Properties

Label 804.2.i.d.565.4
Level $804$
Weight $2$
Character 804.565
Analytic conductor $6.420$
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] = [804,2,Mod(37,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.i (of order \(3\), degree \(2\), 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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 11x^{6} + 4x^{5} + 91x^{4} - 6x^{3} + 129x^{2} + 36x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 565.4
Root \(1.69181 - 2.93030i\) of defining polynomial
Character \(\chi\) \(=\) 804.565
Dual form 804.2.i.d.37.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.38361 q^{5} +(1.69181 + 2.93030i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.38361 q^{5} +(1.69181 + 2.93030i) q^{7} +1.00000 q^{9} +(1.00000 + 1.73205i) q^{11} +(1.53262 - 2.65457i) q^{13} -3.38361 q^{15} +(-1.03262 + 1.78855i) q^{17} +(-3.72442 + 6.45089i) q^{19} +(-1.69181 - 2.93030i) q^{21} +(1.11036 - 1.92321i) q^{23} +6.44885 q^{25} -1.00000 q^{27} +(-2.80217 - 4.85350i) q^{29} +(-2.99398 - 5.18572i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(5.72442 + 9.91499i) q^{35} +(-3.14298 + 5.44380i) q^{37} +(-1.53262 + 2.65457i) q^{39} +(1.92225 + 3.32944i) q^{41} +5.92272 q^{43} +3.38361 q^{45} +(1.92225 + 3.32944i) q^{47} +(-2.22442 + 3.85282i) q^{49} +(1.03262 - 1.78855i) q^{51} +1.93477 q^{53} +(3.38361 + 5.86059i) q^{55} +(3.72442 - 6.45089i) q^{57} +11.6696 q^{59} +(4.98379 - 8.63218i) q^{61} +(1.69181 + 2.93030i) q^{63} +(5.18579 - 8.98204i) q^{65} +(-7.80587 + 2.46342i) q^{67} +(-1.11036 + 1.92321i) q^{69} +(6.65687 + 11.5300i) q^{71} +(2.80587 - 4.85991i) q^{73} -6.44885 q^{75} +(-3.38361 + 5.86059i) q^{77} +(-6.29985 - 10.9117i) q^{79} +1.00000 q^{81} +(-2.03262 + 3.52060i) q^{83} +(-3.49398 + 6.05175i) q^{85} +(2.80217 + 4.85350i) q^{87} -0.837112 q^{89} +10.3716 q^{91} +(2.99398 + 5.18572i) q^{93} +(-12.6020 + 21.8273i) q^{95} +(3.08144 - 5.33722i) q^{97} +(1.00000 + 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 2 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 2 q^{5} + q^{7} + 8 q^{9} + 8 q^{11} - 2 q^{15} + 4 q^{17} - 5 q^{19} - q^{21} + q^{23} + 2 q^{25} - 8 q^{27} - 2 q^{29} + 9 q^{31} - 8 q^{33} + 21 q^{35} - 5 q^{37} + 11 q^{41} + 6 q^{43} + 2 q^{45} + 11 q^{47} + 7 q^{49} - 4 q^{51} + 40 q^{53} + 2 q^{55} + 5 q^{57} + 28 q^{59} + 20 q^{61} + q^{63} - 4 q^{65} - 33 q^{67} - q^{69} + 11 q^{71} - 7 q^{73} - 2 q^{75} - 2 q^{77} + 12 q^{79} + 8 q^{81} - 4 q^{83} + 5 q^{85} + 2 q^{87} - 16 q^{89} - 8 q^{91} - 9 q^{93} - 18 q^{95} + 20 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.38361 1.51320 0.756599 0.653879i \(-0.226859\pi\)
0.756599 + 0.653879i \(0.226859\pi\)
\(6\) 0 0
\(7\) 1.69181 + 2.93030i 0.639443 + 1.10755i 0.985555 + 0.169355i \(0.0541684\pi\)
−0.346112 + 0.938193i \(0.612498\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 1.53262 2.65457i 0.425071 0.736245i −0.571356 0.820703i \(-0.693582\pi\)
0.996427 + 0.0844572i \(0.0269156\pi\)
\(14\) 0 0
\(15\) −3.38361 −0.873646
\(16\) 0 0
\(17\) −1.03262 + 1.78855i −0.250446 + 0.433786i −0.963649 0.267172i \(-0.913911\pi\)
0.713202 + 0.700958i \(0.247244\pi\)
\(18\) 0 0
\(19\) −3.72442 + 6.45089i −0.854442 + 1.47994i 0.0227206 + 0.999742i \(0.492767\pi\)
−0.877162 + 0.480194i \(0.840566\pi\)
\(20\) 0 0
\(21\) −1.69181 2.93030i −0.369183 0.639443i
\(22\) 0 0
\(23\) 1.11036 1.92321i 0.231527 0.401016i −0.726731 0.686922i \(-0.758961\pi\)
0.958258 + 0.285906i \(0.0922946\pi\)
\(24\) 0 0
\(25\) 6.44885 1.28977
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.80217 4.85350i −0.520350 0.901273i −0.999720 0.0236598i \(-0.992468\pi\)
0.479370 0.877613i \(-0.340865\pi\)
\(30\) 0 0
\(31\) −2.99398 5.18572i −0.537734 0.931383i −0.999026 0.0441344i \(-0.985947\pi\)
0.461291 0.887249i \(-0.347386\pi\)
\(32\) 0 0
\(33\) −1.00000 1.73205i −0.174078 0.301511i
\(34\) 0 0
\(35\) 5.72442 + 9.91499i 0.967604 + 1.67594i
\(36\) 0 0
\(37\) −3.14298 + 5.44380i −0.516703 + 0.894955i 0.483109 + 0.875560i \(0.339507\pi\)
−0.999812 + 0.0193953i \(0.993826\pi\)
\(38\) 0 0
\(39\) −1.53262 + 2.65457i −0.245415 + 0.425071i
\(40\) 0 0
\(41\) 1.92225 + 3.32944i 0.300206 + 0.519971i 0.976182 0.216952i \(-0.0696114\pi\)
−0.675977 + 0.736923i \(0.736278\pi\)
\(42\) 0 0
\(43\) 5.92272 0.903207 0.451603 0.892219i \(-0.350852\pi\)
0.451603 + 0.892219i \(0.350852\pi\)
\(44\) 0 0
\(45\) 3.38361 0.504400
\(46\) 0 0
\(47\) 1.92225 + 3.32944i 0.280390 + 0.485649i 0.971481 0.237119i \(-0.0762030\pi\)
−0.691091 + 0.722768i \(0.742870\pi\)
\(48\) 0 0
\(49\) −2.22442 + 3.85282i −0.317775 + 0.550402i
\(50\) 0 0
\(51\) 1.03262 1.78855i 0.144595 0.250446i
\(52\) 0 0
\(53\) 1.93477 0.265761 0.132880 0.991132i \(-0.457577\pi\)
0.132880 + 0.991132i \(0.457577\pi\)
\(54\) 0 0
\(55\) 3.38361 + 5.86059i 0.456247 + 0.790242i
\(56\) 0 0
\(57\) 3.72442 6.45089i 0.493312 0.854442i
\(58\) 0 0
\(59\) 11.6696 1.51925 0.759625 0.650362i \(-0.225383\pi\)
0.759625 + 0.650362i \(0.225383\pi\)
\(60\) 0 0
\(61\) 4.98379 8.63218i 0.638109 1.10524i −0.347738 0.937592i \(-0.613050\pi\)
0.985847 0.167645i \(-0.0536164\pi\)
\(62\) 0 0
\(63\) 1.69181 + 2.93030i 0.213148 + 0.369183i
\(64\) 0 0
\(65\) 5.18579 8.98204i 0.643218 1.11409i
\(66\) 0 0
\(67\) −7.80587 + 2.46342i −0.953639 + 0.300954i
\(68\) 0 0
\(69\) −1.11036 + 1.92321i −0.133672 + 0.231527i
\(70\) 0 0
\(71\) 6.65687 + 11.5300i 0.790025 + 1.36836i 0.925951 + 0.377644i \(0.123266\pi\)
−0.135926 + 0.990719i \(0.543401\pi\)
\(72\) 0 0
\(73\) 2.80587 4.85991i 0.328402 0.568809i −0.653793 0.756674i \(-0.726823\pi\)
0.982195 + 0.187864i \(0.0601566\pi\)
\(74\) 0 0
\(75\) −6.44885 −0.744649
\(76\) 0 0
\(77\) −3.38361 + 5.86059i −0.385599 + 0.667877i
\(78\) 0 0
\(79\) −6.29985 10.9117i −0.708788 1.22766i −0.965307 0.261118i \(-0.915909\pi\)
0.256519 0.966539i \(-0.417424\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.03262 + 3.52060i −0.223109 + 0.386436i −0.955750 0.294179i \(-0.904954\pi\)
0.732642 + 0.680615i \(0.238287\pi\)
\(84\) 0 0
\(85\) −3.49398 + 6.05175i −0.378975 + 0.656404i
\(86\) 0 0
\(87\) 2.80217 + 4.85350i 0.300424 + 0.520350i
\(88\) 0 0
\(89\) −0.837112 −0.0887337 −0.0443668 0.999015i \(-0.514127\pi\)
−0.0443668 + 0.999015i \(0.514127\pi\)
\(90\) 0 0
\(91\) 10.3716 1.08724
\(92\) 0 0
\(93\) 2.99398 + 5.18572i 0.310461 + 0.537734i
\(94\) 0 0
\(95\) −12.6020 + 21.8273i −1.29294 + 2.23944i
\(96\) 0 0
\(97\) 3.08144 5.33722i 0.312873 0.541912i −0.666110 0.745854i \(-0.732042\pi\)
0.978983 + 0.203941i \(0.0653752\pi\)
\(98\) 0 0
\(99\) 1.00000 + 1.73205i 0.100504 + 0.174078i
\(100\) 0 0
\(101\) 5.09785 + 8.82974i 0.507255 + 0.878592i 0.999965 + 0.00839785i \(0.00267315\pi\)
−0.492710 + 0.870194i \(0.663994\pi\)
\(102\) 0 0
\(103\) −1.65919 2.87380i −0.163485 0.283164i 0.772631 0.634855i \(-0.218940\pi\)
−0.936116 + 0.351691i \(0.885607\pi\)
\(104\) 0 0
\(105\) −5.72442 9.91499i −0.558647 0.967604i
\(106\) 0 0
\(107\) 8.15084 0.787972 0.393986 0.919116i \(-0.371096\pi\)
0.393986 + 0.919116i \(0.371096\pi\)
\(108\) 0 0
\(109\) −17.3063 −1.65765 −0.828823 0.559511i \(-0.810989\pi\)
−0.828823 + 0.559511i \(0.810989\pi\)
\(110\) 0 0
\(111\) 3.14298 5.44380i 0.298318 0.516703i
\(112\) 0 0
\(113\) −6.14066 10.6359i −0.577664 1.00054i −0.995747 0.0921348i \(-0.970631\pi\)
0.418082 0.908409i \(-0.362702\pi\)
\(114\) 0 0
\(115\) 3.75704 6.50739i 0.350346 0.606817i
\(116\) 0 0
\(117\) 1.53262 2.65457i 0.141690 0.245415i
\(118\) 0 0
\(119\) −6.98796 −0.640585
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) −1.92225 3.32944i −0.173324 0.300206i
\(124\) 0 0
\(125\) 4.90235 0.438479
\(126\) 0 0
\(127\) −9.49768 16.4505i −0.842782 1.45974i −0.887533 0.460744i \(-0.847583\pi\)
0.0447507 0.998998i \(-0.485751\pi\)
\(128\) 0 0
\(129\) −5.92272 −0.521467
\(130\) 0 0
\(131\) −6.87732 −0.600874 −0.300437 0.953802i \(-0.597133\pi\)
−0.300437 + 0.953802i \(0.597133\pi\)
\(132\) 0 0
\(133\) −25.2040 −2.18547
\(134\) 0 0
\(135\) −3.38361 −0.291215
\(136\) 0 0
\(137\) 1.51873 0.129754 0.0648770 0.997893i \(-0.479335\pi\)
0.0648770 + 0.997893i \(0.479335\pi\)
\(138\) 0 0
\(139\) −10.9301 −0.927080 −0.463540 0.886076i \(-0.653421\pi\)
−0.463540 + 0.886076i \(0.653421\pi\)
\(140\) 0 0
\(141\) −1.92225 3.32944i −0.161883 0.280390i
\(142\) 0 0
\(143\) 6.13047 0.512655
\(144\) 0 0
\(145\) −9.48147 16.4224i −0.787393 1.36380i
\(146\) 0 0
\(147\) 2.22442 3.85282i 0.183467 0.317775i
\(148\) 0 0
\(149\) 24.0365 1.96915 0.984573 0.174973i \(-0.0559837\pi\)
0.984573 + 0.174973i \(0.0559837\pi\)
\(150\) 0 0
\(151\) −1.90604 + 3.30136i −0.155112 + 0.268661i −0.933100 0.359618i \(-0.882907\pi\)
0.777988 + 0.628279i \(0.216240\pi\)
\(152\) 0 0
\(153\) −1.03262 + 1.78855i −0.0834821 + 0.144595i
\(154\) 0 0
\(155\) −10.1305 17.5465i −0.813699 1.40937i
\(156\) 0 0
\(157\) −1.83711 + 3.18197i −0.146617 + 0.253949i −0.929975 0.367622i \(-0.880172\pi\)
0.783358 + 0.621571i \(0.213505\pi\)
\(158\) 0 0
\(159\) −1.93477 −0.153437
\(160\) 0 0
\(161\) 7.51408 0.592193
\(162\) 0 0
\(163\) −7.03631 12.1873i −0.551127 0.954579i −0.998194 0.0600790i \(-0.980865\pi\)
0.447067 0.894501i \(-0.352469\pi\)
\(164\) 0 0
\(165\) −3.38361 5.86059i −0.263414 0.456247i
\(166\) 0 0
\(167\) −11.1835 19.3703i −0.865402 1.49892i −0.866647 0.498921i \(-0.833730\pi\)
0.00124506 0.999999i \(-0.499604\pi\)
\(168\) 0 0
\(169\) 1.80217 + 3.12145i 0.138629 + 0.240112i
\(170\) 0 0
\(171\) −3.72442 + 6.45089i −0.284814 + 0.493312i
\(172\) 0 0
\(173\) 2.30819 3.99791i 0.175489 0.303955i −0.764842 0.644218i \(-0.777183\pi\)
0.940330 + 0.340263i \(0.110516\pi\)
\(174\) 0 0
\(175\) 10.9102 + 18.8970i 0.824734 + 1.42848i
\(176\) 0 0
\(177\) −11.6696 −0.877139
\(178\) 0 0
\(179\) 8.37157 0.625721 0.312860 0.949799i \(-0.398713\pi\)
0.312860 + 0.949799i \(0.398713\pi\)
\(180\) 0 0
\(181\) −1.70802 2.95837i −0.126956 0.219894i 0.795540 0.605901i \(-0.207187\pi\)
−0.922496 + 0.386007i \(0.873854\pi\)
\(182\) 0 0
\(183\) −4.98379 + 8.63218i −0.368412 + 0.638109i
\(184\) 0 0
\(185\) −10.6346 + 18.4197i −0.781874 + 1.35425i
\(186\) 0 0
\(187\) −4.13047 −0.302050
\(188\) 0 0
\(189\) −1.69181 2.93030i −0.123061 0.213148i
\(190\) 0 0
\(191\) 4.36119 7.55380i 0.315564 0.546573i −0.663993 0.747739i \(-0.731140\pi\)
0.979557 + 0.201165i \(0.0644729\pi\)
\(192\) 0 0
\(193\) −4.69460 −0.337925 −0.168962 0.985623i \(-0.554042\pi\)
−0.168962 + 0.985623i \(0.554042\pi\)
\(194\) 0 0
\(195\) −5.18579 + 8.98204i −0.371362 + 0.643218i
\(196\) 0 0
\(197\) −6.47128 11.2086i −0.461059 0.798578i 0.537955 0.842974i \(-0.319197\pi\)
−0.999014 + 0.0443955i \(0.985864\pi\)
\(198\) 0 0
\(199\) 1.87343 3.24487i 0.132804 0.230023i −0.791953 0.610583i \(-0.790935\pi\)
0.924756 + 0.380560i \(0.124269\pi\)
\(200\) 0 0
\(201\) 7.80587 2.46342i 0.550584 0.173756i
\(202\) 0 0
\(203\) 9.48147 16.4224i 0.665468 1.15263i
\(204\) 0 0
\(205\) 6.50417 + 11.2655i 0.454271 + 0.786820i
\(206\) 0 0
\(207\) 1.11036 1.92321i 0.0771756 0.133672i
\(208\) 0 0
\(209\) −14.8977 −1.03050
\(210\) 0 0
\(211\) −7.77325 + 13.4637i −0.535133 + 0.926877i 0.464024 + 0.885823i \(0.346405\pi\)
−0.999157 + 0.0410545i \(0.986928\pi\)
\(212\) 0 0
\(213\) −6.65687 11.5300i −0.456121 0.790025i
\(214\) 0 0
\(215\) 20.0402 1.36673
\(216\) 0 0
\(217\) 10.1305 17.5465i 0.687701 1.19113i
\(218\) 0 0
\(219\) −2.80587 + 4.85991i −0.189603 + 0.328402i
\(220\) 0 0
\(221\) 3.16521 + 5.48231i 0.212915 + 0.368780i
\(222\) 0 0
\(223\) 0.514083 0.0344255 0.0172128 0.999852i \(-0.494521\pi\)
0.0172128 + 0.999852i \(0.494521\pi\)
\(224\) 0 0
\(225\) 6.44885 0.429923
\(226\) 0 0
\(227\) −13.9981 24.2455i −0.929089 1.60923i −0.784849 0.619688i \(-0.787259\pi\)
−0.144241 0.989543i \(-0.546074\pi\)
\(228\) 0 0
\(229\) 3.75334 6.50098i 0.248028 0.429597i −0.714951 0.699175i \(-0.753551\pi\)
0.962979 + 0.269578i \(0.0868842\pi\)
\(230\) 0 0
\(231\) 3.38361 5.86059i 0.222626 0.385599i
\(232\) 0 0
\(233\) 8.95534 + 15.5111i 0.586684 + 1.01617i 0.994663 + 0.103175i \(0.0329002\pi\)
−0.407979 + 0.912991i \(0.633767\pi\)
\(234\) 0 0
\(235\) 6.50417 + 11.2655i 0.424285 + 0.734883i
\(236\) 0 0
\(237\) 6.29985 + 10.9117i 0.409219 + 0.708788i
\(238\) 0 0
\(239\) −3.65919 6.33790i −0.236693 0.409965i 0.723070 0.690775i \(-0.242730\pi\)
−0.959763 + 0.280810i \(0.909397\pi\)
\(240\) 0 0
\(241\) 29.9833 1.93139 0.965697 0.259670i \(-0.0836139\pi\)
0.965697 + 0.259670i \(0.0836139\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −7.52659 + 13.0364i −0.480857 + 0.832868i
\(246\) 0 0
\(247\) 11.4162 + 19.7735i 0.726397 + 1.25816i
\(248\) 0 0
\(249\) 2.03262 3.52060i 0.128812 0.223109i
\(250\) 0 0
\(251\) −13.4917 + 23.3682i −0.851586 + 1.47499i 0.0281911 + 0.999603i \(0.491025\pi\)
−0.879777 + 0.475387i \(0.842308\pi\)
\(252\) 0 0
\(253\) 4.44145 0.279232
\(254\) 0 0
\(255\) 3.49398 6.05175i 0.218801 0.378975i
\(256\) 0 0
\(257\) 12.6221 + 21.8622i 0.787346 + 1.36372i 0.927587 + 0.373606i \(0.121879\pi\)
−0.140241 + 0.990117i \(0.544788\pi\)
\(258\) 0 0
\(259\) −21.2693 −1.32161
\(260\) 0 0
\(261\) −2.80217 4.85350i −0.173450 0.300424i
\(262\) 0 0
\(263\) 11.0532 0.681569 0.340784 0.940141i \(-0.389307\pi\)
0.340784 + 0.940141i \(0.389307\pi\)
\(264\) 0 0
\(265\) 6.54650 0.402148
\(266\) 0 0
\(267\) 0.837112 0.0512304
\(268\) 0 0
\(269\) −20.0161 −1.22040 −0.610202 0.792246i \(-0.708912\pi\)
−0.610202 + 0.792246i \(0.708912\pi\)
\(270\) 0 0
\(271\) −23.8736 −1.45022 −0.725109 0.688634i \(-0.758211\pi\)
−0.725109 + 0.688634i \(0.758211\pi\)
\(272\) 0 0
\(273\) −10.3716 −0.627716
\(274\) 0 0
\(275\) 6.44885 + 11.1697i 0.388880 + 0.673560i
\(276\) 0 0
\(277\) −17.8325 −1.07145 −0.535724 0.844393i \(-0.679961\pi\)
−0.535724 + 0.844393i \(0.679961\pi\)
\(278\) 0 0
\(279\) −2.99398 5.18572i −0.179245 0.310461i
\(280\) 0 0
\(281\) −5.63649 + 9.76269i −0.336245 + 0.582393i −0.983723 0.179691i \(-0.942490\pi\)
0.647478 + 0.762084i \(0.275824\pi\)
\(282\) 0 0
\(283\) −18.5673 −1.10371 −0.551855 0.833940i \(-0.686080\pi\)
−0.551855 + 0.833940i \(0.686080\pi\)
\(284\) 0 0
\(285\) 12.6020 21.8273i 0.746479 1.29294i
\(286\) 0 0
\(287\) −6.50417 + 11.2655i −0.383929 + 0.664984i
\(288\) 0 0
\(289\) 6.36740 + 11.0287i 0.374553 + 0.648745i
\(290\) 0 0
\(291\) −3.08144 + 5.33722i −0.180637 + 0.312873i
\(292\) 0 0
\(293\) −33.8000 −1.97462 −0.987310 0.158806i \(-0.949236\pi\)
−0.987310 + 0.158806i \(0.949236\pi\)
\(294\) 0 0
\(295\) 39.4853 2.29893
\(296\) 0 0
\(297\) −1.00000 1.73205i −0.0580259 0.100504i
\(298\) 0 0
\(299\) −3.40352 5.89507i −0.196831 0.340921i
\(300\) 0 0
\(301\) 10.0201 + 17.3553i 0.577549 + 1.00034i
\(302\) 0 0
\(303\) −5.09785 8.82974i −0.292864 0.507255i
\(304\) 0 0
\(305\) 16.8632 29.2080i 0.965585 1.67244i
\(306\) 0 0
\(307\) 1.95117 3.37953i 0.111359 0.192880i −0.804959 0.593330i \(-0.797813\pi\)
0.916319 + 0.400450i \(0.131146\pi\)
\(308\) 0 0
\(309\) 1.65919 + 2.87380i 0.0943880 + 0.163485i
\(310\) 0 0
\(311\) 26.6071 1.50875 0.754375 0.656444i \(-0.227940\pi\)
0.754375 + 0.656444i \(0.227940\pi\)
\(312\) 0 0
\(313\) 20.8482 1.17841 0.589205 0.807984i \(-0.299441\pi\)
0.589205 + 0.807984i \(0.299441\pi\)
\(314\) 0 0
\(315\) 5.72442 + 9.91499i 0.322535 + 0.558647i
\(316\) 0 0
\(317\) 3.90002 6.75504i 0.219047 0.379401i −0.735470 0.677557i \(-0.763039\pi\)
0.954517 + 0.298157i \(0.0963719\pi\)
\(318\) 0 0
\(319\) 5.60434 9.70700i 0.313783 0.543488i
\(320\) 0 0
\(321\) −8.15084 −0.454936
\(322\) 0 0
\(323\) −7.69181 13.3226i −0.427984 0.741289i
\(324\) 0 0
\(325\) 9.88361 17.1189i 0.548244 0.949587i
\(326\) 0 0
\(327\) 17.3063 0.957043
\(328\) 0 0
\(329\) −6.50417 + 11.2655i −0.358586 + 0.621090i
\(330\) 0 0
\(331\) 10.0569 + 17.4190i 0.552777 + 0.957437i 0.998073 + 0.0620538i \(0.0197650\pi\)
−0.445296 + 0.895383i \(0.646902\pi\)
\(332\) 0 0
\(333\) −3.14298 + 5.44380i −0.172234 + 0.298318i
\(334\) 0 0
\(335\) −26.4121 + 8.33525i −1.44304 + 0.455403i
\(336\) 0 0
\(337\) 12.0819 20.9265i 0.658144 1.13994i −0.322952 0.946415i \(-0.604675\pi\)
0.981096 0.193523i \(-0.0619914\pi\)
\(338\) 0 0
\(339\) 6.14066 + 10.6359i 0.333515 + 0.577664i
\(340\) 0 0
\(341\) 5.98796 10.3714i 0.324266 0.561645i
\(342\) 0 0
\(343\) 8.63211 0.466090
\(344\) 0 0
\(345\) −3.75704 + 6.50739i −0.202272 + 0.350346i
\(346\) 0 0
\(347\) −6.04048 10.4624i −0.324270 0.561652i 0.657094 0.753808i \(-0.271785\pi\)
−0.981364 + 0.192156i \(0.938452\pi\)
\(348\) 0 0
\(349\) 3.09026 0.165418 0.0827089 0.996574i \(-0.473643\pi\)
0.0827089 + 0.996574i \(0.473643\pi\)
\(350\) 0 0
\(351\) −1.53262 + 2.65457i −0.0818050 + 0.141690i
\(352\) 0 0
\(353\) −1.31606 + 2.27948i −0.0700466 + 0.121324i −0.898922 0.438110i \(-0.855648\pi\)
0.828875 + 0.559434i \(0.188981\pi\)
\(354\) 0 0
\(355\) 22.5243 + 39.0132i 1.19546 + 2.07060i
\(356\) 0 0
\(357\) 6.98796 0.369842
\(358\) 0 0
\(359\) −18.1017 −0.955373 −0.477686 0.878530i \(-0.658524\pi\)
−0.477686 + 0.878530i \(0.658524\pi\)
\(360\) 0 0
\(361\) −18.2427 31.5972i −0.960141 1.66301i
\(362\) 0 0
\(363\) −3.50000 + 6.06218i −0.183702 + 0.318182i
\(364\) 0 0
\(365\) 9.49398 16.4441i 0.496938 0.860721i
\(366\) 0 0
\(367\) −1.61685 2.80047i −0.0843991 0.146183i 0.820736 0.571308i \(-0.193564\pi\)
−0.905135 + 0.425124i \(0.860230\pi\)
\(368\) 0 0
\(369\) 1.92225 + 3.32944i 0.100069 + 0.173324i
\(370\) 0 0
\(371\) 3.27325 + 5.66944i 0.169939 + 0.294343i
\(372\) 0 0
\(373\) −14.9241 25.8493i −0.772740 1.33843i −0.936056 0.351852i \(-0.885552\pi\)
0.163315 0.986574i \(-0.447781\pi\)
\(374\) 0 0
\(375\) −4.90235 −0.253156
\(376\) 0 0
\(377\) −17.1786 −0.884744
\(378\) 0 0
\(379\) −11.0717 + 19.1768i −0.568716 + 0.985045i 0.427977 + 0.903790i \(0.359226\pi\)
−0.996693 + 0.0812559i \(0.974107\pi\)
\(380\) 0 0
\(381\) 9.49768 + 16.4505i 0.486581 + 0.842782i
\(382\) 0 0
\(383\) 15.0731 26.1074i 0.770199 1.33402i −0.167254 0.985914i \(-0.553490\pi\)
0.937453 0.348111i \(-0.113177\pi\)
\(384\) 0 0
\(385\) −11.4488 + 19.8300i −0.583487 + 1.01063i
\(386\) 0 0
\(387\) 5.92272 0.301069
\(388\) 0 0
\(389\) 3.31838 5.74760i 0.168249 0.291415i −0.769556 0.638580i \(-0.779522\pi\)
0.937804 + 0.347165i \(0.112856\pi\)
\(390\) 0 0
\(391\) 2.29316 + 3.97187i 0.115970 + 0.200866i
\(392\) 0 0
\(393\) 6.87732 0.346915
\(394\) 0 0
\(395\) −21.3163 36.9208i −1.07254 1.85769i
\(396\) 0 0
\(397\) 0.518731 0.0260344 0.0130172 0.999915i \(-0.495856\pi\)
0.0130172 + 0.999915i \(0.495856\pi\)
\(398\) 0 0
\(399\) 25.2040 1.26178
\(400\) 0 0
\(401\) −10.6973 −0.534200 −0.267100 0.963669i \(-0.586065\pi\)
−0.267100 + 0.963669i \(0.586065\pi\)
\(402\) 0 0
\(403\) −18.3545 −0.914302
\(404\) 0 0
\(405\) 3.38361 0.168133
\(406\) 0 0
\(407\) −12.5719 −0.623167
\(408\) 0 0
\(409\) 5.12795 + 8.88187i 0.253561 + 0.439180i 0.964504 0.264070i \(-0.0850649\pi\)
−0.710943 + 0.703250i \(0.751732\pi\)
\(410\) 0 0
\(411\) −1.51873 −0.0749135
\(412\) 0 0
\(413\) 19.7427 + 34.1953i 0.971473 + 1.68264i
\(414\) 0 0
\(415\) −6.87759 + 11.9123i −0.337608 + 0.584754i
\(416\) 0 0
\(417\) 10.9301 0.535250
\(418\) 0 0
\(419\) 7.41391 12.8413i 0.362193 0.627337i −0.626128 0.779720i \(-0.715361\pi\)
0.988321 + 0.152383i \(0.0486948\pi\)
\(420\) 0 0
\(421\) −5.44468 + 9.43047i −0.265358 + 0.459613i −0.967657 0.252269i \(-0.918823\pi\)
0.702300 + 0.711881i \(0.252157\pi\)
\(422\) 0 0
\(423\) 1.92225 + 3.32944i 0.0934632 + 0.161883i
\(424\) 0 0
\(425\) −6.65919 + 11.5341i −0.323018 + 0.559484i
\(426\) 0 0
\(427\) 33.7264 1.63214
\(428\) 0 0
\(429\) −6.13047 −0.295982
\(430\) 0 0
\(431\) 18.4671 + 31.9860i 0.889529 + 1.54071i 0.840433 + 0.541915i \(0.182301\pi\)
0.0490954 + 0.998794i \(0.484366\pi\)
\(432\) 0 0
\(433\) −19.1913 33.2404i −0.922276 1.59743i −0.795884 0.605449i \(-0.792994\pi\)
−0.126392 0.991980i \(-0.540340\pi\)
\(434\) 0 0
\(435\) 9.48147 + 16.4224i 0.454601 + 0.787393i
\(436\) 0 0
\(437\) 8.27093 + 14.3257i 0.395652 + 0.685290i
\(438\) 0 0
\(439\) 10.5308 18.2398i 0.502606 0.870539i −0.497390 0.867527i \(-0.665708\pi\)
0.999995 0.00301154i \(-0.000958603\pi\)
\(440\) 0 0
\(441\) −2.22442 + 3.85282i −0.105925 + 0.183467i
\(442\) 0 0
\(443\) −6.91993 11.9857i −0.328776 0.569456i 0.653493 0.756932i \(-0.273303\pi\)
−0.982269 + 0.187476i \(0.939969\pi\)
\(444\) 0 0
\(445\) −2.83246 −0.134272
\(446\) 0 0
\(447\) −24.0365 −1.13689
\(448\) 0 0
\(449\) 14.6900 + 25.4437i 0.693262 + 1.20076i 0.970763 + 0.240039i \(0.0771603\pi\)
−0.277502 + 0.960725i \(0.589506\pi\)
\(450\) 0 0
\(451\) −3.84451 + 6.65888i −0.181031 + 0.313555i
\(452\) 0 0
\(453\) 1.90604 3.30136i 0.0895537 0.155112i
\(454\) 0 0
\(455\) 35.0934 1.64520
\(456\) 0 0
\(457\) 1.94079 + 3.36154i 0.0907862 + 0.157246i 0.907842 0.419312i \(-0.137729\pi\)
−0.817056 + 0.576558i \(0.804395\pi\)
\(458\) 0 0
\(459\) 1.03262 1.78855i 0.0481984 0.0834821i
\(460\) 0 0
\(461\) 3.40399 0.158540 0.0792698 0.996853i \(-0.474741\pi\)
0.0792698 + 0.996853i \(0.474741\pi\)
\(462\) 0 0
\(463\) −15.3894 + 26.6551i −0.715204 + 1.23877i 0.247677 + 0.968843i \(0.420333\pi\)
−0.962881 + 0.269927i \(0.913001\pi\)
\(464\) 0 0
\(465\) 10.1305 + 17.5465i 0.469789 + 0.813699i
\(466\) 0 0
\(467\) −13.0983 + 22.6870i −0.606118 + 1.04983i 0.385756 + 0.922601i \(0.373941\pi\)
−0.991874 + 0.127226i \(0.959393\pi\)
\(468\) 0 0
\(469\) −20.4246 18.7059i −0.943119 0.863757i
\(470\) 0 0
\(471\) 1.83711 3.18197i 0.0846496 0.146617i
\(472\) 0 0
\(473\) 5.92272 + 10.2585i 0.272327 + 0.471684i
\(474\) 0 0
\(475\) −24.0182 + 41.6008i −1.10203 + 1.90878i
\(476\) 0 0
\(477\) 1.93477 0.0885869
\(478\) 0 0
\(479\) 18.4666 31.9851i 0.843762 1.46144i −0.0429303 0.999078i \(-0.513669\pi\)
0.886692 0.462360i \(-0.152997\pi\)
\(480\) 0 0
\(481\) 9.63397 + 16.6865i 0.439271 + 0.760840i
\(482\) 0 0
\(483\) −7.51408 −0.341903
\(484\) 0 0
\(485\) 10.4264 18.0591i 0.473439 0.820021i
\(486\) 0 0
\(487\) 6.84313 11.8527i 0.310092 0.537095i −0.668290 0.743901i \(-0.732973\pi\)
0.978382 + 0.206806i \(0.0663068\pi\)
\(488\) 0 0
\(489\) 7.03631 + 12.1873i 0.318193 + 0.551127i
\(490\) 0 0
\(491\) 0.727413 0.0328277 0.0164138 0.999865i \(-0.494775\pi\)
0.0164138 + 0.999865i \(0.494775\pi\)
\(492\) 0 0
\(493\) 11.5743 0.521279
\(494\) 0 0
\(495\) 3.38361 + 5.86059i 0.152082 + 0.263414i
\(496\) 0 0
\(497\) −22.5243 + 39.0132i −1.01035 + 1.74998i
\(498\) 0 0
\(499\) 13.2087 22.8781i 0.591302 1.02416i −0.402756 0.915308i \(-0.631948\pi\)
0.994057 0.108857i \(-0.0347191\pi\)
\(500\) 0 0
\(501\) 11.1835 + 19.3703i 0.499640 + 0.865402i
\(502\) 0 0
\(503\) −15.6346 27.0800i −0.697114 1.20744i −0.969463 0.245238i \(-0.921134\pi\)
0.272349 0.962199i \(-0.412199\pi\)
\(504\) 0 0
\(505\) 17.2492 + 29.8764i 0.767578 + 1.32948i
\(506\) 0 0
\(507\) −1.80217 3.12145i −0.0800372 0.138629i
\(508\) 0 0
\(509\) 37.7505 1.67326 0.836632 0.547766i \(-0.184522\pi\)
0.836632 + 0.547766i \(0.184522\pi\)
\(510\) 0 0
\(511\) 18.9880 0.839978
\(512\) 0 0
\(513\) 3.72442 6.45089i 0.164437 0.284814i
\(514\) 0 0
\(515\) −5.61406 9.72384i −0.247385 0.428484i
\(516\) 0 0
\(517\) −3.84451 + 6.65888i −0.169081 + 0.292857i
\(518\) 0 0
\(519\) −2.30819 + 3.99791i −0.101318 + 0.175489i
\(520\) 0 0
\(521\) −10.2860 −0.450636 −0.225318 0.974285i \(-0.572342\pi\)
−0.225318 + 0.974285i \(0.572342\pi\)
\(522\) 0 0
\(523\) −3.64947 + 6.32107i −0.159580 + 0.276401i −0.934717 0.355392i \(-0.884347\pi\)
0.775137 + 0.631793i \(0.217681\pi\)
\(524\) 0 0
\(525\) −10.9102 18.8970i −0.476161 0.824734i
\(526\) 0 0
\(527\) 12.3665 0.538694
\(528\) 0 0
\(529\) 9.03419 + 15.6477i 0.392791 + 0.680334i
\(530\) 0 0
\(531\) 11.6696 0.506416
\(532\) 0 0
\(533\) 11.7843 0.510435
\(534\) 0 0
\(535\) 27.5793 1.19236
\(536\) 0 0
\(537\) −8.37157 −0.361260
\(538\) 0 0
\(539\) −8.89770 −0.383251
\(540\) 0 0
\(541\) −28.2485 −1.21450 −0.607249 0.794512i \(-0.707727\pi\)
−0.607249 + 0.794512i \(0.707727\pi\)
\(542\) 0 0
\(543\) 1.70802 + 2.95837i 0.0732981 + 0.126956i
\(544\) 0 0
\(545\) −58.5580 −2.50835
\(546\) 0 0
\(547\) −1.13444 1.96490i −0.0485050 0.0840132i 0.840754 0.541418i \(-0.182112\pi\)
−0.889259 + 0.457405i \(0.848779\pi\)
\(548\) 0 0
\(549\) 4.98379 8.63218i 0.212703 0.368412i
\(550\) 0 0
\(551\) 41.7459 1.77843
\(552\) 0 0
\(553\) 21.3163 36.9208i 0.906459 1.57003i
\(554\) 0 0
\(555\) 10.6346 18.4197i 0.451415 0.781874i
\(556\) 0 0
\(557\) 12.2711 + 21.2542i 0.519944 + 0.900570i 0.999731 + 0.0231848i \(0.00738060\pi\)
−0.479787 + 0.877385i \(0.659286\pi\)
\(558\) 0 0
\(559\) 9.07726 15.7223i 0.383927 0.664982i
\(560\) 0 0
\(561\) 4.13047 0.174388
\(562\) 0 0
\(563\) −27.9509 −1.17799 −0.588995 0.808137i \(-0.700476\pi\)
−0.588995 + 0.808137i \(0.700476\pi\)
\(564\) 0 0
\(565\) −20.7776 35.9879i −0.874121 1.51402i
\(566\) 0 0
\(567\) 1.69181 + 2.93030i 0.0710492 + 0.123061i
\(568\) 0 0
\(569\) −7.28317 12.6148i −0.305326 0.528841i 0.672008 0.740544i \(-0.265432\pi\)
−0.977334 + 0.211704i \(0.932099\pi\)
\(570\) 0 0
\(571\) −11.2529 19.4905i −0.470918 0.815653i 0.528529 0.848915i \(-0.322744\pi\)
−0.999447 + 0.0332620i \(0.989410\pi\)
\(572\) 0 0
\(573\) −4.36119 + 7.55380i −0.182191 + 0.315564i
\(574\) 0 0
\(575\) 7.16056 12.4025i 0.298616 0.517218i
\(576\) 0 0
\(577\) 11.5833 + 20.0628i 0.482219 + 0.835227i 0.999792 0.0204120i \(-0.00649779\pi\)
−0.517573 + 0.855639i \(0.673164\pi\)
\(578\) 0 0
\(579\) 4.69460 0.195101
\(580\) 0 0
\(581\) −13.7552 −0.570661
\(582\) 0 0
\(583\) 1.93477 + 3.35111i 0.0801298 + 0.138789i
\(584\) 0 0
\(585\) 5.18579 8.98204i 0.214406 0.371362i
\(586\) 0 0
\(587\) −6.22859 + 10.7882i −0.257081 + 0.445278i −0.965459 0.260556i \(-0.916094\pi\)
0.708377 + 0.705834i \(0.249428\pi\)
\(588\) 0 0
\(589\) 44.6034 1.83785
\(590\) 0 0
\(591\) 6.47128 + 11.2086i 0.266193 + 0.461059i
\(592\) 0 0
\(593\) 16.9181 29.3030i 0.694742 1.20333i −0.275526 0.961294i \(-0.588852\pi\)
0.970268 0.242035i \(-0.0778148\pi\)
\(594\) 0 0
\(595\) −23.6446 −0.969332
\(596\) 0 0
\(597\) −1.87343 + 3.24487i −0.0766742 + 0.132804i
\(598\) 0 0
\(599\) −14.0349 24.3092i −0.573452 0.993248i −0.996208 0.0870046i \(-0.972271\pi\)
0.422756 0.906244i \(-0.361063\pi\)
\(600\) 0 0
\(601\) −14.7059 + 25.4713i −0.599866 + 1.03900i 0.392975 + 0.919549i \(0.371446\pi\)
−0.992840 + 0.119449i \(0.961887\pi\)
\(602\) 0 0
\(603\) −7.80587 + 2.46342i −0.317880 + 0.100318i
\(604\) 0 0
\(605\) 11.8427 20.5121i 0.481472 0.833934i
\(606\) 0 0
\(607\) 9.59553 + 16.6199i 0.389470 + 0.674583i 0.992378 0.123228i \(-0.0393247\pi\)
−0.602908 + 0.797811i \(0.705991\pi\)
\(608\) 0 0
\(609\) −9.48147 + 16.4224i −0.384208 + 0.665468i
\(610\) 0 0
\(611\) 11.7843 0.476742
\(612\) 0 0
\(613\) −15.0574 + 26.0801i −0.608161 + 1.05337i 0.383382 + 0.923590i \(0.374759\pi\)
−0.991543 + 0.129776i \(0.958574\pi\)
\(614\) 0 0
\(615\) −6.50417 11.2655i −0.262273 0.454271i
\(616\) 0 0
\(617\) −20.7579 −0.835683 −0.417841 0.908520i \(-0.637213\pi\)
−0.417841 + 0.908520i \(0.637213\pi\)
\(618\) 0 0
\(619\) 0.359344 0.622402i 0.0144433 0.0250165i −0.858713 0.512456i \(-0.828736\pi\)
0.873157 + 0.487440i \(0.162069\pi\)
\(620\) 0 0
\(621\) −1.11036 + 1.92321i −0.0445573 + 0.0771756i
\(622\) 0 0
\(623\) −1.41623 2.45299i −0.0567401 0.0982768i
\(624\) 0 0
\(625\) −15.6566 −0.626264
\(626\) 0 0
\(627\) 14.8977 0.594957
\(628\) 0 0
\(629\) −6.49099 11.2427i −0.258813 0.448277i
\(630\) 0 0
\(631\) 6.85257 11.8690i 0.272796 0.472497i −0.696780 0.717285i \(-0.745385\pi\)
0.969577 + 0.244787i \(0.0787181\pi\)
\(632\) 0 0
\(633\) 7.77325 13.4637i 0.308959 0.535133i
\(634\) 0 0
\(635\) −32.1365 55.6620i −1.27530 2.20888i
\(636\) 0 0
\(637\) 6.81838 + 11.8098i 0.270154 + 0.467921i
\(638\) 0 0
\(639\) 6.65687 + 11.5300i 0.263342 + 0.456121i
\(640\) 0 0
\(641\) −4.61658 7.99615i −0.182344 0.315829i 0.760334 0.649532i \(-0.225035\pi\)
−0.942678 + 0.333703i \(0.891702\pi\)
\(642\) 0 0
\(643\) −11.2031 −0.441807 −0.220904 0.975296i \(-0.570901\pi\)
−0.220904 + 0.975296i \(0.570901\pi\)
\(644\) 0 0
\(645\) −20.0402 −0.789082
\(646\) 0 0
\(647\) 15.1839 26.2993i 0.596942 1.03393i −0.396328 0.918109i \(-0.629716\pi\)
0.993270 0.115825i \(-0.0369510\pi\)
\(648\) 0 0
\(649\) 11.6696 + 20.2123i 0.458071 + 0.793402i
\(650\) 0 0
\(651\) −10.1305 + 17.5465i −0.397044 + 0.687701i
\(652\) 0 0
\(653\) −21.2388 + 36.7866i −0.831138 + 1.43957i 0.0659991 + 0.997820i \(0.478977\pi\)
−0.897137 + 0.441753i \(0.854357\pi\)
\(654\) 0 0
\(655\) −23.2702 −0.909242
\(656\) 0 0
\(657\) 2.80587 4.85991i 0.109467 0.189603i
\(658\) 0 0
\(659\) −7.62611 13.2088i −0.297071 0.514542i 0.678394 0.734699i \(-0.262676\pi\)
−0.975465 + 0.220157i \(0.929343\pi\)
\(660\) 0 0
\(661\) 31.7218 1.23384 0.616918 0.787027i \(-0.288381\pi\)
0.616918 + 0.787027i \(0.288381\pi\)
\(662\) 0 0
\(663\) −3.16521 5.48231i −0.122927 0.212915i
\(664\) 0 0
\(665\) −85.2807 −3.30705
\(666\) 0 0
\(667\) −12.4457 −0.481900
\(668\) 0 0
\(669\) −0.514083 −0.0198756
\(670\) 0 0
\(671\) 19.9352 0.769588
\(672\) 0 0
\(673\) −22.7904 −0.878504 −0.439252 0.898364i \(-0.644756\pi\)
−0.439252 + 0.898364i \(0.644756\pi\)
\(674\) 0 0
\(675\) −6.44885 −0.248216
\(676\) 0 0
\(677\) 13.5164 + 23.4111i 0.519478 + 0.899762i 0.999744 + 0.0226389i \(0.00720680\pi\)
−0.480266 + 0.877123i \(0.659460\pi\)
\(678\) 0 0
\(679\) 20.8528 0.800259
\(680\) 0 0
\(681\) 13.9981 + 24.2455i 0.536410 + 0.929089i
\(682\) 0 0
\(683\) −2.05784 + 3.56428i −0.0787410 + 0.136383i −0.902707 0.430256i \(-0.858423\pi\)
0.823966 + 0.566639i \(0.191757\pi\)
\(684\) 0 0
\(685\) 5.13880 0.196343
\(686\) 0 0
\(687\) −3.75334 + 6.50098i −0.143199 + 0.248028i
\(688\) 0 0
\(689\) 2.96526 5.13597i 0.112967 0.195665i
\(690\) 0 0
\(691\) −3.82692 6.62843i −0.145583 0.252157i 0.784007 0.620752i \(-0.213172\pi\)
−0.929590 + 0.368594i \(0.879839\pi\)
\(692\) 0 0
\(693\) −3.38361 + 5.86059i −0.128533 + 0.222626i
\(694\) 0 0
\(695\) −36.9833 −1.40286
\(696\) 0 0
\(697\) −7.93981 −0.300742
\(698\) 0 0
\(699\) −8.95534 15.5111i −0.338722 0.586684i
\(700\) 0 0
\(701\) −12.9558 22.4401i −0.489334 0.847551i 0.510591 0.859824i \(-0.329427\pi\)
−0.999925 + 0.0122725i \(0.996093\pi\)
\(702\) 0 0
\(703\) −23.4116 40.5501i −0.882985 1.52937i
\(704\) 0 0
\(705\) −6.50417 11.2655i −0.244961 0.424285i
\(706\) 0 0
\(707\) −17.2492 + 29.8764i −0.648722 + 1.12362i
\(708\) 0 0
\(709\) −11.0204 + 19.0879i −0.413879 + 0.716859i −0.995310 0.0967363i \(-0.969160\pi\)
0.581431 + 0.813596i \(0.302493\pi\)
\(710\) 0 0
\(711\) −6.29985 10.9117i −0.236263 0.409219i
\(712\) 0 0
\(713\) −13.2976 −0.497999
\(714\) 0 0
\(715\) 20.7431 0.775749
\(716\) 0 0
\(717\) 3.65919 + 6.33790i 0.136655 + 0.236693i
\(718\) 0 0
\(719\) 12.1705 21.0799i 0.453882 0.786147i −0.544741 0.838604i \(-0.683372\pi\)
0.998623 + 0.0524572i \(0.0167053\pi\)
\(720\) 0 0
\(721\) 5.61406 9.72384i 0.209079 0.362135i
\(722\) 0 0
\(723\) −29.9833 −1.11509
\(724\) 0 0
\(725\) −18.0708 31.2995i −0.671132 1.16243i
\(726\) 0 0
\(727\) 11.4773 19.8793i 0.425669 0.737281i −0.570813 0.821080i \(-0.693372\pi\)
0.996483 + 0.0837988i \(0.0267053\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.11590 + 10.5931i −0.226205 + 0.391798i
\(732\) 0 0
\(733\) 12.3815 + 21.4454i 0.457321 + 0.792103i 0.998818 0.0485996i \(-0.0154758\pi\)
−0.541498 + 0.840702i \(0.682143\pi\)
\(734\) 0 0
\(735\) 7.52659 13.0364i 0.277623 0.480857i
\(736\) 0 0
\(737\) −12.0726 11.0567i −0.444701 0.407280i
\(738\) 0 0
\(739\) −9.47591 + 16.4128i −0.348577 + 0.603753i −0.985997 0.166763i \(-0.946668\pi\)
0.637420 + 0.770517i \(0.280002\pi\)
\(740\) 0 0
\(741\) −11.4162 19.7735i −0.419386 0.726397i
\(742\) 0 0
\(743\) −10.3637 + 17.9505i −0.380208 + 0.658539i −0.991092 0.133181i \(-0.957481\pi\)
0.610884 + 0.791720i \(0.290814\pi\)
\(744\) 0 0
\(745\) 81.3302 2.97971
\(746\) 0 0
\(747\) −2.03262 + 3.52060i −0.0743696 + 0.128812i
\(748\) 0 0
\(749\) 13.7897 + 23.8844i 0.503863 + 0.872717i
\(750\) 0 0
\(751\) 37.0241 1.35103 0.675514 0.737348i \(-0.263922\pi\)
0.675514 + 0.737348i \(0.263922\pi\)
\(752\) 0 0
\(753\) 13.4917 23.3682i 0.491663 0.851586i
\(754\) 0 0
\(755\) −6.44932 + 11.1705i −0.234715 + 0.406538i
\(756\) 0 0
\(757\) −7.58301 13.1342i −0.275609 0.477369i 0.694679 0.719320i \(-0.255546\pi\)
−0.970289 + 0.241950i \(0.922213\pi\)
\(758\) 0 0
\(759\) −4.44145 −0.161215
\(760\) 0 0
\(761\) −3.28871 −0.119216 −0.0596078 0.998222i \(-0.518985\pi\)
−0.0596078 + 0.998222i \(0.518985\pi\)
\(762\) 0 0
\(763\) −29.2790 50.7127i −1.05997 1.83592i
\(764\) 0 0
\(765\) −3.49398 + 6.05175i −0.126325 + 0.218801i
\(766\) 0 0
\(767\) 17.8850 30.9777i 0.645790 1.11854i
\(768\) 0 0
\(769\) 14.0391 + 24.3164i 0.506263 + 0.876874i 0.999974 + 0.00724723i \(0.00230689\pi\)
−0.493711 + 0.869626i \(0.664360\pi\)
\(770\) 0 0
\(771\) −12.6221 21.8622i −0.454575 0.787346i
\(772\) 0 0
\(773\) −3.88983 6.73739i −0.139908 0.242327i 0.787554 0.616246i \(-0.211347\pi\)
−0.927461 + 0.373919i \(0.878014\pi\)
\(774\) 0 0
\(775\) −19.3077 33.4419i −0.693553 1.20127i
\(776\) 0 0
\(777\) 21.2693 0.763031
\(778\) 0 0
\(779\) −28.6372 −1.02603
\(780\) 0 0
\(781\) −13.3137 + 23.0601i −0.476403 + 0.825154i
\(782\) 0 0
\(783\) 2.80217 + 4.85350i 0.100141 + 0.173450i
\(784\) 0 0
\(785\) −6.21608 + 10.7666i −0.221861 + 0.384275i
\(786\) 0 0
\(787\) 6.27372 10.8664i 0.223634 0.387345i −0.732275 0.681009i \(-0.761541\pi\)
0.955909 + 0.293664i \(0.0948747\pi\)
\(788\) 0 0
\(789\) −11.0532 −0.393504
\(790\) 0 0
\(791\) 20.7776 35.9879i 0.738767 1.27958i
\(792\) 0 0
\(793\) −15.2765 26.4596i −0.542484 0.939609i
\(794\) 0 0
\(795\) −6.54650 −0.232181
\(796\) 0 0
\(797\) 17.5574 + 30.4102i 0.621913 + 1.07719i 0.989129 + 0.147050i \(0.0469778\pi\)
−0.367216 + 0.930136i \(0.619689\pi\)
\(798\) 0 0
\(799\) −7.93981 −0.280890
\(800\) 0 0
\(801\) −0.837112 −0.0295779
\(802\) 0 0
\(803\) 11.2235 0.396068
\(804\) 0 0
\(805\) 25.4248 0.896105
\(806\) 0 0
\(807\) 20.0161 0.704601
\(808\) 0 0
\(809\) −21.5539 −0.757795 −0.378897 0.925439i \(-0.623697\pi\)
−0.378897 + 0.925439i \(0.623697\pi\)
\(810\) 0 0
\(811\) 15.2003 + 26.3278i 0.533756 + 0.924492i 0.999222 + 0.0394270i \(0.0125532\pi\)
−0.465466 + 0.885066i \(0.654113\pi\)
\(812\) 0 0
\(813\) 23.8736 0.837284
\(814\) 0 0
\(815\) −23.8082 41.2370i −0.833964 1.44447i
\(816\) 0 0
\(817\) −22.0587 + 38.2068i −0.771737 + 1.33669i
\(818\) 0 0
\(819\) 10.3716 0.362412
\(820\) 0 0
\(821\) −3.88778 + 6.73383i −0.135684 + 0.235012i −0.925859 0.377870i \(-0.876657\pi\)
0.790174 + 0.612882i \(0.209990\pi\)
\(822\) 0 0
\(823\) −22.2001 + 38.4518i −0.773848 + 1.34034i 0.161591 + 0.986858i \(0.448337\pi\)
−0.935440 + 0.353487i \(0.884996\pi\)
\(824\) 0 0
\(825\) −6.44885 11.1697i −0.224520 0.388880i
\(826\) 0 0
\(827\) −14.8026 + 25.6389i −0.514738 + 0.891553i 0.485116 + 0.874450i \(0.338778\pi\)
−0.999854 + 0.0171026i \(0.994556\pi\)
\(828\) 0 0
\(829\) −40.7352 −1.41479 −0.707396 0.706817i \(-0.750130\pi\)
−0.707396 + 0.706817i \(0.750130\pi\)
\(830\) 0 0
\(831\) 17.8325 0.618601
\(832\) 0 0
\(833\) −4.59396 7.95697i −0.159171 0.275693i
\(834\) 0 0
\(835\) −37.8405 65.5417i −1.30953 2.26816i
\(836\) 0 0
\(837\) 2.99398 + 5.18572i 0.103487 + 0.179245i
\(838\) 0 0
\(839\) 22.1286 + 38.3279i 0.763965 + 1.32323i 0.940792 + 0.338984i \(0.110083\pi\)
−0.176828 + 0.984242i \(0.556584\pi\)
\(840\) 0 0
\(841\) −1.20432 + 2.08594i −0.0415283 + 0.0719291i
\(842\) 0 0
\(843\) 5.63649 9.76269i 0.194131 0.336245i
\(844\) 0 0
\(845\) 6.09785 + 10.5618i 0.209772 + 0.363337i
\(846\) 0 0
\(847\) 23.6853 0.813837
\(848\) 0 0
\(849\) 18.5673 0.637227
\(850\) 0 0
\(851\) 6.97970 + 12.0892i 0.239261 + 0.414412i
\(852\) 0 0
\(853\) −12.4431 + 21.5521i −0.426044 + 0.737929i −0.996517 0.0833868i \(-0.973426\pi\)
0.570474 + 0.821316i \(0.306760\pi\)
\(854\) 0 0
\(855\) −12.6020 + 21.8273i −0.430980 + 0.746479i
\(856\) 0 0
\(857\) 43.9870 1.50257 0.751284 0.659980i \(-0.229435\pi\)
0.751284 + 0.659980i \(0.229435\pi\)
\(858\) 0 0
\(859\) 24.8042 + 42.9622i 0.846309 + 1.46585i 0.884480 + 0.466578i \(0.154513\pi\)
−0.0381711 + 0.999271i \(0.512153\pi\)
\(860\) 0 0
\(861\) 6.50417 11.2655i 0.221661 0.383929i
\(862\) 0 0
\(863\) −49.8616 −1.69731 −0.848654 0.528949i \(-0.822586\pi\)
−0.848654 + 0.528949i \(0.822586\pi\)
\(864\) 0 0
\(865\) 7.81003 13.5274i 0.265549 0.459945i
\(866\) 0 0
\(867\) −6.36740 11.0287i −0.216248 0.374553i
\(868\) 0 0
\(869\) 12.5997 21.8233i 0.427415 0.740305i
\(870\) 0 0
\(871\) −5.42410 + 24.4967i −0.183788 + 0.830039i
\(872\) 0 0
\(873\) 3.08144 5.33722i 0.104291 0.180637i
\(874\) 0 0
\(875\) 8.29382 + 14.3653i 0.280382 + 0.485637i
\(876\) 0 0
\(877\) −3.54699 + 6.14356i −0.119773 + 0.207453i −0.919678 0.392674i \(-0.871550\pi\)
0.799905 + 0.600127i \(0.204883\pi\)
\(878\) 0 0
\(879\) 33.8000 1.14005
\(880\) 0 0
\(881\) 13.6150 23.5819i 0.458701 0.794493i −0.540192 0.841542i \(-0.681648\pi\)
0.998893 + 0.0470487i \(0.0149816\pi\)
\(882\) 0 0
\(883\) −2.42363 4.19785i −0.0815616 0.141269i 0.822359 0.568969i \(-0.192657\pi\)
−0.903921 + 0.427700i \(0.859324\pi\)
\(884\) 0 0
\(885\) −39.4853 −1.32729
\(886\) 0 0
\(887\) 14.7369 25.5251i 0.494818 0.857050i −0.505164 0.863023i \(-0.668568\pi\)
0.999982 + 0.00597347i \(0.00190142\pi\)
\(888\) 0 0
\(889\) 32.1365 55.6620i 1.07782 1.86684i
\(890\) 0 0
\(891\) 1.00000 + 1.73205i 0.0335013 + 0.0580259i
\(892\) 0 0
\(893\) −28.6372 −0.958306
\(894\) 0 0
\(895\) 28.3262 0.946839
\(896\) 0 0
\(897\) 3.40352 + 5.89507i 0.113640 + 0.196831i
\(898\) 0 0
\(899\) −16.7793 + 29.0626i −0.559620 + 0.969290i
\(900\) 0 0
\(901\) −1.99787 + 3.46042i −0.0665588 + 0.115283i
\(902\) 0 0
\(903\) −10.0201 17.3553i −0.333448 0.577549i
\(904\) 0 0
\(905\) −5.77927 10.0100i −0.192110 0.332744i
\(906\) 0 0
\(907\) −28.0039 48.5041i −0.929854 1.61055i −0.783563 0.621312i \(-0.786600\pi\)
−0.146290 0.989242i \(-0.546733\pi\)
\(908\) 0 0
\(909\) 5.09785 + 8.82974i 0.169085 + 0.292864i
\(910\) 0 0
\(911\) 47.1837 1.56326 0.781632 0.623740i \(-0.214387\pi\)
0.781632 + 0.623740i \(0.214387\pi\)
\(912\) 0 0
\(913\) −8.13047 −0.269079
\(914\) 0 0
\(915\) −16.8632 + 29.2080i −0.557481 + 0.965585i
\(916\) 0 0
\(917\) −11.6351 20.1526i −0.384225 0.665497i
\(918\) 0 0
\(919\) −24.0286 + 41.6188i −0.792631 + 1.37288i 0.131701 + 0.991290i \(0.457956\pi\)
−0.924332 + 0.381588i \(0.875377\pi\)
\(920\) 0 0
\(921\) −1.95117 + 3.37953i −0.0642933 + 0.111359i
\(922\) 0 0
\(923\) 40.8097 1.34327
\(924\) 0 0
\(925\) −20.2686 + 35.1063i −0.666428 + 1.15429i
\(926\) 0 0
\(927\) −1.65919 2.87380i −0.0544950 0.0943880i
\(928\) 0 0
\(929\) 20.3314 0.667050 0.333525 0.942741i \(-0.391762\pi\)
0.333525 + 0.942741i \(0.391762\pi\)
\(930\) 0 0
\(931\) −16.5694 28.6990i −0.543040 0.940573i
\(932\) 0 0
\(933\) −26.6071 −0.871077
\(934\) 0 0
\(935\) −13.9759 −0.457061
\(936\) 0 0
\(937\) −34.5904 −1.13002 −0.565009 0.825084i \(-0.691128\pi\)
−0.565009 + 0.825084i \(0.691128\pi\)
\(938\) 0 0
\(939\) −20.8482 −0.680355
\(940\) 0 0
\(941\) 42.9125 1.39891 0.699454 0.714678i \(-0.253427\pi\)
0.699454 + 0.714678i \(0.253427\pi\)
\(942\) 0 0
\(943\) 8.53760 0.278022
\(944\) 0 0
\(945\) −5.72442 9.91499i −0.186216 0.322535i
\(946\) 0 0
\(947\) 34.6482 1.12592 0.562958 0.826485i \(-0.309663\pi\)
0.562958 + 0.826485i \(0.309663\pi\)
\(948\) 0 0
\(949\) −8.60064 14.8968i −0.279189 0.483569i
\(950\) 0 0
\(951\) −3.90002 + 6.75504i −0.126467 + 0.219047i
\(952\) 0 0
\(953\) −28.9023 −0.936239 −0.468119 0.883665i \(-0.655068\pi\)
−0.468119 + 0.883665i \(0.655068\pi\)
\(954\) 0 0
\(955\) 14.7566 25.5591i 0.477511 0.827074i
\(956\) 0 0
\(957\) −5.60434 + 9.70700i −0.181163 + 0.313783i
\(958\) 0 0
\(959\) 2.56940 + 4.45033i 0.0829703 + 0.143709i
\(960\) 0 0
\(961\) −2.42781 + 4.20509i −0.0783164 + 0.135648i
\(962\) 0 0
\(963\) 8.15084 0.262657
\(964\) 0 0
\(965\) −15.8847 −0.511347
\(966\) 0 0
\(967\) −14.6309 25.3415i −0.470499 0.814928i 0.528932 0.848664i \(-0.322593\pi\)
−0.999431 + 0.0337360i \(0.989259\pi\)
\(968\) 0 0
\(969\) 7.69181 + 13.3226i 0.247096 + 0.427984i
\(970\) 0 0
\(971\) −17.0761 29.5766i −0.547998 0.949160i −0.998412 0.0563400i \(-0.982057\pi\)
0.450414 0.892820i \(-0.351276\pi\)
\(972\) 0 0
\(973\) −18.4917 32.0285i −0.592815 1.02679i
\(974\) 0 0
\(975\) −9.88361 + 17.1189i −0.316529 + 0.548244i
\(976\) 0 0
\(977\) −12.4134 + 21.5007i −0.397141 + 0.687869i −0.993372 0.114944i \(-0.963331\pi\)
0.596231 + 0.802813i \(0.296664\pi\)
\(978\) 0 0
\(979\) −0.837112 1.44992i −0.0267542 0.0463396i
\(980\) 0 0
\(981\) −17.3063 −0.552549
\(982\) 0 0
\(983\) −10.1920 −0.325074 −0.162537 0.986702i \(-0.551968\pi\)
−0.162537 + 0.986702i \(0.551968\pi\)
\(984\) 0 0
\(985\) −21.8963 37.9255i −0.697674 1.20841i
\(986\) 0 0
\(987\) 6.50417 11.2655i 0.207030 0.358586i
\(988\) 0 0
\(989\) 6.57637 11.3906i 0.209116 0.362200i
\(990\) 0 0
\(991\) 3.25260 0.103322 0.0516612 0.998665i \(-0.483548\pi\)
0.0516612 + 0.998665i \(0.483548\pi\)
\(992\) 0 0
\(993\) −10.0569 17.4190i −0.319146 0.552777i
\(994\) 0 0
\(995\) 6.33895 10.9794i 0.200958 0.348070i
\(996\) 0 0
\(997\) 44.0619 1.39545 0.697727 0.716363i \(-0.254195\pi\)
0.697727 + 0.716363i \(0.254195\pi\)
\(998\) 0 0
\(999\) 3.14298 5.44380i 0.0994395 0.172234i
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 804.2.i.d.565.4 yes 8
3.2 odd 2 2412.2.l.e.1369.1 8
67.37 even 3 inner 804.2.i.d.37.4 8
201.104 odd 6 2412.2.l.e.37.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
804.2.i.d.37.4 8 67.37 even 3 inner
804.2.i.d.565.4 yes 8 1.1 even 1 trivial
2412.2.l.e.37.1 8 201.104 odd 6
2412.2.l.e.1369.1 8 3.2 odd 2