Properties

Label 1176.2.q.b.961.1
Level $1176$
Weight $2$
Character 1176.961
Analytic conductor $9.390$
Analytic rank $0$
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] = [1176,2,Mod(361,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.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: \(9.39040727770\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1176.961
Dual form 1176.2.q.b.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{9} -2.00000 q^{13} +2.00000 q^{15} +(-3.00000 - 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +6.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(5.00000 - 8.66025i) q^{37} +(1.00000 + 1.73205i) q^{39} -10.0000 q^{41} +12.0000 q^{43} +(-1.00000 - 1.73205i) q^{45} +(4.00000 - 6.92820i) q^{47} +(-3.00000 + 5.19615i) q^{51} +(-3.00000 - 5.19615i) q^{53} -4.00000 q^{57} +(-2.00000 - 3.46410i) q^{59} +(5.00000 - 8.66025i) q^{61} +(2.00000 - 3.46410i) q^{65} +(-6.00000 - 10.3923i) q^{67} -4.00000 q^{69} +4.00000 q^{71} +(-1.00000 - 1.73205i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} +4.00000 q^{83} +12.0000 q^{85} +(-3.00000 - 5.19615i) q^{87} +(-3.00000 + 5.19615i) q^{89} +(4.00000 - 6.92820i) q^{93} +(4.00000 + 6.92820i) q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{5} - q^{9} - 4 q^{13} + 4 q^{15} - 6 q^{17} + 4 q^{19} + 4 q^{23} + q^{25} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 10 q^{37} + 2 q^{39} - 20 q^{41} + 24 q^{43} - 2 q^{45} + 8 q^{47} - 6 q^{51} - 6 q^{53} - 8 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} - 12 q^{67} - 8 q^{69} + 8 q^{71} - 2 q^{73} + q^{75} - 8 q^{79} - q^{81} + 8 q^{83} + 24 q^{85} - 6 q^{87} - 6 q^{89} + 8 q^{93} + 8 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −0.500000 0.866025i −0.288675 0.500000i
\(4\) 0 0
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −3.00000 5.19615i −0.727607 1.26025i −0.957892 0.287129i \(-0.907299\pi\)
0.230285 0.973123i \(-0.426034\pi\)
\(18\) 0 0
\(19\) 2.00000 3.46410i 0.458831 0.794719i −0.540068 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 3.46410i 0.417029 0.722315i −0.578610 0.815604i \(-0.696405\pi\)
0.995639 + 0.0932891i \(0.0297381\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 + 6.92820i 0.718421 + 1.24434i 0.961625 + 0.274367i \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 8.66025i 0.821995 1.42374i −0.0821995 0.996616i \(-0.526194\pi\)
0.904194 0.427121i \(-0.140472\pi\)
\(38\) 0 0
\(39\) 1.00000 + 1.73205i 0.160128 + 0.277350i
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) −1.00000 1.73205i −0.149071 0.258199i
\(46\) 0 0
\(47\) 4.00000 6.92820i 0.583460 1.01058i −0.411606 0.911362i \(-0.635032\pi\)
0.995066 0.0992202i \(-0.0316348\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 0 0
\(53\) −3.00000 5.19615i −0.412082 0.713746i 0.583036 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −2.00000 3.46410i −0.260378 0.450988i 0.705965 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 0 0
\(61\) 5.00000 8.66025i 0.640184 1.10883i −0.345207 0.938527i \(-0.612191\pi\)
0.985391 0.170305i \(-0.0544754\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 3.46410i 0.248069 0.429669i
\(66\) 0 0
\(67\) −6.00000 10.3923i −0.733017 1.26962i −0.955588 0.294706i \(-0.904778\pi\)
0.222571 0.974916i \(-0.428555\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −1.00000 1.73205i −0.117041 0.202721i 0.801553 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 0.500000 0.866025i 0.0577350 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) −3.00000 5.19615i −0.321634 0.557086i
\(88\) 0 0
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 6.92820i 0.414781 0.718421i
\(94\) 0 0
\(95\) 4.00000 + 6.92820i 0.410391 + 0.710819i
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.00000 8.66025i −0.497519 0.861727i 0.502477 0.864590i \(-0.332422\pi\)
−0.999996 + 0.00286291i \(0.999089\pi\)
\(102\) 0 0
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.00000 + 13.8564i −0.773389 + 1.33955i 0.162306 + 0.986740i \(0.448107\pi\)
−0.935695 + 0.352809i \(0.885227\pi\)
\(108\) 0 0
\(109\) −7.00000 12.1244i −0.670478 1.16130i −0.977769 0.209687i \(-0.932756\pi\)
0.307290 0.951616i \(-0.400578\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 4.00000 + 6.92820i 0.373002 + 0.646058i
\(116\) 0 0
\(117\) 1.00000 1.73205i 0.0924500 0.160128i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) 5.00000 + 8.66025i 0.450835 + 0.780869i
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −6.00000 10.3923i −0.528271 0.914991i
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.00000 + 1.73205i −0.0860663 + 0.149071i
\(136\) 0 0
\(137\) 7.00000 + 12.1244i 0.598050 + 1.03585i 0.993109 + 0.117198i \(0.0373911\pi\)
−0.395058 + 0.918656i \(0.629276\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 + 10.3923i −0.498273 + 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 + 5.19615i −0.245770 + 0.425685i −0.962348 0.271821i \(-0.912374\pi\)
0.716578 + 0.697507i \(0.245707\pi\)
\(150\) 0 0
\(151\) 4.00000 + 6.92820i 0.325515 + 0.563809i 0.981617 0.190864i \(-0.0611289\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) 5.00000 + 8.66025i 0.399043 + 0.691164i 0.993608 0.112884i \(-0.0360089\pi\)
−0.594565 + 0.804048i \(0.702676\pi\)
\(158\) 0 0
\(159\) −3.00000 + 5.19615i −0.237915 + 0.412082i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 3.46410i 0.156652 0.271329i −0.777007 0.629492i \(-0.783263\pi\)
0.933659 + 0.358162i \(0.116597\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 + 3.46410i 0.152944 + 0.264906i
\(172\) 0 0
\(173\) −1.00000 + 1.73205i −0.0760286 + 0.131685i −0.901533 0.432710i \(-0.857557\pi\)
0.825505 + 0.564396i \(0.190891\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.00000 + 3.46410i −0.150329 + 0.260378i
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 10.0000 + 17.3205i 0.735215 + 1.27343i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.00000 3.46410i 0.144715 0.250654i −0.784552 0.620063i \(-0.787107\pi\)
0.929267 + 0.369410i \(0.120440\pi\)
\(192\) 0 0
\(193\) −9.00000 15.5885i −0.647834 1.12208i −0.983639 0.180150i \(-0.942342\pi\)
0.335805 0.941932i \(-0.390992\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 12.0000 + 20.7846i 0.850657 + 1.47338i 0.880616 + 0.473831i \(0.157129\pi\)
−0.0299585 + 0.999551i \(0.509538\pi\)
\(200\) 0 0
\(201\) −6.00000 + 10.3923i −0.423207 + 0.733017i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 17.3205i 0.698430 1.20972i
\(206\) 0 0
\(207\) 2.00000 + 3.46410i 0.139010 + 0.240772i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) −2.00000 3.46410i −0.137038 0.237356i
\(214\) 0 0
\(215\) −12.0000 + 20.7846i −0.818393 + 1.41750i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.00000 + 1.73205i −0.0675737 + 0.117041i
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −6.00000 10.3923i −0.398234 0.689761i 0.595274 0.803523i \(-0.297043\pi\)
−0.993508 + 0.113761i \(0.963710\pi\)
\(228\) 0 0
\(229\) −3.00000 + 5.19615i −0.198246 + 0.343371i −0.947960 0.318390i \(-0.896858\pi\)
0.749714 + 0.661762i \(0.230191\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.00000 + 15.5885i −0.589610 + 1.02123i 0.404674 + 0.914461i \(0.367385\pi\)
−0.994283 + 0.106773i \(0.965948\pi\)
\(234\) 0 0
\(235\) 8.00000 + 13.8564i 0.521862 + 0.903892i
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 11.0000 + 19.0526i 0.708572 + 1.22728i 0.965387 + 0.260822i \(0.0839937\pi\)
−0.256814 + 0.966461i \(0.582673\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 + 6.92820i −0.254514 + 0.440831i
\(248\) 0 0
\(249\) −2.00000 3.46410i −0.126745 0.219529i
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −6.00000 10.3923i −0.375735 0.650791i
\(256\) 0 0
\(257\) −7.00000 + 12.1244i −0.436648 + 0.756297i −0.997429 0.0716680i \(-0.977168\pi\)
0.560781 + 0.827964i \(0.310501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 + 5.19615i −0.185695 + 0.321634i
\(262\) 0 0
\(263\) −14.0000 24.2487i −0.863277 1.49524i −0.868748 0.495255i \(-0.835075\pi\)
0.00547092 0.999985i \(-0.498259\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −5.00000 8.66025i −0.304855 0.528025i 0.672374 0.740212i \(-0.265275\pi\)
−0.977229 + 0.212187i \(0.931941\pi\)
\(270\) 0 0
\(271\) 4.00000 6.92820i 0.242983 0.420858i −0.718580 0.695444i \(-0.755208\pi\)
0.961563 + 0.274586i \(0.0885408\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i \(-0.0695395\pi\)
−0.675810 + 0.737075i \(0.736206\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 6.00000 + 10.3923i 0.356663 + 0.617758i 0.987401 0.158237i \(-0.0505811\pi\)
−0.630738 + 0.775996i \(0.717248\pi\)
\(284\) 0 0
\(285\) 4.00000 6.92820i 0.236940 0.410391i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) −5.00000 8.66025i −0.293105 0.507673i
\(292\) 0 0
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 + 6.92820i −0.231326 + 0.400668i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −5.00000 + 8.66025i −0.287242 + 0.497519i
\(304\) 0 0
\(305\) 10.0000 + 17.3205i 0.572598 + 0.991769i
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 3.00000 5.19615i 0.169570 0.293704i −0.768699 0.639611i \(-0.779095\pi\)
0.938269 + 0.345907i \(0.112429\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.00000 1.73205i 0.0561656 0.0972817i −0.836576 0.547852i \(-0.815446\pi\)
0.892741 + 0.450570i \(0.148779\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) −1.00000 1.73205i −0.0554700 0.0960769i
\(326\) 0 0
\(327\) −7.00000 + 12.1244i −0.387101 + 0.670478i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 17.3205i 0.549650 0.952021i −0.448649 0.893708i \(-0.648095\pi\)
0.998298 0.0583130i \(-0.0185721\pi\)
\(332\) 0 0
\(333\) 5.00000 + 8.66025i 0.273998 + 0.474579i
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) 7.00000 + 12.1244i 0.380188 + 0.658505i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 6.92820i 0.215353 0.373002i
\(346\) 0 0
\(347\) 4.00000 + 6.92820i 0.214731 + 0.371925i 0.953189 0.302374i \(-0.0977791\pi\)
−0.738458 + 0.674299i \(0.764446\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 5.00000 + 8.66025i 0.266123 + 0.460939i 0.967857 0.251500i \(-0.0809239\pi\)
−0.701734 + 0.712439i \(0.747591\pi\)
\(354\) 0 0
\(355\) −4.00000 + 6.92820i −0.212298 + 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 10.3923i 0.316668 0.548485i −0.663123 0.748511i \(-0.730769\pi\)
0.979791 + 0.200026i \(0.0641026\pi\)
\(360\) 0 0
\(361\) 1.50000 + 2.59808i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −16.0000 27.7128i −0.835193 1.44660i −0.893873 0.448320i \(-0.852022\pi\)
0.0586798 0.998277i \(-0.481311\pi\)
\(368\) 0 0
\(369\) 5.00000 8.66025i 0.260290 0.450835i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 + 19.0526i −0.569558 + 0.986504i 0.427051 + 0.904227i \(0.359552\pi\)
−0.996610 + 0.0822766i \(0.973781\pi\)
\(374\) 0 0
\(375\) 6.00000 + 10.3923i 0.309839 + 0.536656i
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −4.00000 6.92820i −0.204926 0.354943i
\(382\) 0 0
\(383\) 8.00000 13.8564i 0.408781 0.708029i −0.585973 0.810331i \(-0.699287\pi\)
0.994753 + 0.102302i \(0.0326207\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.00000 + 10.3923i −0.304997 + 0.528271i
\(388\) 0 0
\(389\) −7.00000 12.1244i −0.354914 0.614729i 0.632189 0.774814i \(-0.282157\pi\)
−0.987103 + 0.160085i \(0.948823\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) −8.00000 13.8564i −0.402524 0.697191i
\(396\) 0 0
\(397\) 13.0000 22.5167i 0.652451 1.13008i −0.330075 0.943955i \(-0.607074\pi\)
0.982526 0.186124i \(-0.0595926\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.00000 + 8.66025i −0.249688 + 0.432472i −0.963439 0.267927i \(-0.913661\pi\)
0.713751 + 0.700399i \(0.246995\pi\)
\(402\) 0 0
\(403\) −8.00000 13.8564i −0.398508 0.690237i
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.00000 + 12.1244i 0.346128 + 0.599511i 0.985558 0.169338i \(-0.0541630\pi\)
−0.639430 + 0.768849i \(0.720830\pi\)
\(410\) 0 0
\(411\) 7.00000 12.1244i 0.345285 0.598050i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 + 6.92820i −0.196352 + 0.340092i
\(416\) 0 0
\(417\) 2.00000 + 3.46410i 0.0979404 + 0.169638i
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 4.00000 + 6.92820i 0.194487 + 0.336861i
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 10.3923i −0.289010 0.500580i 0.684564 0.728953i \(-0.259993\pi\)
−0.973574 + 0.228373i \(0.926659\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) −8.00000 13.8564i −0.382692 0.662842i
\(438\) 0 0
\(439\) −12.0000 + 20.7846i −0.572729 + 0.991995i 0.423556 + 0.905870i \(0.360782\pi\)
−0.996284 + 0.0861252i \(0.972552\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000 34.6410i 0.950229 1.64584i 0.205301 0.978699i \(-0.434183\pi\)
0.744927 0.667146i \(-0.232484\pi\)
\(444\) 0 0
\(445\) −6.00000 10.3923i −0.284427 0.492642i
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.00000 6.92820i 0.187936 0.325515i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.00000 + 8.66025i −0.233890 + 0.405110i −0.958950 0.283577i \(-0.908479\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) −3.00000 5.19615i −0.140028 0.242536i
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 8.00000 + 13.8564i 0.370991 + 0.642575i
\(466\) 0 0
\(467\) 2.00000 3.46410i 0.0925490 0.160300i −0.816034 0.578004i \(-0.803832\pi\)
0.908583 + 0.417704i \(0.137165\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.00000 8.66025i 0.230388 0.399043i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 12.0000 + 20.7846i 0.548294 + 0.949673i 0.998392 + 0.0566937i \(0.0180558\pi\)
−0.450098 + 0.892979i \(0.648611\pi\)
\(480\) 0 0
\(481\) −10.0000 + 17.3205i −0.455961 + 0.789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 + 17.3205i −0.454077 + 0.786484i
\(486\) 0 0
\(487\) 4.00000 + 6.92820i 0.181257 + 0.313947i 0.942309 0.334744i \(-0.108650\pi\)
−0.761052 + 0.648691i \(0.775317\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) −18.0000 31.1769i −0.810679 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.00000 + 3.46410i −0.0895323 + 0.155074i −0.907314 0.420455i \(-0.861871\pi\)
0.817781 + 0.575529i \(0.195204\pi\)
\(500\) 0 0
\(501\) −8.00000 13.8564i −0.357414 0.619059i
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 4.50000 + 7.79423i 0.199852 + 0.346154i
\(508\) 0 0
\(509\) 3.00000 5.19615i 0.132973 0.230315i −0.791849 0.610718i \(-0.790881\pi\)
0.924821 + 0.380402i \(0.124214\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2.00000 3.46410i 0.0883022 0.152944i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 9.00000 + 15.5885i 0.394297 + 0.682943i 0.993011 0.118020i \(-0.0376547\pi\)
−0.598714 + 0.800963i \(0.704321\pi\)
\(522\) 0 0
\(523\) −6.00000 + 10.3923i −0.262362 + 0.454424i −0.966869 0.255273i \(-0.917835\pi\)
0.704507 + 0.709697i \(0.251168\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 41.5692i 1.04546 1.81078i
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) −16.0000 27.7128i −0.691740 1.19813i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 1.73205i 0.0429934 0.0744667i −0.843728 0.536771i \(-0.819644\pi\)
0.886721 + 0.462304i \(0.152977\pi\)
\(542\) 0 0
\(543\) −3.00000 5.19615i −0.128742 0.222988i
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 5.00000 + 8.66025i 0.213395 + 0.369611i
\(550\) 0 0
\(551\) 12.0000 20.7846i 0.511217 0.885454i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.0000 17.3205i 0.424476 0.735215i
\(556\) 0 0
\(557\) 9.00000 + 15.5885i 0.381342 + 0.660504i 0.991254 0.131965i \(-0.0421286\pi\)
−0.609912 + 0.792469i \(0.708795\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.00000 10.3923i −0.252870 0.437983i 0.711445 0.702742i \(-0.248041\pi\)
−0.964315 + 0.264758i \(0.914708\pi\)
\(564\) 0 0
\(565\) 14.0000 24.2487i 0.588984 1.02015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.00000 + 15.5885i −0.377300 + 0.653502i −0.990668 0.136295i \(-0.956481\pi\)
0.613369 + 0.789797i \(0.289814\pi\)
\(570\) 0 0
\(571\) −2.00000 3.46410i −0.0836974 0.144968i 0.821138 0.570730i \(-0.193340\pi\)
−0.904835 + 0.425762i \(0.860006\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −9.00000 15.5885i −0.374675 0.648956i 0.615603 0.788056i \(-0.288912\pi\)
−0.990278 + 0.139100i \(0.955579\pi\)
\(578\) 0 0
\(579\) −9.00000 + 15.5885i −0.374027 + 0.647834i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 2.00000 + 3.46410i 0.0826898 + 0.143223i
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) −11.0000 19.0526i −0.452480 0.783718i
\(592\) 0 0
\(593\) −3.00000 + 5.19615i −0.123195 + 0.213380i −0.921026 0.389501i \(-0.872647\pi\)
0.797831 + 0.602881i \(0.205981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.0000 20.7846i 0.491127 0.850657i
\(598\) 0 0
\(599\) −18.0000 31.1769i −0.735460 1.27385i −0.954521 0.298143i \(-0.903633\pi\)
0.219061 0.975711i \(-0.429701\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 11.0000 + 19.0526i 0.447214 + 0.774597i
\(606\) 0 0
\(607\) 16.0000 27.7128i 0.649420 1.12483i −0.333842 0.942629i \(-0.608345\pi\)
0.983262 0.182199i \(-0.0583216\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 + 13.8564i −0.323645 + 0.560570i
\(612\) 0 0
\(613\) 5.00000 + 8.66025i 0.201948 + 0.349784i 0.949156 0.314806i \(-0.101939\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −6.00000 10.3923i −0.241160 0.417702i 0.719885 0.694094i \(-0.244195\pi\)
−0.961045 + 0.276392i \(0.910861\pi\)
\(620\) 0 0
\(621\) 2.00000 3.46410i 0.0802572 0.139010i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 6.00000 + 10.3923i 0.238479 + 0.413057i
\(634\) 0 0
\(635\) −8.00000 + 13.8564i −0.317470 + 0.549875i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.00000 + 3.46410i −0.0791188 + 0.137038i
\(640\) 0 0
\(641\) −13.0000 22.5167i −0.513469 0.889355i −0.999878 0.0156233i \(-0.995027\pi\)
0.486409 0.873731i \(-0.338307\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) 16.0000 + 27.7128i 0.629025 + 1.08950i 0.987748 + 0.156059i \(0.0498790\pi\)
−0.358723 + 0.933444i \(0.616788\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.00000 + 5.19615i −0.117399 + 0.203341i −0.918736 0.394872i \(-0.870789\pi\)
0.801337 + 0.598213i \(0.204122\pi\)
\(654\) 0 0
\(655\) 12.0000 + 20.7846i 0.468879 + 0.812122i
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −7.00000 12.1244i −0.272268 0.471583i 0.697174 0.716902i \(-0.254441\pi\)
−0.969442 + 0.245319i \(0.921107\pi\)
\(662\) 0 0
\(663\) 6.00000 10.3923i 0.233021 0.403604i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000 20.7846i 0.464642 0.804783i
\(668\) 0 0
\(669\) −8.00000 13.8564i −0.309298 0.535720i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) 0.500000 + 0.866025i 0.0192450 + 0.0333333i
\(676\) 0 0
\(677\) 19.0000 32.9090i 0.730229 1.26479i −0.226556 0.973998i \(-0.572747\pi\)
0.956785 0.290796i \(-0.0939201\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.00000 + 10.3923i −0.229920 + 0.398234i
\(682\) 0 0
\(683\) 20.0000 + 34.6410i 0.765279 + 1.32550i 0.940099 + 0.340901i \(0.110732\pi\)
−0.174820 + 0.984600i \(0.555934\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) 6.00000 + 10.3923i 0.228582 + 0.395915i
\(690\) 0 0
\(691\) −18.0000 + 31.1769i −0.684752 + 1.18603i 0.288762 + 0.957401i \(0.406756\pi\)
−0.973515 + 0.228625i \(0.926577\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 6.92820i 0.151729 0.262802i
\(696\) 0 0
\(697\) 30.0000 + 51.9615i 1.13633 + 1.96818i
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) −20.0000 34.6410i −0.754314 1.30651i
\(704\) 0 0
\(705\) 8.00000 13.8564i 0.301297 0.521862i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.00000 + 5.19615i −0.112667 + 0.195146i −0.916845 0.399244i \(-0.869273\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(710\) 0 0
\(711\) −4.00000 6.92820i −0.150012 0.259828i
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000 + 10.3923i 0.224074 + 0.388108i
\(718\) 0 0
\(719\) 8.00000 13.8564i 0.298350 0.516757i −0.677409 0.735607i \(-0.736897\pi\)
0.975759 + 0.218850i \(0.0702305\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.0000 19.0526i 0.409094 0.708572i
\(724\) 0 0
\(725\) 3.00000 + 5.19615i 0.111417 + 0.192980i
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 62.3538i −1.33151 2.30624i
\(732\) 0 0
\(733\) −7.00000 + 12.1244i −0.258551 + 0.447823i −0.965854 0.259087i \(-0.916578\pi\)
0.707303 + 0.706910i \(0.249912\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −14.0000 24.2487i −0.514998 0.892003i −0.999849 0.0174060i \(-0.994459\pi\)
0.484850 0.874597i \(-0.338874\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 52.0000 1.90769 0.953847 0.300291i \(-0.0970839\pi\)
0.953847 + 0.300291i \(0.0970839\pi\)
\(744\) 0 0
\(745\) −6.00000 10.3923i −0.219823 0.380745i
\(746\) 0 0
\(747\) −2.00000 + 3.46410i −0.0731762 + 0.126745i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) −10.0000 17.3205i −0.364420 0.631194i
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 + 5.19615i −0.108750 + 0.188360i −0.915264 0.402854i \(-0.868018\pi\)
0.806514 + 0.591215i \(0.201351\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 + 10.3923i −0.216930 + 0.375735i
\(766\) 0 0
\(767\) 4.00000 + 6.92820i 0.144432 + 0.250163i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 19.0000 + 32.9090i 0.683383 + 1.18365i 0.973942 + 0.226796i \(0.0728252\pi\)
−0.290560 + 0.956857i \(0.593841\pi\)
\(774\) 0 0
\(775\) −4.00000 + 6.92820i −0.143684 + 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0000 + 34.6410i −0.716574 + 1.24114i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) −10.0000 17.3205i −0.356462 0.617409i 0.630905 0.775860i \(-0.282684\pi\)
−0.987367 + 0.158450i \(0.949350\pi\)
\(788\) 0 0
\(789\) −14.0000 + 24.2487i −0.498413 + 0.863277i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 + 17.3205i −0.355110 + 0.615069i
\(794\) 0 0
\(795\) −6.00000 10.3923i −0.212798 0.368577i
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) −3.00000 5.19615i −0.106000 0.183597i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.00000 + 8.66025i −0.176008 + 0.304855i
\(808\) 0 0
\(809\) −13.0000 22.5167i −0.457056 0.791644i 0.541748 0.840541i \(-0.317763\pi\)
−0.998804 + 0.0488972i \(0.984429\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 4.00000 + 6.92820i 0.140114 + 0.242684i
\(816\) 0 0
\(817\) 24.0000 41.5692i 0.839654 1.45432i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.0000 36.3731i 0.732905 1.26943i −0.222731 0.974880i \(-0.571497\pi\)
0.955636 0.294549i \(-0.0951694\pi\)
\(822\) 0 0
\(823\) −16.0000 27.7128i −0.557725 0.966008i −0.997686 0.0679910i \(-0.978341\pi\)
0.439961 0.898017i \(-0.354992\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) −7.00000 12.1244i −0.243120 0.421096i 0.718481 0.695546i \(-0.244838\pi\)
−0.961601 + 0.274450i \(0.911504\pi\)
\(830\) 0 0
\(831\) 5.00000 8.66025i 0.173448 0.300421i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −16.0000 + 27.7128i −0.553703 + 0.959041i
\(836\) 0 0
\(837\) 4.00000 + 6.92820i 0.138260 + 0.239474i
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 15.0000 + 25.9808i 0.516627 + 0.894825i
\(844\) 0 0
\(845\) 9.00000 15.5885i 0.309609 0.536259i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 6.00000 10.3923i 0.205919 0.356663i
\(850\) 0 0
\(851\) −20.0000 34.6410i −0.685591 1.18748i
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −19.0000 32.9090i −0.649028 1.12415i −0.983355 0.181692i \(-0.941843\pi\)
0.334328 0.942457i \(-0.391491\pi\)
\(858\) 0 0
\(859\) 22.0000 38.1051i 0.750630 1.30013i −0.196887 0.980426i \(-0.563083\pi\)
0.947518 0.319704i \(-0.103583\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.00000 + 3.46410i −0.0680808 + 0.117919i −0.898056 0.439880i \(-0.855021\pi\)
0.829976 + 0.557800i \(0.188354\pi\)
\(864\) 0 0
\(865\) −2.00000 3.46410i −0.0680020 0.117783i
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 12.0000 + 20.7846i 0.406604 + 0.704260i
\(872\) 0 0
\(873\) −5.00000 + 8.66025i −0.169224 + 0.293105i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.00000 1.73205i 0.0337676 0.0584872i −0.848648 0.528958i \(-0.822583\pi\)
0.882415 + 0.470471i \(0.155916\pi\)
\(878\) 0 0
\(879\) 11.0000 + 19.0526i 0.371021 + 0.642627i
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) −4.00000 6.92820i −0.134459 0.232889i
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −16.0000 27.7128i −0.535420 0.927374i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) 24.0000 + 41.5692i 0.800445 + 1.38641i
\(900\) 0 0
\(901\) −18.0000 + 31.1769i −0.599667 + 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 + 10.3923i −0.199447 + 0.345452i
\(906\) 0 0
\(907\) 26.0000 + 45.0333i 0.863316 + 1.49531i 0.868710 + 0.495321i \(0.164950\pi\)
−0.00539395 + 0.999985i \(0.501717\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 10.0000 17.3205i 0.330590 0.572598i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.00000 + 6.92820i −0.131948 + 0.228540i −0.924427 0.381358i \(-0.875456\pi\)
0.792480 + 0.609898i \(0.208790\pi\)
\(920\) 0 0
\(921\) 2.00000 + 3.46410i 0.0659022 + 0.114146i
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.0000 + 25.9808i −0.492134 + 0.852401i −0.999959 0.00905914i \(-0.997116\pi\)
0.507825 + 0.861460i \(0.330450\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 11.0000 + 19.0526i 0.358590 + 0.621096i 0.987725 0.156200i \(-0.0499244\pi\)
−0.629136 + 0.777295i \(0.716591\pi\)
\(942\) 0 0
\(943\) −20.0000 + 34.6410i −0.651290 + 1.12807i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 20.7846i 0.389948 0.675409i −0.602494 0.798123i \(-0.705826\pi\)
0.992442 + 0.122714i \(0.0391598\pi\)
\(948\) 0 0
\(949\) 2.00000 + 3.46410i 0.0649227 + 0.112449i
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) 4.00000 + 6.92820i 0.129437 + 0.224191i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) −8.00000 13.8564i −0.257796 0.446516i
\(964\) 0 0
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 12.0000 + 20.7846i 0.385496 + 0.667698i
\(970\) 0 0
\(971\) −14.0000 + 24.2487i −0.449281 + 0.778178i −0.998339 0.0576061i \(-0.981653\pi\)
0.549058 + 0.835784i \(0.314987\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.00000 + 1.73205i −0.0320256 + 0.0554700i
\(976\) 0 0
\(977\) 11.0000 + 19.0526i 0.351921 + 0.609545i 0.986586 0.163242i \(-0.0521952\pi\)
−0.634665 + 0.772787i \(0.718862\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 20.0000 + 34.6410i 0.637901 + 1.10488i 0.985893 + 0.167379i \(0.0535304\pi\)
−0.347992 + 0.937498i \(0.613136\pi\)
\(984\) 0 0
\(985\) −22.0000 + 38.1051i −0.700978 + 1.21413i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 41.5692i 0.763156 1.32182i
\(990\) 0 0
\(991\) 24.0000 + 41.5692i 0.762385 + 1.32049i 0.941618 + 0.336683i \(0.109305\pi\)
−0.179233 + 0.983807i \(0.557362\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) 0 0
\(997\) 1.00000 + 1.73205i 0.0316703 + 0.0548546i 0.881426 0.472322i \(-0.156584\pi\)
−0.849756 + 0.527176i \(0.823251\pi\)
\(998\) 0 0
\(999\) 5.00000 8.66025i 0.158193 0.273998i
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 1176.2.q.b.961.1 2
3.2 odd 2 3528.2.s.v.3313.1 2
4.3 odd 2 2352.2.q.o.961.1 2
7.2 even 3 168.2.a.b.1.1 1
7.3 odd 6 1176.2.q.j.361.1 2
7.4 even 3 inner 1176.2.q.b.361.1 2
7.5 odd 6 1176.2.a.a.1.1 1
7.6 odd 2 1176.2.q.j.961.1 2
21.2 odd 6 504.2.a.b.1.1 1
21.5 even 6 3528.2.a.w.1.1 1
21.11 odd 6 3528.2.s.v.361.1 2
21.17 even 6 3528.2.s.h.361.1 2
21.20 even 2 3528.2.s.h.3313.1 2
28.3 even 6 2352.2.q.j.1537.1 2
28.11 odd 6 2352.2.q.o.1537.1 2
28.19 even 6 2352.2.a.q.1.1 1
28.23 odd 6 336.2.a.c.1.1 1
28.27 even 2 2352.2.q.j.961.1 2
35.2 odd 12 4200.2.t.m.1849.1 2
35.9 even 6 4200.2.a.i.1.1 1
35.23 odd 12 4200.2.t.m.1849.2 2
56.5 odd 6 9408.2.a.cy.1.1 1
56.19 even 6 9408.2.a.bc.1.1 1
56.37 even 6 1344.2.a.c.1.1 1
56.51 odd 6 1344.2.a.n.1.1 1
84.23 even 6 1008.2.a.e.1.1 1
84.47 odd 6 7056.2.a.br.1.1 1
112.37 even 12 5376.2.c.bd.2689.1 2
112.51 odd 12 5376.2.c.f.2689.1 2
112.93 even 12 5376.2.c.bd.2689.2 2
112.107 odd 12 5376.2.c.f.2689.2 2
140.79 odd 6 8400.2.a.bx.1.1 1
168.107 even 6 4032.2.a.bj.1.1 1
168.149 odd 6 4032.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.a.b.1.1 1 7.2 even 3
336.2.a.c.1.1 1 28.23 odd 6
504.2.a.b.1.1 1 21.2 odd 6
1008.2.a.e.1.1 1 84.23 even 6
1176.2.a.a.1.1 1 7.5 odd 6
1176.2.q.b.361.1 2 7.4 even 3 inner
1176.2.q.b.961.1 2 1.1 even 1 trivial
1176.2.q.j.361.1 2 7.3 odd 6
1176.2.q.j.961.1 2 7.6 odd 2
1344.2.a.c.1.1 1 56.37 even 6
1344.2.a.n.1.1 1 56.51 odd 6
2352.2.a.q.1.1 1 28.19 even 6
2352.2.q.j.961.1 2 28.27 even 2
2352.2.q.j.1537.1 2 28.3 even 6
2352.2.q.o.961.1 2 4.3 odd 2
2352.2.q.o.1537.1 2 28.11 odd 6
3528.2.a.w.1.1 1 21.5 even 6
3528.2.s.h.361.1 2 21.17 even 6
3528.2.s.h.3313.1 2 21.20 even 2
3528.2.s.v.361.1 2 21.11 odd 6
3528.2.s.v.3313.1 2 3.2 odd 2
4032.2.a.be.1.1 1 168.149 odd 6
4032.2.a.bj.1.1 1 168.107 even 6
4200.2.a.i.1.1 1 35.9 even 6
4200.2.t.m.1849.1 2 35.2 odd 12
4200.2.t.m.1849.2 2 35.23 odd 12
5376.2.c.f.2689.1 2 112.51 odd 12
5376.2.c.f.2689.2 2 112.107 odd 12
5376.2.c.bd.2689.1 2 112.37 even 12
5376.2.c.bd.2689.2 2 112.93 even 12
7056.2.a.br.1.1 1 84.47 odd 6
8400.2.a.bx.1.1 1 140.79 odd 6
9408.2.a.bc.1.1 1 56.19 even 6
9408.2.a.cy.1.1 1 56.5 odd 6