Properties

Label 475.2.b.e.324.6
Level $475$
Weight $2$
Character 475.324
Analytic conductor $3.793$
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] = [475,2,Mod(324,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2058981376.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.6
Root \(0.148421 - 0.148421i\) of defining polynomial
Character \(\chi\) \(=\) 475.324
Dual form 475.2.b.e.324.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.95594i q^{2} -0.296842i q^{3} -1.82571 q^{4} +0.580605 q^{6} -3.56331i q^{7} +0.340899i q^{8} +2.91188 q^{9} +O(q^{10})\) \(q+1.95594i q^{2} -0.296842i q^{3} -1.82571 q^{4} +0.580605 q^{6} -3.56331i q^{7} +0.340899i q^{8} +2.91188 q^{9} +5.56331 q^{11} +0.541947i q^{12} -5.26647i q^{13} +6.96962 q^{14} -4.31820 q^{16} +1.40632i q^{17} +5.69548i q^{18} -1.00000 q^{19} -1.05774 q^{21} +10.8815i q^{22} +6.96962i q^{23} +0.101193 q^{24} +10.3009 q^{26} -1.75489i q^{27} +6.50557i q^{28} -1.40632 q^{29} +1.75489 q^{31} -7.76435i q^{32} -1.65142i q^{33} -2.75067 q^{34} -5.31626 q^{36} +3.61504i q^{37} -1.95594i q^{38} -1.56331 q^{39} +4.34858 q^{41} -2.06888i q^{42} +3.56331i q^{43} -10.1570 q^{44} -13.6322 q^{46} -8.26046i q^{47} +1.28182i q^{48} -5.69716 q^{49} +0.417453 q^{51} +9.61504i q^{52} +7.61504i q^{53} +3.43247 q^{54} +1.21473 q^{56} +0.296842i q^{57} -2.75067i q^{58} -9.47519 q^{59} +9.21473 q^{61} +3.43247i q^{62} -10.3759i q^{63} +6.55023 q^{64} +3.23009 q^{66} -4.76090i q^{67} -2.56753i q^{68} +2.06888 q^{69} -14.0689 q^{71} +0.992660i q^{72} -6.59368i q^{73} -7.07082 q^{74} +1.82571 q^{76} -19.8238i q^{77} -3.05774i q^{78} -5.47519 q^{79} +8.21473 q^{81} +8.50557i q^{82} +4.15699i q^{83} +1.93112 q^{84} -6.96962 q^{86} +0.417453i q^{87} +1.89653i q^{88} +9.23009 q^{89} -18.7660 q^{91} -12.7245i q^{92} -0.520926i q^{93} +16.1570 q^{94} -2.30478 q^{96} +11.5116i q^{97} -11.1433i q^{98} +16.1997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51} + 40 q^{54} - 24 q^{56} + 40 q^{61} + 8 q^{64} - 56 q^{66} - 56 q^{69} - 40 q^{71} - 64 q^{74} + 16 q^{76} + 32 q^{79} + 32 q^{81} + 88 q^{84} - 16 q^{86} - 8 q^{89} - 72 q^{91} + 96 q^{94} - 104 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.95594i 1.38306i 0.722348 + 0.691530i \(0.243063\pi\)
−0.722348 + 0.691530i \(0.756937\pi\)
\(3\) − 0.296842i − 0.171382i −0.996322 0.0856908i \(-0.972690\pi\)
0.996322 0.0856908i \(-0.0273097\pi\)
\(4\) −1.82571 −0.912855
\(5\) 0 0
\(6\) 0.580605 0.237031
\(7\) − 3.56331i − 1.34680i −0.739277 0.673402i \(-0.764832\pi\)
0.739277 0.673402i \(-0.235168\pi\)
\(8\) 0.340899i 0.120526i
\(9\) 2.91188 0.970628
\(10\) 0 0
\(11\) 5.56331 1.67740 0.838700 0.544594i \(-0.183316\pi\)
0.838700 + 0.544594i \(0.183316\pi\)
\(12\) 0.541947i 0.156447i
\(13\) − 5.26647i − 1.46065i −0.683097 0.730327i \(-0.739368\pi\)
0.683097 0.730327i \(-0.260632\pi\)
\(14\) 6.96962 1.86271
\(15\) 0 0
\(16\) −4.31820 −1.07955
\(17\) 1.40632i 0.341082i 0.985351 + 0.170541i \(0.0545515\pi\)
−0.985351 + 0.170541i \(0.945448\pi\)
\(18\) 5.69548i 1.34244i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.05774 −0.230817
\(22\) 10.8815i 2.31995i
\(23\) 6.96962i 1.45327i 0.687025 + 0.726633i \(0.258916\pi\)
−0.687025 + 0.726633i \(0.741084\pi\)
\(24\) 0.101193 0.0206560
\(25\) 0 0
\(26\) 10.3009 2.02017
\(27\) − 1.75489i − 0.337730i
\(28\) 6.50557i 1.22944i
\(29\) −1.40632 −0.261146 −0.130573 0.991439i \(-0.541682\pi\)
−0.130573 + 0.991439i \(0.541682\pi\)
\(30\) 0 0
\(31\) 1.75489 0.315188 0.157594 0.987504i \(-0.449626\pi\)
0.157594 + 0.987504i \(0.449626\pi\)
\(32\) − 7.76435i − 1.37256i
\(33\) − 1.65142i − 0.287476i
\(34\) −2.75067 −0.471737
\(35\) 0 0
\(36\) −5.31626 −0.886043
\(37\) 3.61504i 0.594309i 0.954829 + 0.297155i \(0.0960377\pi\)
−0.954829 + 0.297155i \(0.903962\pi\)
\(38\) − 1.95594i − 0.317296i
\(39\) −1.56331 −0.250329
\(40\) 0 0
\(41\) 4.34858 0.679134 0.339567 0.940582i \(-0.389720\pi\)
0.339567 + 0.940582i \(0.389720\pi\)
\(42\) − 2.06888i − 0.319234i
\(43\) 3.56331i 0.543399i 0.962382 + 0.271700i \(0.0875857\pi\)
−0.962382 + 0.271700i \(0.912414\pi\)
\(44\) −10.1570 −1.53122
\(45\) 0 0
\(46\) −13.6322 −2.00996
\(47\) − 8.26046i − 1.20491i −0.798152 0.602456i \(-0.794189\pi\)
0.798152 0.602456i \(-0.205811\pi\)
\(48\) 1.28182i 0.185015i
\(49\) −5.69716 −0.813879
\(50\) 0 0
\(51\) 0.417453 0.0584552
\(52\) 9.61504i 1.33337i
\(53\) 7.61504i 1.04601i 0.852331 + 0.523003i \(0.175188\pi\)
−0.852331 + 0.523003i \(0.824812\pi\)
\(54\) 3.43247 0.467100
\(55\) 0 0
\(56\) 1.21473 0.162325
\(57\) 0.296842i 0.0393177i
\(58\) − 2.75067i − 0.361181i
\(59\) −9.47519 −1.23356 −0.616782 0.787134i \(-0.711564\pi\)
−0.616782 + 0.787134i \(0.711564\pi\)
\(60\) 0 0
\(61\) 9.21473 1.17983 0.589913 0.807467i \(-0.299162\pi\)
0.589913 + 0.807467i \(0.299162\pi\)
\(62\) 3.43247i 0.435924i
\(63\) − 10.3759i − 1.30725i
\(64\) 6.55023 0.818779
\(65\) 0 0
\(66\) 3.23009 0.397596
\(67\) − 4.76090i − 0.581636i −0.956778 0.290818i \(-0.906073\pi\)
0.956778 0.290818i \(-0.0939274\pi\)
\(68\) − 2.56753i − 0.311358i
\(69\) 2.06888 0.249063
\(70\) 0 0
\(71\) −14.0689 −1.66967 −0.834834 0.550502i \(-0.814437\pi\)
−0.834834 + 0.550502i \(0.814437\pi\)
\(72\) 0.992660i 0.116986i
\(73\) − 6.59368i − 0.771732i −0.922555 0.385866i \(-0.873903\pi\)
0.922555 0.385866i \(-0.126097\pi\)
\(74\) −7.07082 −0.821966
\(75\) 0 0
\(76\) 1.82571 0.209423
\(77\) − 19.8238i − 2.25913i
\(78\) − 3.05774i − 0.346221i
\(79\) −5.47519 −0.616007 −0.308004 0.951385i \(-0.599661\pi\)
−0.308004 + 0.951385i \(0.599661\pi\)
\(80\) 0 0
\(81\) 8.21473 0.912748
\(82\) 8.50557i 0.939283i
\(83\) 4.15699i 0.456289i 0.973627 + 0.228144i \(0.0732659\pi\)
−0.973627 + 0.228144i \(0.926734\pi\)
\(84\) 1.93112 0.210703
\(85\) 0 0
\(86\) −6.96962 −0.751554
\(87\) 0.417453i 0.0447557i
\(88\) 1.89653i 0.202171i
\(89\) 9.23009 0.978387 0.489194 0.872175i \(-0.337291\pi\)
0.489194 + 0.872175i \(0.337291\pi\)
\(90\) 0 0
\(91\) −18.7660 −1.96721
\(92\) − 12.7245i − 1.32662i
\(93\) − 0.520926i − 0.0540175i
\(94\) 16.1570 1.66647
\(95\) 0 0
\(96\) −2.30478 −0.235231
\(97\) 11.5116i 1.16882i 0.811457 + 0.584411i \(0.198675\pi\)
−0.811457 + 0.584411i \(0.801325\pi\)
\(98\) − 11.1433i − 1.12564i
\(99\) 16.1997 1.62813
\(100\) 0 0
\(101\) −11.8511 −1.17923 −0.589616 0.807684i \(-0.700721\pi\)
−0.589616 + 0.807684i \(0.700721\pi\)
\(102\) 0.816515i 0.0808470i
\(103\) − 1.35458i − 0.133471i −0.997771 0.0667354i \(-0.978742\pi\)
0.997771 0.0667354i \(-0.0212583\pi\)
\(104\) 1.79533 0.176047
\(105\) 0 0
\(106\) −14.8946 −1.44669
\(107\) − 7.06287i − 0.682794i −0.939919 0.341397i \(-0.889100\pi\)
0.939919 0.341397i \(-0.110900\pi\)
\(108\) 3.20393i 0.308298i
\(109\) −10.1844 −0.975484 −0.487742 0.872988i \(-0.662179\pi\)
−0.487742 + 0.872988i \(0.662179\pi\)
\(110\) 0 0
\(111\) 1.07310 0.101854
\(112\) 15.3871i 1.45394i
\(113\) − 1.86015i − 0.174988i −0.996165 0.0874940i \(-0.972114\pi\)
0.996165 0.0874940i \(-0.0278859\pi\)
\(114\) −0.580605 −0.0543787
\(115\) 0 0
\(116\) 2.56753 0.238389
\(117\) − 15.3353i − 1.41775i
\(118\) − 18.5329i − 1.70609i
\(119\) 5.01114 0.459370
\(120\) 0 0
\(121\) 19.9504 1.81367
\(122\) 18.0235i 1.63177i
\(123\) − 1.29084i − 0.116391i
\(124\) −3.20393 −0.287721
\(125\) 0 0
\(126\) 20.2947 1.80800
\(127\) − 4.89053i − 0.433964i −0.976176 0.216982i \(-0.930379\pi\)
0.976176 0.216982i \(-0.0696213\pi\)
\(128\) − 2.71684i − 0.240137i
\(129\) 1.05774 0.0931287
\(130\) 0 0
\(131\) 2.81263 0.245741 0.122870 0.992423i \(-0.460790\pi\)
0.122870 + 0.992423i \(0.460790\pi\)
\(132\) 3.01502i 0.262424i
\(133\) 3.56331i 0.308978i
\(134\) 9.31204 0.804438
\(135\) 0 0
\(136\) −0.479412 −0.0411093
\(137\) 9.23009i 0.788579i 0.918986 + 0.394290i \(0.129009\pi\)
−0.918986 + 0.394290i \(0.870991\pi\)
\(138\) 4.04660i 0.344470i
\(139\) 3.67878 0.312030 0.156015 0.987755i \(-0.450135\pi\)
0.156015 + 0.987755i \(0.450135\pi\)
\(140\) 0 0
\(141\) −2.45205 −0.206500
\(142\) − 27.5179i − 2.30925i
\(143\) − 29.2990i − 2.45010i
\(144\) −12.5741 −1.04784
\(145\) 0 0
\(146\) 12.8969 1.06735
\(147\) 1.69115i 0.139484i
\(148\) − 6.60002i − 0.542519i
\(149\) −7.09925 −0.581593 −0.290797 0.956785i \(-0.593920\pi\)
−0.290797 + 0.956785i \(0.593920\pi\)
\(150\) 0 0
\(151\) −18.3567 −1.49385 −0.746924 0.664910i \(-0.768470\pi\)
−0.746924 + 0.664910i \(0.768470\pi\)
\(152\) − 0.340899i − 0.0276506i
\(153\) 4.09503i 0.331064i
\(154\) 38.7742 3.12451
\(155\) 0 0
\(156\) 2.85415 0.228515
\(157\) 17.2301i 1.37511i 0.726132 + 0.687555i \(0.241316\pi\)
−0.726132 + 0.687555i \(0.758684\pi\)
\(158\) − 10.7092i − 0.851975i
\(159\) 2.26046 0.179266
\(160\) 0 0
\(161\) 24.8349 1.95726
\(162\) 16.0675i 1.26238i
\(163\) − 10.8662i − 0.851103i −0.904934 0.425551i \(-0.860080\pi\)
0.904934 0.425551i \(-0.139920\pi\)
\(164\) −7.93925 −0.619951
\(165\) 0 0
\(166\) −8.13083 −0.631075
\(167\) 2.82977i 0.218974i 0.993988 + 0.109487i \(0.0349209\pi\)
−0.993988 + 0.109487i \(0.965079\pi\)
\(168\) − 0.360582i − 0.0278195i
\(169\) −14.7357 −1.13351
\(170\) 0 0
\(171\) −2.91188 −0.222677
\(172\) − 6.50557i − 0.496045i
\(173\) 9.26647i 0.704516i 0.935903 + 0.352258i \(0.114586\pi\)
−0.935903 + 0.352258i \(0.885414\pi\)
\(174\) −0.816515 −0.0618998
\(175\) 0 0
\(176\) −24.0235 −1.81084
\(177\) 2.81263i 0.211410i
\(178\) 18.0535i 1.35317i
\(179\) 3.59067 0.268379 0.134190 0.990956i \(-0.457157\pi\)
0.134190 + 0.990956i \(0.457157\pi\)
\(180\) 0 0
\(181\) −19.7630 −1.46897 −0.734487 0.678623i \(-0.762577\pi\)
−0.734487 + 0.678623i \(0.762577\pi\)
\(182\) − 36.7053i − 2.72078i
\(183\) − 2.73532i − 0.202200i
\(184\) −2.37594 −0.175157
\(185\) 0 0
\(186\) 1.01890 0.0747095
\(187\) 7.82377i 0.572131i
\(188\) 15.0812i 1.09991i
\(189\) −6.25323 −0.454855
\(190\) 0 0
\(191\) 8.31398 0.601579 0.300789 0.953691i \(-0.402750\pi\)
0.300789 + 0.953691i \(0.402750\pi\)
\(192\) − 1.94438i − 0.140324i
\(193\) 22.2514i 1.60169i 0.598869 + 0.800847i \(0.295617\pi\)
−0.598869 + 0.800847i \(0.704383\pi\)
\(194\) −22.5160 −1.61655
\(195\) 0 0
\(196\) 10.4014 0.742954
\(197\) − 8.81263i − 0.627874i −0.949444 0.313937i \(-0.898352\pi\)
0.949444 0.313937i \(-0.101648\pi\)
\(198\) 31.6857i 2.25180i
\(199\) −21.0659 −1.49332 −0.746660 0.665206i \(-0.768344\pi\)
−0.746660 + 0.665206i \(0.768344\pi\)
\(200\) 0 0
\(201\) −1.41323 −0.0996818
\(202\) − 23.1801i − 1.63095i
\(203\) 5.01114i 0.351713i
\(204\) −0.762149 −0.0533611
\(205\) 0 0
\(206\) 2.64948 0.184598
\(207\) 20.2947i 1.41058i
\(208\) 22.7417i 1.57685i
\(209\) −5.56331 −0.384822
\(210\) 0 0
\(211\) 5.34556 0.368004 0.184002 0.982926i \(-0.441095\pi\)
0.184002 + 0.982926i \(0.441095\pi\)
\(212\) − 13.9029i − 0.954853i
\(213\) 4.17623i 0.286151i
\(214\) 13.8146 0.944345
\(215\) 0 0
\(216\) 0.598242 0.0407052
\(217\) − 6.25323i − 0.424497i
\(218\) − 19.9200i − 1.34915i
\(219\) −1.95728 −0.132261
\(220\) 0 0
\(221\) 7.40632 0.498203
\(222\) 2.09891i 0.140870i
\(223\) 3.10947i 0.208226i 0.994565 + 0.104113i \(0.0332003\pi\)
−0.994565 + 0.104113i \(0.966800\pi\)
\(224\) −27.6668 −1.84856
\(225\) 0 0
\(226\) 3.63834 0.242019
\(227\) 14.4692i 0.960354i 0.877172 + 0.480177i \(0.159428\pi\)
−0.877172 + 0.480177i \(0.840572\pi\)
\(228\) − 0.541947i − 0.0358913i
\(229\) 5.21473 0.344599 0.172299 0.985045i \(-0.444880\pi\)
0.172299 + 0.985045i \(0.444880\pi\)
\(230\) 0 0
\(231\) −5.88452 −0.387173
\(232\) − 0.479412i − 0.0314750i
\(233\) 3.18737i 0.208811i 0.994535 + 0.104406i \(0.0332940\pi\)
−0.994535 + 0.104406i \(0.966706\pi\)
\(234\) 29.9950 1.96084
\(235\) 0 0
\(236\) 17.2990 1.12607
\(237\) 1.62527i 0.105572i
\(238\) 9.80150i 0.635337i
\(239\) −16.5209 −1.06865 −0.534325 0.845279i \(-0.679434\pi\)
−0.534325 + 0.845279i \(0.679434\pi\)
\(240\) 0 0
\(241\) −12.2271 −0.787615 −0.393807 0.919193i \(-0.628842\pi\)
−0.393807 + 0.919193i \(0.628842\pi\)
\(242\) 39.0218i 2.50842i
\(243\) − 7.70316i − 0.494158i
\(244\) −16.8234 −1.07701
\(245\) 0 0
\(246\) 2.52481 0.160976
\(247\) 5.26647i 0.335097i
\(248\) 0.598242i 0.0379884i
\(249\) 1.23397 0.0781996
\(250\) 0 0
\(251\) 4.52093 0.285358 0.142679 0.989769i \(-0.454428\pi\)
0.142679 + 0.989769i \(0.454428\pi\)
\(252\) 18.9435i 1.19333i
\(253\) 38.7742i 2.43771i
\(254\) 9.56559 0.600198
\(255\) 0 0
\(256\) 18.4144 1.15090
\(257\) 15.9290i 0.993625i 0.867858 + 0.496813i \(0.165496\pi\)
−0.867858 + 0.496813i \(0.834504\pi\)
\(258\) 2.06888i 0.128803i
\(259\) 12.8815 0.800418
\(260\) 0 0
\(261\) −4.09503 −0.253476
\(262\) 5.50135i 0.339874i
\(263\) − 0.854147i − 0.0526689i −0.999653 0.0263345i \(-0.991617\pi\)
0.999653 0.0263345i \(-0.00838349\pi\)
\(264\) 0.562969 0.0346483
\(265\) 0 0
\(266\) −6.96962 −0.427335
\(267\) − 2.73988i − 0.167678i
\(268\) 8.69202i 0.530950i
\(269\) −10.3913 −0.633569 −0.316784 0.948498i \(-0.602603\pi\)
−0.316784 + 0.948498i \(0.602603\pi\)
\(270\) 0 0
\(271\) −8.19971 −0.498097 −0.249048 0.968491i \(-0.580118\pi\)
−0.249048 + 0.968491i \(0.580118\pi\)
\(272\) − 6.07276i − 0.368215i
\(273\) 5.57054i 0.337145i
\(274\) −18.0535 −1.09065
\(275\) 0 0
\(276\) −3.77717 −0.227359
\(277\) 14.6484i 0.880137i 0.897964 + 0.440069i \(0.145046\pi\)
−0.897964 + 0.440069i \(0.854954\pi\)
\(278\) 7.19549i 0.431557i
\(279\) 5.11005 0.305931
\(280\) 0 0
\(281\) −31.6129 −1.88587 −0.942935 0.332977i \(-0.891947\pi\)
−0.942935 + 0.332977i \(0.891947\pi\)
\(282\) − 4.79607i − 0.285602i
\(283\) − 24.7326i − 1.47020i −0.677957 0.735101i \(-0.737135\pi\)
0.677957 0.735101i \(-0.262865\pi\)
\(284\) 25.6857 1.52417
\(285\) 0 0
\(286\) 57.3071 3.38864
\(287\) − 15.4953i − 0.914660i
\(288\) − 22.6089i − 1.33224i
\(289\) 15.0223 0.883663
\(290\) 0 0
\(291\) 3.41712 0.200315
\(292\) 12.0382i 0.704480i
\(293\) − 30.2857i − 1.76931i −0.466245 0.884655i \(-0.654394\pi\)
0.466245 0.884655i \(-0.345606\pi\)
\(294\) −3.30780 −0.192915
\(295\) 0 0
\(296\) −1.23237 −0.0716298
\(297\) − 9.76302i − 0.566508i
\(298\) − 13.8857i − 0.804379i
\(299\) 36.7053 2.12272
\(300\) 0 0
\(301\) 12.6972 0.731852
\(302\) − 35.9046i − 2.06608i
\(303\) 3.51791i 0.202099i
\(304\) 4.31820 0.247666
\(305\) 0 0
\(306\) −8.00965 −0.457881
\(307\) − 23.0629i − 1.31627i −0.752901 0.658134i \(-0.771346\pi\)
0.752901 0.658134i \(-0.228654\pi\)
\(308\) 36.1925i 2.06226i
\(309\) −0.402096 −0.0228744
\(310\) 0 0
\(311\) −10.3152 −0.584921 −0.292460 0.956278i \(-0.594474\pi\)
−0.292460 + 0.956278i \(0.594474\pi\)
\(312\) − 0.532930i − 0.0301712i
\(313\) 4.95038i 0.279812i 0.990165 + 0.139906i \(0.0446801\pi\)
−0.990165 + 0.139906i \(0.955320\pi\)
\(314\) −33.7011 −1.90186
\(315\) 0 0
\(316\) 9.99612 0.562326
\(317\) − 14.2433i − 0.799985i −0.916518 0.399992i \(-0.869013\pi\)
0.916518 0.399992i \(-0.130987\pi\)
\(318\) 4.42134i 0.247936i
\(319\) −7.82377 −0.438047
\(320\) 0 0
\(321\) −2.09656 −0.117018
\(322\) 48.5756i 2.70702i
\(323\) − 1.40632i − 0.0782495i
\(324\) −14.9977 −0.833207
\(325\) 0 0
\(326\) 21.2536 1.17713
\(327\) 3.02314i 0.167180i
\(328\) 1.48243i 0.0818534i
\(329\) −29.4346 −1.62278
\(330\) 0 0
\(331\) 2.04272 0.112278 0.0561390 0.998423i \(-0.482121\pi\)
0.0561390 + 0.998423i \(0.482121\pi\)
\(332\) − 7.58946i − 0.416526i
\(333\) 10.5266i 0.576854i
\(334\) −5.53487 −0.302855
\(335\) 0 0
\(336\) 4.56753 0.249179
\(337\) 34.6951i 1.88996i 0.327128 + 0.944980i \(0.393919\pi\)
−0.327128 + 0.944980i \(0.606081\pi\)
\(338\) − 28.8221i − 1.56772i
\(339\) −0.552170 −0.0299898
\(340\) 0 0
\(341\) 9.76302 0.528697
\(342\) − 5.69548i − 0.307976i
\(343\) − 4.64243i − 0.250668i
\(344\) −1.21473 −0.0654938
\(345\) 0 0
\(346\) −18.1247 −0.974388
\(347\) 7.35280i 0.394719i 0.980331 + 0.197359i \(0.0632366\pi\)
−0.980331 + 0.197359i \(0.936763\pi\)
\(348\) − 0.762149i − 0.0408555i
\(349\) 31.7630 1.70024 0.850118 0.526593i \(-0.176531\pi\)
0.850118 + 0.526593i \(0.176531\pi\)
\(350\) 0 0
\(351\) −9.24209 −0.493306
\(352\) − 43.1955i − 2.30233i
\(353\) 7.52179i 0.400345i 0.979761 + 0.200172i \(0.0641502\pi\)
−0.979761 + 0.200172i \(0.935850\pi\)
\(354\) −5.50135 −0.292393
\(355\) 0 0
\(356\) −16.8515 −0.893126
\(357\) − 1.48751i − 0.0787276i
\(358\) 7.02314i 0.371185i
\(359\) 30.3982 1.60436 0.802178 0.597085i \(-0.203674\pi\)
0.802178 + 0.597085i \(0.203674\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 38.6553i − 2.03168i
\(363\) − 5.92211i − 0.310830i
\(364\) 34.2613 1.79578
\(365\) 0 0
\(366\) 5.35012 0.279655
\(367\) 3.91577i 0.204401i 0.994764 + 0.102201i \(0.0325884\pi\)
−0.994764 + 0.102201i \(0.967412\pi\)
\(368\) − 30.0962i − 1.56887i
\(369\) 12.6626 0.659186
\(370\) 0 0
\(371\) 27.1347 1.40877
\(372\) 0.951060i 0.0493102i
\(373\) − 26.7759i − 1.38641i −0.720743 0.693203i \(-0.756199\pi\)
0.720743 0.693203i \(-0.243801\pi\)
\(374\) −15.3028 −0.791291
\(375\) 0 0
\(376\) 2.81599 0.145223
\(377\) 7.40632i 0.381445i
\(378\) − 12.2310i − 0.629092i
\(379\) −14.9504 −0.767950 −0.383975 0.923344i \(-0.625445\pi\)
−0.383975 + 0.923344i \(0.625445\pi\)
\(380\) 0 0
\(381\) −1.45171 −0.0743735
\(382\) 16.2617i 0.832019i
\(383\) 27.9910i 1.43027i 0.698985 + 0.715136i \(0.253635\pi\)
−0.698985 + 0.715136i \(0.746365\pi\)
\(384\) −0.806471 −0.0411551
\(385\) 0 0
\(386\) −43.5225 −2.21524
\(387\) 10.3759i 0.527439i
\(388\) − 21.0168i − 1.06697i
\(389\) 35.2036 1.78489 0.892447 0.451152i \(-0.148987\pi\)
0.892447 + 0.451152i \(0.148987\pi\)
\(390\) 0 0
\(391\) −9.80150 −0.495683
\(392\) − 1.94216i − 0.0980937i
\(393\) − 0.834907i − 0.0421155i
\(394\) 17.2370 0.868388
\(395\) 0 0
\(396\) −29.5760 −1.48625
\(397\) − 35.9735i − 1.80546i −0.430208 0.902730i \(-0.641560\pi\)
0.430208 0.902730i \(-0.358440\pi\)
\(398\) − 41.2036i − 2.06535i
\(399\) 1.05774 0.0529532
\(400\) 0 0
\(401\) 23.2421 1.16065 0.580327 0.814383i \(-0.302925\pi\)
0.580327 + 0.814383i \(0.302925\pi\)
\(402\) − 2.76420i − 0.137866i
\(403\) − 9.24209i − 0.460381i
\(404\) 21.6367 1.07647
\(405\) 0 0
\(406\) −9.80150 −0.486440
\(407\) 20.1116i 0.996895i
\(408\) 0.142310i 0.00704538i
\(409\) 31.8926 1.57699 0.788495 0.615041i \(-0.210861\pi\)
0.788495 + 0.615041i \(0.210861\pi\)
\(410\) 0 0
\(411\) 2.73988 0.135148
\(412\) 2.47307i 0.121840i
\(413\) 33.7630i 1.66137i
\(414\) −39.6953 −1.95092
\(415\) 0 0
\(416\) −40.8907 −2.00483
\(417\) − 1.09202i − 0.0534763i
\(418\) − 10.8815i − 0.532232i
\(419\) −31.8238 −1.55469 −0.777346 0.629073i \(-0.783435\pi\)
−0.777346 + 0.629073i \(0.783435\pi\)
\(420\) 0 0
\(421\) 0.348578 0.0169887 0.00849433 0.999964i \(-0.497296\pi\)
0.00849433 + 0.999964i \(0.497296\pi\)
\(422\) 10.4556i 0.508971i
\(423\) − 24.0535i − 1.16952i
\(424\) −2.59596 −0.126071
\(425\) 0 0
\(426\) −8.16847 −0.395763
\(427\) − 32.8349i − 1.58899i
\(428\) 12.8948i 0.623292i
\(429\) −8.69716 −0.419903
\(430\) 0 0
\(431\) 29.2764 1.41019 0.705097 0.709111i \(-0.250904\pi\)
0.705097 + 0.709111i \(0.250904\pi\)
\(432\) 7.57799i 0.364596i
\(433\) 0.883290i 0.0424482i 0.999775 + 0.0212241i \(0.00675635\pi\)
−0.999775 + 0.0212241i \(0.993244\pi\)
\(434\) 12.2310 0.587105
\(435\) 0 0
\(436\) 18.5937 0.890476
\(437\) − 6.96962i − 0.333402i
\(438\) − 3.82833i − 0.182925i
\(439\) 13.8584 0.661424 0.330712 0.943732i \(-0.392711\pi\)
0.330712 + 0.943732i \(0.392711\pi\)
\(440\) 0 0
\(441\) −16.5895 −0.789974
\(442\) 14.4863i 0.689044i
\(443\) 13.5753i 0.644982i 0.946572 + 0.322491i \(0.104520\pi\)
−0.946572 + 0.322491i \(0.895480\pi\)
\(444\) −1.95916 −0.0929778
\(445\) 0 0
\(446\) −6.08195 −0.287989
\(447\) 2.10735i 0.0996745i
\(448\) − 23.3405i − 1.10273i
\(449\) 15.6334 0.737785 0.368893 0.929472i \(-0.379737\pi\)
0.368893 + 0.929472i \(0.379737\pi\)
\(450\) 0 0
\(451\) 24.1925 1.13918
\(452\) 3.39609i 0.159739i
\(453\) 5.44904i 0.256018i
\(454\) −28.3009 −1.32823
\(455\) 0 0
\(456\) −0.101193 −0.00473880
\(457\) 5.30284i 0.248057i 0.992279 + 0.124028i \(0.0395814\pi\)
−0.992279 + 0.124028i \(0.960419\pi\)
\(458\) 10.1997i 0.476601i
\(459\) 2.46794 0.115193
\(460\) 0 0
\(461\) 0.374734 0.0174531 0.00872656 0.999962i \(-0.497222\pi\)
0.00872656 + 0.999962i \(0.497222\pi\)
\(462\) − 11.5098i − 0.535484i
\(463\) − 6.65564i − 0.309314i −0.987968 0.154657i \(-0.950573\pi\)
0.987968 0.154657i \(-0.0494272\pi\)
\(464\) 6.07276 0.281921
\(465\) 0 0
\(466\) −6.23431 −0.288799
\(467\) − 0.854147i − 0.0395252i −0.999805 0.0197626i \(-0.993709\pi\)
0.999805 0.0197626i \(-0.00629104\pi\)
\(468\) 27.9979i 1.29420i
\(469\) −16.9645 −0.783349
\(470\) 0 0
\(471\) 5.11461 0.235669
\(472\) − 3.23009i − 0.148677i
\(473\) 19.8238i 0.911498i
\(474\) −3.17893 −0.146013
\(475\) 0 0
\(476\) −9.14889 −0.419339
\(477\) 22.1741i 1.01528i
\(478\) − 32.3140i − 1.47801i
\(479\) 17.0731 0.780090 0.390045 0.920796i \(-0.372460\pi\)
0.390045 + 0.920796i \(0.372460\pi\)
\(480\) 0 0
\(481\) 19.0385 0.868081
\(482\) − 23.9154i − 1.08932i
\(483\) − 7.37204i − 0.335439i
\(484\) −36.4236 −1.65562
\(485\) 0 0
\(486\) 15.0669 0.683450
\(487\) 12.8259i 0.581197i 0.956845 + 0.290598i \(0.0938543\pi\)
−0.956845 + 0.290598i \(0.906146\pi\)
\(488\) 3.14129i 0.142200i
\(489\) −3.22553 −0.145863
\(490\) 0 0
\(491\) −10.4054 −0.469591 −0.234796 0.972045i \(-0.575442\pi\)
−0.234796 + 0.972045i \(0.575442\pi\)
\(492\) 2.35670i 0.106248i
\(493\) − 1.97773i − 0.0890723i
\(494\) −10.3009 −0.463460
\(495\) 0 0
\(496\) −7.57799 −0.340262
\(497\) 50.1317i 2.24872i
\(498\) 2.41357i 0.108155i
\(499\) −36.1612 −1.61880 −0.809399 0.587258i \(-0.800207\pi\)
−0.809399 + 0.587258i \(0.800207\pi\)
\(500\) 0 0
\(501\) 0.839995 0.0375282
\(502\) 8.84267i 0.394668i
\(503\) − 33.2536i − 1.48270i −0.671117 0.741352i \(-0.734185\pi\)
0.671117 0.741352i \(-0.265815\pi\)
\(504\) 3.53715 0.157557
\(505\) 0 0
\(506\) −75.8400 −3.37150
\(507\) 4.37416i 0.194263i
\(508\) 8.92869i 0.396146i
\(509\) −7.29084 −0.323161 −0.161580 0.986860i \(-0.551659\pi\)
−0.161580 + 0.986860i \(0.551659\pi\)
\(510\) 0 0
\(511\) −23.4953 −1.03937
\(512\) 30.5839i 1.35163i
\(513\) 1.75489i 0.0774805i
\(514\) −31.1563 −1.37424
\(515\) 0 0
\(516\) −1.93112 −0.0850130
\(517\) − 45.9555i − 2.02112i
\(518\) 25.1955i 1.10703i
\(519\) 2.75067 0.120741
\(520\) 0 0
\(521\) −9.87849 −0.432785 −0.216392 0.976306i \(-0.569429\pi\)
−0.216392 + 0.976306i \(0.569429\pi\)
\(522\) − 8.00965i − 0.350573i
\(523\) 11.1925i 0.489414i 0.969597 + 0.244707i \(0.0786918\pi\)
−0.969597 + 0.244707i \(0.921308\pi\)
\(524\) −5.13505 −0.224326
\(525\) 0 0
\(526\) 1.67066 0.0728443
\(527\) 2.46794i 0.107505i
\(528\) 7.13117i 0.310344i
\(529\) −25.5756 −1.11198
\(530\) 0 0
\(531\) −27.5907 −1.19733
\(532\) − 6.50557i − 0.282052i
\(533\) − 22.9016i − 0.991980i
\(534\) 5.35904 0.231908
\(535\) 0 0
\(536\) 1.62299 0.0701023
\(537\) − 1.06586i − 0.0459953i
\(538\) − 20.3248i − 0.876263i
\(539\) −31.6950 −1.36520
\(540\) 0 0
\(541\) 2.22587 0.0956974 0.0478487 0.998855i \(-0.484763\pi\)
0.0478487 + 0.998855i \(0.484763\pi\)
\(542\) − 16.0382i − 0.688898i
\(543\) 5.86649i 0.251755i
\(544\) 10.9191 0.468154
\(545\) 0 0
\(546\) −10.8957 −0.466291
\(547\) − 34.9675i − 1.49510i −0.664204 0.747552i \(-0.731229\pi\)
0.664204 0.747552i \(-0.268771\pi\)
\(548\) − 16.8515i − 0.719859i
\(549\) 26.8322 1.14517
\(550\) 0 0
\(551\) 1.40632 0.0599111
\(552\) 0.705278i 0.0300186i
\(553\) 19.5098i 0.829641i
\(554\) −28.6514 −1.21728
\(555\) 0 0
\(556\) −6.71640 −0.284839
\(557\) − 8.88539i − 0.376486i −0.982122 0.188243i \(-0.939721\pi\)
0.982122 0.188243i \(-0.0602793\pi\)
\(558\) 9.99497i 0.423121i
\(559\) 18.7660 0.793719
\(560\) 0 0
\(561\) 2.32242 0.0980527
\(562\) − 61.8331i − 2.60827i
\(563\) 29.6767i 1.25072i 0.780335 + 0.625362i \(0.215048\pi\)
−0.780335 + 0.625362i \(0.784952\pi\)
\(564\) 4.47674 0.188505
\(565\) 0 0
\(566\) 48.3756 2.03338
\(567\) − 29.2716i − 1.22929i
\(568\) − 4.79607i − 0.201239i
\(569\) 22.1152 0.927116 0.463558 0.886067i \(-0.346573\pi\)
0.463558 + 0.886067i \(0.346573\pi\)
\(570\) 0 0
\(571\) −33.2656 −1.39212 −0.696060 0.717983i \(-0.745065\pi\)
−0.696060 + 0.717983i \(0.745065\pi\)
\(572\) 53.4914i 2.23659i
\(573\) − 2.46794i − 0.103100i
\(574\) 30.3080 1.26503
\(575\) 0 0
\(576\) 19.0735 0.794730
\(577\) 0.934140i 0.0388887i 0.999811 + 0.0194444i \(0.00618972\pi\)
−0.999811 + 0.0194444i \(0.993810\pi\)
\(578\) 29.3827i 1.22216i
\(579\) 6.60516 0.274501
\(580\) 0 0
\(581\) 14.8126 0.614532
\(582\) 6.68368i 0.277047i
\(583\) 42.3648i 1.75457i
\(584\) 2.24778 0.0930139
\(585\) 0 0
\(586\) 59.2371 2.44706
\(587\) 1.57531i 0.0650200i 0.999471 + 0.0325100i \(0.0103501\pi\)
−0.999471 + 0.0325100i \(0.989650\pi\)
\(588\) − 3.08756i − 0.127329i
\(589\) −1.75489 −0.0723092
\(590\) 0 0
\(591\) −2.61596 −0.107606
\(592\) − 15.6105i − 0.641587i
\(593\) − 26.0162i − 1.06836i −0.845371 0.534180i \(-0.820621\pi\)
0.845371 0.534180i \(-0.179379\pi\)
\(594\) 19.0959 0.783514
\(595\) 0 0
\(596\) 12.9612 0.530911
\(597\) 6.25323i 0.255928i
\(598\) 71.7934i 2.93585i
\(599\) −19.5359 −0.798217 −0.399109 0.916904i \(-0.630680\pi\)
−0.399109 + 0.916904i \(0.630680\pi\)
\(600\) 0 0
\(601\) 43.5299 1.77562 0.887811 0.460208i \(-0.152225\pi\)
0.887811 + 0.460208i \(0.152225\pi\)
\(602\) 24.8349i 1.01220i
\(603\) − 13.8632i − 0.564552i
\(604\) 33.5140 1.36367
\(605\) 0 0
\(606\) −6.88083 −0.279515
\(607\) − 12.5847i − 0.510796i −0.966836 0.255398i \(-0.917794\pi\)
0.966836 0.255398i \(-0.0822065\pi\)
\(608\) 7.76435i 0.314886i
\(609\) 1.48751 0.0602771
\(610\) 0 0
\(611\) −43.5034 −1.75996
\(612\) − 7.47634i − 0.302213i
\(613\) 18.2412i 0.736756i 0.929676 + 0.368378i \(0.120087\pi\)
−0.929676 + 0.368378i \(0.879913\pi\)
\(614\) 45.1097 1.82048
\(615\) 0 0
\(616\) 6.75791 0.272284
\(617\) 26.7314i 1.07617i 0.842892 + 0.538084i \(0.180852\pi\)
−0.842892 + 0.538084i \(0.819148\pi\)
\(618\) − 0.786477i − 0.0316367i
\(619\) −2.92690 −0.117642 −0.0588211 0.998269i \(-0.518734\pi\)
−0.0588211 + 0.998269i \(0.518734\pi\)
\(620\) 0 0
\(621\) 12.2310 0.490811
\(622\) − 20.1759i − 0.808980i
\(623\) − 32.8896i − 1.31770i
\(624\) 6.75067 0.270243
\(625\) 0 0
\(626\) −9.68267 −0.386997
\(627\) 1.65142i 0.0659514i
\(628\) − 31.4572i − 1.25528i
\(629\) −5.08389 −0.202708
\(630\) 0 0
\(631\) −13.2493 −0.527447 −0.263724 0.964598i \(-0.584951\pi\)
−0.263724 + 0.964598i \(0.584951\pi\)
\(632\) − 1.86649i − 0.0742450i
\(633\) − 1.58679i − 0.0630691i
\(634\) 27.8591 1.10643
\(635\) 0 0
\(636\) −4.12695 −0.163644
\(637\) 30.0039i 1.18880i
\(638\) − 15.3028i − 0.605845i
\(639\) −40.9669 −1.62063
\(640\) 0 0
\(641\) 6.93026 0.273729 0.136864 0.990590i \(-0.456298\pi\)
0.136864 + 0.990590i \(0.456298\pi\)
\(642\) − 4.10074i − 0.161843i
\(643\) − 14.4794i − 0.571012i −0.958377 0.285506i \(-0.907838\pi\)
0.958377 0.285506i \(-0.0921617\pi\)
\(644\) −45.3414 −1.78670
\(645\) 0 0
\(646\) 2.75067 0.108224
\(647\) 6.35549i 0.249860i 0.992166 + 0.124930i \(0.0398707\pi\)
−0.992166 + 0.124930i \(0.960129\pi\)
\(648\) 2.80040i 0.110010i
\(649\) −52.7134 −2.06918
\(650\) 0 0
\(651\) −1.85622 −0.0727510
\(652\) 19.8385i 0.776934i
\(653\) 34.4030i 1.34629i 0.739509 + 0.673146i \(0.235058\pi\)
−0.739509 + 0.673146i \(0.764942\pi\)
\(654\) −5.91309 −0.231220
\(655\) 0 0
\(656\) −18.7780 −0.733159
\(657\) − 19.2000i − 0.749065i
\(658\) − 57.5723i − 2.24440i
\(659\) 19.6214 0.764341 0.382170 0.924092i \(-0.375177\pi\)
0.382170 + 0.924092i \(0.375177\pi\)
\(660\) 0 0
\(661\) 39.8054 1.54825 0.774126 0.633032i \(-0.218190\pi\)
0.774126 + 0.633032i \(0.218190\pi\)
\(662\) 3.99544i 0.155287i
\(663\) − 2.19850i − 0.0853828i
\(664\) −1.41712 −0.0549947
\(665\) 0 0
\(666\) −20.5894 −0.797823
\(667\) − 9.80150i − 0.379515i
\(668\) − 5.16635i − 0.199892i
\(669\) 0.923022 0.0356861
\(670\) 0 0
\(671\) 51.2644 1.97904
\(672\) 8.21266i 0.316810i
\(673\) 8.90374i 0.343214i 0.985166 + 0.171607i \(0.0548959\pi\)
−0.985166 + 0.171607i \(0.945104\pi\)
\(674\) −67.8615 −2.61393
\(675\) 0 0
\(676\) 26.9030 1.03473
\(677\) 22.2695i 0.855886i 0.903806 + 0.427943i \(0.140762\pi\)
−0.903806 + 0.427943i \(0.859238\pi\)
\(678\) − 1.08001i − 0.0414776i
\(679\) 41.0193 1.57417
\(680\) 0 0
\(681\) 4.29506 0.164587
\(682\) 19.0959i 0.731220i
\(683\) − 15.4054i − 0.589472i −0.955579 0.294736i \(-0.904768\pi\)
0.955579 0.294736i \(-0.0952317\pi\)
\(684\) 5.31626 0.203272
\(685\) 0 0
\(686\) 9.08033 0.346689
\(687\) − 1.54795i − 0.0590580i
\(688\) − 15.3871i − 0.586627i
\(689\) 40.1044 1.52785
\(690\) 0 0
\(691\) −17.8773 −0.680084 −0.340042 0.940410i \(-0.610441\pi\)
−0.340042 + 0.940410i \(0.610441\pi\)
\(692\) − 16.9179i − 0.643122i
\(693\) − 57.7245i − 2.19277i
\(694\) −14.3817 −0.545920
\(695\) 0 0
\(696\) −0.142310 −0.00539423
\(697\) 6.11548i 0.231640i
\(698\) 62.1266i 2.35153i
\(699\) 0.946144 0.0357864
\(700\) 0 0
\(701\) 35.3609 1.33556 0.667782 0.744357i \(-0.267244\pi\)
0.667782 + 0.744357i \(0.267244\pi\)
\(702\) − 18.0770i − 0.682272i
\(703\) − 3.61504i − 0.136344i
\(704\) 36.4409 1.37342
\(705\) 0 0
\(706\) −14.7122 −0.553701
\(707\) 42.2292i 1.58819i
\(708\) − 5.13505i − 0.192987i
\(709\) −6.90410 −0.259289 −0.129644 0.991561i \(-0.541384\pi\)
−0.129644 + 0.991561i \(0.541384\pi\)
\(710\) 0 0
\(711\) −15.9431 −0.597914
\(712\) 3.14653i 0.117921i
\(713\) 12.2310i 0.458053i
\(714\) 2.90949 0.108885
\(715\) 0 0
\(716\) −6.55552 −0.244991
\(717\) 4.90410i 0.183147i
\(718\) 59.4572i 2.21892i
\(719\) 38.9431 1.45233 0.726167 0.687518i \(-0.241300\pi\)
0.726167 + 0.687518i \(0.241300\pi\)
\(720\) 0 0
\(721\) −4.82678 −0.179759
\(722\) 1.95594i 0.0727926i
\(723\) 3.62951i 0.134983i
\(724\) 36.0816 1.34096
\(725\) 0 0
\(726\) 11.5833 0.429897
\(727\) − 6.18347i − 0.229332i −0.993404 0.114666i \(-0.963420\pi\)
0.993404 0.114666i \(-0.0365798\pi\)
\(728\) − 6.39733i − 0.237101i
\(729\) 22.3576 0.828058
\(730\) 0 0
\(731\) −5.01114 −0.185344
\(732\) 4.99390i 0.184580i
\(733\) 0.688715i 0.0254383i 0.999919 + 0.0127191i \(0.00404874\pi\)
−0.999919 + 0.0127191i \(0.995951\pi\)
\(734\) −7.65902 −0.282699
\(735\) 0 0
\(736\) 54.1146 1.99469
\(737\) − 26.4863i − 0.975636i
\(738\) 24.7672i 0.911695i
\(739\) 26.2532 0.965741 0.482870 0.875692i \(-0.339594\pi\)
0.482870 + 0.875692i \(0.339594\pi\)
\(740\) 0 0
\(741\) 1.56331 0.0574295
\(742\) 53.0740i 1.94841i
\(743\) 32.5688i 1.19483i 0.801931 + 0.597416i \(0.203806\pi\)
−0.801931 + 0.597416i \(0.796194\pi\)
\(744\) 0.177583 0.00651052
\(745\) 0 0
\(746\) 52.3722 1.91748
\(747\) 12.1047i 0.442887i
\(748\) − 14.2839i − 0.522273i
\(749\) −25.1672 −0.919589
\(750\) 0 0
\(751\) 45.4833 1.65971 0.829855 0.557979i \(-0.188423\pi\)
0.829855 + 0.557979i \(0.188423\pi\)
\(752\) 35.6703i 1.30076i
\(753\) − 1.34200i − 0.0489052i
\(754\) −14.4863 −0.527561
\(755\) 0 0
\(756\) 11.4166 0.415217
\(757\) 2.74947i 0.0999311i 0.998751 + 0.0499656i \(0.0159112\pi\)
−0.998751 + 0.0499656i \(0.984089\pi\)
\(758\) − 29.2421i − 1.06212i
\(759\) 11.5098 0.417779
\(760\) 0 0
\(761\) −33.2978 −1.20704 −0.603521 0.797347i \(-0.706236\pi\)
−0.603521 + 0.797347i \(0.706236\pi\)
\(762\) − 2.83947i − 0.102863i
\(763\) 36.2900i 1.31379i
\(764\) −15.1789 −0.549154
\(765\) 0 0
\(766\) −54.7488 −1.97815
\(767\) 49.9008i 1.80181i
\(768\) − 5.46617i − 0.197244i
\(769\) 19.1540 0.690710 0.345355 0.938472i \(-0.387759\pi\)
0.345355 + 0.938472i \(0.387759\pi\)
\(770\) 0 0
\(771\) 4.72840 0.170289
\(772\) − 40.6247i − 1.46212i
\(773\) − 0.569309i − 0.0204766i −0.999948 0.0102383i \(-0.996741\pi\)
0.999948 0.0102383i \(-0.00325901\pi\)
\(774\) −20.2947 −0.729479
\(775\) 0 0
\(776\) −3.92429 −0.140874
\(777\) − 3.82377i − 0.137177i
\(778\) 68.8562i 2.46862i
\(779\) −4.34858 −0.155804
\(780\) 0 0
\(781\) −78.2695 −2.80070
\(782\) − 19.1712i − 0.685559i
\(783\) 2.46794i 0.0881969i
\(784\) 24.6015 0.878624
\(785\) 0 0
\(786\) 1.63303 0.0582483
\(787\) 15.9991i 0.570307i 0.958482 + 0.285153i \(0.0920445\pi\)
−0.958482 + 0.285153i \(0.907956\pi\)
\(788\) 16.0893i 0.573158i
\(789\) −0.253546 −0.00902649
\(790\) 0 0
\(791\) −6.62828 −0.235675
\(792\) 5.52247i 0.196232i
\(793\) − 48.5290i − 1.72332i
\(794\) 70.3621 2.49706
\(795\) 0 0
\(796\) 38.4602 1.36318
\(797\) − 35.7528i − 1.26643i −0.773976 0.633214i \(-0.781735\pi\)
0.773976 0.633214i \(-0.218265\pi\)
\(798\) 2.06888i 0.0732374i
\(799\) 11.6168 0.410974
\(800\) 0 0
\(801\) 26.8769 0.949650
\(802\) 45.4602i 1.60526i
\(803\) − 36.6827i − 1.29450i
\(804\) 2.58016 0.0909951
\(805\) 0 0
\(806\) 18.0770 0.636735
\(807\) 3.08457i 0.108582i
\(808\) − 4.04004i − 0.142128i
\(809\) −23.2036 −0.815796 −0.407898 0.913028i \(-0.633738\pi\)
−0.407898 + 0.913028i \(0.633738\pi\)
\(810\) 0 0
\(811\) −21.7549 −0.763918 −0.381959 0.924179i \(-0.624750\pi\)
−0.381959 + 0.924179i \(0.624750\pi\)
\(812\) − 9.14889i − 0.321063i
\(813\) 2.43402i 0.0853647i
\(814\) −39.3371 −1.37877
\(815\) 0 0
\(816\) −1.80265 −0.0631053
\(817\) − 3.56331i − 0.124664i
\(818\) 62.3802i 2.18107i
\(819\) −54.6445 −1.90943
\(820\) 0 0
\(821\) 52.2532 1.82365 0.911825 0.410579i \(-0.134673\pi\)
0.911825 + 0.410579i \(0.134673\pi\)
\(822\) 5.35904i 0.186918i
\(823\) 9.44783i 0.329331i 0.986349 + 0.164665i \(0.0526544\pi\)
−0.986349 + 0.164665i \(0.947346\pi\)
\(824\) 0.461775 0.0160867
\(825\) 0 0
\(826\) −66.0385 −2.29777
\(827\) − 50.2216i − 1.74638i −0.487383 0.873188i \(-0.662048\pi\)
0.487383 0.873188i \(-0.337952\pi\)
\(828\) − 37.0523i − 1.28766i
\(829\) −44.2066 −1.53536 −0.767680 0.640834i \(-0.778589\pi\)
−0.767680 + 0.640834i \(0.778589\pi\)
\(830\) 0 0
\(831\) 4.34826 0.150839
\(832\) − 34.4966i − 1.19595i
\(833\) − 8.01200i − 0.277599i
\(834\) 2.13592 0.0739609
\(835\) 0 0
\(836\) 10.1570 0.351287
\(837\) − 3.07965i − 0.106448i
\(838\) − 62.2455i − 2.15023i
\(839\) 30.4033 1.04964 0.524819 0.851214i \(-0.324133\pi\)
0.524819 + 0.851214i \(0.324133\pi\)
\(840\) 0 0
\(841\) −27.0223 −0.931803
\(842\) 0.681799i 0.0234963i
\(843\) 9.38404i 0.323204i
\(844\) −9.75945 −0.335934
\(845\) 0 0
\(846\) 47.0473 1.61752
\(847\) − 71.0893i − 2.44266i
\(848\) − 32.8833i − 1.12922i
\(849\) −7.34168 −0.251966
\(850\) 0 0
\(851\) −25.1955 −0.863690
\(852\) − 7.62459i − 0.261214i
\(853\) − 3.93925i − 0.134877i −0.997723 0.0674386i \(-0.978517\pi\)
0.997723 0.0674386i \(-0.0214827\pi\)
\(854\) 64.2232 2.19767
\(855\) 0 0
\(856\) 2.40773 0.0822945
\(857\) 27.4388i 0.937292i 0.883386 + 0.468646i \(0.155258\pi\)
−0.883386 + 0.468646i \(0.844742\pi\)
\(858\) − 17.0111i − 0.580751i
\(859\) −32.4517 −1.10724 −0.553619 0.832770i \(-0.686754\pi\)
−0.553619 + 0.832770i \(0.686754\pi\)
\(860\) 0 0
\(861\) −4.59966 −0.156756
\(862\) 57.2629i 1.95038i
\(863\) 2.10861i 0.0717778i 0.999356 + 0.0358889i \(0.0114263\pi\)
−0.999356 + 0.0358889i \(0.988574\pi\)
\(864\) −13.6256 −0.463553
\(865\) 0 0
\(866\) −1.72766 −0.0587084
\(867\) − 4.45924i − 0.151444i
\(868\) 11.4166i 0.387504i
\(869\) −30.4602 −1.03329
\(870\) 0 0
\(871\) −25.0731 −0.849569
\(872\) − 3.47184i − 0.117571i
\(873\) 33.5204i 1.13449i
\(874\) 13.6322 0.461115
\(875\) 0 0
\(876\) 3.57343 0.120735
\(877\) − 37.6613i − 1.27173i −0.771799 0.635866i \(-0.780643\pi\)
0.771799 0.635866i \(-0.219357\pi\)
\(878\) 27.1062i 0.914789i
\(879\) −8.99007 −0.303227
\(880\) 0 0
\(881\) −39.8818 −1.34365 −0.671827 0.740708i \(-0.734490\pi\)
−0.671827 + 0.740708i \(0.734490\pi\)
\(882\) − 32.4480i − 1.09258i
\(883\) − 36.0458i − 1.21304i −0.795070 0.606518i \(-0.792566\pi\)
0.795070 0.606518i \(-0.207434\pi\)
\(884\) −13.5218 −0.454787
\(885\) 0 0
\(886\) −26.5525 −0.892050
\(887\) − 26.2057i − 0.879902i −0.898022 0.439951i \(-0.854996\pi\)
0.898022 0.439951i \(-0.145004\pi\)
\(888\) 0.365818i 0.0122760i
\(889\) −17.4264 −0.584464
\(890\) 0 0
\(891\) 45.7011 1.53104
\(892\) − 5.67700i − 0.190080i
\(893\) 8.26046i 0.276426i
\(894\) −4.12186 −0.137856
\(895\) 0 0
\(896\) −9.68093 −0.323417
\(897\) − 10.8957i − 0.363796i
\(898\) 30.5780i 1.02040i
\(899\) −2.46794 −0.0823103
\(900\) 0 0
\(901\) −10.7092 −0.356774
\(902\) 47.3191i 1.57555i
\(903\) − 3.76905i − 0.125426i
\(904\) 0.634123 0.0210906
\(905\) 0 0
\(906\) −10.6580 −0.354088
\(907\) − 22.8036i − 0.757182i −0.925564 0.378591i \(-0.876409\pi\)
0.925564 0.378591i \(-0.123591\pi\)
\(908\) − 26.4166i − 0.876664i
\(909\) −34.5091 −1.14460
\(910\) 0 0
\(911\) 4.44146 0.147152 0.0735761 0.997290i \(-0.476559\pi\)
0.0735761 + 0.997290i \(0.476559\pi\)
\(912\) − 1.28182i − 0.0424454i
\(913\) 23.1266i 0.765379i
\(914\) −10.3721 −0.343077
\(915\) 0 0
\(916\) −9.52059 −0.314569
\(917\) − 10.0223i − 0.330965i
\(918\) 4.82714i 0.159319i
\(919\) −37.0111 −1.22088 −0.610442 0.792061i \(-0.709008\pi\)
−0.610442 + 0.792061i \(0.709008\pi\)
\(920\) 0 0
\(921\) −6.84602 −0.225584
\(922\) 0.732959i 0.0241387i
\(923\) 74.0932i 2.43881i
\(924\) 10.7434 0.353433
\(925\) 0 0
\(926\) 13.0181 0.427800
\(927\) − 3.94438i − 0.129550i
\(928\) 10.9191i 0.358438i
\(929\) 3.04185 0.0997999 0.0499000 0.998754i \(-0.484110\pi\)
0.0499000 + 0.998754i \(0.484110\pi\)
\(930\) 0 0
\(931\) 5.69716 0.186717
\(932\) − 5.81921i − 0.190615i
\(933\) 3.06198i 0.100245i
\(934\) 1.67066 0.0546657
\(935\) 0 0
\(936\) 5.22781 0.170876
\(937\) 38.6099i 1.26133i 0.776055 + 0.630666i \(0.217218\pi\)
−0.776055 + 0.630666i \(0.782782\pi\)
\(938\) − 33.1817i − 1.08342i
\(939\) 1.46948 0.0479547
\(940\) 0 0
\(941\) 2.69716 0.0879248 0.0439624 0.999033i \(-0.486002\pi\)
0.0439624 + 0.999033i \(0.486002\pi\)
\(942\) 10.0039i 0.325944i
\(943\) 30.3080i 0.986963i
\(944\) 40.9158 1.33170
\(945\) 0 0
\(946\) −38.7742 −1.26066
\(947\) − 20.1877i − 0.656012i −0.944676 0.328006i \(-0.893623\pi\)
0.944676 0.328006i \(-0.106377\pi\)
\(948\) − 2.96727i − 0.0963723i
\(949\) −34.7254 −1.12723
\(950\) 0 0
\(951\) −4.22801 −0.137103
\(952\) 1.70829i 0.0553661i
\(953\) − 5.18559i − 0.167978i −0.996467 0.0839888i \(-0.973234\pi\)
0.996467 0.0839888i \(-0.0267660\pi\)
\(954\) −43.3713 −1.40420
\(955\) 0 0
\(956\) 30.1624 0.975523
\(957\) 2.32242i 0.0750732i
\(958\) 33.3940i 1.07891i
\(959\) 32.8896 1.06206
\(960\) 0 0
\(961\) −27.9203 −0.900656
\(962\) 37.2382i 1.20061i
\(963\) − 20.5663i − 0.662739i
\(964\) 22.3231 0.718979
\(965\) 0 0
\(966\) 14.4193 0.463933
\(967\) 58.0054i 1.86533i 0.360744 + 0.932665i \(0.382523\pi\)
−0.360744 + 0.932665i \(0.617477\pi\)
\(968\) 6.80107i 0.218595i
\(969\) −0.417453 −0.0134105
\(970\) 0 0
\(971\) −24.3221 −0.780533 −0.390267 0.920702i \(-0.627617\pi\)
−0.390267 + 0.920702i \(0.627617\pi\)
\(972\) 14.0637i 0.451095i
\(973\) − 13.1086i − 0.420244i
\(974\) −25.0867 −0.803830
\(975\) 0 0
\(976\) −39.7911 −1.27368
\(977\) − 36.1134i − 1.15537i −0.816260 0.577685i \(-0.803956\pi\)
0.816260 0.577685i \(-0.196044\pi\)
\(978\) − 6.30895i − 0.201738i
\(979\) 51.3498 1.64115
\(980\) 0 0
\(981\) −29.6557 −0.946832
\(982\) − 20.3525i − 0.649473i
\(983\) − 21.1603i − 0.674909i −0.941342 0.337455i \(-0.890434\pi\)
0.941342 0.337455i \(-0.109566\pi\)
\(984\) 0.440046 0.0140282
\(985\) 0 0
\(986\) 3.86832 0.123192
\(987\) 8.73741i 0.278115i
\(988\) − 9.61504i − 0.305895i
\(989\) −24.8349 −0.789704
\(990\) 0 0
\(991\) 20.2652 0.643746 0.321873 0.946783i \(-0.395688\pi\)
0.321873 + 0.946783i \(0.395688\pi\)
\(992\) − 13.6256i − 0.432614i
\(993\) − 0.606364i − 0.0192424i
\(994\) −98.0548 −3.11011
\(995\) 0 0
\(996\) −2.25287 −0.0713849
\(997\) 24.2875i 0.769193i 0.923085 + 0.384597i \(0.125659\pi\)
−0.923085 + 0.384597i \(0.874341\pi\)
\(998\) − 70.7293i − 2.23890i
\(999\) 6.34402 0.200716
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 475.2.b.e.324.6 8
5.2 odd 4 475.2.a.i.1.1 4
5.3 odd 4 95.2.a.b.1.4 4
5.4 even 2 inner 475.2.b.e.324.3 8
15.2 even 4 4275.2.a.bo.1.4 4
15.8 even 4 855.2.a.m.1.1 4
20.3 even 4 1520.2.a.t.1.3 4
20.7 even 4 7600.2.a.cf.1.2 4
35.13 even 4 4655.2.a.y.1.4 4
40.3 even 4 6080.2.a.ch.1.2 4
40.13 odd 4 6080.2.a.cc.1.3 4
95.18 even 4 1805.2.a.p.1.1 4
95.37 even 4 9025.2.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.4 4 5.3 odd 4
475.2.a.i.1.1 4 5.2 odd 4
475.2.b.e.324.3 8 5.4 even 2 inner
475.2.b.e.324.6 8 1.1 even 1 trivial
855.2.a.m.1.1 4 15.8 even 4
1520.2.a.t.1.3 4 20.3 even 4
1805.2.a.p.1.1 4 95.18 even 4
4275.2.a.bo.1.4 4 15.2 even 4
4655.2.a.y.1.4 4 35.13 even 4
6080.2.a.cc.1.3 4 40.13 odd 4
6080.2.a.ch.1.2 4 40.3 even 4
7600.2.a.cf.1.2 4 20.7 even 4
9025.2.a.bf.1.4 4 95.37 even 4