96.4972i − 4.31548i
\(501\) 9.79864i 0.437771i
\(502\) 68.3121i 3.04892i
\(503\) 33.1452 1.47787 0.738936 0.673776i \(-0.235329\pi\)
0.738936 + 0.673776i \(0.235329\pi\)
\(504\) −5.56092 −0.247703
\(505\) 24.3036i 1.08149i
\(506\) 7.09910 0.315594
\(507\) 0 0
\(508\) 5.74170 0.254747
\(509\) − 27.6580i − 1.22592i −0.790114 0.612961i \(-0.789978\pi\)
0.790114 0.612961i \(-0.210022\pi\)
\(510\) 12.2475 0.542330
\(511\) 4.90621 0.217038
\(512\) 14.2061i 0.627828i
\(513\) 11.3063i 0.499184i
\(514\) 3.27143i 0.144296i
\(515\) 2.41290i 0.106325i
\(516\) 5.18204 0.228126
\(517\) 4.51158 0.198419
\(518\) − 0.731589i − 0.0321442i
\(519\) 4.16085 0.182641
\(520\) 0 0
\(521\) 1.42217 0.0623062 0.0311531 0.999515i \(-0.490082\pi\)
0.0311531 + 0.999515i \(0.490082\pi\)
\(522\) 21.5558i 0.943472i
\(523\) −3.36178 −0.147000 −0.0735002 0.997295i \(-0.523417\pi\)
−0.0735002 + 0.997295i \(0.523417\pi\)
\(524\) 19.0354 0.831567
\(525\) − 7.03891i − 0.307203i
\(526\) 14.8229i 0.646307i
\(527\) − 12.2475i − 0.533511i
\(528\) − 0.386913i − 0.0168382i
\(529\) 10.8596 0.472156
\(530\) 24.9076 1.08192
\(531\) − 4.87670i − 0.211631i
\(532\) −10.5736 −0.458426
\(533\) 0 0
\(534\) −9.11047 −0.394249
\(535\) − 17.1820i − 0.742844i
\(536\) 21.3735 0.923197
\(537\) 12.5316 0.540779
\(538\) 16.8182i 0.725083i
\(539\) − 0.549551i − 0.0236708i
\(540\) − 38.6971i − 1.66526i
\(541\) − 7.76289i − 0.333753i −0.985978 0.166876i \(-0.946632\pi\)
0.985978 0.166876i \(-0.0533681\pi\)
\(542\) 45.6612 1.96131
\(543\) −7.65490 −0.328503
\(544\) − 16.5723i − 0.710530i
\(545\) 64.6027 2.76728
\(546\) 0 0
\(547\) −6.19247 −0.264771 −0.132385 0.991198i \(-0.542264\pi\)
−0.132385 + 0.991198i \(0.542264\pi\)
\(548\) − 44.6808i − 1.90867i
\(549\) −1.64395 −0.0701622
\(550\) 15.6264 0.666313
\(551\) 12.9945i 0.553582i
\(552\) 6.59105i 0.280534i
\(553\) − 14.0171i − 0.596068i
\(554\) − 12.6661i − 0.538129i
\(555\) 0.764247 0.0324405
\(556\) 51.2360 2.17289
\(557\) − 29.5458i − 1.25190i −0.779865 0.625948i \(-0.784712\pi\)
0.779865 0.625948i \(-0.215288\pi\)
\(558\) −30.8369 −1.30543
\(559\) 0 0
\(560\) −5.40642 −0.228463
\(561\) 0.718438i 0.0303325i
\(562\) −14.1484 −0.596816
\(563\) −6.46736 −0.272567 −0.136283 0.990670i \(-0.543516\pi\)
−0.136283 + 0.990670i \(0.543516\pi\)
\(564\) 13.2119i 0.556320i
\(565\) − 51.3867i − 2.16185i
\(566\) − 59.9833i − 2.52129i
\(567\) 6.37315i 0.267647i
\(568\) −23.0801 −0.968420
\(569\) −21.6956 −0.909526 −0.454763 0.890612i \(-0.650276\pi\)
−0.454763 + 0.890612i \(0.650276\pi\)
\(570\) − 18.5894i − 0.778624i
\(571\) 16.6418 0.696436 0.348218 0.937414i \(-0.386787\pi\)
0.348218 + 0.937414i \(0.386787\pi\)
\(572\) 0 0
\(573\) −6.96381 −0.290917
\(574\) 13.9673i 0.582986i
\(575\) 74.5312 3.10816
\(576\) −34.8127 −1.45053
\(577\) − 2.64240i − 0.110005i −0.998486 0.0550024i \(-0.982483\pi\)
0.998486 0.0550024i \(-0.0175166\pi\)
\(578\) − 25.1769i − 1.04722i
\(579\) 2.28334i 0.0948925i
\(580\) − 44.4752i − 1.84673i
\(581\) 5.73159 0.237786
\(582\) 8.34571 0.345941
\(583\) 1.46107i 0.0605114i
\(584\) −10.1123 −0.418452
\(585\) 0 0
\(586\) −10.8219 −0.447049
\(587\) 7.38815i 0.304942i 0.988308 + 0.152471i \(0.0487230\pi\)
−0.988308 + 0.152471i \(0.951277\pi\)
\(588\) 1.60932 0.0663674
\(589\) −18.5894 −0.765963
\(590\) 16.9337i 0.697151i
\(591\) − 3.76453i − 0.154852i
\(592\) − 0.422191i − 0.0173519i
\(593\) − 46.9030i − 1.92607i −0.269370 0.963037i \(-0.586815\pi\)
0.269370 0.963037i \(-0.413185\pi\)
\(594\) 3.82026 0.156747
\(595\) 10.0389 0.411555
\(596\) − 13.3386i − 0.546371i
\(597\) 0.447108 0.0182989
\(598\) 0 0
\(599\) 1.62290 0.0663098 0.0331549 0.999450i \(-0.489445\pi\)
0.0331549 + 0.999450i \(0.489445\pi\)
\(600\) 14.5081i 0.592291i
\(601\) −47.0347 −1.91859 −0.959293 0.282412i \(-0.908865\pi\)
−0.959293 + 0.282412i \(0.908865\pi\)
\(602\) 7.14844 0.291349
\(603\) − 27.9777i − 1.13934i
\(604\) − 18.5344i − 0.754155i
\(605\) − 45.1456i − 1.83543i
\(606\) − 7.02618i − 0.285419i
\(607\) −28.3869 −1.15219 −0.576095 0.817383i \(-0.695424\pi\)
−0.576095 + 0.817383i \(0.695424\pi\)
\(608\) −25.1535 −1.02011
\(609\) − 1.97777i − 0.0801435i
\(610\) 5.70843 0.231127
\(611\) 0 0
\(612\) 18.7953 0.759756
\(613\) − 47.5564i − 1.92079i −0.278650 0.960393i \(-0.589887\pi\)
0.278650 0.960393i \(-0.410113\pi\)
\(614\) 35.9108 1.44924
\(615\) −14.5909 −0.588360
\(616\) 1.13270i 0.0456377i
\(617\) − 16.4868i − 0.663732i −0.943327 0.331866i \(-0.892322\pi\)
0.943327 0.331866i \(-0.107678\pi\)
\(618\) − 0.697572i − 0.0280604i
\(619\) 31.9412i 1.28382i 0.766778 + 0.641912i \(0.221859\pi\)
−0.766778 + 0.641912i \(0.778141\pi\)
\(620\) 63.6245 2.55522
\(621\) 18.2209 0.731181
\(622\) 2.90391i 0.116436i
\(623\) −7.46755 −0.299181
\(624\) 0 0
\(625\) 75.0145 3.00058
\(626\) 29.2992i 1.17103i
\(627\) 1.09045 0.0435484
\(628\) −47.7629 −1.90595
\(629\) 0.783945i 0.0312579i
\(630\) − 25.2760i − 1.00702i
\(631\) − 13.1915i − 0.525147i −0.964912 0.262573i \(-0.915429\pi\)
0.964912 0.262573i \(-0.0845712\pi\)
\(632\) 28.8911i 1.14923i
\(633\) −7.67910 −0.305217
\(634\) 17.9277 0.712000
\(635\) 8.27405i 0.328346i
\(636\) −4.27865 −0.169660
\(637\) 0 0
\(638\) 4.39068 0.173829
\(639\) 30.2115i 1.19515i
\(640\) 62.0864 2.45418
\(641\) 47.1627 1.86282 0.931408 0.363978i \(-0.118582\pi\)
0.931408 + 0.363978i \(0.118582\pi\)
\(642\) 4.96734i 0.196045i
\(643\) − 2.81359i − 0.110957i −0.998460 0.0554785i \(-0.982332\pi\)
0.998460 0.0554785i \(-0.0176684\pi\)
\(644\) 17.0403i 0.671481i
\(645\) 7.46755i 0.294035i
\(646\) 19.0685 0.750241
\(647\) −25.9783 −1.02131 −0.510656 0.859785i \(-0.670597\pi\)
−0.510656 + 0.859785i \(0.670597\pi\)
\(648\) − 13.1359i − 0.516027i
\(649\) −0.993330 −0.0389916
\(650\) 0 0
\(651\) 2.82933 0.110890
\(652\) 69.1106i 2.70658i
\(653\) −26.8426 −1.05043 −0.525216 0.850969i \(-0.676015\pi\)
−0.525216 + 0.850969i \(0.676015\pi\)
\(654\) −18.6767 −0.730317
\(655\) 27.4309i 1.07182i
\(656\) 8.06039i 0.314705i
\(657\) 13.2369i 0.516421i
\(658\) 18.2253i 0.710497i
\(659\) 15.5733 0.606651 0.303325 0.952887i \(-0.401903\pi\)
0.303325 + 0.952887i \(0.401903\pi\)
\(660\) −3.73220 −0.145276
\(661\) − 33.3804i − 1.29835i −0.760640 0.649174i \(-0.775115\pi\)
0.760640 0.649174i \(-0.224885\pi\)
\(662\) 33.1783 1.28951
\(663\) 0 0
\(664\) −11.8136 −0.458455
\(665\) − 15.2371i − 0.590870i
\(666\) 1.97382 0.0764840
\(667\) 20.9416 0.810861
\(668\) − 52.2148i − 2.02025i
\(669\) − 7.43762i − 0.287555i
\(670\) 97.1490i 3.75319i
\(671\) 0.334855i 0.0129269i
\(672\) 3.82840 0.147684
\(673\) −0.854152 −0.0329251 −0.0164626 0.999864i \(-0.505240\pi\)
−0.0164626 + 0.999864i \(0.505240\pi\)
\(674\) 38.1164i 1.46819i
\(675\) 40.1076 1.54374
\(676\) 0 0
\(677\) −24.5449 −0.943339 −0.471669 0.881775i \(-0.656348\pi\)
−0.471669 + 0.881775i \(0.656348\pi\)
\(678\) 14.8559i 0.570539i
\(679\) 6.84070 0.262522
\(680\) −20.6915 −0.793483
\(681\) − 2.94972i − 0.113034i
\(682\) 6.28114i 0.240517i
\(683\) 43.0372i 1.64677i 0.567480 + 0.823387i \(0.307918\pi\)
−0.567480 + 0.823387i \(0.692082\pi\)
\(684\) − 28.5276i − 1.09078i
\(685\) 64.3870 2.46010
\(686\) 2.22001 0.0847603
\(687\) − 1.70312i − 0.0649779i
\(688\) 4.12528 0.157275
\(689\) 0 0
\(690\) −29.9583 −1.14049
\(691\) − 25.8195i − 0.982220i −0.871097 0.491110i \(-0.836591\pi\)
0.871097 0.491110i \(-0.163409\pi\)
\(692\) −22.1722 −0.842861
\(693\) 1.48269 0.0563226
\(694\) − 8.74338i − 0.331894i
\(695\) 73.8334i 2.80066i
\(696\) 4.07646i 0.154518i
\(697\) − 14.9669i − 0.566913i
\(698\) 38.1345 1.44341
\(699\) 11.1950 0.423434
\(700\) 37.5088i 1.41770i
\(701\) −16.3178 −0.616313 −0.308156 0.951336i \(-0.599712\pi\)
−0.308156 + 0.951336i \(0.599712\pi\)
\(702\) 0 0
\(703\) 1.18988 0.0448770
\(704\) 7.09096i 0.267251i
\(705\) −19.0389 −0.717047
\(706\) 40.3962 1.52033
\(707\) − 5.75913i − 0.216594i
\(708\) − 2.90890i − 0.109323i
\(709\) 22.3794i 0.840476i 0.907414 + 0.420238i \(0.138053\pi\)
−0.907414 + 0.420238i \(0.861947\pi\)
\(710\) − 104.906i − 3.93705i
\(711\) 37.8181 1.41829
\(712\) 15.3916 0.576825
\(713\) 29.9583i 1.12195i
\(714\) −2.90226 −0.108614
\(715\) 0 0
\(716\) −66.7781 −2.49561
\(717\) − 0.709785i − 0.0265074i
\(718\) 36.2142 1.35150
\(719\) 22.7445 0.848227 0.424113 0.905609i \(-0.360586\pi\)
0.424113 + 0.905609i \(0.360586\pi\)
\(720\) − 14.5865i − 0.543606i
\(721\) − 0.571776i − 0.0212941i
\(722\) 13.2378i 0.492661i
\(723\) − 1.17203i − 0.0435881i
\(724\) 40.7913 1.51599
\(725\) 46.0963 1.71197
\(726\) 13.0516i 0.484392i
\(727\) −18.7274 −0.694561 −0.347280 0.937761i \(-0.612895\pi\)
−0.347280 + 0.937761i \(0.612895\pi\)
\(728\) 0 0
\(729\) −12.0322 −0.445638
\(730\) − 45.9636i − 1.70119i
\(731\) −7.66002 −0.283316
\(732\) −0.980601 −0.0362441
\(733\) − 1.69268i − 0.0625206i −0.999511 0.0312603i \(-0.990048\pi\)
0.999511 0.0312603i \(-0.00995209\pi\)
\(734\) − 80.3561i − 2.96600i
\(735\) 2.31911i 0.0855417i
\(736\) 40.5368i 1.49421i
\(737\) −5.69874 −0.209916
\(738\) −37.6838 −1.38716
\(739\) − 46.9161i − 1.72584i −0.505343 0.862919i \(-0.668634\pi\)
0.505343 0.862919i \(-0.331366\pi\)
\(740\) −4.07250 −0.149708
\(741\) 0 0
\(742\) −5.90226 −0.216679
\(743\) 12.8966i 0.473131i 0.971616 + 0.236566i \(0.0760218\pi\)
−0.971616 + 0.236566i \(0.923978\pi\)
\(744\) −5.83163 −0.213798
\(745\) 19.2216 0.704223
\(746\) 21.7513i 0.796371i
\(747\) 15.4638i 0.565790i
\(748\) − 3.82840i − 0.139980i
\(749\) 4.07157i 0.148772i
\(750\) −40.2014 −1.46795
\(751\) 45.6333 1.66518 0.832591 0.553888i \(-0.186856\pi\)
0.832591 + 0.553888i \(0.186856\pi\)
\(752\) 10.5176i 0.383538i
\(753\) 16.9103 0.616245
\(754\) 0 0
\(755\) 26.7089 0.972038
\(756\) 9.16992i 0.333507i
\(757\) −38.1565 −1.38682 −0.693410 0.720543i \(-0.743893\pi\)
−0.693410 + 0.720543i \(0.743893\pi\)
\(758\) −29.0055 −1.05353
\(759\) − 1.75735i − 0.0637876i
\(760\) 31.4057i 1.13920i
\(761\) − 42.7345i − 1.54912i −0.632498 0.774562i \(-0.717970\pi\)
0.632498 0.774562i \(-0.282030\pi\)
\(762\) − 2.39203i − 0.0866543i
\(763\) −15.3087 −0.554211
\(764\) 37.1086 1.34254
\(765\) 27.0849i 0.979257i
\(766\) −61.7005 −2.22933
\(767\) 0 0
\(768\) −3.76729 −0.135940
\(769\) 21.6177i 0.779553i 0.920909 + 0.389777i \(0.127448\pi\)
−0.920909 + 0.389777i \(0.872552\pi\)
\(770\) −5.14844 −0.185537
\(771\) 0.809824 0.0291651
\(772\) − 12.1674i − 0.437915i
\(773\) 10.0011i 0.359716i 0.983693 + 0.179858i \(0.0575638\pi\)
−0.983693 + 0.179858i \(0.942436\pi\)
\(774\) 19.2865i 0.693237i
\(775\) 65.9436i 2.36877i
\(776\) −14.0996 −0.506146
\(777\) −0.181101 −0.00649696
\(778\) − 30.4245i − 1.09077i
\(779\) −22.7169 −0.813917
\(780\) 0 0
\(781\) 6.15375 0.220199
\(782\) − 30.7304i − 1.09892i
\(783\) 11.2693 0.402734
\(784\) 1.28114 0.0457550
\(785\) − 68.8285i − 2.45659i
\(786\) − 7.93031i − 0.282865i
\(787\) − 41.7878i − 1.48957i −0.667302 0.744787i \(-0.732551\pi\)
0.667302 0.744787i \(-0.267449\pi\)
\(788\) 20.0604i 0.714621i
\(789\) 3.66932 0.130631
\(790\) −131.319 −4.67210
\(791\) 12.1769i 0.432961i
\(792\) −3.05601 −0.108591
\(793\) 0 0
\(794\) 17.5646 0.623343
\(795\) − 6.16574i − 0.218676i
\(796\) −2.38254 −0.0844468
\(797\) 22.7711 0.806594 0.403297 0.915069i \(-0.367864\pi\)
0.403297 + 0.915069i \(0.367864\pi\)
\(798\) 4.40506i 0.155937i
\(799\) − 19.5296i − 0.690908i
\(800\) 89.2290i 3.15472i
\(801\) − 20.1474i − 0.711874i
\(802\) −36.7283 −1.29692
\(803\) 2.69621 0.0951473
\(804\) − 16.6884i − 0.588554i
\(805\) −24.5558 −0.865478
\(806\) 0 0
\(807\) 4.16325 0.146553
\(808\) 11.8703i 0.417596i
\(809\) −37.5702 −1.32090 −0.660449 0.750871i \(-0.729634\pi\)
−0.660449 + 0.750871i \(0.729634\pi\)
\(810\) 59.7065 2.09787
\(811\) 11.5936i 0.407106i 0.979064 + 0.203553i \(0.0652489\pi\)
−0.979064 + 0.203553i \(0.934751\pi\)
\(812\) 10.5391i 0.369851i
\(813\) − 11.3032i − 0.396420i
\(814\) − 0.402045i − 0.0140917i
\(815\) −99.5915 −3.48854
\(816\) −1.67486 −0.0586317
\(817\) 11.6264i 0.406757i
\(818\) −57.2359 −2.00121
\(819\) 0 0
\(820\) 77.7514 2.71520
\(821\) 31.0243i 1.08276i 0.840780 + 0.541378i \(0.182097\pi\)
−0.840780 + 0.541378i \(0.817903\pi\)
\(822\) −18.6143 −0.649250
\(823\) −29.0775 −1.01358 −0.506789 0.862070i \(-0.669168\pi\)
−0.506789 + 0.862070i \(0.669168\pi\)
\(824\) 1.17851i 0.0410552i
\(825\) − 3.86824i − 0.134675i
\(826\) − 4.01273i − 0.139621i
\(827\) − 14.8920i − 0.517846i −0.965898 0.258923i \(-0.916632\pi\)
0.965898 0.258923i \(-0.0833676\pi\)
\(828\) −45.9745 −1.59773
\(829\) 4.37189 0.151842 0.0759210 0.997114i \(-0.475810\pi\)
0.0759210 + 0.997114i \(0.475810\pi\)
\(830\) − 53.6961i − 1.86382i
\(831\) −3.13541 −0.108766
\(832\) 0 0
\(833\) −2.37888 −0.0824234
\(834\) − 21.3453i − 0.739127i
\(835\) 75.2439 2.60392
\(836\) −5.81076 −0.200969
\(837\) 16.1215i 0.557241i
\(838\) 52.5856i 1.81654i
\(839\) − 22.8218i − 0.787896i −0.919133 0.393948i \(-0.871109\pi\)
0.919133 0.393948i \(-0.128891\pi\)
\(840\) − 4.77999i − 0.164925i
\(841\) −16.0480 −0.553379
\(842\) 46.2308 1.59322
\(843\) 3.50237i 0.120628i
\(844\) 40.9202 1.40853
\(845\) 0 0
\(846\) −49.1718 −1.69056
\(847\) 10.6980i 0.367587i
\(848\) −3.40612 −0.116967
\(849\) −14.8485 −0.509601
\(850\) − 67.6433i − 2.32015i
\(851\) − 1.91758i − 0.0657338i
\(852\) 18.0209i 0.617385i
\(853\) − 23.3549i − 0.799656i −0.916590 0.399828i \(-0.869070\pi\)
0.916590 0.399828i \(-0.130930\pi\)
\(854\) −1.35271 −0.0462886
\(855\) 41.1096 1.40592
\(856\) − 8.39203i − 0.286834i
\(857\) 43.5306 1.48698 0.743488 0.668750i \(-0.233170\pi\)
0.743488 + 0.668750i \(0.233170\pi\)
\(858\) 0 0
\(859\) 20.5113 0.699838 0.349919 0.936780i \(-0.386209\pi\)
0.349919 + 0.936780i \(0.386209\pi\)
\(860\) − 39.7929i − 1.35693i
\(861\) 3.45754 0.117833
\(862\) −44.2901 −1.50853
\(863\) − 50.6678i − 1.72475i −0.506268 0.862376i \(-0.668975\pi\)
0.506268 0.862376i \(-0.331025\pi\)
\(864\) 21.8142i 0.742133i
\(865\) − 31.9512i − 1.08637i
\(866\) − 0.0370334i − 0.00125845i
\(867\) −6.23241 −0.211664
\(868\) −15.0769 −0.511743
\(869\) − 7.70312i − 0.261310i
\(870\) −18.5287 −0.628181
\(871\) 0 0
\(872\) 31.5532 1.06853
\(873\) 18.4562i 0.624647i
\(874\) −46.6428 −1.57772
\(875\) −32.9518 −1.11397
\(876\) 7.89568i 0.266770i
\(877\) − 47.0361i − 1.58830i −0.607725 0.794148i \(-0.707918\pi\)
0.607725 0.794148i \(-0.292082\pi\)
\(878\) − 29.9613i − 1.01114i
\(879\) 2.67891i 0.0903573i
\(880\) −2.97110 −0.100156
\(881\) 16.1135 0.542877 0.271439 0.962456i \(-0.412501\pi\)
0.271439 + 0.962456i \(0.412501\pi\)
\(882\) 5.98957i 0.201679i
\(883\) −42.0733 −1.41588 −0.707940 0.706273i \(-0.750375\pi\)
−0.707940 + 0.706273i \(0.750375\pi\)
\(884\) 0 0
\(885\) 4.19186 0.140908
\(886\) − 33.3246i − 1.11956i
\(887\) −41.7628 −1.40226 −0.701128 0.713035i \(-0.747320\pi\)
−0.701128 + 0.713035i \(0.747320\pi\)
\(888\) 0.373273 0.0125262
\(889\) − 1.96067i − 0.0657588i
\(890\) 69.9594i 2.34504i
\(891\) 3.50237i 0.117334i
\(892\) 39.6334i 1.32703i
\(893\) −29.6422 −0.991938
\(894\) −5.55697 −0.185853
\(895\) − 96.2303i − 3.21662i
\(896\) −14.7124 −0.491506
\(897\) 0 0
\(898\) −52.7999 −1.76195
\(899\) 18.5287i 0.617966i
\(900\) −101.198 −3.37328
\(901\) 6.32465 0.210705
\(902\) 7.67577i 0.255575i
\(903\) − 1.76956i − 0.0588872i
\(904\) − 25.0982i − 0.834755i
\(905\) 58.7821i 1.95398i
\(906\) −7.72158 −0.256532
\(907\) 15.4225 0.512095 0.256048 0.966664i \(-0.417580\pi\)
0.256048 + 0.966664i \(0.417580\pi\)
\(908\) 15.7184i 0.521634i
\(909\) 15.5381 0.515366
\(910\) 0 0
\(911\) 37.5462 1.24396 0.621981 0.783033i \(-0.286328\pi\)
0.621981 + 0.783033i \(0.286328\pi\)
\(912\) 2.54211i 0.0841776i
\(913\) 3.14980 0.104243
\(914\) −40.2517 −1.33141
\(915\) − 1.41309i − 0.0467153i
\(916\) 9.07552i 0.299864i
\(917\) − 6.50021i − 0.214656i
\(918\) − 16.5370i − 0.545804i
\(919\) −9.47464 −0.312540 −0.156270 0.987714i \(-0.549947\pi\)
−0.156270 + 0.987714i \(0.549947\pi\)
\(920\) 50.6127 1.66865
\(921\) − 8.88953i − 0.292920i
\(922\) 13.4967 0.444492
\(923\) 0 0
\(924\) 0.884406 0.0290948
\(925\) − 4.22094i − 0.138784i
\(926\) 11.5283 0.378842
\(927\) 1.54265 0.0506672
\(928\) 25.0713i 0.823007i
\(929\) 35.8439i 1.17600i 0.808860 + 0.588001i \(0.200085\pi\)
−0.808860 + 0.588001i \(0.799915\pi\)
\(930\) − 26.5065i − 0.869181i
\(931\) 3.61068i 0.118335i
\(932\) −59.6556 −1.95409
\(933\) 0.718848 0.0235340
\(934\) 19.3134i 0.631952i
\(935\) 5.51689 0.180422
\(936\) 0 0
\(937\) −31.3709 −1.02484 −0.512422 0.858734i \(-0.671252\pi\)
−0.512422 + 0.858734i \(0.671252\pi\)
\(938\) − 23.0211i − 0.751664i
\(939\) 7.25286 0.236688
\(940\) 101.454 3.30907
\(941\) 44.7844i 1.45993i 0.683486 + 0.729964i \(0.260463\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(942\) 19.8984i 0.648324i
\(943\) 36.6100i 1.19219i
\(944\) − 2.31570i − 0.0753695i
\(945\) −13.2143 −0.429860
\(946\) 3.92843 0.127724
\(947\) 35.0674i 1.13954i 0.821805 + 0.569768i \(0.192967\pi\)
−0.821805 + 0.569768i \(0.807033\pi\)
\(948\) 22.5581 0.732652
\(949\) 0 0
\(950\) −102.669 −3.33104
\(951\) − 4.43791i − 0.143909i
\(952\) 4.90319 0.158913
\(953\) −58.9704 −1.91024 −0.955120 0.296219i \(-0.904274\pi\)
−0.955120 + 0.296219i \(0.904274\pi\)
\(954\) − 15.9242i − 0.515567i
\(955\) 53.4752i 1.73042i
\(956\) 3.78229i 0.122328i
\(957\) − 1.08689i − 0.0351341i
\(958\) 53.7660 1.73710
\(959\) −15.2576 −0.492692
\(960\) − 29.9239i − 0.965791i
\(961\) 4.49354 0.144953
\(962\) 0 0
\(963\) −10.9851 −0.353989
\(964\) 6.24547i 0.201153i
\(965\) 17.5338 0.564433
\(966\) 7.09910 0.228410
\(967\) 30.3671i 0.976540i 0.872693 + 0.488270i \(0.162372\pi\)
−0.872693 + 0.488270i \(0.837628\pi\)
\(968\) − 22.0500i − 0.708713i
\(969\) − 4.72031i − 0.151638i
\(970\) − 64.0868i − 2.05770i
\(971\) −49.5175 −1.58909 −0.794546 0.607204i \(-0.792291\pi\)
−0.794546 + 0.607204i \(0.792291\pi\)
\(972\) −37.7662 −1.21135
\(973\) − 17.4960i − 0.560897i
\(974\) −3.93815 −0.126186
\(975\) 0 0
\(976\) −0.780630 −0.0249874
\(977\) − 10.8671i − 0.347670i −0.984775 0.173835i \(-0.944384\pi\)
0.984775 0.173835i \(-0.0556159\pi\)
\(978\) 28.7920 0.920666
\(979\) −4.10380 −0.131158
\(980\) − 12.3580i − 0.394762i
\(981\) − 41.3027i − 1.31869i
\(982\) 14.8511i 0.473918i
\(983\) − 2.34833i − 0.0749001i −0.999299 0.0374501i \(-0.988076\pi\)
0.999299 0.0374501i \(-0.0119235\pi\)
\(984\) −7.12645 −0.227183
\(985\) −28.9079 −0.921082
\(986\) − 19.0062i − 0.605282i
\(987\) 4.51158 0.143605
\(988\) 0 0
\(989\) 18.7369 0.595799
\(990\) − 13.8905i − 0.441468i
\(991\) 24.4815 0.777681 0.388841 0.921305i \(-0.372876\pi\)
0.388841 + 0.921305i \(0.372876\pi\)
\(992\) −35.8661 −1.13875
\(993\) − 8.21311i − 0.260635i
\(994\) 24.8592i 0.788485i
\(995\) − 3.43335i − 0.108844i
\(996\) 9.22399i 0.292273i
\(997\) −6.62341 −0.209766 −0.104883 0.994485i \(-0.533447\pi\)
−0.104883 + 0.994485i \(0.533447\pi\)
\(998\) −54.6982 −1.73144
\(999\) − 1.03191i − 0.0326482i
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 1183.2.c.g.337.7 8
13.2 odd 12 91.2.f.c.22.1 8
13.5 odd 4 1183.2.a.k.1.4 4
13.6 odd 12 91.2.f.c.29.1 yes 8
13.8 odd 4 1183.2.a.l.1.1 4
13.12 even 2 inner 1183.2.c.g.337.2 8
39.2 even 12 819.2.o.h.568.4 8
39.32 even 12 819.2.o.h.757.4 8
52.15 even 12 1456.2.s.q.113.2 8
52.19 even 12 1456.2.s.q.1121.2 8
91.2 odd 12 637.2.h.h.165.4 8
91.6 even 12 637.2.f.i.393.1 8
91.19 even 12 637.2.g.j.263.1 8
91.32 odd 12 637.2.h.h.471.4 8
91.34 even 4 8281.2.a.bt.1.1 4
91.41 even 12 637.2.f.i.295.1 8
91.45 even 12 637.2.h.i.471.4 8
91.54 even 12 637.2.h.i.165.4 8
91.58 odd 12 637.2.g.k.263.1 8
91.67 odd 12 637.2.g.k.373.1 8
91.80 even 12 637.2.g.j.373.1 8
91.83 even 4 8281.2.a.bp.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.1 8 13.2 odd 12
91.2.f.c.29.1 yes 8 13.6 odd 12
637.2.f.i.295.1 8 91.41 even 12
637.2.f.i.393.1 8 91.6 even 12
637.2.g.j.263.1 8 91.19 even 12
637.2.g.j.373.1 8 91.80 even 12
637.2.g.k.263.1 8 91.58 odd 12
637.2.g.k.373.1 8 91.67 odd 12
637.2.h.h.165.4 8 91.2 odd 12
637.2.h.h.471.4 8 91.32 odd 12
637.2.h.i.165.4 8 91.54 even 12
637.2.h.i.471.4 8 91.45 even 12
819.2.o.h.568.4 8 39.2 even 12
819.2.o.h.757.4 8 39.32 even 12
1183.2.a.k.1.4 4 13.5 odd 4
1183.2.a.l.1.1 4 13.8 odd 4
1183.2.c.g.337.2 8 13.12 even 2 inner
1183.2.c.g.337.7 8 1.1 even 1 trivial
1456.2.s.q.113.2 8 52.15 even 12
1456.2.s.q.1121.2 8 52.19 even 12
8281.2.a.bp.1.4 4 91.83 even 4
8281.2.a.bt.1.1 4 91.34 even 4