Properties

Label 1008.2.q.b.529.1
Level $1008$
Weight $2$
Character 1008.529
Analytic conductor $8.049$
Analytic rank $1$
Dimension $2$
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] = [1008,2,Mod(529,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.q (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: \(8.04892052375\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.529
Dual form 1008.2.q.b.625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000 q^{9} +(-1.50000 - 2.59808i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(1.50000 + 0.866025i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(2.50000 + 4.33013i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(0.500000 - 0.866025i) q^{23} +(2.00000 + 3.46410i) q^{25} +5.19615i q^{27} +(-4.50000 + 7.79423i) q^{29} -4.00000 q^{31} +(-4.50000 + 2.59808i) q^{33} +(2.50000 - 0.866025i) q^{35} +(-2.50000 - 4.33013i) q^{37} +(-1.50000 + 0.866025i) q^{39} +(-3.50000 - 6.06218i) q^{41} +(1.50000 - 2.59808i) q^{43} +(1.50000 - 2.59808i) q^{45} -8.00000 q^{47} +(1.00000 + 6.92820i) q^{49} +(4.50000 + 2.59808i) q^{51} +(-4.50000 + 7.79423i) q^{53} +3.00000 q^{55} +(7.50000 - 4.33013i) q^{57} +4.00000 q^{59} +2.00000 q^{61} +(6.00000 + 5.19615i) q^{63} +1.00000 q^{65} -12.0000 q^{67} +(-1.50000 - 0.866025i) q^{69} -8.00000 q^{71} +(6.50000 - 11.2583i) q^{73} +(6.00000 - 3.46410i) q^{75} +(-1.50000 + 7.79423i) q^{77} -8.00000 q^{79} +9.00000 q^{81} +(-6.50000 + 11.2583i) q^{83} +(-1.50000 - 2.59808i) q^{85} +(13.5000 + 7.79423i) q^{87} +(4.50000 + 7.79423i) q^{89} +(-0.500000 + 2.59808i) q^{91} +6.92820i q^{93} -5.00000 q^{95} +(8.50000 - 14.7224i) q^{97} +(4.50000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} - 4 q^{7} - 6 q^{9} - 3 q^{11} - q^{13} + 3 q^{15} - 3 q^{17} + 5 q^{19} - 6 q^{21} + q^{23} + 4 q^{25} - 9 q^{29} - 8 q^{31} - 9 q^{33} + 5 q^{35} - 5 q^{37} - 3 q^{39} - 7 q^{41} + 3 q^{43} + 3 q^{45} - 16 q^{47} + 2 q^{49} + 9 q^{51} - 9 q^{53} + 6 q^{55} + 15 q^{57} + 8 q^{59} + 4 q^{61} + 12 q^{63} + 2 q^{65} - 24 q^{67} - 3 q^{69} - 16 q^{71} + 13 q^{73} + 12 q^{75} - 3 q^{77} - 16 q^{79} + 18 q^{81} - 13 q^{83} - 3 q^{85} + 27 q^{87} + 9 q^{89} - q^{91} - 10 q^{95} + 17 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i −0.955901 0.293691i \(-0.905116\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) −0.500000 0.866025i −0.138675 0.240192i 0.788320 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 1.50000 + 0.866025i 0.387298 + 0.223607i
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) 2.50000 + 4.33013i 0.573539 + 0.993399i 0.996199 + 0.0871106i \(0.0277634\pi\)
−0.422659 + 0.906289i \(0.638903\pi\)
\(20\) 0 0
\(21\) −3.00000 + 3.46410i −0.654654 + 0.755929i
\(22\) 0 0
\(23\) 0.500000 0.866025i 0.104257 0.180579i −0.809177 0.587565i \(-0.800087\pi\)
0.913434 + 0.406986i \(0.133420\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −4.50000 + 7.79423i −0.835629 + 1.44735i 0.0578882 + 0.998323i \(0.481563\pi\)
−0.893517 + 0.449029i \(0.851770\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −4.50000 + 2.59808i −0.783349 + 0.452267i
\(34\) 0 0
\(35\) 2.50000 0.866025i 0.422577 0.146385i
\(36\) 0 0
\(37\) −2.50000 4.33013i −0.410997 0.711868i 0.584002 0.811752i \(-0.301486\pi\)
−0.994999 + 0.0998840i \(0.968153\pi\)
\(38\) 0 0
\(39\) −1.50000 + 0.866025i −0.240192 + 0.138675i
\(40\) 0 0
\(41\) −3.50000 6.06218i −0.546608 0.946753i −0.998504 0.0546823i \(-0.982585\pi\)
0.451896 0.892071i \(-0.350748\pi\)
\(42\) 0 0
\(43\) 1.50000 2.59808i 0.228748 0.396203i −0.728689 0.684844i \(-0.759870\pi\)
0.957437 + 0.288641i \(0.0932035\pi\)
\(44\) 0 0
\(45\) 1.50000 2.59808i 0.223607 0.387298i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 4.50000 + 2.59808i 0.630126 + 0.363803i
\(52\) 0 0
\(53\) −4.50000 + 7.79423i −0.618123 + 1.07062i 0.371706 + 0.928351i \(0.378773\pi\)
−0.989828 + 0.142269i \(0.954560\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 7.50000 4.33013i 0.993399 0.573539i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 6.00000 + 5.19615i 0.755929 + 0.654654i
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −1.50000 0.866025i −0.180579 0.104257i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 6.50000 11.2583i 0.760767 1.31769i −0.181688 0.983356i \(-0.558156\pi\)
0.942455 0.334332i \(-0.108511\pi\)
\(74\) 0 0
\(75\) 6.00000 3.46410i 0.692820 0.400000i
\(76\) 0 0
\(77\) −1.50000 + 7.79423i −0.170941 + 0.888235i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.50000 + 11.2583i −0.713468 + 1.23576i 0.250080 + 0.968225i \(0.419543\pi\)
−0.963548 + 0.267537i \(0.913790\pi\)
\(84\) 0 0
\(85\) −1.50000 2.59808i −0.162698 0.281801i
\(86\) 0 0
\(87\) 13.5000 + 7.79423i 1.44735 + 0.835629i
\(88\) 0 0
\(89\) 4.50000 + 7.79423i 0.476999 + 0.826187i 0.999653 0.0263586i \(-0.00839118\pi\)
−0.522654 + 0.852545i \(0.675058\pi\)
\(90\) 0 0
\(91\) −0.500000 + 2.59808i −0.0524142 + 0.272352i
\(92\) 0 0
\(93\) 6.92820i 0.718421i
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) 8.50000 14.7224i 0.863044 1.49484i −0.00593185 0.999982i \(-0.501888\pi\)
0.868976 0.494854i \(-0.164778\pi\)
\(98\) 0 0
\(99\) 4.50000 + 7.79423i 0.452267 + 0.783349i
\(100\) 0 0
\(101\) 3.50000 + 6.06218i 0.348263 + 0.603209i 0.985941 0.167094i \(-0.0534383\pi\)
−0.637678 + 0.770303i \(0.720105\pi\)
\(102\) 0 0
\(103\) 4.50000 7.79423i 0.443398 0.767988i −0.554541 0.832156i \(-0.687106\pi\)
0.997939 + 0.0641683i \(0.0204394\pi\)
\(104\) 0 0
\(105\) −1.50000 4.33013i −0.146385 0.422577i
\(106\) 0 0
\(107\) −3.50000 6.06218i −0.338358 0.586053i 0.645766 0.763535i \(-0.276538\pi\)
−0.984124 + 0.177482i \(0.943205\pi\)
\(108\) 0 0
\(109\) −2.50000 + 4.33013i −0.239457 + 0.414751i −0.960558 0.278078i \(-0.910303\pi\)
0.721102 + 0.692829i \(0.243636\pi\)
\(110\) 0 0
\(111\) −7.50000 + 4.33013i −0.711868 + 0.410997i
\(112\) 0 0
\(113\) 0.500000 + 0.866025i 0.0470360 + 0.0814688i 0.888585 0.458712i \(-0.151689\pi\)
−0.841549 + 0.540181i \(0.818356\pi\)
\(114\) 0 0
\(115\) 0.500000 + 0.866025i 0.0466252 + 0.0807573i
\(116\) 0 0
\(117\) 1.50000 + 2.59808i 0.138675 + 0.240192i
\(118\) 0 0
\(119\) 7.50000 2.59808i 0.687524 0.238165i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) −10.5000 + 6.06218i −0.946753 + 0.546608i
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −4.50000 2.59808i −0.396203 0.228748i
\(130\) 0 0
\(131\) 1.50000 2.59808i 0.131056 0.226995i −0.793028 0.609185i \(-0.791497\pi\)
0.924084 + 0.382190i \(0.124830\pi\)
\(132\) 0 0
\(133\) 2.50000 12.9904i 0.216777 1.12641i
\(134\) 0 0
\(135\) −4.50000 2.59808i −0.387298 0.223607i
\(136\) 0 0
\(137\) −5.50000 9.52628i −0.469897 0.813885i 0.529511 0.848303i \(-0.322376\pi\)
−0.999408 + 0.0344182i \(0.989042\pi\)
\(138\) 0 0
\(139\) 2.50000 + 4.33013i 0.212047 + 0.367277i 0.952355 0.304991i \(-0.0986536\pi\)
−0.740308 + 0.672268i \(0.765320\pi\)
\(140\) 0 0
\(141\) 13.8564i 1.16692i
\(142\) 0 0
\(143\) −1.50000 + 2.59808i −0.125436 + 0.217262i
\(144\) 0 0
\(145\) −4.50000 7.79423i −0.373705 0.647275i
\(146\) 0 0
\(147\) 12.0000 1.73205i 0.989743 0.142857i
\(148\) 0 0
\(149\) 7.50000 12.9904i 0.614424 1.06421i −0.376061 0.926595i \(-0.622722\pi\)
0.990485 0.137619i \(-0.0439449\pi\)
\(150\) 0 0
\(151\) −8.50000 14.7224i −0.691720 1.19809i −0.971274 0.237964i \(-0.923520\pi\)
0.279554 0.960130i \(-0.409814\pi\)
\(152\) 0 0
\(153\) 4.50000 7.79423i 0.363803 0.630126i
\(154\) 0 0
\(155\) 2.00000 3.46410i 0.160644 0.278243i
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 13.5000 + 7.79423i 1.07062 + 0.618123i
\(160\) 0 0
\(161\) −2.50000 + 0.866025i −0.197028 + 0.0682524i
\(162\) 0 0
\(163\) 8.50000 + 14.7224i 0.665771 + 1.15315i 0.979076 + 0.203497i \(0.0652307\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 5.19615i 0.404520i
\(166\) 0 0
\(167\) −0.500000 0.866025i −0.0386912 0.0670151i 0.846031 0.533133i \(-0.178986\pi\)
−0.884723 + 0.466118i \(0.845652\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) −7.50000 12.9904i −0.573539 0.993399i
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 2.00000 10.3923i 0.151186 0.785584i
\(176\) 0 0
\(177\) 6.92820i 0.520756i
\(178\) 0 0
\(179\) −10.5000 + 18.1865i −0.784807 + 1.35933i 0.144308 + 0.989533i \(0.453905\pi\)
−0.929114 + 0.369792i \(0.879429\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 3.46410i 0.256074i
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 9.00000 10.3923i 0.654654 0.755929i
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 1.73205i 0.124035i
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 10.5000 18.1865i 0.744325 1.28921i −0.206184 0.978513i \(-0.566105\pi\)
0.950509 0.310696i \(-0.100562\pi\)
\(200\) 0 0
\(201\) 20.7846i 1.46603i
\(202\) 0 0
\(203\) 22.5000 7.79423i 1.57919 0.547048i
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 0 0
\(207\) −1.50000 + 2.59808i −0.104257 + 0.180579i
\(208\) 0 0
\(209\) 7.50000 12.9904i 0.518786 0.898563i
\(210\) 0 0
\(211\) −13.5000 23.3827i −0.929378 1.60973i −0.784364 0.620301i \(-0.787010\pi\)
−0.145014 0.989430i \(-0.546323\pi\)
\(212\) 0 0
\(213\) 13.8564i 0.949425i
\(214\) 0 0
\(215\) 1.50000 + 2.59808i 0.102299 + 0.177187i
\(216\) 0 0
\(217\) 8.00000 + 6.92820i 0.543075 + 0.470317i
\(218\) 0 0
\(219\) −19.5000 11.2583i −1.31769 0.760767i
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 6.50000 11.2583i 0.435272 0.753914i −0.562046 0.827106i \(-0.689985\pi\)
0.997318 + 0.0731927i \(0.0233188\pi\)
\(224\) 0 0
\(225\) −6.00000 10.3923i −0.400000 0.692820i
\(226\) 0 0
\(227\) −7.50000 12.9904i −0.497792 0.862202i 0.502204 0.864749i \(-0.332523\pi\)
−0.999997 + 0.00254715i \(0.999189\pi\)
\(228\) 0 0
\(229\) −10.5000 + 18.1865i −0.693860 + 1.20180i 0.276704 + 0.960955i \(0.410758\pi\)
−0.970564 + 0.240845i \(0.922576\pi\)
\(230\) 0 0
\(231\) 13.5000 + 2.59808i 0.888235 + 0.170941i
\(232\) 0 0
\(233\) −1.50000 2.59808i −0.0982683 0.170206i 0.812700 0.582683i \(-0.197997\pi\)
−0.910968 + 0.412477i \(0.864664\pi\)
\(234\) 0 0
\(235\) 4.00000 6.92820i 0.260931 0.451946i
\(236\) 0 0
\(237\) 13.8564i 0.900070i
\(238\) 0 0
\(239\) −4.50000 7.79423i −0.291081 0.504167i 0.682985 0.730433i \(-0.260682\pi\)
−0.974066 + 0.226266i \(0.927348\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.0322078 + 0.0557856i 0.881680 0.471848i \(-0.156413\pi\)
−0.849472 + 0.527633i \(0.823079\pi\)
\(242\) 0 0
\(243\) 15.5885i 1.00000i
\(244\) 0 0
\(245\) −6.50000 2.59808i −0.415270 0.165985i
\(246\) 0 0
\(247\) 2.50000 4.33013i 0.159071 0.275519i
\(248\) 0 0
\(249\) 19.5000 + 11.2583i 1.23576 + 0.713468i
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) −4.50000 + 2.59808i −0.281801 + 0.162698i
\(256\) 0 0
\(257\) 8.50000 14.7224i 0.530215 0.918360i −0.469163 0.883112i \(-0.655444\pi\)
0.999379 0.0352486i \(-0.0112223\pi\)
\(258\) 0 0
\(259\) −2.50000 + 12.9904i −0.155342 + 0.807183i
\(260\) 0 0
\(261\) 13.5000 23.3827i 0.835629 1.44735i
\(262\) 0 0
\(263\) 5.50000 + 9.52628i 0.339145 + 0.587416i 0.984272 0.176659i \(-0.0565291\pi\)
−0.645128 + 0.764075i \(0.723196\pi\)
\(264\) 0 0
\(265\) −4.50000 7.79423i −0.276433 0.478796i
\(266\) 0 0
\(267\) 13.5000 7.79423i 0.826187 0.476999i
\(268\) 0 0
\(269\) 3.50000 6.06218i 0.213399 0.369618i −0.739377 0.673291i \(-0.764880\pi\)
0.952776 + 0.303674i \(0.0982133\pi\)
\(270\) 0 0
\(271\) 15.5000 + 26.8468i 0.941558 + 1.63083i 0.762501 + 0.646988i \(0.223971\pi\)
0.179057 + 0.983839i \(0.442695\pi\)
\(272\) 0 0
\(273\) 4.50000 + 0.866025i 0.272352 + 0.0524142i
\(274\) 0 0
\(275\) 6.00000 10.3923i 0.361814 0.626680i
\(276\) 0 0
\(277\) 9.50000 + 16.4545i 0.570800 + 0.988654i 0.996484 + 0.0837823i \(0.0267000\pi\)
−0.425684 + 0.904872i \(0.639967\pi\)
\(278\) 0 0
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) −13.5000 + 23.3827i −0.805342 + 1.39489i 0.110717 + 0.993852i \(0.464685\pi\)
−0.916060 + 0.401042i \(0.868648\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 0 0
\(285\) 8.66025i 0.512989i
\(286\) 0 0
\(287\) −3.50000 + 18.1865i −0.206598 + 1.07352i
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) −25.5000 14.7224i −1.49484 0.863044i
\(292\) 0 0
\(293\) −0.500000 0.866025i −0.0292103 0.0505937i 0.851051 0.525084i \(-0.175966\pi\)
−0.880261 + 0.474490i \(0.842633\pi\)
\(294\) 0 0
\(295\) −2.00000 + 3.46410i −0.116445 + 0.201688i
\(296\) 0 0
\(297\) 13.5000 7.79423i 0.783349 0.452267i
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) −7.50000 + 2.59808i −0.432293 + 0.149751i
\(302\) 0 0
\(303\) 10.5000 6.06218i 0.603209 0.348263i
\(304\) 0 0
\(305\) −1.00000 + 1.73205i −0.0572598 + 0.0991769i
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −13.5000 7.79423i −0.767988 0.443398i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) −7.50000 + 2.59808i −0.422577 + 0.146385i
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) −10.5000 + 6.06218i −0.586053 + 0.338358i
\(322\) 0 0
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) 2.00000 3.46410i 0.110940 0.192154i
\(326\) 0 0
\(327\) 7.50000 + 4.33013i 0.414751 + 0.239457i
\(328\) 0 0
\(329\) 16.0000 + 13.8564i 0.882109 + 0.763928i
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 7.50000 + 12.9904i 0.410997 + 0.711868i
\(334\) 0 0
\(335\) 6.00000 10.3923i 0.327815 0.567792i
\(336\) 0 0
\(337\) −7.50000 12.9904i −0.408551 0.707631i 0.586177 0.810183i \(-0.300632\pi\)
−0.994728 + 0.102552i \(0.967299\pi\)
\(338\) 0 0
\(339\) 1.50000 0.866025i 0.0814688 0.0470360i
\(340\) 0 0
\(341\) 6.00000 + 10.3923i 0.324918 + 0.562775i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 1.50000 0.866025i 0.0807573 0.0466252i
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −8.50000 + 14.7224i −0.454995 + 0.788074i −0.998688 0.0512103i \(-0.983692\pi\)
0.543693 + 0.839284i \(0.317025\pi\)
\(350\) 0 0
\(351\) 4.50000 2.59808i 0.240192 0.138675i
\(352\) 0 0
\(353\) −15.5000 26.8468i −0.824982 1.42891i −0.901933 0.431875i \(-0.857852\pi\)
0.0769515 0.997035i \(-0.475481\pi\)
\(354\) 0 0
\(355\) 4.00000 6.92820i 0.212298 0.367711i
\(356\) 0 0
\(357\) −4.50000 12.9904i −0.238165 0.687524i
\(358\) 0 0
\(359\) 13.5000 + 23.3827i 0.712503 + 1.23409i 0.963915 + 0.266211i \(0.0857717\pi\)
−0.251412 + 0.967880i \(0.580895\pi\)
\(360\) 0 0
\(361\) −3.00000 + 5.19615i −0.157895 + 0.273482i
\(362\) 0 0
\(363\) −3.00000 1.73205i −0.157459 0.0909091i
\(364\) 0 0
\(365\) 6.50000 + 11.2583i 0.340226 + 0.589288i
\(366\) 0 0
\(367\) 5.50000 + 9.52628i 0.287098 + 0.497268i 0.973116 0.230317i \(-0.0739762\pi\)
−0.686018 + 0.727585i \(0.740643\pi\)
\(368\) 0 0
\(369\) 10.5000 + 18.1865i 0.546608 + 0.946753i
\(370\) 0 0
\(371\) 22.5000 7.79423i 1.16814 0.404656i
\(372\) 0 0
\(373\) −6.50000 + 11.2583i −0.336557 + 0.582934i −0.983783 0.179364i \(-0.942596\pi\)
0.647225 + 0.762299i \(0.275929\pi\)
\(374\) 0 0
\(375\) 15.5885i 0.804984i
\(376\) 0 0
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) 13.8564i 0.709885i
\(382\) 0 0
\(383\) −17.5000 + 30.3109i −0.894208 + 1.54881i −0.0594268 + 0.998233i \(0.518927\pi\)
−0.834781 + 0.550581i \(0.814406\pi\)
\(384\) 0 0
\(385\) −6.00000 5.19615i −0.305788 0.264820i
\(386\) 0 0
\(387\) −4.50000 + 7.79423i −0.228748 + 0.396203i
\(388\) 0 0
\(389\) −18.5000 32.0429i −0.937987 1.62464i −0.769218 0.638986i \(-0.779354\pi\)
−0.168769 0.985656i \(-0.553979\pi\)
\(390\) 0 0
\(391\) 1.50000 + 2.59808i 0.0758583 + 0.131390i
\(392\) 0 0
\(393\) −4.50000 2.59808i −0.226995 0.131056i
\(394\) 0 0
\(395\) 4.00000 6.92820i 0.201262 0.348596i
\(396\) 0 0
\(397\) −12.5000 21.6506i −0.627357 1.08661i −0.988080 0.153941i \(-0.950803\pi\)
0.360723 0.932673i \(-0.382530\pi\)
\(398\) 0 0
\(399\) −22.5000 4.33013i −1.12641 0.216777i
\(400\) 0 0
\(401\) 6.50000 11.2583i 0.324595 0.562214i −0.656836 0.754034i \(-0.728105\pi\)
0.981430 + 0.191820i \(0.0614388\pi\)
\(402\) 0 0
\(403\) 2.00000 + 3.46410i 0.0996271 + 0.172559i
\(404\) 0 0
\(405\) −4.50000 + 7.79423i −0.223607 + 0.387298i
\(406\) 0 0
\(407\) −7.50000 + 12.9904i −0.371761 + 0.643909i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −16.5000 + 9.52628i −0.813885 + 0.469897i
\(412\) 0 0
\(413\) −8.00000 6.92820i −0.393654 0.340915i
\(414\) 0 0
\(415\) −6.50000 11.2583i −0.319072 0.552650i
\(416\) 0 0
\(417\) 7.50000 4.33013i 0.367277 0.212047i
\(418\) 0 0
\(419\) 10.5000 + 18.1865i 0.512959 + 0.888470i 0.999887 + 0.0150285i \(0.00478389\pi\)
−0.486928 + 0.873442i \(0.661883\pi\)
\(420\) 0 0
\(421\) 1.50000 2.59808i 0.0731055 0.126622i −0.827155 0.561973i \(-0.810042\pi\)
0.900261 + 0.435351i \(0.143376\pi\)
\(422\) 0 0
\(423\) 24.0000 1.16692
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −4.00000 3.46410i −0.193574 0.167640i
\(428\) 0 0
\(429\) 4.50000 + 2.59808i 0.217262 + 0.125436i
\(430\) 0 0
\(431\) −7.50000 + 12.9904i −0.361262 + 0.625725i −0.988169 0.153370i \(-0.950987\pi\)
0.626907 + 0.779094i \(0.284321\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) −13.5000 + 7.79423i −0.647275 + 0.373705i
\(436\) 0 0
\(437\) 5.00000 0.239182
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −3.00000 20.7846i −0.142857 0.989743i
\(442\) 0 0
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 0 0
\(447\) −22.5000 12.9904i −1.06421 0.614424i
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) −10.5000 + 18.1865i −0.494426 + 0.856370i
\(452\) 0 0
\(453\) −25.5000 + 14.7224i −1.19809 + 0.691720i
\(454\) 0 0
\(455\) −2.00000 1.73205i −0.0937614 0.0811998i
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) −13.5000 7.79423i −0.630126 0.363803i
\(460\) 0 0
\(461\) 13.5000 23.3827i 0.628758 1.08904i −0.359044 0.933321i \(-0.616897\pi\)
0.987801 0.155719i \(-0.0497696\pi\)
\(462\) 0 0
\(463\) −0.500000 0.866025i −0.0232370 0.0402476i 0.854173 0.519989i \(-0.174064\pi\)
−0.877410 + 0.479741i \(0.840731\pi\)
\(464\) 0 0
\(465\) −6.00000 3.46410i −0.278243 0.160644i
\(466\) 0 0
\(467\) 6.50000 + 11.2583i 0.300784 + 0.520973i 0.976314 0.216359i \(-0.0694183\pi\)
−0.675530 + 0.737333i \(0.736085\pi\)
\(468\) 0 0
\(469\) 24.0000 + 20.7846i 1.10822 + 0.959744i
\(470\) 0 0
\(471\) 24.2487i 1.11732i
\(472\) 0 0
\(473\) −9.00000 −0.413820
\(474\) 0 0
\(475\) −10.0000 + 17.3205i −0.458831 + 0.794719i
\(476\) 0 0
\(477\) 13.5000 23.3827i 0.618123 1.07062i
\(478\) 0 0
\(479\) −10.5000 18.1865i −0.479757 0.830964i 0.519973 0.854183i \(-0.325942\pi\)
−0.999730 + 0.0232187i \(0.992609\pi\)
\(480\) 0 0
\(481\) −2.50000 + 4.33013i −0.113990 + 0.197437i
\(482\) 0 0
\(483\) 1.50000 + 4.33013i 0.0682524 + 0.197028i
\(484\) 0 0
\(485\) 8.50000 + 14.7224i 0.385965 + 0.668511i
\(486\) 0 0
\(487\) 18.5000 32.0429i 0.838315 1.45200i −0.0529875 0.998595i \(-0.516874\pi\)
0.891303 0.453409i \(-0.149792\pi\)
\(488\) 0 0
\(489\) 25.5000 14.7224i 1.15315 0.665771i
\(490\) 0 0
\(491\) 4.50000 + 7.79423i 0.203082 + 0.351749i 0.949520 0.313707i \(-0.101571\pi\)
−0.746438 + 0.665455i \(0.768237\pi\)
\(492\) 0 0
\(493\) −13.5000 23.3827i −0.608009 1.05310i
\(494\) 0 0
\(495\) −9.00000 −0.404520
\(496\) 0 0
\(497\) 16.0000 + 13.8564i 0.717698 + 0.621545i
\(498\) 0 0
\(499\) −14.5000 + 25.1147i −0.649109 + 1.12429i 0.334227 + 0.942493i \(0.391525\pi\)
−0.983336 + 0.181797i \(0.941809\pi\)
\(500\) 0 0
\(501\) −1.50000 + 0.866025i −0.0670151 + 0.0386912i
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −7.00000 −0.311496
\(506\) 0 0
\(507\) −18.0000 10.3923i −0.799408 0.461538i
\(508\) 0 0
\(509\) −16.5000 + 28.5788i −0.731350 + 1.26673i 0.224957 + 0.974369i \(0.427776\pi\)
−0.956306 + 0.292366i \(0.905557\pi\)
\(510\) 0 0
\(511\) −32.5000 + 11.2583i −1.43772 + 0.498039i
\(512\) 0 0
\(513\) −22.5000 + 12.9904i −0.993399 + 0.573539i
\(514\) 0 0
\(515\) 4.50000 + 7.79423i 0.198294 + 0.343455i
\(516\) 0 0
\(517\) 12.0000 + 20.7846i 0.527759 + 0.914106i
\(518\) 0 0
\(519\) 3.46410i 0.152057i
\(520\) 0 0
\(521\) −1.50000 + 2.59808i −0.0657162 + 0.113824i −0.897011 0.442007i \(-0.854267\pi\)
0.831295 + 0.555831i \(0.187600\pi\)
\(522\) 0 0
\(523\) 16.5000 + 28.5788i 0.721495 + 1.24967i 0.960401 + 0.278623i \(0.0898779\pi\)
−0.238906 + 0.971043i \(0.576789\pi\)
\(524\) 0 0
\(525\) −18.0000 3.46410i −0.785584 0.151186i
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) 11.0000 + 19.0526i 0.478261 + 0.828372i
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −3.50000 + 6.06218i −0.151602 + 0.262582i
\(534\) 0 0
\(535\) 7.00000 0.302636
\(536\) 0 0
\(537\) 31.5000 + 18.1865i 1.35933 + 0.784807i
\(538\) 0 0
\(539\) 16.5000 12.9904i 0.710705 0.559535i
\(540\) 0 0
\(541\) 5.50000 + 9.52628i 0.236463 + 0.409567i 0.959697 0.281037i \(-0.0906783\pi\)
−0.723234 + 0.690604i \(0.757345\pi\)
\(542\) 0 0
\(543\) 31.1769i 1.33793i
\(544\) 0 0
\(545\) −2.50000 4.33013i −0.107088 0.185482i
\(546\) 0 0
\(547\) −2.50000 + 4.33013i −0.106892 + 0.185143i −0.914510 0.404564i \(-0.867423\pi\)
0.807617 + 0.589707i \(0.200757\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −45.0000 −1.91706
\(552\) 0 0
\(553\) 16.0000 + 13.8564i 0.680389 + 0.589234i
\(554\) 0 0
\(555\) 8.66025i 0.367607i
\(556\) 0 0
\(557\) 19.5000 33.7750i 0.826242 1.43109i −0.0747252 0.997204i \(-0.523808\pi\)
0.900967 0.433888i \(-0.142859\pi\)
\(558\) 0 0
\(559\) −3.00000 −0.126886
\(560\) 0 0
\(561\) 15.5885i 0.658145i
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −1.00000 −0.0420703
\(566\) 0 0
\(567\) −18.0000 15.5885i −0.755929 0.654654i
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 6.92820i 0.289430i
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −1.50000 + 2.59808i −0.0624458 + 0.108159i −0.895558 0.444945i \(-0.853223\pi\)
0.833112 + 0.553104i \(0.186557\pi\)
\(578\) 0 0
\(579\) 31.1769i 1.29567i
\(580\) 0 0
\(581\) 32.5000 11.2583i 1.34833 0.467074i
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) 0 0
\(585\) −3.00000 −0.124035
\(586\) 0 0
\(587\) 7.50000 12.9904i 0.309558 0.536170i −0.668708 0.743525i \(-0.733152\pi\)
0.978266 + 0.207355i \(0.0664855\pi\)
\(588\) 0 0
\(589\) −10.0000 17.3205i −0.412043 0.713679i
\(590\) 0 0
\(591\) 38.1051i 1.56744i
\(592\) 0 0
\(593\) 10.5000 + 18.1865i 0.431183 + 0.746831i 0.996976 0.0777165i \(-0.0247629\pi\)
−0.565792 + 0.824548i \(0.691430\pi\)
\(594\) 0 0
\(595\) −1.50000 + 7.79423i −0.0614940 + 0.319532i
\(596\) 0 0
\(597\) −31.5000 18.1865i −1.28921 0.744325i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −11.5000 + 19.9186i −0.469095 + 0.812496i −0.999376 0.0353259i \(-0.988753\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 36.0000 1.46603
\(604\) 0 0
\(605\) 1.00000 + 1.73205i 0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) −11.5000 + 19.9186i −0.466771 + 0.808470i −0.999279 0.0379540i \(-0.987916\pi\)
0.532509 + 0.846424i \(0.321249\pi\)
\(608\) 0 0
\(609\) −13.5000 38.9711i −0.547048 1.57919i
\(610\) 0 0
\(611\) 4.00000 + 6.92820i 0.161823 + 0.280285i
\(612\) 0 0
\(613\) −2.50000 + 4.33013i −0.100974 + 0.174892i −0.912086 0.409998i \(-0.865529\pi\)
0.811112 + 0.584891i \(0.198863\pi\)
\(614\) 0 0
\(615\) 12.1244i 0.488901i
\(616\) 0 0
\(617\) −1.50000 2.59808i −0.0603877 0.104595i 0.834251 0.551385i \(-0.185900\pi\)
−0.894639 + 0.446790i \(0.852567\pi\)
\(618\) 0 0
\(619\) 10.5000 + 18.1865i 0.422031 + 0.730978i 0.996138 0.0878015i \(-0.0279841\pi\)
−0.574107 + 0.818780i \(0.694651\pi\)
\(620\) 0 0
\(621\) 4.50000 + 2.59808i 0.180579 + 0.104257i
\(622\) 0 0
\(623\) 4.50000 23.3827i 0.180289 0.936808i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) −22.5000 12.9904i −0.898563 0.518786i
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) −40.5000 + 23.3827i −1.60973 + 0.929378i
\(634\) 0 0
\(635\) 4.00000 6.92820i 0.158735 0.274937i
\(636\) 0 0
\(637\) 5.50000 4.33013i 0.217918 0.171566i
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) 2.50000 + 4.33013i 0.0987441 + 0.171030i 0.911165 0.412042i \(-0.135184\pi\)
−0.812421 + 0.583071i \(0.801851\pi\)
\(642\) 0 0
\(643\) 18.5000 + 32.0429i 0.729569 + 1.26365i 0.957066 + 0.289871i \(0.0936125\pi\)
−0.227497 + 0.973779i \(0.573054\pi\)
\(644\) 0 0
\(645\) 4.50000 2.59808i 0.177187 0.102299i
\(646\) 0 0
\(647\) 20.5000 35.5070i 0.805938 1.39593i −0.109718 0.993963i \(-0.534995\pi\)
0.915656 0.401963i \(-0.131672\pi\)
\(648\) 0 0
\(649\) −6.00000 10.3923i −0.235521 0.407934i
\(650\) 0 0
\(651\) 12.0000 13.8564i 0.470317 0.543075i
\(652\) 0 0
\(653\) 1.50000 2.59808i 0.0586995 0.101671i −0.835182 0.549973i \(-0.814638\pi\)
0.893882 + 0.448303i \(0.147971\pi\)
\(654\) 0 0
\(655\) 1.50000 + 2.59808i 0.0586098 + 0.101515i
\(656\) 0 0
\(657\) −19.5000 + 33.7750i −0.760767 + 1.31769i
\(658\) 0 0
\(659\) 7.50000 12.9904i 0.292159 0.506033i −0.682161 0.731202i \(-0.738960\pi\)
0.974320 + 0.225168i \(0.0722932\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 5.19615i 0.201802i
\(664\) 0 0
\(665\) 10.0000 + 8.66025i 0.387783 + 0.335830i
\(666\) 0 0
\(667\) 4.50000 + 7.79423i 0.174241 + 0.301794i
\(668\) 0 0
\(669\) −19.5000 11.2583i −0.753914 0.435272i
\(670\) 0 0
\(671\) −3.00000 5.19615i −0.115814 0.200595i
\(672\) 0 0
\(673\) −17.5000 + 30.3109i −0.674575 + 1.16840i 0.302017 + 0.953302i \(0.402340\pi\)
−0.976593 + 0.215096i \(0.930993\pi\)
\(674\) 0 0
\(675\) −18.0000 + 10.3923i −0.692820 + 0.400000i
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −42.5000 + 14.7224i −1.63100 + 0.564995i
\(680\) 0 0
\(681\) −22.5000 + 12.9904i −0.862202 + 0.497792i
\(682\) 0 0
\(683\) 15.5000 26.8468i 0.593091 1.02726i −0.400722 0.916200i \(-0.631241\pi\)
0.993813 0.111064i \(-0.0354259\pi\)
\(684\) 0 0
\(685\) 11.0000 0.420288
\(686\) 0 0
\(687\) 31.5000 + 18.1865i 1.20180 + 0.693860i
\(688\) 0 0
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) 4.50000 23.3827i 0.170941 0.888235i
\(694\) 0 0
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) 21.0000 0.795432
\(698\) 0 0
\(699\) −4.50000 + 2.59808i −0.170206 + 0.0982683i
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) 12.5000 21.6506i 0.471446 0.816569i
\(704\) 0 0
\(705\) −12.0000 6.92820i −0.451946 0.260931i
\(706\) 0 0
\(707\) 3.50000 18.1865i 0.131631 0.683975i
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) −2.00000 + 3.46410i −0.0749006 + 0.129732i
\(714\) 0 0
\(715\) −1.50000 2.59808i −0.0560968 0.0971625i
\(716\) 0 0
\(717\) −13.5000 + 7.79423i −0.504167 + 0.291081i
\(718\) 0 0
\(719\) 9.50000 + 16.4545i 0.354290 + 0.613649i 0.986996 0.160743i \(-0.0513892\pi\)
−0.632706 + 0.774392i \(0.718056\pi\)
\(720\) 0 0
\(721\) −22.5000 + 7.79423i −0.837944 + 0.290272i
\(722\) 0 0
\(723\) 1.50000 0.866025i 0.0557856 0.0322078i
\(724\) 0 0
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) 8.50000 14.7224i 0.315248 0.546025i −0.664243 0.747517i \(-0.731246\pi\)
0.979490 + 0.201492i \(0.0645791\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 4.50000 + 7.79423i 0.166439 + 0.288280i
\(732\) 0 0
\(733\) 21.5000 37.2391i 0.794121 1.37546i −0.129275 0.991609i \(-0.541265\pi\)
0.923396 0.383849i \(-0.125402\pi\)
\(734\) 0 0
\(735\) −4.50000 + 11.2583i −0.165985 + 0.415270i
\(736\) 0 0
\(737\) 18.0000 + 31.1769i 0.663039 + 1.14842i
\(738\) 0 0
\(739\) −8.50000 + 14.7224i −0.312678 + 0.541573i −0.978941 0.204143i \(-0.934559\pi\)
0.666264 + 0.745716i \(0.267893\pi\)
\(740\) 0 0
\(741\) −7.50000 4.33013i −0.275519 0.159071i
\(742\) 0 0
\(743\) −0.500000 0.866025i −0.0183432 0.0317714i 0.856708 0.515802i \(-0.172506\pi\)
−0.875051 + 0.484030i \(0.839172\pi\)
\(744\) 0 0
\(745\) 7.50000 + 12.9904i 0.274779 + 0.475931i
\(746\) 0 0
\(747\) 19.5000 33.7750i 0.713468 1.23576i
\(748\) 0 0
\(749\) −3.50000 + 18.1865i −0.127887 + 0.664521i
\(750\) 0 0
\(751\) −7.50000 + 12.9904i −0.273679 + 0.474026i −0.969801 0.243898i \(-0.921574\pi\)
0.696122 + 0.717923i \(0.254907\pi\)
\(752\) 0 0
\(753\) 6.92820i 0.252478i
\(754\) 0 0
\(755\) 17.0000 0.618693
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 0 0
\(759\) 5.19615i 0.188608i
\(760\) 0 0
\(761\) 22.5000 38.9711i 0.815624 1.41270i −0.0932544 0.995642i \(-0.529727\pi\)
0.908879 0.417061i \(-0.136940\pi\)
\(762\) 0 0
\(763\) 12.5000 4.33013i 0.452530 0.156761i
\(764\) 0 0
\(765\) 4.50000 + 7.79423i 0.162698 + 0.281801i
\(766\) 0 0
\(767\) −2.00000 3.46410i −0.0722158 0.125081i
\(768\) 0 0
\(769\) 20.5000 + 35.5070i 0.739249 + 1.28042i 0.952834 + 0.303492i \(0.0981526\pi\)
−0.213585 + 0.976924i \(0.568514\pi\)
\(770\) 0 0
\(771\) −25.5000 14.7224i −0.918360 0.530215i
\(772\) 0 0
\(773\) 3.50000 6.06218i 0.125886 0.218041i −0.796193 0.605043i \(-0.793156\pi\)
0.922079 + 0.387002i \(0.126489\pi\)
\(774\) 0 0
\(775\) −8.00000 13.8564i −0.287368 0.497737i
\(776\) 0 0
\(777\) 22.5000 + 4.33013i 0.807183 + 0.155342i
\(778\) 0 0
\(779\) 17.5000 30.3109i 0.627003 1.08600i
\(780\) 0 0
\(781\) 12.0000 + 20.7846i 0.429394 + 0.743732i
\(782\) 0 0
\(783\) −40.5000 23.3827i −1.44735 0.835629i
\(784\) 0 0
\(785\) −7.00000 + 12.1244i −0.249841 + 0.432737i
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 0 0
\(789\) 16.5000 9.52628i 0.587416 0.339145i
\(790\) 0 0
\(791\) 0.500000 2.59808i 0.0177780 0.0923770i
\(792\) 0 0
\(793\) −1.00000 1.73205i −0.0355110 0.0615069i
\(794\) 0 0
\(795\) −13.5000 + 7.79423i −0.478796 + 0.276433i
\(796\) 0 0
\(797\) 19.5000 + 33.7750i 0.690725 + 1.19637i 0.971601 + 0.236627i \(0.0760420\pi\)
−0.280875 + 0.959744i \(0.590625\pi\)
\(798\) 0 0
\(799\) 12.0000 20.7846i 0.424529 0.735307i
\(800\) 0 0
\(801\) −13.5000 23.3827i −0.476999 0.826187i
\(802\) 0 0
\(803\) −39.0000 −1.37628
\(804\) 0 0
\(805\) 0.500000 2.59808i 0.0176227 0.0915702i
\(806\) 0 0
\(807\) −10.5000 6.06218i −0.369618 0.213399i
\(808\) 0 0
\(809\) −27.5000 + 47.6314i −0.966849 + 1.67463i −0.262284 + 0.964991i \(0.584476\pi\)
−0.704564 + 0.709640i \(0.748858\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 46.5000 26.8468i 1.63083 0.941558i
\(814\) 0 0
\(815\) −17.0000 −0.595484
\(816\) 0 0
\(817\) 15.0000 0.524784
\(818\) 0 0
\(819\) 1.50000 7.79423i 0.0524142 0.272352i
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) −18.0000 10.3923i −0.626680 0.361814i
\(826\) 0 0
\(827\) 40.0000 1.39094 0.695468 0.718557i \(-0.255197\pi\)
0.695468 + 0.718557i \(0.255197\pi\)
\(828\) 0 0
\(829\) −2.50000 + 4.33013i −0.0868286 + 0.150392i −0.906169 0.422916i \(-0.861007\pi\)
0.819340 + 0.573307i \(0.194340\pi\)
\(830\) 0 0
\(831\) 28.5000 16.4545i 0.988654 0.570800i
\(832\) 0 0
\(833\) −19.5000 7.79423i −0.675635 0.270054i
\(834\) 0 0
\(835\) 1.00000 0.0346064
\(836\) 0 0
\(837\) 20.7846i 0.718421i
\(838\) 0 0
\(839\) 22.5000 38.9711i 0.776786 1.34543i −0.156999 0.987599i \(-0.550182\pi\)
0.933785 0.357834i \(-0.116485\pi\)
\(840\) 0 0
\(841\) −26.0000 45.0333i −0.896552 1.55287i
\(842\) 0 0
\(843\) 40.5000 + 23.3827i 1.39489 + 0.805342i
\(844\) 0 0
\(845\) 6.00000 + 10.3923i 0.206406 + 0.357506i
\(846\) 0 0
\(847\) −5.00000 + 1.73205i −0.171802 + 0.0595140i
\(848\) 0 0
\(849\) 41.5692i 1.42665i
\(850\) 0 0
\(851\) −5.00000 −0.171398
\(852\) 0 0
\(853\) −14.5000 + 25.1147i −0.496471 + 0.859912i −0.999992 0.00407068i \(-0.998704\pi\)
0.503521 + 0.863983i \(0.332038\pi\)
\(854\) 0 0
\(855\) 15.0000 0.512989
\(856\) 0 0
\(857\) −3.50000 6.06218i −0.119558 0.207080i 0.800035 0.599954i \(-0.204814\pi\)
−0.919592 + 0.392874i \(0.871481\pi\)
\(858\) 0 0
\(859\) 9.50000 16.4545i 0.324136 0.561420i −0.657201 0.753715i \(-0.728260\pi\)
0.981337 + 0.192295i \(0.0615932\pi\)
\(860\) 0 0
\(861\) 31.5000 + 6.06218i 1.07352 + 0.206598i
\(862\) 0 0
\(863\) 5.50000 + 9.52628i 0.187222 + 0.324278i 0.944323 0.329020i \(-0.106718\pi\)
−0.757101 + 0.653298i \(0.773385\pi\)
\(864\) 0 0
\(865\) 1.00000 1.73205i 0.0340010 0.0588915i
\(866\) 0 0
\(867\) 12.0000 6.92820i 0.407541 0.235294i
\(868\) 0 0
\(869\) 12.0000 + 20.7846i 0.407072 + 0.705070i
\(870\) 0 0
\(871\) 6.00000 + 10.3923i 0.203302 + 0.352130i
\(872\) 0 0
\(873\) −25.5000 + 44.1673i −0.863044 + 1.49484i
\(874\) 0 0
\(875\) 18.0000 + 15.5885i 0.608511 + 0.526986i
\(876\) 0 0
\(877\) 11.5000 19.9186i 0.388327 0.672603i −0.603897 0.797062i \(-0.706386\pi\)
0.992225 + 0.124459i \(0.0397196\pi\)
\(878\) 0 0
\(879\) −1.50000 + 0.866025i −0.0505937 + 0.0292103i
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 6.00000 + 3.46410i 0.201688 + 0.116445i
\(886\) 0 0
\(887\) −13.5000 + 23.3827i −0.453286 + 0.785114i −0.998588 0.0531258i \(-0.983082\pi\)
0.545302 + 0.838240i \(0.316415\pi\)
\(888\) 0 0
\(889\) 16.0000 + 13.8564i 0.536623 + 0.464729i
\(890\) 0 0
\(891\) −13.5000 23.3827i −0.452267 0.783349i
\(892\) 0 0
\(893\) −20.0000 34.6410i −0.669274 1.15922i
\(894\) 0 0
\(895\) −10.5000 18.1865i −0.350976 0.607909i
\(896\) 0 0
\(897\) 1.73205i 0.0578315i
\(898\) 0 0
\(899\) 18.0000 31.1769i 0.600334 1.03981i
\(900\) 0 0
\(901\) −13.5000 23.3827i −0.449750 0.778990i
\(902\) 0 0
\(903\) 4.50000 + 12.9904i 0.149751 + 0.432293i
\(904\) 0 0
\(905\) 9.00000 15.5885i 0.299170 0.518178i
\(906\) 0 0
\(907\) −3.50000 6.06218i −0.116216 0.201291i 0.802049 0.597258i \(-0.203743\pi\)
−0.918265 + 0.395966i \(0.870410\pi\)
\(908\) 0 0
\(909\) −10.5000 18.1865i −0.348263 0.603209i
\(910\) 0 0
\(911\) −13.5000 + 23.3827i −0.447275 + 0.774703i −0.998208 0.0598468i \(-0.980939\pi\)
0.550933 + 0.834550i \(0.314272\pi\)
\(912\) 0 0
\(913\) 39.0000 1.29071
\(914\) 0 0
\(915\) 3.00000 + 1.73205i 0.0991769 + 0.0572598i
\(916\) 0 0
\(917\) −7.50000 + 2.59808i −0.247672 + 0.0857960i
\(918\) 0 0
\(919\) −6.50000 11.2583i −0.214415 0.371378i 0.738676 0.674060i \(-0.235451\pi\)
−0.953092 + 0.302682i \(0.902118\pi\)
\(920\) 0 0
\(921\) 6.92820i 0.228292i
\(922\) 0 0
\(923\) 4.00000 + 6.92820i 0.131662 + 0.228045i
\(924\) 0 0
\(925\) 10.0000 17.3205i 0.328798 0.569495i
\(926\) 0 0
\(927\) −13.5000 + 23.3827i −0.443398 + 0.767988i
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −27.5000 + 21.6506i −0.901276 + 0.709571i
\(932\) 0 0
\(933\) 41.5692i 1.36092i
\(934\) 0 0
\(935\) −4.50000 + 7.79423i −0.147166 + 0.254899i
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 10.3923i 0.339140i
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) −7.00000 −0.227951
\(944\) 0 0
\(945\) 4.50000 + 12.9904i 0.146385 + 0.422577i
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −13.0000 −0.421998
\(950\) 0 0
\(951\) 24.2487i 0.786318i
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 2.00000 3.46410i 0.0647185 0.112096i
\(956\) 0 0
\(957\) 46.7654i 1.51171i
\(958\) 0 0
\(959\) −5.50000 + 28.5788i −0.177604 + 0.922859i
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 10.5000 + 18.1865i 0.338358 + 0.586053i
\(964\) 0 0
\(965\) 9.00000 15.5885i 0.289720 0.501810i
\(966\) 0 0
\(967\) 13.5000 + 23.3827i 0.434131 + 0.751936i 0.997224 0.0744567i \(-0.0237223\pi\)
−0.563094 + 0.826393i \(0.690389\pi\)
\(968\) 0 0
\(969\) 25.9808i 0.834622i
\(970\) 0 0
\(971\) −21.5000 37.2391i −0.689968 1.19506i −0.971848 0.235610i \(-0.924291\pi\)
0.281880 0.959450i \(-0.409042\pi\)
\(972\) 0 0
\(973\) 2.50000 12.9904i 0.0801463 0.416452i
\(974\) 0 0
\(975\) −6.00000 3.46410i −0.192154 0.110940i
\(976\) 0 0
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 0 0
\(979\) 13.5000 23.3827i 0.431462 0.747314i
\(980\) 0 0
\(981\) 7.50000 12.9904i 0.239457 0.414751i
\(982\) 0 0
\(983\) −20.5000 35.5070i −0.653848 1.13250i −0.982181 0.187937i \(-0.939820\pi\)
0.328333 0.944562i \(-0.393513\pi\)
\(984\) 0 0
\(985\) 11.0000 19.0526i 0.350489 0.607065i
\(986\) 0 0
\(987\) 24.0000 27.7128i 0.763928 0.882109i
\(988\) 0 0
\(989\) −1.50000 2.59808i −0.0476972 0.0826140i
\(990\) 0 0
\(991\) −7.50000 + 12.9904i −0.238245 + 0.412653i −0.960211 0.279276i \(-0.909906\pi\)
0.721966 + 0.691929i \(0.243239\pi\)
\(992\) 0 0
\(993\) 6.92820i 0.219860i
\(994\) 0 0
\(995\) 10.5000 + 18.1865i 0.332872 + 0.576552i
\(996\) 0 0
\(997\) −6.50000 11.2583i −0.205857 0.356555i 0.744548 0.667568i \(-0.232665\pi\)
−0.950405 + 0.311014i \(0.899332\pi\)
\(998\) 0 0
\(999\) 22.5000 12.9904i 0.711868 0.410997i
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 1008.2.q.b.529.1 2
3.2 odd 2 3024.2.q.d.2881.1 2
4.3 odd 2 504.2.q.a.25.1 2
7.2 even 3 1008.2.t.e.961.1 2
9.4 even 3 1008.2.t.e.193.1 2
9.5 odd 6 3024.2.t.c.1873.1 2
12.11 even 2 1512.2.q.b.1369.1 2
21.2 odd 6 3024.2.t.c.289.1 2
28.23 odd 6 504.2.t.a.457.1 yes 2
36.23 even 6 1512.2.t.a.361.1 2
36.31 odd 6 504.2.t.a.193.1 yes 2
63.23 odd 6 3024.2.q.d.2305.1 2
63.58 even 3 inner 1008.2.q.b.625.1 2
84.23 even 6 1512.2.t.a.289.1 2
252.23 even 6 1512.2.q.b.793.1 2
252.247 odd 6 504.2.q.a.121.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.a.25.1 2 4.3 odd 2
504.2.q.a.121.1 yes 2 252.247 odd 6
504.2.t.a.193.1 yes 2 36.31 odd 6
504.2.t.a.457.1 yes 2 28.23 odd 6
1008.2.q.b.529.1 2 1.1 even 1 trivial
1008.2.q.b.625.1 2 63.58 even 3 inner
1008.2.t.e.193.1 2 9.4 even 3
1008.2.t.e.961.1 2 7.2 even 3
1512.2.q.b.793.1 2 252.23 even 6
1512.2.q.b.1369.1 2 12.11 even 2
1512.2.t.a.289.1 2 84.23 even 6
1512.2.t.a.361.1 2 36.23 even 6
3024.2.q.d.2305.1 2 63.23 odd 6
3024.2.q.d.2881.1 2 3.2 odd 2
3024.2.t.c.289.1 2 21.2 odd 6
3024.2.t.c.1873.1 2 9.5 odd 6