Properties

Label 625.2.b.c.624.8
Level $625$
Weight $2$
Character 625.624
Analytic conductor $4.991$
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] = [625,2,Mod(624,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.624");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.58140625.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 624.8
Root \(-0.357358 + 1.86824i\) of defining polynomial
Character \(\chi\) \(=\) 625.624
Dual form 625.2.b.c.624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.30927i q^{2} -0.474903i q^{3} -3.33275 q^{4} +1.09668 q^{6} +3.03582i q^{7} -3.07768i q^{8} +2.77447 q^{9} +O(q^{10})\) \(q+2.30927i q^{2} -0.474903i q^{3} -3.33275 q^{4} +1.09668 q^{6} +3.03582i q^{7} -3.07768i q^{8} +2.77447 q^{9} +2.00000 q^{11} +1.58273i q^{12} +1.42721i q^{13} -7.01054 q^{14} +0.441718 q^{16} +1.86025i q^{17} +6.40701i q^{18} -0.903319 q^{19} +1.44172 q^{21} +4.61855i q^{22} +3.32932i q^{23} -1.46160 q^{24} -3.29582 q^{26} -2.74231i q^{27} -10.1176i q^{28} -3.96307 q^{29} -6.43997 q^{31} -5.13532i q^{32} -0.949806i q^{33} -4.29582 q^{34} -9.24660 q^{36} +3.82022i q^{37} -2.08601i q^{38} +0.677786 q^{39} -1.83422 q^{41} +3.32932i q^{42} +3.59445i q^{43} -6.66550 q^{44} -7.68832 q^{46} -4.79995i q^{47} -0.209773i q^{48} -2.21619 q^{49} +0.883436 q^{51} -4.75653i q^{52} +9.50473i q^{53} +6.33275 q^{54} +9.34328 q^{56} +0.428989i q^{57} -9.15182i q^{58} -10.6456 q^{59} +14.2742 q^{61} -14.8716i q^{62} +8.42278i q^{63} +12.7423 q^{64} +2.19336 q^{66} -10.6902i q^{67} -6.19974i q^{68} +1.58111 q^{69} +12.4598 q^{71} -8.53893i q^{72} -0.267631i q^{73} -8.82193 q^{74} +3.01054 q^{76} +6.07163i q^{77} +1.56519i q^{78} +8.57176 q^{79} +7.02107 q^{81} -4.23572i q^{82} +12.6182i q^{83} -4.80489 q^{84} -8.30058 q^{86} +1.88207i q^{87} -6.15537i q^{88} +4.76796 q^{89} -4.33275 q^{91} -11.0958i q^{92} +3.05836i q^{93} +11.0844 q^{94} -2.43878 q^{96} -9.95805i q^{97} -5.11778i q^{98} +5.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9} + 16 q^{11} - 12 q^{14} - 2 q^{16} - 10 q^{19} + 6 q^{21} - 20 q^{24} + 6 q^{26} - 20 q^{29} + 16 q^{31} - 2 q^{34} - 12 q^{36} - 18 q^{39} + 26 q^{41} - 12 q^{44} + 6 q^{46} + 14 q^{49} - 4 q^{51} + 30 q^{54} + 10 q^{56} - 30 q^{59} + 6 q^{61} + 44 q^{64} + 12 q^{66} - 8 q^{69} + 46 q^{71} - 12 q^{74} - 20 q^{76} - 10 q^{79} - 32 q^{81} + 18 q^{84} - 14 q^{86} - 30 q^{89} - 14 q^{91} + 68 q^{94} - 54 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30927i 1.63290i 0.577414 + 0.816452i \(0.304062\pi\)
−0.577414 + 0.816452i \(0.695938\pi\)
\(3\) − 0.474903i − 0.274185i −0.990558 0.137093i \(-0.956224\pi\)
0.990558 0.137093i \(-0.0437758\pi\)
\(4\) −3.33275 −1.66637
\(5\) 0 0
\(6\) 1.09668 0.447718
\(7\) 3.03582i 1.14743i 0.819055 + 0.573716i \(0.194498\pi\)
−0.819055 + 0.573716i \(0.805502\pi\)
\(8\) − 3.07768i − 1.08813i
\(9\) 2.77447 0.924822
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.58273i 0.456895i
\(13\) 1.42721i 0.395837i 0.980218 + 0.197918i \(0.0634181\pi\)
−0.980218 + 0.197918i \(0.936582\pi\)
\(14\) −7.01054 −1.87364
\(15\) 0 0
\(16\) 0.441718 0.110430
\(17\) 1.86025i 0.451176i 0.974223 + 0.225588i \(0.0724303\pi\)
−0.974223 + 0.225588i \(0.927570\pi\)
\(18\) 6.40701i 1.51015i
\(19\) −0.903319 −0.207236 −0.103618 0.994617i \(-0.533042\pi\)
−0.103618 + 0.994617i \(0.533042\pi\)
\(20\) 0 0
\(21\) 1.44172 0.314609
\(22\) 4.61855i 0.984678i
\(23\) 3.32932i 0.694212i 0.937826 + 0.347106i \(0.112836\pi\)
−0.937826 + 0.347106i \(0.887164\pi\)
\(24\) −1.46160 −0.298348
\(25\) 0 0
\(26\) −3.29582 −0.646364
\(27\) − 2.74231i − 0.527758i
\(28\) − 10.1176i − 1.91205i
\(29\) −3.96307 −0.735924 −0.367962 0.929841i \(-0.619944\pi\)
−0.367962 + 0.929841i \(0.619944\pi\)
\(30\) 0 0
\(31\) −6.43997 −1.15665 −0.578326 0.815806i \(-0.696294\pi\)
−0.578326 + 0.815806i \(0.696294\pi\)
\(32\) − 5.13532i − 0.907805i
\(33\) − 0.949806i − 0.165340i
\(34\) −4.29582 −0.736727
\(35\) 0 0
\(36\) −9.24660 −1.54110
\(37\) 3.82022i 0.628040i 0.949416 + 0.314020i \(0.101676\pi\)
−0.949416 + 0.314020i \(0.898324\pi\)
\(38\) − 2.08601i − 0.338396i
\(39\) 0.677786 0.108533
\(40\) 0 0
\(41\) −1.83422 −0.286457 −0.143228 0.989690i \(-0.545748\pi\)
−0.143228 + 0.989690i \(0.545748\pi\)
\(42\) 3.32932i 0.513726i
\(43\) 3.59445i 0.548149i 0.961708 + 0.274074i \(0.0883715\pi\)
−0.961708 + 0.274074i \(0.911629\pi\)
\(44\) −6.66550 −1.00486
\(45\) 0 0
\(46\) −7.68832 −1.13358
\(47\) − 4.79995i − 0.700144i −0.936723 0.350072i \(-0.886157\pi\)
0.936723 0.350072i \(-0.113843\pi\)
\(48\) − 0.209773i − 0.0302782i
\(49\) −2.21619 −0.316598
\(50\) 0 0
\(51\) 0.883436 0.123706
\(52\) − 4.75653i − 0.659613i
\(53\) 9.50473i 1.30558i 0.757541 + 0.652788i \(0.226401\pi\)
−0.757541 + 0.652788i \(0.773599\pi\)
\(54\) 6.33275 0.861778
\(55\) 0 0
\(56\) 9.34328 1.24855
\(57\) 0.428989i 0.0568209i
\(58\) − 9.15182i − 1.20169i
\(59\) −10.6456 −1.38594 −0.692971 0.720966i \(-0.743698\pi\)
−0.692971 + 0.720966i \(0.743698\pi\)
\(60\) 0 0
\(61\) 14.2742 1.82762 0.913811 0.406140i \(-0.133125\pi\)
0.913811 + 0.406140i \(0.133125\pi\)
\(62\) − 14.8716i − 1.88870i
\(63\) 8.42278i 1.06117i
\(64\) 12.7423 1.59279
\(65\) 0 0
\(66\) 2.19336 0.269984
\(67\) − 10.6902i − 1.30601i −0.757352 0.653007i \(-0.773507\pi\)
0.757352 0.653007i \(-0.226493\pi\)
\(68\) − 6.19974i − 0.751828i
\(69\) 1.58111 0.190343
\(70\) 0 0
\(71\) 12.4598 1.47871 0.739356 0.673315i \(-0.235130\pi\)
0.739356 + 0.673315i \(0.235130\pi\)
\(72\) − 8.53893i − 1.00632i
\(73\) − 0.267631i − 0.0313239i −0.999877 0.0156619i \(-0.995014\pi\)
0.999877 0.0156619i \(-0.00498555\pi\)
\(74\) −8.82193 −1.02553
\(75\) 0 0
\(76\) 3.01054 0.345332
\(77\) 6.07163i 0.691927i
\(78\) 1.56519i 0.177223i
\(79\) 8.57176 0.964398 0.482199 0.876062i \(-0.339838\pi\)
0.482199 + 0.876062i \(0.339838\pi\)
\(80\) 0 0
\(81\) 7.02107 0.780119
\(82\) − 4.23572i − 0.467757i
\(83\) 12.6182i 1.38502i 0.721406 + 0.692512i \(0.243496\pi\)
−0.721406 + 0.692512i \(0.756504\pi\)
\(84\) −4.80489 −0.524256
\(85\) 0 0
\(86\) −8.30058 −0.895074
\(87\) 1.88207i 0.201779i
\(88\) − 6.15537i − 0.656164i
\(89\) 4.76796 0.505402 0.252701 0.967544i \(-0.418681\pi\)
0.252701 + 0.967544i \(0.418681\pi\)
\(90\) 0 0
\(91\) −4.33275 −0.454196
\(92\) − 11.0958i − 1.15682i
\(93\) 3.05836i 0.317137i
\(94\) 11.0844 1.14327
\(95\) 0 0
\(96\) −2.43878 −0.248907
\(97\) − 9.95805i − 1.01109i −0.862801 0.505543i \(-0.831292\pi\)
0.862801 0.505543i \(-0.168708\pi\)
\(98\) − 5.11778i − 0.516974i
\(99\) 5.54893 0.557689
\(100\) 0 0
\(101\) 9.34612 0.929974 0.464987 0.885318i \(-0.346059\pi\)
0.464987 + 0.885318i \(0.346059\pi\)
\(102\) 2.04010i 0.202000i
\(103\) − 9.08408i − 0.895081i −0.894264 0.447540i \(-0.852300\pi\)
0.894264 0.447540i \(-0.147700\pi\)
\(104\) 4.39250 0.430720
\(105\) 0 0
\(106\) −21.9490 −2.13188
\(107\) 5.62871i 0.544148i 0.962276 + 0.272074i \(0.0877096\pi\)
−0.962276 + 0.272074i \(0.912290\pi\)
\(108\) 9.13943i 0.879442i
\(109\) 10.1130 0.968649 0.484325 0.874888i \(-0.339065\pi\)
0.484325 + 0.874888i \(0.339065\pi\)
\(110\) 0 0
\(111\) 1.81423 0.172199
\(112\) 1.34098i 0.126710i
\(113\) − 10.7120i − 1.00770i −0.863791 0.503851i \(-0.831916\pi\)
0.863791 0.503851i \(-0.168084\pi\)
\(114\) −0.990653 −0.0927831
\(115\) 0 0
\(116\) 13.2079 1.22632
\(117\) 3.95975i 0.366079i
\(118\) − 24.5836i − 2.26311i
\(119\) −5.64737 −0.517693
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 32.9630i 2.98433i
\(123\) 0.871076i 0.0785423i
\(124\) 21.4628 1.92742
\(125\) 0 0
\(126\) −19.4505 −1.73279
\(127\) − 11.3609i − 1.00812i −0.863669 0.504060i \(-0.831839\pi\)
0.863669 0.504060i \(-0.168161\pi\)
\(128\) 19.1548i 1.69306i
\(129\) 1.70702 0.150294
\(130\) 0 0
\(131\) 7.98771 0.697890 0.348945 0.937143i \(-0.386540\pi\)
0.348945 + 0.937143i \(0.386540\pi\)
\(132\) 3.16546i 0.275518i
\(133\) − 2.74231i − 0.237789i
\(134\) 24.6866 2.13259
\(135\) 0 0
\(136\) 5.72525 0.490936
\(137\) 9.33726i 0.797736i 0.917008 + 0.398868i \(0.130597\pi\)
−0.917008 + 0.398868i \(0.869403\pi\)
\(138\) 3.65121i 0.310811i
\(139\) −17.9150 −1.51953 −0.759767 0.650195i \(-0.774687\pi\)
−0.759767 + 0.650195i \(0.774687\pi\)
\(140\) 0 0
\(141\) −2.27951 −0.191969
\(142\) 28.7732i 2.41459i
\(143\) 2.85442i 0.238699i
\(144\) 1.22553 0.102128
\(145\) 0 0
\(146\) 0.618034 0.0511489
\(147\) 1.05247i 0.0868065i
\(148\) − 12.7318i − 1.04655i
\(149\) 6.31395 0.517259 0.258629 0.965977i \(-0.416729\pi\)
0.258629 + 0.965977i \(0.416729\pi\)
\(150\) 0 0
\(151\) 4.71947 0.384065 0.192033 0.981389i \(-0.438492\pi\)
0.192033 + 0.981389i \(0.438492\pi\)
\(152\) 2.78013i 0.225498i
\(153\) 5.16119i 0.417258i
\(154\) −14.0211 −1.12985
\(155\) 0 0
\(156\) −2.25889 −0.180856
\(157\) 1.46908i 0.117245i 0.998280 + 0.0586225i \(0.0186708\pi\)
−0.998280 + 0.0586225i \(0.981329\pi\)
\(158\) 19.7945i 1.57477i
\(159\) 4.51382 0.357969
\(160\) 0 0
\(161\) −10.1072 −0.796560
\(162\) 16.2136i 1.27386i
\(163\) − 4.45969i − 0.349310i −0.984630 0.174655i \(-0.944119\pi\)
0.984630 0.174655i \(-0.0558810\pi\)
\(164\) 6.11299 0.477345
\(165\) 0 0
\(166\) −29.1388 −2.26161
\(167\) 10.4337i 0.807381i 0.914896 + 0.403691i \(0.132273\pi\)
−0.914896 + 0.403691i \(0.867727\pi\)
\(168\) − 4.43715i − 0.342334i
\(169\) 10.9631 0.843313
\(170\) 0 0
\(171\) −2.50623 −0.191656
\(172\) − 11.9794i − 0.913421i
\(173\) 7.67619i 0.583610i 0.956478 + 0.291805i \(0.0942559\pi\)
−0.956478 + 0.291805i \(0.905744\pi\)
\(174\) −4.34623 −0.329486
\(175\) 0 0
\(176\) 0.883436 0.0665915
\(177\) 5.05563i 0.380005i
\(178\) 11.0105i 0.825273i
\(179\) −15.5168 −1.15978 −0.579889 0.814696i \(-0.696904\pi\)
−0.579889 + 0.814696i \(0.696904\pi\)
\(180\) 0 0
\(181\) −1.59056 −0.118225 −0.0591126 0.998251i \(-0.518827\pi\)
−0.0591126 + 0.998251i \(0.518827\pi\)
\(182\) − 10.0055i − 0.741658i
\(183\) − 6.77885i − 0.501107i
\(184\) 10.2466 0.755390
\(185\) 0 0
\(186\) −7.06259 −0.517854
\(187\) 3.72049i 0.272069i
\(188\) 15.9970i 1.16670i
\(189\) 8.32515 0.605566
\(190\) 0 0
\(191\) 19.6684 1.42316 0.711579 0.702606i \(-0.247980\pi\)
0.711579 + 0.702606i \(0.247980\pi\)
\(192\) − 6.05135i − 0.436719i
\(193\) − 13.1100i − 0.943680i −0.881684 0.471840i \(-0.843590\pi\)
0.881684 0.471840i \(-0.156410\pi\)
\(194\) 22.9959 1.65101
\(195\) 0 0
\(196\) 7.38599 0.527571
\(197\) − 3.42949i − 0.244341i −0.992509 0.122170i \(-0.961015\pi\)
0.992509 0.122170i \(-0.0389855\pi\)
\(198\) 12.8140i 0.910652i
\(199\) −17.6959 −1.25443 −0.627215 0.778846i \(-0.715805\pi\)
−0.627215 + 0.778846i \(0.715805\pi\)
\(200\) 0 0
\(201\) −5.07680 −0.358090
\(202\) 21.5828i 1.51856i
\(203\) − 12.0312i − 0.844422i
\(204\) −2.94427 −0.206140
\(205\) 0 0
\(206\) 20.9776 1.46158
\(207\) 9.23710i 0.642023i
\(208\) 0.630425i 0.0437121i
\(209\) −1.80664 −0.124968
\(210\) 0 0
\(211\) −3.24366 −0.223303 −0.111651 0.993747i \(-0.535614\pi\)
−0.111651 + 0.993747i \(0.535614\pi\)
\(212\) − 31.6769i − 2.17558i
\(213\) − 5.91722i − 0.405441i
\(214\) −12.9982 −0.888542
\(215\) 0 0
\(216\) −8.43997 −0.574267
\(217\) − 19.5506i − 1.32718i
\(218\) 23.3537i 1.58171i
\(219\) −0.127099 −0.00858854
\(220\) 0 0
\(221\) −2.65496 −0.178592
\(222\) 4.18956i 0.281185i
\(223\) − 28.7148i − 1.92288i −0.275010 0.961441i \(-0.588681\pi\)
0.275010 0.961441i \(-0.411319\pi\)
\(224\) 15.5899 1.04164
\(225\) 0 0
\(226\) 24.7370 1.64548
\(227\) 11.7206i 0.777926i 0.921253 + 0.388963i \(0.127167\pi\)
−0.921253 + 0.388963i \(0.872833\pi\)
\(228\) − 1.42971i − 0.0946850i
\(229\) 16.4013 1.08383 0.541914 0.840434i \(-0.317700\pi\)
0.541914 + 0.840434i \(0.317700\pi\)
\(230\) 0 0
\(231\) 2.88344 0.189716
\(232\) 12.1971i 0.800777i
\(233\) − 22.5146i − 1.47498i −0.675358 0.737490i \(-0.736011\pi\)
0.675358 0.737490i \(-0.263989\pi\)
\(234\) −9.14415 −0.597771
\(235\) 0 0
\(236\) 35.4792 2.30950
\(237\) − 4.07075i − 0.264424i
\(238\) − 13.0413i − 0.845344i
\(239\) 6.63333 0.429074 0.214537 0.976716i \(-0.431176\pi\)
0.214537 + 0.976716i \(0.431176\pi\)
\(240\) 0 0
\(241\) 26.2261 1.68937 0.844684 0.535265i \(-0.179788\pi\)
0.844684 + 0.535265i \(0.179788\pi\)
\(242\) − 16.1649i − 1.03912i
\(243\) − 11.5613i − 0.741655i
\(244\) −47.5723 −3.04550
\(245\) 0 0
\(246\) −2.01155 −0.128252
\(247\) − 1.28923i − 0.0820315i
\(248\) 19.8202i 1.25858i
\(249\) 5.99241 0.379753
\(250\) 0 0
\(251\) −10.9121 −0.688766 −0.344383 0.938829i \(-0.611912\pi\)
−0.344383 + 0.938829i \(0.611912\pi\)
\(252\) − 28.0710i − 1.76831i
\(253\) 6.65865i 0.418626i
\(254\) 26.2355 1.64616
\(255\) 0 0
\(256\) −18.7492 −1.17182
\(257\) 6.58051i 0.410481i 0.978712 + 0.205240i \(0.0657976\pi\)
−0.978712 + 0.205240i \(0.934202\pi\)
\(258\) 3.94197i 0.245416i
\(259\) −11.5975 −0.720632
\(260\) 0 0
\(261\) −10.9954 −0.680599
\(262\) 18.4458i 1.13959i
\(263\) 27.1073i 1.67151i 0.549106 + 0.835753i \(0.314968\pi\)
−0.549106 + 0.835753i \(0.685032\pi\)
\(264\) −2.92320 −0.179911
\(265\) 0 0
\(266\) 6.33275 0.388286
\(267\) − 2.26432i − 0.138574i
\(268\) 35.6277i 2.17631i
\(269\) −1.00945 −0.0615474 −0.0307737 0.999526i \(-0.509797\pi\)
−0.0307737 + 0.999526i \(0.509797\pi\)
\(270\) 0 0
\(271\) 6.25203 0.379784 0.189892 0.981805i \(-0.439186\pi\)
0.189892 + 0.981805i \(0.439186\pi\)
\(272\) 0.821705i 0.0498232i
\(273\) 2.05763i 0.124534i
\(274\) −21.5623 −1.30263
\(275\) 0 0
\(276\) −5.26943 −0.317182
\(277\) − 24.6703i − 1.48230i −0.671342 0.741148i \(-0.734282\pi\)
0.671342 0.741148i \(-0.265718\pi\)
\(278\) − 41.3708i − 2.48125i
\(279\) −17.8675 −1.06970
\(280\) 0 0
\(281\) 1.83891 0.109700 0.0548502 0.998495i \(-0.482532\pi\)
0.0548502 + 0.998495i \(0.482532\pi\)
\(282\) − 5.26401i − 0.313467i
\(283\) 8.64116i 0.513664i 0.966456 + 0.256832i \(0.0826786\pi\)
−0.966456 + 0.256832i \(0.917321\pi\)
\(284\) −41.5255 −2.46409
\(285\) 0 0
\(286\) −6.59164 −0.389772
\(287\) − 5.56835i − 0.328690i
\(288\) − 14.2478i − 0.839558i
\(289\) 13.5395 0.796440
\(290\) 0 0
\(291\) −4.72910 −0.277225
\(292\) 0.891948i 0.0521973i
\(293\) − 6.29156i − 0.367557i −0.982968 0.183779i \(-0.941167\pi\)
0.982968 0.183779i \(-0.0588329\pi\)
\(294\) −2.43045 −0.141747
\(295\) 0 0
\(296\) 11.7574 0.683386
\(297\) − 5.48462i − 0.318250i
\(298\) 14.5806i 0.844634i
\(299\) −4.75164 −0.274795
\(300\) 0 0
\(301\) −10.9121 −0.628963
\(302\) 10.8986i 0.627142i
\(303\) − 4.43850i − 0.254985i
\(304\) −0.399012 −0.0228849
\(305\) 0 0
\(306\) −11.9186 −0.681342
\(307\) − 28.6661i − 1.63606i −0.575175 0.818030i \(-0.695066\pi\)
0.575175 0.818030i \(-0.304934\pi\)
\(308\) − 20.2352i − 1.15301i
\(309\) −4.31405 −0.245418
\(310\) 0 0
\(311\) −7.83649 −0.444367 −0.222183 0.975005i \(-0.571318\pi\)
−0.222183 + 0.975005i \(0.571318\pi\)
\(312\) − 2.08601i − 0.118097i
\(313\) − 21.4093i − 1.21012i −0.796179 0.605061i \(-0.793149\pi\)
0.796179 0.605061i \(-0.206851\pi\)
\(314\) −3.39250 −0.191450
\(315\) 0 0
\(316\) −28.5675 −1.60705
\(317\) 4.01983i 0.225776i 0.993608 + 0.112888i \(0.0360102\pi\)
−0.993608 + 0.112888i \(0.963990\pi\)
\(318\) 10.4237i 0.584530i
\(319\) −7.92614 −0.443779
\(320\) 0 0
\(321\) 2.67309 0.149197
\(322\) − 23.3403i − 1.30071i
\(323\) − 1.68040i − 0.0934997i
\(324\) −23.3995 −1.29997
\(325\) 0 0
\(326\) 10.2987 0.570390
\(327\) − 4.80269i − 0.265589i
\(328\) 5.64515i 0.311701i
\(329\) 14.5718 0.803367
\(330\) 0 0
\(331\) 11.6439 0.640005 0.320002 0.947417i \(-0.396316\pi\)
0.320002 + 0.947417i \(0.396316\pi\)
\(332\) − 42.0532i − 2.30797i
\(333\) 10.5991i 0.580825i
\(334\) −24.0942 −1.31838
\(335\) 0 0
\(336\) 0.636833 0.0347421
\(337\) − 21.5348i − 1.17307i −0.809922 0.586537i \(-0.800491\pi\)
0.809922 0.586537i \(-0.199509\pi\)
\(338\) 25.3167i 1.37705i
\(339\) −5.08716 −0.276297
\(340\) 0 0
\(341\) −12.8799 −0.697487
\(342\) − 5.78757i − 0.312956i
\(343\) 14.5228i 0.784157i
\(344\) 11.0626 0.596455
\(345\) 0 0
\(346\) −17.7264 −0.952979
\(347\) 15.5972i 0.837303i 0.908147 + 0.418652i \(0.137497\pi\)
−0.908147 + 0.418652i \(0.862503\pi\)
\(348\) − 6.27248i − 0.336240i
\(349\) −5.56598 −0.297940 −0.148970 0.988842i \(-0.547596\pi\)
−0.148970 + 0.988842i \(0.547596\pi\)
\(350\) 0 0
\(351\) 3.91385 0.208906
\(352\) − 10.2706i − 0.547427i
\(353\) − 8.02216i − 0.426977i −0.976946 0.213488i \(-0.931517\pi\)
0.976946 0.213488i \(-0.0684825\pi\)
\(354\) −11.6748 −0.620511
\(355\) 0 0
\(356\) −15.8904 −0.842190
\(357\) 2.68195i 0.141944i
\(358\) − 35.8325i − 1.89380i
\(359\) 12.3427 0.651424 0.325712 0.945469i \(-0.394396\pi\)
0.325712 + 0.945469i \(0.394396\pi\)
\(360\) 0 0
\(361\) −18.1840 −0.957053
\(362\) − 3.67303i − 0.193050i
\(363\) 3.32432i 0.174482i
\(364\) 14.4400 0.756860
\(365\) 0 0
\(366\) 15.6542 0.818260
\(367\) 26.8749i 1.40286i 0.712738 + 0.701430i \(0.247455\pi\)
−0.712738 + 0.701430i \(0.752545\pi\)
\(368\) 1.47062i 0.0766615i
\(369\) −5.08898 −0.264922
\(370\) 0 0
\(371\) −28.8546 −1.49806
\(372\) − 10.1927i − 0.528469i
\(373\) 27.6389i 1.43109i 0.698567 + 0.715544i \(0.253821\pi\)
−0.698567 + 0.715544i \(0.746179\pi\)
\(374\) −8.59164 −0.444263
\(375\) 0 0
\(376\) −14.7727 −0.761845
\(377\) − 5.65614i − 0.291306i
\(378\) 19.2251i 0.988831i
\(379\) 3.47462 0.178479 0.0892397 0.996010i \(-0.471556\pi\)
0.0892397 + 0.996010i \(0.471556\pi\)
\(380\) 0 0
\(381\) −5.39534 −0.276412
\(382\) 45.4198i 2.32388i
\(383\) − 27.3719i − 1.39864i −0.714810 0.699319i \(-0.753487\pi\)
0.714810 0.699319i \(-0.246513\pi\)
\(384\) 9.09668 0.464213
\(385\) 0 0
\(386\) 30.2746 1.54094
\(387\) 9.97269i 0.506940i
\(388\) 33.1877i 1.68485i
\(389\) −10.8845 −0.551867 −0.275934 0.961177i \(-0.588987\pi\)
−0.275934 + 0.961177i \(0.588987\pi\)
\(390\) 0 0
\(391\) −6.19336 −0.313212
\(392\) 6.82072i 0.344498i
\(393\) − 3.79339i − 0.191351i
\(394\) 7.91963 0.398985
\(395\) 0 0
\(396\) −18.4932 −0.929319
\(397\) 16.2212i 0.814120i 0.913401 + 0.407060i \(0.133446\pi\)
−0.913401 + 0.407060i \(0.866554\pi\)
\(398\) − 40.8647i − 2.04836i
\(399\) −1.30233 −0.0651981
\(400\) 0 0
\(401\) 3.78686 0.189107 0.0945534 0.995520i \(-0.469858\pi\)
0.0945534 + 0.995520i \(0.469858\pi\)
\(402\) − 11.7237i − 0.584726i
\(403\) − 9.19118i − 0.457845i
\(404\) −31.1483 −1.54968
\(405\) 0 0
\(406\) 27.7832 1.37886
\(407\) 7.64044i 0.378722i
\(408\) − 2.71894i − 0.134607i
\(409\) −1.85585 −0.0917661 −0.0458831 0.998947i \(-0.514610\pi\)
−0.0458831 + 0.998947i \(0.514610\pi\)
\(410\) 0 0
\(411\) 4.43429 0.218728
\(412\) 30.2750i 1.49154i
\(413\) − 32.3181i − 1.59027i
\(414\) −21.3310 −1.04836
\(415\) 0 0
\(416\) 7.32918 0.359343
\(417\) 8.50790i 0.416634i
\(418\) − 4.17202i − 0.204060i
\(419\) −14.3472 −0.700907 −0.350453 0.936580i \(-0.613972\pi\)
−0.350453 + 0.936580i \(0.613972\pi\)
\(420\) 0 0
\(421\) 15.4545 0.753207 0.376604 0.926374i \(-0.377092\pi\)
0.376604 + 0.926374i \(0.377092\pi\)
\(422\) − 7.49051i − 0.364632i
\(423\) − 13.3173i − 0.647509i
\(424\) 29.2525 1.42063
\(425\) 0 0
\(426\) 13.6645 0.662046
\(427\) 43.3338i 2.09707i
\(428\) − 18.7591i − 0.906755i
\(429\) 1.35557 0.0654477
\(430\) 0 0
\(431\) 12.5043 0.602311 0.301156 0.953575i \(-0.402628\pi\)
0.301156 + 0.953575i \(0.402628\pi\)
\(432\) − 1.21133i − 0.0582801i
\(433\) 22.4951i 1.08105i 0.841329 + 0.540524i \(0.181774\pi\)
−0.841329 + 0.540524i \(0.818226\pi\)
\(434\) 45.1476 2.16715
\(435\) 0 0
\(436\) −33.7041 −1.61413
\(437\) − 3.00744i − 0.143865i
\(438\) − 0.293506i − 0.0140243i
\(439\) −12.1776 −0.581204 −0.290602 0.956844i \(-0.593855\pi\)
−0.290602 + 0.956844i \(0.593855\pi\)
\(440\) 0 0
\(441\) −6.14873 −0.292797
\(442\) − 6.13104i − 0.291624i
\(443\) 20.7101i 0.983968i 0.870604 + 0.491984i \(0.163728\pi\)
−0.870604 + 0.491984i \(0.836272\pi\)
\(444\) −6.04638 −0.286949
\(445\) 0 0
\(446\) 66.3103 3.13988
\(447\) − 2.99851i − 0.141825i
\(448\) 38.6833i 1.82761i
\(449\) 25.9539 1.22484 0.612420 0.790533i \(-0.290196\pi\)
0.612420 + 0.790533i \(0.290196\pi\)
\(450\) 0 0
\(451\) −3.66844 −0.172740
\(452\) 35.7004i 1.67921i
\(453\) − 2.24129i − 0.105305i
\(454\) −27.0662 −1.27028
\(455\) 0 0
\(456\) 1.32029 0.0618283
\(457\) − 8.50150i − 0.397684i −0.980032 0.198842i \(-0.936282\pi\)
0.980032 0.198842i \(-0.0637180\pi\)
\(458\) 37.8751i 1.76979i
\(459\) 5.10137 0.238112
\(460\) 0 0
\(461\) 14.7851 0.688609 0.344305 0.938858i \(-0.388115\pi\)
0.344305 + 0.938858i \(0.388115\pi\)
\(462\) 6.65865i 0.309788i
\(463\) 22.1921i 1.03135i 0.856783 + 0.515677i \(0.172460\pi\)
−0.856783 + 0.515677i \(0.827540\pi\)
\(464\) −1.75056 −0.0812677
\(465\) 0 0
\(466\) 51.9924 2.40850
\(467\) − 28.5014i − 1.31889i −0.751754 0.659443i \(-0.770792\pi\)
0.751754 0.659443i \(-0.229208\pi\)
\(468\) − 13.1968i − 0.610024i
\(469\) 32.4534 1.49856
\(470\) 0 0
\(471\) 0.697669 0.0321469
\(472\) 32.7638i 1.50808i
\(473\) 7.18891i 0.330546i
\(474\) 9.40048 0.431779
\(475\) 0 0
\(476\) 18.8213 0.862671
\(477\) 26.3706i 1.20743i
\(478\) 15.3182i 0.700637i
\(479\) −25.0569 −1.14488 −0.572440 0.819947i \(-0.694003\pi\)
−0.572440 + 0.819947i \(0.694003\pi\)
\(480\) 0 0
\(481\) −5.45225 −0.248601
\(482\) 60.5632i 2.75858i
\(483\) 4.79995i 0.218405i
\(484\) 23.3292 1.06042
\(485\) 0 0
\(486\) 26.6981 1.21105
\(487\) − 1.46479i − 0.0663758i −0.999449 0.0331879i \(-0.989434\pi\)
0.999449 0.0331879i \(-0.0105660\pi\)
\(488\) − 43.9314i − 1.98868i
\(489\) −2.11792 −0.0957757
\(490\) 0 0
\(491\) −20.0686 −0.905685 −0.452843 0.891591i \(-0.649590\pi\)
−0.452843 + 0.891591i \(0.649590\pi\)
\(492\) − 2.90308i − 0.130881i
\(493\) − 7.37229i − 0.332031i
\(494\) 2.97718 0.133950
\(495\) 0 0
\(496\) −2.84465 −0.127729
\(497\) 37.8258i 1.69672i
\(498\) 13.8381i 0.620101i
\(499\) −0.624999 −0.0279788 −0.0139894 0.999902i \(-0.504453\pi\)
−0.0139894 + 0.999902i \(0.504453\pi\)
\(500\) 0 0
\(501\) 4.95498 0.221372
\(502\) − 25.1990i − 1.12469i
\(503\) 19.3052i 0.860776i 0.902644 + 0.430388i \(0.141623\pi\)
−0.902644 + 0.430388i \(0.858377\pi\)
\(504\) 25.9226 1.15469
\(505\) 0 0
\(506\) −15.3766 −0.683575
\(507\) − 5.20639i − 0.231224i
\(508\) 37.8631i 1.67990i
\(509\) 10.5202 0.466298 0.233149 0.972441i \(-0.425097\pi\)
0.233149 + 0.972441i \(0.425097\pi\)
\(510\) 0 0
\(511\) 0.812479 0.0359420
\(512\) − 4.98730i − 0.220410i
\(513\) 2.47718i 0.109370i
\(514\) −15.1962 −0.670276
\(515\) 0 0
\(516\) −5.68906 −0.250447
\(517\) − 9.59989i − 0.422203i
\(518\) − 26.7818i − 1.17672i
\(519\) 3.64545 0.160017
\(520\) 0 0
\(521\) −10.0070 −0.438413 −0.219207 0.975678i \(-0.570347\pi\)
−0.219207 + 0.975678i \(0.570347\pi\)
\(522\) − 25.3914i − 1.11135i
\(523\) − 22.7830i − 0.996233i −0.867110 0.498117i \(-0.834025\pi\)
0.867110 0.498117i \(-0.165975\pi\)
\(524\) −26.6210 −1.16295
\(525\) 0 0
\(526\) −62.5981 −2.72941
\(527\) − 11.9799i − 0.521854i
\(528\) − 0.419546i − 0.0182584i
\(529\) 11.9156 0.518070
\(530\) 0 0
\(531\) −29.5359 −1.28175
\(532\) 9.13943i 0.396245i
\(533\) − 2.61782i − 0.113390i
\(534\) 5.22893 0.226278
\(535\) 0 0
\(536\) −32.9010 −1.42111
\(537\) 7.36896i 0.317994i
\(538\) − 2.33110i − 0.100501i
\(539\) −4.43237 −0.190916
\(540\) 0 0
\(541\) 3.25900 0.140115 0.0700576 0.997543i \(-0.477682\pi\)
0.0700576 + 0.997543i \(0.477682\pi\)
\(542\) 14.4377i 0.620150i
\(543\) 0.755360i 0.0324156i
\(544\) 9.55296 0.409580
\(545\) 0 0
\(546\) −4.75164 −0.203352
\(547\) − 13.5883i − 0.580993i −0.956876 0.290497i \(-0.906179\pi\)
0.956876 0.290497i \(-0.0938205\pi\)
\(548\) − 31.1188i − 1.32933i
\(549\) 39.6033 1.69023
\(550\) 0 0
\(551\) 3.57992 0.152510
\(552\) − 4.86614i − 0.207117i
\(553\) 26.0223i 1.10658i
\(554\) 56.9706 2.42045
\(555\) 0 0
\(556\) 59.7063 2.53211
\(557\) − 27.6399i − 1.17114i −0.810621 0.585571i \(-0.800870\pi\)
0.810621 0.585571i \(-0.199130\pi\)
\(558\) − 41.2609i − 1.74671i
\(559\) −5.13004 −0.216978
\(560\) 0 0
\(561\) 1.76687 0.0745974
\(562\) 4.24656i 0.179130i
\(563\) 1.65925i 0.0699291i 0.999389 + 0.0349646i \(0.0111318\pi\)
−0.999389 + 0.0349646i \(0.988868\pi\)
\(564\) 7.59703 0.319893
\(565\) 0 0
\(566\) −19.9548 −0.838763
\(567\) 21.3147i 0.895133i
\(568\) − 38.3475i − 1.60902i
\(569\) −17.8828 −0.749686 −0.374843 0.927088i \(-0.622303\pi\)
−0.374843 + 0.927088i \(0.622303\pi\)
\(570\) 0 0
\(571\) −36.8723 −1.54306 −0.771530 0.636193i \(-0.780508\pi\)
−0.771530 + 0.636193i \(0.780508\pi\)
\(572\) − 9.51307i − 0.397761i
\(573\) − 9.34060i − 0.390209i
\(574\) 12.8589 0.536718
\(575\) 0 0
\(576\) 35.3531 1.47305
\(577\) − 22.8137i − 0.949746i −0.880054 0.474873i \(-0.842494\pi\)
0.880054 0.474873i \(-0.157506\pi\)
\(578\) 31.2664i 1.30051i
\(579\) −6.22599 −0.258743
\(580\) 0 0
\(581\) −38.3065 −1.58922
\(582\) − 10.9208i − 0.452682i
\(583\) 19.0095i 0.787291i
\(584\) −0.823684 −0.0340843
\(585\) 0 0
\(586\) 14.5289 0.600185
\(587\) − 11.0855i − 0.457546i −0.973480 0.228773i \(-0.926529\pi\)
0.973480 0.228773i \(-0.0734714\pi\)
\(588\) − 3.50763i − 0.144652i
\(589\) 5.81734 0.239699
\(590\) 0 0
\(591\) −1.62867 −0.0669947
\(592\) 1.68746i 0.0693542i
\(593\) − 11.1321i − 0.457139i −0.973528 0.228570i \(-0.926595\pi\)
0.973528 0.228570i \(-0.0734049\pi\)
\(594\) 12.6655 0.519672
\(595\) 0 0
\(596\) −21.0428 −0.861947
\(597\) 8.40384i 0.343946i
\(598\) − 10.9729i − 0.448713i
\(599\) −36.2736 −1.48210 −0.741049 0.671451i \(-0.765671\pi\)
−0.741049 + 0.671451i \(0.765671\pi\)
\(600\) 0 0
\(601\) −15.1051 −0.616150 −0.308075 0.951362i \(-0.599685\pi\)
−0.308075 + 0.951362i \(0.599685\pi\)
\(602\) − 25.1990i − 1.02704i
\(603\) − 29.6596i − 1.20783i
\(604\) −15.7288 −0.639997
\(605\) 0 0
\(606\) 10.2497 0.416366
\(607\) − 33.5066i − 1.35999i −0.733216 0.679996i \(-0.761982\pi\)
0.733216 0.679996i \(-0.238018\pi\)
\(608\) 4.63883i 0.188129i
\(609\) −5.71363 −0.231528
\(610\) 0 0
\(611\) 6.85053 0.277143
\(612\) − 17.2010i − 0.695308i
\(613\) 28.0289i 1.13208i 0.824379 + 0.566038i \(0.191524\pi\)
−0.824379 + 0.566038i \(0.808476\pi\)
\(614\) 66.1979 2.67153
\(615\) 0 0
\(616\) 18.6866 0.752903
\(617\) 30.5223i 1.22878i 0.789002 + 0.614391i \(0.210598\pi\)
−0.789002 + 0.614391i \(0.789402\pi\)
\(618\) − 9.96234i − 0.400744i
\(619\) 21.6971 0.872080 0.436040 0.899927i \(-0.356381\pi\)
0.436040 + 0.899927i \(0.356381\pi\)
\(620\) 0 0
\(621\) 9.13004 0.366376
\(622\) − 18.0966i − 0.725608i
\(623\) 14.4746i 0.579914i
\(624\) 0.299391 0.0119852
\(625\) 0 0
\(626\) 49.4399 1.97601
\(627\) 0.857977i 0.0342643i
\(628\) − 4.89606i − 0.195374i
\(629\) −7.10655 −0.283357
\(630\) 0 0
\(631\) −16.2277 −0.646015 −0.323007 0.946396i \(-0.604694\pi\)
−0.323007 + 0.946396i \(0.604694\pi\)
\(632\) − 26.3812i − 1.04939i
\(633\) 1.54042i 0.0612264i
\(634\) −9.28290 −0.368671
\(635\) 0 0
\(636\) −15.0434 −0.596511
\(637\) − 3.16296i − 0.125321i
\(638\) − 18.3036i − 0.724648i
\(639\) 34.5694 1.36755
\(640\) 0 0
\(641\) −22.1774 −0.875956 −0.437978 0.898986i \(-0.644305\pi\)
−0.437978 + 0.898986i \(0.644305\pi\)
\(642\) 6.17290i 0.243625i
\(643\) 13.2767i 0.523583i 0.965124 + 0.261792i \(0.0843133\pi\)
−0.965124 + 0.261792i \(0.915687\pi\)
\(644\) 33.6848 1.32737
\(645\) 0 0
\(646\) 3.88050 0.152676
\(647\) − 11.2853i − 0.443671i −0.975084 0.221835i \(-0.928795\pi\)
0.975084 0.221835i \(-0.0712047\pi\)
\(648\) − 21.6086i − 0.848867i
\(649\) −21.2912 −0.835754
\(650\) 0 0
\(651\) −9.28462 −0.363893
\(652\) 14.8630i 0.582082i
\(653\) 35.8134i 1.40149i 0.713414 + 0.700743i \(0.247148\pi\)
−0.713414 + 0.700743i \(0.752852\pi\)
\(654\) 11.0907 0.433682
\(655\) 0 0
\(656\) −0.810208 −0.0316333
\(657\) − 0.742534i − 0.0289690i
\(658\) 33.6502i 1.31182i
\(659\) −39.7655 −1.54905 −0.774523 0.632546i \(-0.782010\pi\)
−0.774523 + 0.632546i \(0.782010\pi\)
\(660\) 0 0
\(661\) −6.24734 −0.242993 −0.121497 0.992592i \(-0.538769\pi\)
−0.121497 + 0.992592i \(0.538769\pi\)
\(662\) 26.8889i 1.04507i
\(663\) 1.26085i 0.0489673i
\(664\) 38.8347 1.50708
\(665\) 0 0
\(666\) −24.4762 −0.948432
\(667\) − 13.1943i − 0.510887i
\(668\) − 34.7728i − 1.34540i
\(669\) −13.6367 −0.527226
\(670\) 0 0
\(671\) 28.5484 1.10210
\(672\) − 7.40368i − 0.285603i
\(673\) 41.4627i 1.59827i 0.601151 + 0.799135i \(0.294709\pi\)
−0.601151 + 0.799135i \(0.705291\pi\)
\(674\) 49.7297 1.91552
\(675\) 0 0
\(676\) −36.5372 −1.40528
\(677\) − 1.43915i − 0.0553112i −0.999618 0.0276556i \(-0.991196\pi\)
0.999618 0.0276556i \(-0.00880417\pi\)
\(678\) − 11.7477i − 0.451166i
\(679\) 30.2308 1.16015
\(680\) 0 0
\(681\) 5.56616 0.213296
\(682\) − 29.7433i − 1.13893i
\(683\) 8.48623i 0.324716i 0.986732 + 0.162358i \(0.0519100\pi\)
−0.986732 + 0.162358i \(0.948090\pi\)
\(684\) 8.35263 0.319371
\(685\) 0 0
\(686\) −33.5371 −1.28045
\(687\) − 7.78902i − 0.297170i
\(688\) 1.58774i 0.0605318i
\(689\) −13.5652 −0.516795
\(690\) 0 0
\(691\) −43.7797 −1.66546 −0.832730 0.553679i \(-0.813223\pi\)
−0.832730 + 0.553679i \(0.813223\pi\)
\(692\) − 25.5828i − 0.972513i
\(693\) 16.8456i 0.639910i
\(694\) −36.0183 −1.36724
\(695\) 0 0
\(696\) 5.79243 0.219561
\(697\) − 3.41210i − 0.129243i
\(698\) − 12.8534i − 0.486508i
\(699\) −10.6922 −0.404418
\(700\) 0 0
\(701\) 0.840795 0.0317564 0.0158782 0.999874i \(-0.494946\pi\)
0.0158782 + 0.999874i \(0.494946\pi\)
\(702\) 9.03816i 0.341124i
\(703\) − 3.45087i − 0.130152i
\(704\) 25.4846 0.960487
\(705\) 0 0
\(706\) 18.5254 0.697212
\(707\) 28.3731i 1.06708i
\(708\) − 16.8492i − 0.633230i
\(709\) 13.3812 0.502543 0.251271 0.967917i \(-0.419151\pi\)
0.251271 + 0.967917i \(0.419151\pi\)
\(710\) 0 0
\(711\) 23.7821 0.891897
\(712\) − 14.6743i − 0.549941i
\(713\) − 21.4407i − 0.802962i
\(714\) −6.19336 −0.231781
\(715\) 0 0
\(716\) 51.7135 1.93262
\(717\) − 3.15019i − 0.117646i
\(718\) 28.5027i 1.06371i
\(719\) −43.4148 −1.61910 −0.809550 0.587051i \(-0.800289\pi\)
−0.809550 + 0.587051i \(0.800289\pi\)
\(720\) 0 0
\(721\) 27.5776 1.02704
\(722\) − 41.9919i − 1.56278i
\(723\) − 12.4548i − 0.463200i
\(724\) 5.30093 0.197007
\(725\) 0 0
\(726\) −7.67677 −0.284912
\(727\) − 32.0480i − 1.18859i −0.804245 0.594297i \(-0.797430\pi\)
0.804245 0.594297i \(-0.202570\pi\)
\(728\) 13.3348i 0.494222i
\(729\) 15.5727 0.576768
\(730\) 0 0
\(731\) −6.68657 −0.247312
\(732\) 22.5922i 0.835032i
\(733\) − 8.13928i − 0.300631i −0.988638 0.150316i \(-0.951971\pi\)
0.988638 0.150316i \(-0.0480289\pi\)
\(734\) −62.0616 −2.29074
\(735\) 0 0
\(736\) 17.0971 0.630209
\(737\) − 21.3804i − 0.787556i
\(738\) − 11.7519i − 0.432592i
\(739\) 7.12714 0.262176 0.131088 0.991371i \(-0.458153\pi\)
0.131088 + 0.991371i \(0.458153\pi\)
\(740\) 0 0
\(741\) −0.612257 −0.0224918
\(742\) − 66.6332i − 2.44618i
\(743\) − 21.9040i − 0.803578i −0.915732 0.401789i \(-0.868388\pi\)
0.915732 0.401789i \(-0.131612\pi\)
\(744\) 9.41266 0.345085
\(745\) 0 0
\(746\) −63.8258 −2.33683
\(747\) 35.0087i 1.28090i
\(748\) − 12.3995i − 0.453370i
\(749\) −17.0877 −0.624373
\(750\) 0 0
\(751\) 9.21909 0.336409 0.168205 0.985752i \(-0.446203\pi\)
0.168205 + 0.985752i \(0.446203\pi\)
\(752\) − 2.12022i − 0.0773166i
\(753\) 5.18219i 0.188849i
\(754\) 13.0616 0.475674
\(755\) 0 0
\(756\) −27.7457 −1.00910
\(757\) 45.6524i 1.65926i 0.558311 + 0.829632i \(0.311450\pi\)
−0.558311 + 0.829632i \(0.688550\pi\)
\(758\) 8.02386i 0.291440i
\(759\) 3.16221 0.114781
\(760\) 0 0
\(761\) 39.9058 1.44658 0.723291 0.690543i \(-0.242628\pi\)
0.723291 + 0.690543i \(0.242628\pi\)
\(762\) − 12.4593i − 0.451353i
\(763\) 30.7012i 1.11146i
\(764\) −65.5500 −2.37151
\(765\) 0 0
\(766\) 63.2092 2.28384
\(767\) − 15.1935i − 0.548607i
\(768\) 8.90403i 0.321296i
\(769\) 44.3420 1.59901 0.799506 0.600658i \(-0.205094\pi\)
0.799506 + 0.600658i \(0.205094\pi\)
\(770\) 0 0
\(771\) 3.12510 0.112548
\(772\) 43.6924i 1.57252i
\(773\) 38.4944i 1.38455i 0.721635 + 0.692273i \(0.243391\pi\)
−0.721635 + 0.692273i \(0.756609\pi\)
\(774\) −23.0297 −0.827785
\(775\) 0 0
\(776\) −30.6477 −1.10019
\(777\) 5.50768i 0.197587i
\(778\) − 25.1353i − 0.901146i
\(779\) 1.65689 0.0593641
\(780\) 0 0
\(781\) 24.9197 0.891697
\(782\) − 14.3022i − 0.511445i
\(783\) 10.8680i 0.388390i
\(784\) −0.978929 −0.0349618
\(785\) 0 0
\(786\) 8.75997 0.312458
\(787\) − 53.0202i − 1.88997i −0.327120 0.944983i \(-0.606078\pi\)
0.327120 0.944983i \(-0.393922\pi\)
\(788\) 11.4296i 0.407164i
\(789\) 12.8733 0.458302
\(790\) 0 0
\(791\) 32.5197 1.15627
\(792\) − 17.0779i − 0.606835i
\(793\) 20.3723i 0.723440i
\(794\) −37.4593 −1.32938
\(795\) 0 0
\(796\) 58.9760 2.09035
\(797\) − 12.6769i − 0.449037i −0.974470 0.224519i \(-0.927919\pi\)
0.974470 0.224519i \(-0.0720810\pi\)
\(798\) − 3.00744i − 0.106462i
\(799\) 8.92908 0.315888
\(800\) 0 0
\(801\) 13.2285 0.467407
\(802\) 8.74490i 0.308793i
\(803\) − 0.535262i − 0.0188890i
\(804\) 16.9197 0.596712
\(805\) 0 0
\(806\) 21.2250 0.747618
\(807\) 0.479392i 0.0168754i
\(808\) − 28.7644i − 1.01193i
\(809\) 41.8935 1.47290 0.736449 0.676493i \(-0.236501\pi\)
0.736449 + 0.676493i \(0.236501\pi\)
\(810\) 0 0
\(811\) −34.5486 −1.21317 −0.606583 0.795020i \(-0.707460\pi\)
−0.606583 + 0.795020i \(0.707460\pi\)
\(812\) 40.0968i 1.40712i
\(813\) − 2.96911i − 0.104131i
\(814\) −17.6439 −0.618417
\(815\) 0 0
\(816\) 0.390230 0.0136608
\(817\) − 3.24694i − 0.113596i
\(818\) − 4.28568i − 0.149845i
\(819\) −12.0211 −0.420050
\(820\) 0 0
\(821\) −21.1349 −0.737612 −0.368806 0.929506i \(-0.620233\pi\)
−0.368806 + 0.929506i \(0.620233\pi\)
\(822\) 10.2400i 0.357161i
\(823\) − 3.17714i − 0.110748i −0.998466 0.0553740i \(-0.982365\pi\)
0.998466 0.0553740i \(-0.0176351\pi\)
\(824\) −27.9579 −0.973960
\(825\) 0 0
\(826\) 74.6315 2.59676
\(827\) 9.74482i 0.338860i 0.985542 + 0.169430i \(0.0541927\pi\)
−0.985542 + 0.169430i \(0.945807\pi\)
\(828\) − 30.7849i − 1.06985i
\(829\) −23.4352 −0.813938 −0.406969 0.913442i \(-0.633414\pi\)
−0.406969 + 0.913442i \(0.633414\pi\)
\(830\) 0 0
\(831\) −11.7160 −0.406424
\(832\) 18.1859i 0.630484i
\(833\) − 4.12265i − 0.142841i
\(834\) −19.6471 −0.680323
\(835\) 0 0
\(836\) 6.02107 0.208243
\(837\) 17.6604i 0.610432i
\(838\) − 33.1316i − 1.14451i
\(839\) −42.6819 −1.47354 −0.736771 0.676142i \(-0.763650\pi\)
−0.736771 + 0.676142i \(0.763650\pi\)
\(840\) 0 0
\(841\) −13.2941 −0.458416
\(842\) 35.6887i 1.22992i
\(843\) − 0.873305i − 0.0300782i
\(844\) 10.8103 0.372106
\(845\) 0 0
\(846\) 30.7533 1.05732
\(847\) − 21.2507i − 0.730183i
\(848\) 4.19841i 0.144174i
\(849\) 4.10371 0.140839
\(850\) 0 0
\(851\) −12.7187 −0.435993
\(852\) 19.7206i 0.675617i
\(853\) − 16.9610i − 0.580733i −0.956915 0.290367i \(-0.906223\pi\)
0.956915 0.290367i \(-0.0937773\pi\)
\(854\) −100.070 −3.42431
\(855\) 0 0
\(856\) 17.3234 0.592102
\(857\) − 39.3176i − 1.34306i −0.740976 0.671531i \(-0.765637\pi\)
0.740976 0.671531i \(-0.234363\pi\)
\(858\) 3.13039i 0.106870i
\(859\) −0.707056 −0.0241244 −0.0120622 0.999927i \(-0.503840\pi\)
−0.0120622 + 0.999927i \(0.503840\pi\)
\(860\) 0 0
\(861\) −2.64443 −0.0901219
\(862\) 28.8759i 0.983516i
\(863\) − 0.909409i − 0.0309567i −0.999880 0.0154783i \(-0.995073\pi\)
0.999880 0.0154783i \(-0.00492710\pi\)
\(864\) −14.0826 −0.479101
\(865\) 0 0
\(866\) −51.9474 −1.76525
\(867\) − 6.42994i − 0.218372i
\(868\) 65.1571i 2.21158i
\(869\) 17.1435 0.581554
\(870\) 0 0
\(871\) 15.2571 0.516968
\(872\) − 31.1246i − 1.05401i
\(873\) − 27.6283i − 0.935075i
\(874\) 6.94501 0.234918
\(875\) 0 0
\(876\) 0.423589 0.0143117
\(877\) 33.6613i 1.13666i 0.822800 + 0.568331i \(0.192411\pi\)
−0.822800 + 0.568331i \(0.807589\pi\)
\(878\) − 28.1213i − 0.949050i
\(879\) −2.98788 −0.100779
\(880\) 0 0
\(881\) −19.9283 −0.671402 −0.335701 0.941969i \(-0.608973\pi\)
−0.335701 + 0.941969i \(0.608973\pi\)
\(882\) − 14.1991i − 0.478109i
\(883\) − 15.6081i − 0.525255i −0.964897 0.262627i \(-0.915411\pi\)
0.964897 0.262627i \(-0.0845890\pi\)
\(884\) 8.84833 0.297601
\(885\) 0 0
\(886\) −47.8254 −1.60673
\(887\) 46.8968i 1.57464i 0.616544 + 0.787320i \(0.288532\pi\)
−0.616544 + 0.787320i \(0.711468\pi\)
\(888\) − 5.58363i − 0.187374i
\(889\) 34.4897 1.15675
\(890\) 0 0
\(891\) 14.0421 0.470429
\(892\) 95.6991i 3.20424i
\(893\) 4.33588i 0.145095i
\(894\) 6.92439 0.231586
\(895\) 0 0
\(896\) −58.1505 −1.94267
\(897\) 2.25657i 0.0753447i
\(898\) 59.9347i 2.00005i
\(899\) 25.5220 0.851208
\(900\) 0 0
\(901\) −17.6811 −0.589044
\(902\) − 8.47143i − 0.282068i
\(903\) 5.18219i 0.172452i
\(904\) −32.9682 −1.09651
\(905\) 0 0
\(906\) 5.17576 0.171953
\(907\) 1.43447i 0.0476308i 0.999716 + 0.0238154i \(0.00758139\pi\)
−0.999716 + 0.0238154i \(0.992419\pi\)
\(908\) − 39.0619i − 1.29632i
\(909\) 25.9305 0.860061
\(910\) 0 0
\(911\) −2.81129 −0.0931422 −0.0465711 0.998915i \(-0.514829\pi\)
−0.0465711 + 0.998915i \(0.514829\pi\)
\(912\) 0.189492i 0.00627471i
\(913\) 25.2363i 0.835201i
\(914\) 19.6323 0.649379
\(915\) 0 0
\(916\) −54.6614 −1.80606
\(917\) 24.2492i 0.800780i
\(918\) 11.7805i 0.388814i
\(919\) −0.992236 −0.0327308 −0.0163654 0.999866i \(-0.505210\pi\)
−0.0163654 + 0.999866i \(0.505210\pi\)
\(920\) 0 0
\(921\) −13.6136 −0.448584
\(922\) 34.1428i 1.12443i
\(923\) 17.7828i 0.585329i
\(924\) −9.60977 −0.316138
\(925\) 0 0
\(926\) −51.2477 −1.68410
\(927\) − 25.2035i − 0.827791i
\(928\) 20.3516i 0.668075i
\(929\) −33.3227 −1.09328 −0.546642 0.837367i \(-0.684094\pi\)
−0.546642 + 0.837367i \(0.684094\pi\)
\(930\) 0 0
\(931\) 2.00192 0.0656104
\(932\) 75.0355i 2.45787i
\(933\) 3.72157i 0.121839i
\(934\) 65.8175 2.15361
\(935\) 0 0
\(936\) 12.1869 0.398340
\(937\) 13.3675i 0.436698i 0.975871 + 0.218349i \(0.0700671\pi\)
−0.975871 + 0.218349i \(0.929933\pi\)
\(938\) 74.9439i 2.44701i
\(939\) −10.1673 −0.331798
\(940\) 0 0
\(941\) −2.14982 −0.0700820 −0.0350410 0.999386i \(-0.511156\pi\)
−0.0350410 + 0.999386i \(0.511156\pi\)
\(942\) 1.61111i 0.0524928i
\(943\) − 6.10671i − 0.198862i
\(944\) −4.70236 −0.153049
\(945\) 0 0
\(946\) −16.6012 −0.539750
\(947\) 34.9183i 1.13469i 0.823479 + 0.567346i \(0.192030\pi\)
−0.823479 + 0.567346i \(0.807970\pi\)
\(948\) 13.5668i 0.440629i
\(949\) 0.381966 0.0123991
\(950\) 0 0
\(951\) 1.90903 0.0619046
\(952\) 17.3808i 0.563315i
\(953\) 8.27883i 0.268178i 0.990969 + 0.134089i \(0.0428108\pi\)
−0.990969 + 0.134089i \(0.957189\pi\)
\(954\) −60.8969 −1.97161
\(955\) 0 0
\(956\) −22.1072 −0.714998
\(957\) 3.76415i 0.121678i
\(958\) − 57.8633i − 1.86948i
\(959\) −28.3462 −0.915347
\(960\) 0 0
\(961\) 10.4732 0.337844
\(962\) − 12.5908i − 0.405942i
\(963\) 15.6167i 0.503241i
\(964\) −87.4048 −2.81512
\(965\) 0 0
\(966\) −11.0844 −0.356635
\(967\) 29.4871i 0.948241i 0.880460 + 0.474121i \(0.157234\pi\)
−0.880460 + 0.474121i \(0.842766\pi\)
\(968\) 21.5438i 0.692443i
\(969\) −0.798025 −0.0256363
\(970\) 0 0
\(971\) 21.8666 0.701733 0.350867 0.936425i \(-0.385887\pi\)
0.350867 + 0.936425i \(0.385887\pi\)
\(972\) 38.5308i 1.23588i
\(973\) − 54.3868i − 1.74356i
\(974\) 3.38260 0.108385
\(975\) 0 0
\(976\) 6.30517 0.201823
\(977\) 13.8482i 0.443043i 0.975155 + 0.221521i \(0.0711023\pi\)
−0.975155 + 0.221521i \(0.928898\pi\)
\(978\) − 4.89086i − 0.156392i
\(979\) 9.53591 0.304769
\(980\) 0 0
\(981\) 28.0582 0.895828
\(982\) − 46.3440i − 1.47890i
\(983\) − 2.93538i − 0.0936241i −0.998904 0.0468120i \(-0.985094\pi\)
0.998904 0.0468120i \(-0.0149062\pi\)
\(984\) 2.68090 0.0854639
\(985\) 0 0
\(986\) 17.0246 0.542175
\(987\) − 6.92017i − 0.220271i
\(988\) 4.29667i 0.136695i
\(989\) −11.9671 −0.380531
\(990\) 0 0
\(991\) 31.6137 1.00424 0.502122 0.864797i \(-0.332553\pi\)
0.502122 + 0.864797i \(0.332553\pi\)
\(992\) 33.0713i 1.05001i
\(993\) − 5.52970i − 0.175480i
\(994\) −87.3502 −2.77058
\(995\) 0 0
\(996\) −19.9712 −0.632811
\(997\) − 12.0885i − 0.382845i −0.981508 0.191423i \(-0.938690\pi\)
0.981508 0.191423i \(-0.0613101\pi\)
\(998\) − 1.44329i − 0.0456867i
\(999\) 10.4762 0.331453
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 625.2.b.c.624.8 8
5.2 odd 4 625.2.a.f.1.1 8
5.3 odd 4 625.2.a.f.1.8 8
5.4 even 2 inner 625.2.b.c.624.1 8
15.2 even 4 5625.2.a.x.1.8 8
15.8 even 4 5625.2.a.x.1.1 8
20.3 even 4 10000.2.a.bj.1.4 8
20.7 even 4 10000.2.a.bj.1.5 8
25.2 odd 20 625.2.d.o.501.1 16
25.3 odd 20 125.2.d.b.76.1 16
25.4 even 10 25.2.e.a.9.1 8
25.6 even 5 25.2.e.a.14.1 yes 8
25.8 odd 20 125.2.d.b.51.1 16
25.9 even 10 625.2.e.a.499.2 8
25.11 even 5 625.2.e.a.124.2 8
25.12 odd 20 625.2.d.o.126.1 16
25.13 odd 20 625.2.d.o.126.4 16
25.14 even 10 625.2.e.i.124.1 8
25.16 even 5 625.2.e.i.499.1 8
25.17 odd 20 125.2.d.b.51.4 16
25.19 even 10 125.2.e.b.74.2 8
25.21 even 5 125.2.e.b.49.2 8
25.22 odd 20 125.2.d.b.76.4 16
25.23 odd 20 625.2.d.o.501.4 16
75.29 odd 10 225.2.m.a.109.2 8
75.56 odd 10 225.2.m.a.64.2 8
100.31 odd 10 400.2.y.c.289.1 8
100.79 odd 10 400.2.y.c.209.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.e.a.9.1 8 25.4 even 10
25.2.e.a.14.1 yes 8 25.6 even 5
125.2.d.b.51.1 16 25.8 odd 20
125.2.d.b.51.4 16 25.17 odd 20
125.2.d.b.76.1 16 25.3 odd 20
125.2.d.b.76.4 16 25.22 odd 20
125.2.e.b.49.2 8 25.21 even 5
125.2.e.b.74.2 8 25.19 even 10
225.2.m.a.64.2 8 75.56 odd 10
225.2.m.a.109.2 8 75.29 odd 10
400.2.y.c.209.1 8 100.79 odd 10
400.2.y.c.289.1 8 100.31 odd 10
625.2.a.f.1.1 8 5.2 odd 4
625.2.a.f.1.8 8 5.3 odd 4
625.2.b.c.624.1 8 5.4 even 2 inner
625.2.b.c.624.8 8 1.1 even 1 trivial
625.2.d.o.126.1 16 25.12 odd 20
625.2.d.o.126.4 16 25.13 odd 20
625.2.d.o.501.1 16 25.2 odd 20
625.2.d.o.501.4 16 25.23 odd 20
625.2.e.a.124.2 8 25.11 even 5
625.2.e.a.499.2 8 25.9 even 10
625.2.e.i.124.1 8 25.14 even 10
625.2.e.i.499.1 8 25.16 even 5
5625.2.a.x.1.1 8 15.8 even 4
5625.2.a.x.1.8 8 15.2 even 4
10000.2.a.bj.1.4 8 20.3 even 4
10000.2.a.bj.1.5 8 20.7 even 4