Properties

Label 1680.2.bg.u.1201.3
Level $1680$
Weight $2$
Character 1680.1201
Analytic conductor $13.415$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(961,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1680.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.bg (of order \(3\), degree \(2\), 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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.38363328.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 3x^{4} - 2x^{3} - 21x^{2} - 49x + 343 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1201.3
Root \(0.247423 - 2.63416i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1201
Dual form 1680.2.bg.u.961.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(2.40496 - 1.10280i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(2.40496 - 1.10280i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.90496 - 3.29948i) q^{11} +4.31507 q^{13} -1.00000 q^{15} +(0.657535 - 1.13888i) q^{17} +(3.81507 + 6.60790i) q^{19} +(0.247423 - 2.63416i) q^{21} +(3.41011 + 5.90649i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} -4.30476 q^{29} +(0.747423 - 1.29457i) q^{31} +(-1.90496 - 3.29948i) q^{33} +(-2.15754 - 1.53135i) q^{35} +(-4.65238 - 8.05816i) q^{37} +(2.15754 - 3.73696i) q^{39} -10.4401 q^{41} +4.80992 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-1.75258 - 3.03555i) q^{47} +(4.56765 - 5.30439i) q^{49} +(-0.657535 - 1.13888i) q^{51} +(4.06765 - 7.04537i) q^{53} -3.80992 q^{55} +7.63014 q^{57} +(4.15238 - 7.19214i) q^{59} +(3.31507 + 5.74187i) q^{61} +(-2.15754 - 1.53135i) q^{63} +(-2.15754 - 3.73696i) q^{65} +(-7.72003 + 13.3715i) q^{67} +6.82022 q^{69} +4.32538 q^{71} +(-1.91011 + 3.30841i) q^{73} +(0.500000 + 0.866025i) q^{75} +(0.942661 - 10.0359i) q^{77} +(-1.06249 - 1.84029i) q^{79} +(-0.500000 + 0.866025i) q^{81} +10.9349 q^{83} -1.31507 q^{85} +(-2.15238 + 3.72803i) q^{87} +(-5.65754 - 9.79914i) q^{89} +(10.3776 - 4.75868i) q^{91} +(-0.747423 - 1.29457i) q^{93} +(3.81507 - 6.60790i) q^{95} +10.6301 q^{97} -3.80992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 3 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 3 q^{5} + 2 q^{7} - 3 q^{9} - q^{11} + 2 q^{13} - 6 q^{15} - 8 q^{17} - q^{19} + q^{21} + 9 q^{23} - 3 q^{25} - 6 q^{27} + 4 q^{31} + q^{33} - q^{35} - 15 q^{37} + q^{39} + 10 q^{41} + 4 q^{43} - 3 q^{45} - 11 q^{47} + 4 q^{49} + 8 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 12 q^{59} - 4 q^{61} - q^{63} - q^{65} - 10 q^{67} + 18 q^{69} + 4 q^{71} + 3 q^{75} + 31 q^{77} + 18 q^{79} - 3 q^{81} - 8 q^{83} + 16 q^{85} - 22 q^{89} + 14 q^{91} - 4 q^{93} - q^{95} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 2.40496 1.10280i 0.908989 0.416821i
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 1.90496 3.29948i 0.574366 0.994832i −0.421744 0.906715i \(-0.638582\pi\)
0.996110 0.0881168i \(-0.0280849\pi\)
\(12\) 0 0
\(13\) 4.31507 1.19679 0.598393 0.801203i \(-0.295806\pi\)
0.598393 + 0.801203i \(0.295806\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.657535 1.13888i 0.159476 0.276220i −0.775204 0.631711i \(-0.782353\pi\)
0.934680 + 0.355491i \(0.115686\pi\)
\(18\) 0 0
\(19\) 3.81507 + 6.60790i 0.875237 + 1.51596i 0.856510 + 0.516131i \(0.172628\pi\)
0.0187273 + 0.999825i \(0.494039\pi\)
\(20\) 0 0
\(21\) 0.247423 2.63416i 0.0539921 0.574820i
\(22\) 0 0
\(23\) 3.41011 + 5.90649i 0.711058 + 1.23159i 0.964461 + 0.264227i \(0.0851168\pi\)
−0.253403 + 0.967361i \(0.581550\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.30476 −0.799374 −0.399687 0.916652i \(-0.630881\pi\)
−0.399687 + 0.916652i \(0.630881\pi\)
\(30\) 0 0
\(31\) 0.747423 1.29457i 0.134241 0.232512i −0.791066 0.611731i \(-0.790474\pi\)
0.925307 + 0.379218i \(0.123807\pi\)
\(32\) 0 0
\(33\) −1.90496 3.29948i −0.331611 0.574366i
\(34\) 0 0
\(35\) −2.15754 1.53135i −0.364690 0.258846i
\(36\) 0 0
\(37\) −4.65238 8.05816i −0.764847 1.32475i −0.940328 0.340271i \(-0.889481\pi\)
0.175481 0.984483i \(-0.443852\pi\)
\(38\) 0 0
\(39\) 2.15754 3.73696i 0.345482 0.598393i
\(40\) 0 0
\(41\) −10.4401 −1.63046 −0.815231 0.579135i \(-0.803390\pi\)
−0.815231 + 0.579135i \(0.803390\pi\)
\(42\) 0 0
\(43\) 4.80992 0.733505 0.366753 0.930318i \(-0.380470\pi\)
0.366753 + 0.930318i \(0.380470\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) 0 0
\(47\) −1.75258 3.03555i −0.255640 0.442781i 0.709429 0.704776i \(-0.248953\pi\)
−0.965069 + 0.261996i \(0.915619\pi\)
\(48\) 0 0
\(49\) 4.56765 5.30439i 0.652521 0.757771i
\(50\) 0 0
\(51\) −0.657535 1.13888i −0.0920733 0.159476i
\(52\) 0 0
\(53\) 4.06765 7.04537i 0.558734 0.967756i −0.438868 0.898551i \(-0.644620\pi\)
0.997602 0.0692048i \(-0.0220462\pi\)
\(54\) 0 0
\(55\) −3.80992 −0.513729
\(56\) 0 0
\(57\) 7.63014 1.01064
\(58\) 0 0
\(59\) 4.15238 7.19214i 0.540594 0.936336i −0.458276 0.888810i \(-0.651533\pi\)
0.998870 0.0475263i \(-0.0151338\pi\)
\(60\) 0 0
\(61\) 3.31507 + 5.74187i 0.424451 + 0.735171i 0.996369 0.0851398i \(-0.0271337\pi\)
−0.571918 + 0.820311i \(0.693800\pi\)
\(62\) 0 0
\(63\) −2.15754 1.53135i −0.271824 0.192932i
\(64\) 0 0
\(65\) −2.15754 3.73696i −0.267609 0.463513i
\(66\) 0 0
\(67\) −7.72003 + 13.3715i −0.943152 + 1.63359i −0.183741 + 0.982975i \(0.558821\pi\)
−0.759410 + 0.650612i \(0.774513\pi\)
\(68\) 0 0
\(69\) 6.82022 0.821059
\(70\) 0 0
\(71\) 4.32538 0.513328 0.256664 0.966501i \(-0.417377\pi\)
0.256664 + 0.966501i \(0.417377\pi\)
\(72\) 0 0
\(73\) −1.91011 + 3.30841i −0.223562 + 0.387220i −0.955887 0.293735i \(-0.905102\pi\)
0.732325 + 0.680955i \(0.238435\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 0.942661 10.0359i 0.107426 1.14370i
\(78\) 0 0
\(79\) −1.06249 1.84029i −0.119540 0.207049i 0.800046 0.599939i \(-0.204809\pi\)
−0.919585 + 0.392890i \(0.871475\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 10.9349 1.20026 0.600131 0.799902i \(-0.295115\pi\)
0.600131 + 0.799902i \(0.295115\pi\)
\(84\) 0 0
\(85\) −1.31507 −0.142639
\(86\) 0 0
\(87\) −2.15238 + 3.72803i −0.230759 + 0.399687i
\(88\) 0 0
\(89\) −5.65754 9.79914i −0.599698 1.03871i −0.992865 0.119240i \(-0.961954\pi\)
0.393168 0.919467i \(-0.371379\pi\)
\(90\) 0 0
\(91\) 10.3776 4.75868i 1.08786 0.498845i
\(92\) 0 0
\(93\) −0.747423 1.29457i −0.0775041 0.134241i
\(94\) 0 0
\(95\) 3.81507 6.60790i 0.391418 0.677956i
\(96\) 0 0
\(97\) 10.6301 1.07933 0.539664 0.841881i \(-0.318551\pi\)
0.539664 + 0.841881i \(0.318551\pi\)
\(98\) 0 0
\(99\) −3.80992 −0.382911
\(100\) 0 0
\(101\) −5.34246 + 9.25342i −0.531595 + 0.920750i 0.467725 + 0.883874i \(0.345074\pi\)
−0.999320 + 0.0368756i \(0.988259\pi\)
\(102\) 0 0
\(103\) −9.72003 16.8356i −0.957743 1.65886i −0.727963 0.685617i \(-0.759533\pi\)
−0.229780 0.973243i \(-0.573801\pi\)
\(104\) 0 0
\(105\) −2.40496 + 1.10280i −0.234700 + 0.107623i
\(106\) 0 0
\(107\) 5.15238 + 8.92419i 0.498099 + 0.862734i 0.999998 0.00219314i \(-0.000698099\pi\)
−0.501898 + 0.864927i \(0.667365\pi\)
\(108\) 0 0
\(109\) 3.55734 6.16149i 0.340731 0.590164i −0.643837 0.765162i \(-0.722659\pi\)
0.984569 + 0.174998i \(0.0559920\pi\)
\(110\) 0 0
\(111\) −9.30476 −0.883169
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 3.41011 5.90649i 0.317995 0.550783i
\(116\) 0 0
\(117\) −2.15754 3.73696i −0.199464 0.345482i
\(118\) 0 0
\(119\) 0.325379 3.46410i 0.0298274 0.317554i
\(120\) 0 0
\(121\) −1.75773 3.04448i −0.159794 0.276771i
\(122\) 0 0
\(123\) −5.22003 + 9.04135i −0.470674 + 0.815231i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.9452 −1.50364 −0.751822 0.659366i \(-0.770825\pi\)
−0.751822 + 0.659366i \(0.770825\pi\)
\(128\) 0 0
\(129\) 2.40496 4.16551i 0.211745 0.366753i
\(130\) 0 0
\(131\) −6.55219 11.3487i −0.572467 0.991542i −0.996312 0.0858072i \(-0.972653\pi\)
0.423845 0.905735i \(-0.360680\pi\)
\(132\) 0 0
\(133\) 16.4623 + 11.6844i 1.42746 + 1.01317i
\(134\) 0 0
\(135\) 0.500000 + 0.866025i 0.0430331 + 0.0745356i
\(136\) 0 0
\(137\) −6.46745 + 11.2020i −0.552552 + 0.957048i 0.445538 + 0.895263i \(0.353012\pi\)
−0.998090 + 0.0617845i \(0.980321\pi\)
\(138\) 0 0
\(139\) 11.7448 0.996183 0.498091 0.867125i \(-0.334034\pi\)
0.498091 + 0.867125i \(0.334034\pi\)
\(140\) 0 0
\(141\) −3.50515 −0.295187
\(142\) 0 0
\(143\) 8.22003 14.2375i 0.687393 1.19060i
\(144\) 0 0
\(145\) 2.15238 + 3.72803i 0.178746 + 0.309596i
\(146\) 0 0
\(147\) −2.30992 6.60790i −0.190519 0.545010i
\(148\) 0 0
\(149\) 2.00000 + 3.46410i 0.163846 + 0.283790i 0.936245 0.351348i \(-0.114277\pi\)
−0.772399 + 0.635138i \(0.780943\pi\)
\(150\) 0 0
\(151\) 0.315070 0.545718i 0.0256401 0.0444099i −0.852921 0.522041i \(-0.825171\pi\)
0.878561 + 0.477631i \(0.158504\pi\)
\(152\) 0 0
\(153\) −1.31507 −0.106317
\(154\) 0 0
\(155\) −1.49485 −0.120069
\(156\) 0 0
\(157\) −4.43751 + 7.68599i −0.354152 + 0.613409i −0.986972 0.160890i \(-0.948564\pi\)
0.632821 + 0.774298i \(0.281897\pi\)
\(158\) 0 0
\(159\) −4.06765 7.04537i −0.322585 0.558734i
\(160\) 0 0
\(161\) 14.7149 + 10.4442i 1.15969 + 0.823116i
\(162\) 0 0
\(163\) 8.00000 + 13.8564i 0.626608 + 1.08532i 0.988227 + 0.152992i \(0.0488907\pi\)
−0.361619 + 0.932326i \(0.617776\pi\)
\(164\) 0 0
\(165\) −1.90496 + 3.29948i −0.148301 + 0.256865i
\(166\) 0 0
\(167\) −12.0599 −0.933222 −0.466611 0.884463i \(-0.654525\pi\)
−0.466611 + 0.884463i \(0.654525\pi\)
\(168\) 0 0
\(169\) 5.61983 0.432295
\(170\) 0 0
\(171\) 3.81507 6.60790i 0.291746 0.505318i
\(172\) 0 0
\(173\) −8.56249 14.8307i −0.650994 1.12756i −0.982882 0.184236i \(-0.941019\pi\)
0.331888 0.943319i \(-0.392315\pi\)
\(174\) 0 0
\(175\) −0.247423 + 2.63416i −0.0187034 + 0.199124i
\(176\) 0 0
\(177\) −4.15238 7.19214i −0.312112 0.540594i
\(178\) 0 0
\(179\) −5.75258 + 9.96376i −0.429968 + 0.744726i −0.996870 0.0790591i \(-0.974808\pi\)
0.566902 + 0.823785i \(0.308142\pi\)
\(180\) 0 0
\(181\) 6.48454 0.481992 0.240996 0.970526i \(-0.422526\pi\)
0.240996 + 0.970526i \(0.422526\pi\)
\(182\) 0 0
\(183\) 6.63014 0.490114
\(184\) 0 0
\(185\) −4.65238 + 8.05816i −0.342050 + 0.592448i
\(186\) 0 0
\(187\) −2.50515 4.33905i −0.183195 0.317303i
\(188\) 0 0
\(189\) −2.40496 + 1.10280i −0.174935 + 0.0802172i
\(190\) 0 0
\(191\) −6.61983 11.4659i −0.478994 0.829642i 0.520716 0.853730i \(-0.325665\pi\)
−0.999710 + 0.0240878i \(0.992332\pi\)
\(192\) 0 0
\(193\) −7.73034 + 13.3893i −0.556442 + 0.963785i 0.441348 + 0.897336i \(0.354500\pi\)
−0.997790 + 0.0664495i \(0.978833\pi\)
\(194\) 0 0
\(195\) −4.31507 −0.309009
\(196\) 0 0
\(197\) −14.4401 −1.02881 −0.514406 0.857547i \(-0.671987\pi\)
−0.514406 + 0.857547i \(0.671987\pi\)
\(198\) 0 0
\(199\) 1.19008 2.06129i 0.0843628 0.146121i −0.820757 0.571278i \(-0.806448\pi\)
0.905120 + 0.425157i \(0.139781\pi\)
\(200\) 0 0
\(201\) 7.72003 + 13.3715i 0.544529 + 0.943152i
\(202\) 0 0
\(203\) −10.3528 + 4.74731i −0.726622 + 0.333196i
\(204\) 0 0
\(205\) 5.22003 + 9.04135i 0.364583 + 0.631476i
\(206\) 0 0
\(207\) 3.41011 5.90649i 0.237019 0.410529i
\(208\) 0 0
\(209\) 29.0702 2.01083
\(210\) 0 0
\(211\) 16.1147 1.10938 0.554690 0.832057i \(-0.312837\pi\)
0.554690 + 0.832057i \(0.312837\pi\)
\(212\) 0 0
\(213\) 2.16269 3.74589i 0.148185 0.256664i
\(214\) 0 0
\(215\) −2.40496 4.16551i −0.164017 0.284085i
\(216\) 0 0
\(217\) 0.369859 3.93766i 0.0251077 0.267306i
\(218\) 0 0
\(219\) 1.91011 + 3.30841i 0.129073 + 0.223562i
\(220\) 0 0
\(221\) 2.83731 4.91437i 0.190858 0.330576i
\(222\) 0 0
\(223\) −12.6095 −0.844396 −0.422198 0.906504i \(-0.638741\pi\)
−0.422198 + 0.906504i \(0.638741\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −1.46745 + 2.54170i −0.0973982 + 0.168699i −0.910607 0.413273i \(-0.864385\pi\)
0.813209 + 0.581972i \(0.197719\pi\)
\(228\) 0 0
\(229\) −6.24227 10.8119i −0.412501 0.714472i 0.582662 0.812715i \(-0.302011\pi\)
−0.995163 + 0.0982425i \(0.968678\pi\)
\(230\) 0 0
\(231\) −8.22003 5.83433i −0.540838 0.383871i
\(232\) 0 0
\(233\) 12.3151 + 21.3303i 0.806787 + 1.39740i 0.915078 + 0.403277i \(0.132129\pi\)
−0.108291 + 0.994119i \(0.534538\pi\)
\(234\) 0 0
\(235\) −1.75258 + 3.03555i −0.114326 + 0.198018i
\(236\) 0 0
\(237\) −2.12499 −0.138033
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −7.87756 + 13.6443i −0.507438 + 0.878909i 0.492524 + 0.870299i \(0.336074\pi\)
−0.999963 + 0.00861063i \(0.997259\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) −6.87756 1.30350i −0.439391 0.0832777i
\(246\) 0 0
\(247\) 16.4623 + 28.5135i 1.04747 + 1.81427i
\(248\) 0 0
\(249\) 5.46745 9.46990i 0.346486 0.600131i
\(250\) 0 0
\(251\) 8.82022 0.556728 0.278364 0.960476i \(-0.410208\pi\)
0.278364 + 0.960476i \(0.410208\pi\)
\(252\) 0 0
\(253\) 25.9845 1.63363
\(254\) 0 0
\(255\) −0.657535 + 1.13888i −0.0411765 + 0.0713197i
\(256\) 0 0
\(257\) −7.46745 12.9340i −0.465807 0.806801i 0.533431 0.845844i \(-0.320903\pi\)
−0.999238 + 0.0390427i \(0.987569\pi\)
\(258\) 0 0
\(259\) −20.0754 14.2489i −1.24742 0.885382i
\(260\) 0 0
\(261\) 2.15238 + 3.72803i 0.133229 + 0.230759i
\(262\) 0 0
\(263\) 1.31507 2.27777i 0.0810907 0.140453i −0.822628 0.568580i \(-0.807493\pi\)
0.903719 + 0.428127i \(0.140826\pi\)
\(264\) 0 0
\(265\) −8.13529 −0.499747
\(266\) 0 0
\(267\) −11.3151 −0.692471
\(268\) 0 0
\(269\) −9.82022 + 17.0091i −0.598750 + 1.03706i 0.394256 + 0.919001i \(0.371002\pi\)
−0.993006 + 0.118064i \(0.962331\pi\)
\(270\) 0 0
\(271\) 7.44006 + 12.8866i 0.451951 + 0.782803i 0.998507 0.0546200i \(-0.0173947\pi\)
−0.546556 + 0.837423i \(0.684061\pi\)
\(272\) 0 0
\(273\) 1.06765 11.3666i 0.0646170 0.687936i
\(274\) 0 0
\(275\) 1.90496 + 3.29948i 0.114873 + 0.198966i
\(276\) 0 0
\(277\) 2.41527 4.18336i 0.145119 0.251354i −0.784298 0.620384i \(-0.786977\pi\)
0.929417 + 0.369030i \(0.120310\pi\)
\(278\) 0 0
\(279\) −1.49485 −0.0894941
\(280\) 0 0
\(281\) −9.50515 −0.567030 −0.283515 0.958968i \(-0.591501\pi\)
−0.283515 + 0.958968i \(0.591501\pi\)
\(282\) 0 0
\(283\) −13.8553 + 23.9981i −0.823613 + 1.42654i 0.0793609 + 0.996846i \(0.474712\pi\)
−0.902974 + 0.429694i \(0.858621\pi\)
\(284\) 0 0
\(285\) −3.81507 6.60790i −0.225985 0.391418i
\(286\) 0 0
\(287\) −25.1079 + 11.5133i −1.48207 + 0.679611i
\(288\) 0 0
\(289\) 7.63529 + 13.2247i 0.449135 + 0.777925i
\(290\) 0 0
\(291\) 5.31507 9.20597i 0.311575 0.539664i
\(292\) 0 0
\(293\) −2.49485 −0.145750 −0.0728752 0.997341i \(-0.523217\pi\)
−0.0728752 + 0.997341i \(0.523217\pi\)
\(294\) 0 0
\(295\) −8.30476 −0.483522
\(296\) 0 0
\(297\) −1.90496 + 3.29948i −0.110537 + 0.191455i
\(298\) 0 0
\(299\) 14.7149 + 25.4869i 0.850983 + 1.47395i
\(300\) 0 0
\(301\) 11.5676 5.30439i 0.666748 0.305740i
\(302\) 0 0
\(303\) 5.34246 + 9.25342i 0.306917 + 0.531595i
\(304\) 0 0
\(305\) 3.31507 5.74187i 0.189820 0.328779i
\(306\) 0 0
\(307\) −1.21070 −0.0690983 −0.0345492 0.999403i \(-0.511000\pi\)
−0.0345492 + 0.999403i \(0.511000\pi\)
\(308\) 0 0
\(309\) −19.4401 −1.10591
\(310\) 0 0
\(311\) 14.6027 25.2927i 0.828046 1.43422i −0.0715233 0.997439i \(-0.522786\pi\)
0.899569 0.436778i \(-0.143881\pi\)
\(312\) 0 0
\(313\) 0.404958 + 0.701408i 0.0228896 + 0.0396459i 0.877243 0.480046i \(-0.159380\pi\)
−0.854354 + 0.519692i \(0.826047\pi\)
\(314\) 0 0
\(315\) −0.247423 + 2.63416i −0.0139407 + 0.148418i
\(316\) 0 0
\(317\) 4.14207 + 7.17428i 0.232642 + 0.402948i 0.958585 0.284807i \(-0.0919296\pi\)
−0.725943 + 0.687755i \(0.758596\pi\)
\(318\) 0 0
\(319\) −8.20039 + 14.2035i −0.459134 + 0.795243i
\(320\) 0 0
\(321\) 10.3048 0.575156
\(322\) 0 0
\(323\) 10.0342 0.558316
\(324\) 0 0
\(325\) −2.15754 + 3.73696i −0.119679 + 0.207289i
\(326\) 0 0
\(327\) −3.55734 6.16149i −0.196721 0.340731i
\(328\) 0 0
\(329\) −7.56249 5.36763i −0.416934 0.295927i
\(330\) 0 0
\(331\) 12.9401 + 22.4128i 0.711250 + 1.23192i 0.964388 + 0.264491i \(0.0852039\pi\)
−0.253138 + 0.967430i \(0.581463\pi\)
\(332\) 0 0
\(333\) −4.65238 + 8.05816i −0.254949 + 0.441584i
\(334\) 0 0
\(335\) 15.4401 0.843580
\(336\) 0 0
\(337\) −30.6797 −1.67123 −0.835615 0.549315i \(-0.814889\pi\)
−0.835615 + 0.549315i \(0.814889\pi\)
\(338\) 0 0
\(339\) −3.00000 + 5.19615i −0.162938 + 0.282216i
\(340\) 0 0
\(341\) −2.84762 4.93222i −0.154207 0.267095i
\(342\) 0 0
\(343\) 5.13529 17.7941i 0.277280 0.960789i
\(344\) 0 0
\(345\) −3.41011 5.90649i −0.183594 0.317995i
\(346\) 0 0
\(347\) −15.1250 + 26.1972i −0.811952 + 1.40634i 0.0995443 + 0.995033i \(0.468262\pi\)
−0.911496 + 0.411309i \(0.865072\pi\)
\(348\) 0 0
\(349\) 26.9897 1.44473 0.722363 0.691515i \(-0.243056\pi\)
0.722363 + 0.691515i \(0.243056\pi\)
\(350\) 0 0
\(351\) −4.31507 −0.230321
\(352\) 0 0
\(353\) 7.46745 12.9340i 0.397452 0.688408i −0.595959 0.803015i \(-0.703228\pi\)
0.993411 + 0.114608i \(0.0365611\pi\)
\(354\) 0 0
\(355\) −2.16269 3.74589i −0.114784 0.198811i
\(356\) 0 0
\(357\) −2.83731 2.01384i −0.150166 0.106584i
\(358\) 0 0
\(359\) −7.84762 13.5925i −0.414181 0.717383i 0.581161 0.813789i \(-0.302599\pi\)
−0.995342 + 0.0964055i \(0.969265\pi\)
\(360\) 0 0
\(361\) −19.6095 + 33.9647i −1.03208 + 1.78762i
\(362\) 0 0
\(363\) −3.51546 −0.184514
\(364\) 0 0
\(365\) 3.82022 0.199960
\(366\) 0 0
\(367\) 10.1575 17.5934i 0.530219 0.918366i −0.469159 0.883113i \(-0.655443\pi\)
0.999378 0.0352528i \(-0.0112236\pi\)
\(368\) 0 0
\(369\) 5.22003 + 9.04135i 0.271744 + 0.470674i
\(370\) 0 0
\(371\) 2.01286 21.4296i 0.104502 1.11257i
\(372\) 0 0
\(373\) 3.40496 + 5.89756i 0.176302 + 0.305364i 0.940611 0.339486i \(-0.110253\pi\)
−0.764309 + 0.644850i \(0.776920\pi\)
\(374\) 0 0
\(375\) 0.500000 0.866025i 0.0258199 0.0447214i
\(376\) 0 0
\(377\) −18.5754 −0.956679
\(378\) 0 0
\(379\) 30.2397 1.55331 0.776654 0.629928i \(-0.216916\pi\)
0.776654 + 0.629928i \(0.216916\pi\)
\(380\) 0 0
\(381\) −8.47261 + 14.6750i −0.434065 + 0.751822i
\(382\) 0 0
\(383\) 4.06765 + 7.04537i 0.207847 + 0.360002i 0.951036 0.309080i \(-0.100021\pi\)
−0.743189 + 0.669082i \(0.766688\pi\)
\(384\) 0 0
\(385\) −9.16269 + 4.20159i −0.466974 + 0.214133i
\(386\) 0 0
\(387\) −2.40496 4.16551i −0.122251 0.211745i
\(388\) 0 0
\(389\) 14.4675 25.0584i 0.733529 1.27051i −0.221837 0.975084i \(-0.571205\pi\)
0.955366 0.295426i \(-0.0954614\pi\)
\(390\) 0 0
\(391\) 8.96908 0.453586
\(392\) 0 0
\(393\) −13.1044 −0.661028
\(394\) 0 0
\(395\) −1.06249 + 1.84029i −0.0534598 + 0.0925952i
\(396\) 0 0
\(397\) 0.214874 + 0.372173i 0.0107842 + 0.0186789i 0.871367 0.490631i \(-0.163234\pi\)
−0.860583 + 0.509310i \(0.829901\pi\)
\(398\) 0 0
\(399\) 18.3502 8.41455i 0.918658 0.421254i
\(400\) 0 0
\(401\) −1.55219 2.68846i −0.0775124 0.134255i 0.824664 0.565623i \(-0.191364\pi\)
−0.902176 + 0.431368i \(0.858031\pi\)
\(402\) 0 0
\(403\) 3.22518 5.58618i 0.160658 0.278267i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −35.4504 −1.75721
\(408\) 0 0
\(409\) 5.75773 9.97268i 0.284701 0.493117i −0.687835 0.725867i \(-0.741439\pi\)
0.972537 + 0.232749i \(0.0747722\pi\)
\(410\) 0 0
\(411\) 6.46745 + 11.2020i 0.319016 + 0.552552i
\(412\) 0 0
\(413\) 2.05479 21.8760i 0.101110 1.07645i
\(414\) 0 0
\(415\) −5.46745 9.46990i −0.268387 0.464859i
\(416\) 0 0
\(417\) 5.87241 10.1713i 0.287573 0.498091i
\(418\) 0 0
\(419\) 9.86471 0.481922 0.240961 0.970535i \(-0.422537\pi\)
0.240961 + 0.970535i \(0.422537\pi\)
\(420\) 0 0
\(421\) −21.1353 −1.03007 −0.515036 0.857169i \(-0.672221\pi\)
−0.515036 + 0.857169i \(0.672221\pi\)
\(422\) 0 0
\(423\) −1.75258 + 3.03555i −0.0852132 + 0.147594i
\(424\) 0 0
\(425\) 0.657535 + 1.13888i 0.0318951 + 0.0552440i
\(426\) 0 0
\(427\) 14.3048 + 10.1531i 0.692256 + 0.491342i
\(428\) 0 0
\(429\) −8.22003 14.2375i −0.396867 0.687393i
\(430\) 0 0
\(431\) −19.0599 + 33.0127i −0.918083 + 1.59017i −0.115758 + 0.993277i \(0.536930\pi\)
−0.802324 + 0.596888i \(0.796404\pi\)
\(432\) 0 0
\(433\) 16.8099 0.807833 0.403917 0.914796i \(-0.367649\pi\)
0.403917 + 0.914796i \(0.367649\pi\)
\(434\) 0 0
\(435\) 4.30476 0.206398
\(436\) 0 0
\(437\) −26.0196 + 45.0673i −1.24469 + 2.15586i
\(438\) 0 0
\(439\) 4.50515 + 7.80316i 0.215019 + 0.372424i 0.953279 0.302093i \(-0.0976852\pi\)
−0.738259 + 0.674517i \(0.764352\pi\)
\(440\) 0 0
\(441\) −6.87756 1.30350i −0.327503 0.0620715i
\(442\) 0 0
\(443\) 6.30476 + 10.9202i 0.299548 + 0.518833i 0.976033 0.217624i \(-0.0698307\pi\)
−0.676484 + 0.736457i \(0.736497\pi\)
\(444\) 0 0
\(445\) −5.65754 + 9.79914i −0.268193 + 0.464524i
\(446\) 0 0
\(447\) 4.00000 0.189194
\(448\) 0 0
\(449\) 6.74482 0.318308 0.159154 0.987254i \(-0.449123\pi\)
0.159154 + 0.987254i \(0.449123\pi\)
\(450\) 0 0
\(451\) −19.8879 + 34.4468i −0.936483 + 1.62204i
\(452\) 0 0
\(453\) −0.315070 0.545718i −0.0148033 0.0256401i
\(454\) 0 0
\(455\) −9.30992 6.60790i −0.436456 0.309783i
\(456\) 0 0
\(457\) −17.0454 29.5235i −0.797351 1.38105i −0.921336 0.388768i \(-0.872901\pi\)
0.123985 0.992284i \(-0.460432\pi\)
\(458\) 0 0
\(459\) −0.657535 + 1.13888i −0.0306911 + 0.0531586i
\(460\) 0 0
\(461\) 31.5857 1.47109 0.735545 0.677475i \(-0.236926\pi\)
0.735545 + 0.677475i \(0.236926\pi\)
\(462\) 0 0
\(463\) 30.5857 1.42144 0.710718 0.703477i \(-0.248370\pi\)
0.710718 + 0.703477i \(0.248370\pi\)
\(464\) 0 0
\(465\) −0.747423 + 1.29457i −0.0346609 + 0.0600345i
\(466\) 0 0
\(467\) −3.51546 6.08896i −0.162676 0.281763i 0.773151 0.634221i \(-0.218679\pi\)
−0.935828 + 0.352458i \(0.885346\pi\)
\(468\) 0 0
\(469\) −3.82022 + 40.6715i −0.176402 + 1.87804i
\(470\) 0 0
\(471\) 4.43751 + 7.68599i 0.204470 + 0.354152i
\(472\) 0 0
\(473\) 9.16269 15.8702i 0.421301 0.729715i
\(474\) 0 0
\(475\) −7.63014 −0.350095
\(476\) 0 0
\(477\) −8.13529 −0.372490
\(478\) 0 0
\(479\) −15.7551 + 27.2887i −0.719870 + 1.24685i 0.241181 + 0.970480i \(0.422465\pi\)
−0.961051 + 0.276371i \(0.910868\pi\)
\(480\) 0 0
\(481\) −20.0754 34.7715i −0.915357 1.58545i
\(482\) 0 0
\(483\) 16.4024 7.52137i 0.746333 0.342234i
\(484\) 0 0
\(485\) −5.31507 9.20597i −0.241345 0.418022i
\(486\) 0 0
\(487\) −10.0248 + 17.3634i −0.454267 + 0.786813i −0.998646 0.0520264i \(-0.983432\pi\)
0.544379 + 0.838839i \(0.316765\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −4.60952 −0.208025 −0.104012 0.994576i \(-0.533168\pi\)
−0.104012 + 0.994576i \(0.533168\pi\)
\(492\) 0 0
\(493\) −2.83053 + 4.90263i −0.127481 + 0.220803i
\(494\) 0 0
\(495\) 1.90496 + 3.29948i 0.0856215 + 0.148301i
\(496\) 0 0
\(497\) 10.4024 4.77004i 0.466609 0.213966i
\(498\) 0 0
\(499\) 21.1327 + 36.6029i 0.946029 + 1.63857i 0.753677 + 0.657245i \(0.228278\pi\)
0.192352 + 0.981326i \(0.438389\pi\)
\(500\) 0 0
\(501\) −6.02994 + 10.4442i −0.269398 + 0.466611i
\(502\) 0 0
\(503\) −5.61983 −0.250576 −0.125288 0.992120i \(-0.539985\pi\)
−0.125288 + 0.992120i \(0.539985\pi\)
\(504\) 0 0
\(505\) 10.6849 0.475473
\(506\) 0 0
\(507\) 2.80992 4.86692i 0.124793 0.216147i
\(508\) 0 0
\(509\) 14.5650 + 25.2274i 0.645584 + 1.11818i 0.984166 + 0.177248i \(0.0567194\pi\)
−0.338582 + 0.940937i \(0.609947\pi\)
\(510\) 0 0
\(511\) −0.945211 + 10.0631i −0.0418137 + 0.445164i
\(512\) 0 0
\(513\) −3.81507 6.60790i −0.168439 0.291746i
\(514\) 0 0
\(515\) −9.72003 + 16.8356i −0.428316 + 0.741864i
\(516\) 0 0
\(517\) −13.3543 −0.587323
\(518\) 0 0
\(519\) −17.1250 −0.751703
\(520\) 0 0
\(521\) 7.68748 13.3151i 0.336795 0.583345i −0.647033 0.762462i \(-0.723991\pi\)
0.983828 + 0.179116i \(0.0573239\pi\)
\(522\) 0 0
\(523\) −2.59504 4.49474i −0.113473 0.196541i 0.803695 0.595041i \(-0.202864\pi\)
−0.917168 + 0.398500i \(0.869531\pi\)
\(524\) 0 0
\(525\) 2.15754 + 1.53135i 0.0941626 + 0.0668337i
\(526\) 0 0
\(527\) −0.982914 1.70246i −0.0428164 0.0741602i
\(528\) 0 0
\(529\) −11.7577 + 20.3650i −0.511206 + 0.885434i
\(530\) 0 0
\(531\) −8.30476 −0.360396
\(532\) 0 0
\(533\) −45.0496 −1.95131
\(534\) 0 0
\(535\) 5.15238 8.92419i 0.222757 0.385826i
\(536\) 0 0
\(537\) 5.75258 + 9.96376i 0.248242 + 0.429968i
\(538\) 0 0
\(539\) −8.80059 25.1755i −0.379068 1.08439i
\(540\) 0 0
\(541\) −2.24227 3.88372i −0.0964027 0.166974i 0.813790 0.581158i \(-0.197400\pi\)
−0.910193 + 0.414184i \(0.864067\pi\)
\(542\) 0 0
\(543\) 3.24227 5.61577i 0.139139 0.240996i
\(544\) 0 0
\(545\) −7.11468 −0.304759
\(546\) 0 0
\(547\) 0.650757 0.0278244 0.0139122 0.999903i \(-0.495571\pi\)
0.0139122 + 0.999903i \(0.495571\pi\)
\(548\) 0 0
\(549\) 3.31507 5.74187i 0.141484 0.245057i
\(550\) 0 0
\(551\) −16.4230 28.4454i −0.699642 1.21182i
\(552\) 0 0
\(553\) −4.58473 3.25410i −0.194963 0.138379i
\(554\) 0 0
\(555\) 4.65238 + 8.05816i 0.197483 + 0.342050i
\(556\) 0 0
\(557\) −17.5077 + 30.3242i −0.741825 + 1.28488i 0.209838 + 0.977736i \(0.432706\pi\)
−0.951663 + 0.307143i \(0.900627\pi\)
\(558\) 0 0
\(559\) 20.7551 0.877848
\(560\) 0 0
\(561\) −5.01031 −0.211535
\(562\) 0 0
\(563\) −15.6301 + 27.0722i −0.658732 + 1.14096i 0.322212 + 0.946667i \(0.395574\pi\)
−0.980944 + 0.194290i \(0.937760\pi\)
\(564\) 0 0
\(565\) 3.00000 + 5.19615i 0.126211 + 0.218604i
\(566\) 0 0
\(567\) −0.247423 + 2.63416i −0.0103908 + 0.110624i
\(568\) 0 0
\(569\) 0.779971 + 1.35095i 0.0326981 + 0.0566348i 0.881911 0.471415i \(-0.156257\pi\)
−0.849213 + 0.528050i \(0.822923\pi\)
\(570\) 0 0
\(571\) −4.57796 + 7.92925i −0.191581 + 0.331829i −0.945775 0.324824i \(-0.894695\pi\)
0.754193 + 0.656653i \(0.228028\pi\)
\(572\) 0 0
\(573\) −13.2397 −0.553095
\(574\) 0 0
\(575\) −6.82022 −0.284423
\(576\) 0 0
\(577\) 0.415266 0.719262i 0.0172878 0.0299433i −0.857252 0.514897i \(-0.827830\pi\)
0.874540 + 0.484954i \(0.161164\pi\)
\(578\) 0 0
\(579\) 7.73034 + 13.3893i 0.321262 + 0.556442i
\(580\) 0 0
\(581\) 26.2980 12.0591i 1.09102 0.500294i
\(582\) 0 0
\(583\) −15.4974 26.8423i −0.641837 1.11169i
\(584\) 0 0
\(585\) −2.15754 + 3.73696i −0.0892031 + 0.154504i
\(586\) 0 0
\(587\) 23.0651 0.951998 0.475999 0.879446i \(-0.342087\pi\)
0.475999 + 0.879446i \(0.342087\pi\)
\(588\) 0 0
\(589\) 11.4059 0.469971
\(590\) 0 0
\(591\) −7.22003 + 12.5055i −0.296992 + 0.514406i
\(592\) 0 0
\(593\) 11.7928 + 20.4258i 0.484273 + 0.838786i 0.999837 0.0180652i \(-0.00575064\pi\)
−0.515563 + 0.856851i \(0.672417\pi\)
\(594\) 0 0
\(595\) −3.16269 + 1.45026i −0.129658 + 0.0594551i
\(596\) 0 0
\(597\) −1.19008 2.06129i −0.0487069 0.0843628i
\(598\) 0 0
\(599\) −0.190084 + 0.329235i −0.00776661 + 0.0134522i −0.869883 0.493259i \(-0.835806\pi\)
0.862116 + 0.506711i \(0.169139\pi\)
\(600\) 0 0
\(601\) −6.10437 −0.249002 −0.124501 0.992219i \(-0.539733\pi\)
−0.124501 + 0.992219i \(0.539733\pi\)
\(602\) 0 0
\(603\) 15.4401 0.628768
\(604\) 0 0
\(605\) −1.75773 + 3.04448i −0.0714619 + 0.123776i
\(606\) 0 0
\(607\) −4.71748 8.17091i −0.191477 0.331647i 0.754263 0.656572i \(-0.227994\pi\)
−0.945740 + 0.324925i \(0.894661\pi\)
\(608\) 0 0
\(609\) −1.06510 + 11.3394i −0.0431599 + 0.459496i
\(610\) 0 0
\(611\) −7.56249 13.0986i −0.305946 0.529914i
\(612\) 0 0
\(613\) −12.0676 + 20.9018i −0.487408 + 0.844215i −0.999895 0.0144798i \(-0.995391\pi\)
0.512487 + 0.858695i \(0.328724\pi\)
\(614\) 0 0
\(615\) 10.4401 0.420984
\(616\) 0 0
\(617\) 36.8595 1.48391 0.741954 0.670451i \(-0.233899\pi\)
0.741954 + 0.670451i \(0.233899\pi\)
\(618\) 0 0
\(619\) 16.0650 27.8255i 0.645709 1.11840i −0.338429 0.940992i \(-0.609895\pi\)
0.984137 0.177408i \(-0.0567713\pi\)
\(620\) 0 0
\(621\) −3.41011 5.90649i −0.136843 0.237019i
\(622\) 0 0
\(623\) −24.4127 17.3274i −0.978073 0.694206i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 14.5351 25.1755i 0.580476 1.00541i
\(628\) 0 0
\(629\) −12.2364 −0.487898
\(630\) 0 0
\(631\) −22.2912 −0.887399 −0.443699 0.896176i \(-0.646334\pi\)
−0.443699 + 0.896176i \(0.646334\pi\)
\(632\) 0 0
\(633\) 8.05734 13.9557i 0.320250 0.554690i
\(634\) 0 0
\(635\) 8.47261 + 14.6750i 0.336225 + 0.582359i
\(636\) 0 0
\(637\) 19.7097 22.8888i 0.780928 0.906889i
\(638\) 0 0
\(639\) −2.16269 3.74589i −0.0855547 0.148185i
\(640\) 0 0
\(641\) 17.0196 29.4789i 0.672235 1.16435i −0.305034 0.952342i \(-0.598668\pi\)
0.977269 0.212004i \(-0.0679990\pi\)
\(642\) 0 0
\(643\) −36.4297 −1.43665 −0.718325 0.695708i \(-0.755091\pi\)
−0.718325 + 0.695708i \(0.755091\pi\)
\(644\) 0 0
\(645\) −4.80992 −0.189390
\(646\) 0 0
\(647\) −11.0196 + 19.0866i −0.433227 + 0.750371i −0.997149 0.0754573i \(-0.975958\pi\)
0.563922 + 0.825828i \(0.309292\pi\)
\(648\) 0 0
\(649\) −15.8202 27.4014i −0.620998 1.07560i
\(650\) 0 0
\(651\) −3.22518 2.28914i −0.126405 0.0897183i
\(652\) 0 0
\(653\) 4.79028 + 8.29701i 0.187458 + 0.324687i 0.944402 0.328793i \(-0.106642\pi\)
−0.756944 + 0.653480i \(0.773308\pi\)
\(654\) 0 0
\(655\) −6.55219 + 11.3487i −0.256015 + 0.443431i
\(656\) 0 0
\(657\) 3.82022 0.149041
\(658\) 0 0
\(659\) 25.2603 0.984001 0.492000 0.870595i \(-0.336266\pi\)
0.492000 + 0.870595i \(0.336266\pi\)
\(660\) 0 0
\(661\) 12.8827 22.3135i 0.501080 0.867895i −0.498920 0.866648i \(-0.666270\pi\)
0.999999 0.00124712i \(-0.000396971\pi\)
\(662\) 0 0
\(663\) −2.83731 4.91437i −0.110192 0.190858i
\(664\) 0 0
\(665\) 1.88787 20.0990i 0.0732085 0.779405i
\(666\) 0 0
\(667\) −14.6797 25.4260i −0.568401 0.984500i
\(668\) 0 0
\(669\) −6.30476 + 10.9202i −0.243756 + 0.422198i
\(670\) 0 0
\(671\) 25.2603 0.975162
\(672\) 0 0
\(673\) −5.82022 −0.224353 −0.112177 0.993688i \(-0.535782\pi\)
−0.112177 + 0.993688i \(0.535782\pi\)
\(674\) 0 0
\(675\) 0.500000 0.866025i 0.0192450 0.0333333i
\(676\) 0 0
\(677\) −9.05056 15.6760i −0.347841 0.602479i 0.638024 0.770016i \(-0.279752\pi\)
−0.985866 + 0.167537i \(0.946419\pi\)
\(678\) 0 0
\(679\) 25.5650 11.7230i 0.981096 0.449886i
\(680\) 0 0
\(681\) 1.46745 + 2.54170i 0.0562329 + 0.0973982i
\(682\) 0 0
\(683\) −4.65754 + 8.06709i −0.178216 + 0.308679i −0.941269 0.337656i \(-0.890366\pi\)
0.763054 + 0.646335i \(0.223699\pi\)
\(684\) 0 0
\(685\) 12.9349 0.494217
\(686\) 0 0
\(687\) −12.4845 −0.476315
\(688\) 0 0
\(689\) 17.5522 30.4013i 0.668685 1.15820i
\(690\) 0 0
\(691\) −19.1875 33.2337i −0.729926 1.26427i −0.956914 0.290372i \(-0.906221\pi\)
0.226988 0.973898i \(-0.427112\pi\)
\(692\) 0 0
\(693\) −9.16269 + 4.20159i −0.348062 + 0.159605i
\(694\) 0 0
\(695\) −5.87241 10.1713i −0.222753 0.385820i
\(696\) 0 0
\(697\) −6.86471 + 11.8900i −0.260019 + 0.450367i
\(698\) 0 0
\(699\) 24.6301 0.931597
\(700\) 0 0
\(701\) 0.575352 0.0217307 0.0108654 0.999941i \(-0.496541\pi\)
0.0108654 + 0.999941i \(0.496541\pi\)
\(702\) 0 0
\(703\) 35.4983 61.4849i 1.33884 2.31895i
\(704\) 0 0
\(705\) 1.75258 + 3.03555i 0.0660059 + 0.114326i
\(706\) 0 0
\(707\) −2.64370 + 28.1458i −0.0994265 + 1.05853i
\(708\) 0 0
\(709\) −11.3151 19.5983i −0.424946 0.736029i 0.571469 0.820624i \(-0.306374\pi\)
−0.996415 + 0.0845949i \(0.973040\pi\)
\(710\) 0 0
\(711\) −1.06249 + 1.84029i −0.0398466 + 0.0690164i
\(712\) 0 0
\(713\) 10.1952 0.381813
\(714\) 0 0
\(715\) −16.4401 −0.614823
\(716\) 0 0
\(717\) 3.00000 5.19615i 0.112037 0.194054i
\(718\) 0 0
\(719\) 10.3151 + 17.8662i 0.384687 + 0.666298i 0.991726 0.128375i \(-0.0409760\pi\)
−0.607039 + 0.794672i \(0.707643\pi\)
\(720\) 0 0
\(721\) −41.9426 29.7696i −1.56202 1.10868i
\(722\) 0 0
\(723\) 7.87756 + 13.6443i 0.292970 + 0.507438i
\(724\) 0 0
\(725\) 2.15238 3.72803i 0.0799374 0.138456i
\(726\) 0 0
\(727\) −6.27384 −0.232684 −0.116342 0.993209i \(-0.537117\pi\)
−0.116342 + 0.993209i \(0.537117\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.16269 5.47794i 0.116976 0.202609i
\(732\) 0 0
\(733\) −3.33731 5.78039i −0.123266 0.213504i 0.797788 0.602939i \(-0.206004\pi\)
−0.921054 + 0.389435i \(0.872670\pi\)
\(734\) 0 0
\(735\) −4.56765 + 5.30439i −0.168480 + 0.195656i
\(736\) 0 0
\(737\) 29.4127 + 50.9442i 1.08343 + 1.87655i
\(738\) 0 0
\(739\) 5.13014 8.88566i 0.188715 0.326864i −0.756107 0.654448i \(-0.772901\pi\)
0.944822 + 0.327584i \(0.106234\pi\)
\(740\) 0 0
\(741\) 32.9246 1.20952
\(742\) 0 0
\(743\) 1.85115 0.0679121 0.0339560 0.999423i \(-0.489189\pi\)
0.0339560 + 0.999423i \(0.489189\pi\)
\(744\) 0 0
\(745\) 2.00000 3.46410i 0.0732743 0.126915i
\(746\) 0 0
\(747\) −5.46745 9.46990i −0.200044 0.346486i
\(748\) 0 0
\(749\) 22.2329 + 15.7802i 0.812372 + 0.576597i
\(750\) 0 0
\(751\) 7.68233 + 13.3062i 0.280332 + 0.485549i 0.971466 0.237177i \(-0.0762221\pi\)
−0.691134 + 0.722726i \(0.742889\pi\)
\(752\) 0 0
\(753\) 4.41011 7.63854i 0.160713 0.278364i
\(754\) 0 0
\(755\) −0.630141 −0.0229332
\(756\) 0 0
\(757\) 32.0206 1.16381 0.581905 0.813257i \(-0.302308\pi\)
0.581905 + 0.813257i \(0.302308\pi\)
\(758\) 0 0
\(759\) 12.9922 22.5032i 0.471589 0.816815i
\(760\) 0 0
\(761\) −7.70457 13.3447i −0.279290 0.483745i 0.691918 0.721976i \(-0.256766\pi\)
−0.971209 + 0.238231i \(0.923433\pi\)
\(762\) 0 0
\(763\) 1.76033 18.7412i 0.0637284 0.678476i
\(764\) 0 0
\(765\) 0.657535 + 1.13888i 0.0237732 + 0.0411765i
\(766\) 0 0
\(767\) 17.9178 31.0346i 0.646975 1.12059i
\(768\) 0 0
\(769\) 39.2500 1.41539 0.707695 0.706518i \(-0.249735\pi\)
0.707695 + 0.706518i \(0.249735\pi\)
\(770\) 0 0
\(771\) −14.9349 −0.537867
\(772\) 0 0
\(773\) −11.0196 + 19.0866i −0.396349 + 0.686496i −0.993272 0.115802i \(-0.963056\pi\)
0.596924 + 0.802298i \(0.296390\pi\)
\(774\) 0 0
\(775\) 0.747423 + 1.29457i 0.0268482 + 0.0465025i
\(776\) 0 0
\(777\) −22.3776 + 10.2613i −0.802791 + 0.368123i
\(778\) 0 0
\(779\) −39.8296 68.9868i −1.42704 2.47171i
\(780\) 0 0
\(781\) 8.23967 14.2715i 0.294838 0.510675i
\(782\) 0 0
\(783\) 4.30476 0.153840
\(784\) 0 0
\(785\) 8.87501 0.316763
\(786\) 0 0
\(787\) 24.4401 42.3314i 0.871194 1.50895i 0.0104313 0.999946i \(-0.496680\pi\)
0.860763 0.509007i \(-0.169987\pi\)
\(788\) 0 0
\(789\) −1.31507 2.27777i −0.0468177 0.0810907i
\(790\) 0 0
\(791\) −14.4297 + 6.61682i −0.513063 + 0.235267i
\(792\) 0 0
\(793\) 14.3048 + 24.7766i 0.507977 + 0.879842i
\(794\) 0 0
\(795\) −4.06765 + 7.04537i −0.144265 + 0.249874i
\(796\) 0 0
\(797\) 28.1746 0.997994 0.498997 0.866604i \(-0.333702\pi\)
0.498997 + 0.866604i \(0.333702\pi\)
\(798\) 0 0
\(799\) −4.60952 −0.163073
\(800\) 0 0
\(801\) −5.65754 + 9.79914i −0.199899 + 0.346236i
\(802\) 0 0
\(803\) 7.27737 + 12.6048i 0.256813 + 0.444813i
\(804\) 0 0
\(805\) 1.68748 17.9655i 0.0594759 0.633202i
\(806\) 0 0
\(807\) 9.82022 + 17.0091i 0.345688 + 0.598750i
\(808\) 0 0
\(809\) −27.1275 + 46.9863i −0.953753 + 1.65195i −0.216558 + 0.976270i \(0.569483\pi\)
−0.737195 + 0.675680i \(0.763850\pi\)
\(810\) 0 0
\(811\) 13.5258 0.474954 0.237477 0.971393i \(-0.423680\pi\)
0.237477 + 0.971393i \(0.423680\pi\)
\(812\) 0 0
\(813\) 14.8801 0.521868
\(814\) 0 0
\(815\) 8.00000 13.8564i 0.280228 0.485369i
\(816\) 0 0
\(817\) 18.3502 + 31.7834i 0.641991 + 1.11196i
\(818\) 0 0
\(819\) −9.30992 6.60790i −0.325315 0.230899i
\(820\) 0 0
\(821\) 5.85793 + 10.1462i 0.204443 + 0.354106i 0.949955 0.312386i \(-0.101128\pi\)
−0.745512 + 0.666492i \(0.767795\pi\)
\(822\) 0 0
\(823\) −8.50515 + 14.7314i −0.296471 + 0.513503i −0.975326 0.220769i \(-0.929143\pi\)
0.678855 + 0.734272i \(0.262476\pi\)
\(824\) 0 0
\(825\) 3.80992 0.132644
\(826\) 0 0
\(827\) −25.3905 −0.882913 −0.441457 0.897283i \(-0.645538\pi\)
−0.441457 + 0.897283i \(0.645538\pi\)
\(828\) 0 0
\(829\) 26.6275 46.1202i 0.924813 1.60182i 0.132950 0.991123i \(-0.457555\pi\)
0.791862 0.610700i \(-0.209112\pi\)
\(830\) 0 0
\(831\) −2.41527 4.18336i −0.0837847 0.145119i
\(832\) 0 0
\(833\) −3.03770 8.68985i −0.105250 0.301085i
\(834\) 0 0
\(835\) 6.02994 + 10.4442i 0.208675 + 0.361435i
\(836\) 0 0
\(837\) −0.747423 + 1.29457i −0.0258347 + 0.0447470i
\(838\) 0 0
\(839\) 23.6952 0.818050 0.409025 0.912523i \(-0.365869\pi\)
0.409025 + 0.912523i \(0.365869\pi\)
\(840\) 0 0
\(841\) −10.4690 −0.361001
\(842\) 0 0
\(843\) −4.75258 + 8.23170i −0.163687 + 0.283515i
\(844\) 0 0
\(845\) −2.80992 4.86692i −0.0966641 0.167427i
\(846\) 0 0
\(847\) −7.58473 5.38341i −0.260615 0.184976i
\(848\) 0 0
\(849\) 13.8553 + 23.9981i 0.475513 + 0.823613i
\(850\) 0 0
\(851\) 31.7303 54.9585i 1.08770 1.88395i
\(852\) 0 0
\(853\) 22.5238 0.771201 0.385600 0.922666i \(-0.373994\pi\)
0.385600 + 0.922666i \(0.373994\pi\)
\(854\) 0 0
\(855\) −7.63014 −0.260945
\(856\) 0 0
\(857\) −22.7380 + 39.3834i −0.776717 + 1.34531i 0.157108 + 0.987581i \(0.449783\pi\)
−0.933825 + 0.357731i \(0.883550\pi\)
\(858\) 0 0
\(859\) −20.7345 35.9132i −0.707452 1.22534i −0.965799 0.259291i \(-0.916511\pi\)
0.258347 0.966052i \(-0.416822\pi\)
\(860\) 0 0
\(861\) −2.58311 + 27.5007i −0.0880321 + 0.937223i
\(862\) 0 0
\(863\) 21.8331 + 37.8160i 0.743207 + 1.28727i 0.951028 + 0.309105i \(0.100029\pi\)
−0.207821 + 0.978167i \(0.566637\pi\)
\(864\) 0 0
\(865\) −8.56249 + 14.8307i −0.291134 + 0.504258i
\(866\) 0 0
\(867\) 15.2706 0.518616
\(868\) 0 0
\(869\) −8.09602 −0.274639
\(870\) 0 0
\(871\) −33.3125 + 57.6989i −1.12875 + 1.95505i
\(872\) 0 0
\(873\) −5.31507 9.20597i −0.179888 0.311575i
\(874\) 0 0
\(875\) 2.40496 1.10280i 0.0813024 0.0372816i
\(876\) 0 0
\(877\) −20.7423 35.9267i −0.700417 1.21316i −0.968320 0.249712i \(-0.919664\pi\)
0.267904 0.963446i \(-0.413669\pi\)
\(878\) 0 0
\(879\) −1.24742 + 2.16060i −0.0420745 + 0.0728752i
\(880\) 0 0
\(881\) 13.0154 0.438500 0.219250 0.975669i \(-0.429639\pi\)
0.219250 + 0.975669i \(0.429639\pi\)
\(882\) 0 0
\(883\) −22.2996 −0.750440 −0.375220 0.926936i \(-0.622433\pi\)
−0.375220 + 0.926936i \(0.622433\pi\)
\(884\) 0 0
\(885\) −4.15238 + 7.19214i −0.139581 + 0.241761i
\(886\) 0 0
\(887\) −13.7722 23.8542i −0.462426 0.800945i 0.536656 0.843801i \(-0.319687\pi\)
−0.999081 + 0.0428566i \(0.986354\pi\)
\(888\) 0 0
\(889\) −40.7525 + 18.6872i −1.36680 + 0.626750i
\(890\) 0 0
\(891\) 1.90496 + 3.29948i 0.0638185 + 0.110537i
\(892\) 0 0
\(893\) 13.3724 23.1617i 0.447491 0.775077i
\(894\) 0 0
\(895\) 11.5052 0.384575
\(896\) 0 0
\(897\) 29.4297 0.982631
\(898\) 0 0
\(899\) −3.21748 + 5.57284i −0.107309 + 0.185864i
\(900\) 0 0
\(901\) −5.34924 9.26516i −0.178209 0.308667i
\(902\) 0 0
\(903\) 1.19008 12.6701i 0.0396035 0.421634i
\(904\) 0 0
\(905\) −3.24227 5.61577i −0.107777 0.186675i
\(906\) 0 0
\(907\) 16.1601 27.9901i 0.536587 0.929396i −0.462498 0.886620i \(-0.653047\pi\)
0.999085 0.0427753i \(-0.0136199\pi\)
\(908\) 0 0
\(909\) 10.6849 0.354397
\(910\) 0 0
\(911\) 2.55474 0.0846422 0.0423211 0.999104i \(-0.486525\pi\)
0.0423211 + 0.999104i \(0.486525\pi\)
\(912\) 0 0
\(913\) 20.8305 36.0795i 0.689390 1.19406i
\(914\) 0 0
\(915\) −3.31507 5.74187i −0.109593 0.189820i
\(916\) 0 0
\(917\) −28.2731 20.0674i −0.933661 0.662684i
\(918\) 0 0
\(919\) −20.5573 35.6064i −0.678124 1.17455i −0.975545 0.219799i \(-0.929460\pi\)
0.297421 0.954746i \(-0.403873\pi\)
\(920\) 0 0
\(921\) −0.605350 + 1.04850i −0.0199470 + 0.0345492i
\(922\) 0 0
\(923\) 18.6643 0.614343
\(924\) 0 0
\(925\) 9.30476 0.305939
\(926\) 0 0
\(927\) −9.72003 + 16.8356i −0.319248 + 0.552953i
\(928\) 0 0
\(929\) 1.10535 + 1.91452i 0.0362654 + 0.0628134i 0.883588 0.468264i \(-0.155121\pi\)
−0.847323 + 0.531078i \(0.821787\pi\)
\(930\) 0 0
\(931\) 52.4768 + 9.94590i 1.71986 + 0.325964i
\(932\) 0 0
\(933\) −14.6027 25.2927i −0.478072 0.828046i
\(934\) 0 0
\(935\) −2.50515 + 4.33905i −0.0819273 + 0.141902i
\(936\) 0 0
\(937\) −48.7003 −1.59097 −0.795485 0.605973i \(-0.792784\pi\)
−0.795485 + 0.605973i \(0.792784\pi\)
\(938\) 0 0
\(939\) 0.809916 0.0264306
\(940\) 0 0
\(941\) 13.9075 24.0885i 0.453372 0.785263i −0.545221 0.838292i \(-0.683554\pi\)
0.998593 + 0.0530292i \(0.0168876\pi\)
\(942\) 0 0
\(943\) −35.6018 61.6641i −1.15935 2.00806i
\(944\) 0 0
\(945\) 2.15754 + 1.53135i 0.0701846 + 0.0498149i
\(946\) 0 0
\(947\) −18.9520 32.8258i −0.615857 1.06670i −0.990234 0.139419i \(-0.955477\pi\)
0.374377 0.927277i \(-0.377857\pi\)
\(948\) 0 0
\(949\) −8.24227 + 14.2760i −0.267555 + 0.463419i
\(950\) 0 0
\(951\) 8.28415 0.268632
\(952\) 0 0
\(953\) −13.9110 −0.450623 −0.225311 0.974287i \(-0.572340\pi\)
−0.225311 + 0.974287i \(0.572340\pi\)
\(954\) 0 0
\(955\) −6.61983 + 11.4659i −0.214213 + 0.371027i
\(956\) 0 0
\(957\) 8.20039 + 14.2035i 0.265081 + 0.459134i
\(958\) 0 0
\(959\) −3.20039 + 34.0726i −0.103346 + 1.10026i
\(960\) 0 0
\(961\) 14.3827 + 24.9116i 0.463959 + 0.803600i
\(962\) 0 0
\(963\) 5.15238 8.92419i 0.166033 0.287578i
\(964\) 0 0
\(965\) 15.4607 0.497697
\(966\) 0 0
\(967\) −4.80992 −0.154676 −0.0773382 0.997005i \(-0.524642\pi\)
−0.0773382 + 0.997005i \(0.524642\pi\)
\(968\) 0 0
\(969\) 5.01709 8.68985i 0.161172 0.279158i
\(970\) 0 0
\(971\) 2.25773 + 3.91051i 0.0724540 + 0.125494i 0.899976 0.435939i \(-0.143584\pi\)
−0.827522 + 0.561433i \(0.810250\pi\)
\(972\) 0 0
\(973\) 28.2458 12.9522i 0.905519 0.415229i
\(974\) 0 0
\(975\) 2.15754 + 3.73696i 0.0690964 + 0.119679i
\(976\) 0 0
\(977\) 21.5480 37.3222i 0.689380 1.19404i −0.282658 0.959221i \(-0.591216\pi\)
0.972039 0.234821i \(-0.0754504\pi\)
\(978\) 0 0
\(979\) −43.1095 −1.37778
\(980\) 0 0
\(981\) −7.11468 −0.227154
\(982\) 0 0
\(983\) 7.57958 13.1282i 0.241751 0.418725i −0.719462 0.694532i \(-0.755612\pi\)
0.961213 + 0.275807i \(0.0889449\pi\)
\(984\) 0 0
\(985\) 7.22003 + 12.5055i 0.230049 + 0.398457i
\(986\) 0 0
\(987\) −8.42975 + 3.86550i −0.268322 + 0.123040i
\(988\) 0 0
\(989\) 16.4024 + 28.4097i 0.521565 + 0.903376i
\(990\) 0 0
\(991\) −7.68233 + 13.3062i −0.244037 + 0.422685i −0.961860 0.273541i \(-0.911805\pi\)
0.717823 + 0.696225i \(0.245139\pi\)
\(992\) 0 0
\(993\) 25.8801 0.821281
\(994\) 0 0
\(995\) −2.38017 −0.0754564
\(996\) 0 0
\(997\) −22.4153 + 38.8244i −0.709899 + 1.22958i 0.254996 + 0.966942i \(0.417926\pi\)
−0.964894 + 0.262638i \(0.915407\pi\)
\(998\) 0 0
\(999\) 4.65238 + 8.05816i 0.147195 + 0.254949i
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 1680.2.bg.u.1201.3 6
4.3 odd 2 840.2.bg.i.361.1 yes 6
7.2 even 3 inner 1680.2.bg.u.961.3 6
12.11 even 2 2520.2.bi.o.361.1 6
28.3 even 6 5880.2.a.bt.1.3 3
28.11 odd 6 5880.2.a.bw.1.3 3
28.23 odd 6 840.2.bg.i.121.1 6
84.23 even 6 2520.2.bi.o.1801.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bg.i.121.1 6 28.23 odd 6
840.2.bg.i.361.1 yes 6 4.3 odd 2
1680.2.bg.u.961.3 6 7.2 even 3 inner
1680.2.bg.u.1201.3 6 1.1 even 1 trivial
2520.2.bi.o.361.1 6 12.11 even 2
2520.2.bi.o.1801.1 6 84.23 even 6
5880.2.a.bt.1.3 3 28.3 even 6
5880.2.a.bw.1.3 3 28.11 odd 6