Properties

Label 980.2.i.g.361.1
Level $980$
Weight $2$
Character 980.361
Analytic conductor $7.825$
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] = [980,2,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("980.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (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: \(7.82533939809\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 980.361
Dual form 980.2.i.g.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{9} +(0.500000 - 0.866025i) q^{11} +5.00000 q^{13} -1.00000 q^{15} +(0.500000 - 0.866025i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.00000 q^{27} +3.00000 q^{29} +(1.00000 - 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{33} +(-4.00000 - 6.92820i) q^{37} +(2.50000 - 4.33013i) q^{39} +10.0000 q^{41} -2.00000 q^{43} +(1.00000 - 1.73205i) q^{45} +(-3.50000 - 6.06218i) q^{47} +(-0.500000 - 0.866025i) q^{51} +(1.00000 - 1.73205i) q^{53} -1.00000 q^{55} -6.00000 q^{57} +(7.00000 - 12.1244i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(-2.50000 - 4.33013i) q^{65} +(-7.00000 + 12.1244i) q^{67} +4.00000 q^{69} +(-5.00000 + 8.66025i) q^{73} +(0.500000 + 0.866025i) q^{75} +(5.50000 + 9.52628i) q^{79} +(-0.500000 + 0.866025i) q^{81} +4.00000 q^{83} -1.00000 q^{85} +(1.50000 - 2.59808i) q^{87} +(2.00000 + 3.46410i) q^{89} +(-1.00000 - 1.73205i) q^{93} +(-3.00000 + 5.19615i) q^{95} +3.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - q^{5} + 2 q^{9} + q^{11} + 10 q^{13} - 2 q^{15} + q^{17} - 6 q^{19} + 4 q^{23} - q^{25} + 10 q^{27} + 6 q^{29} + 2 q^{31} - q^{33} - 8 q^{37} + 5 q^{39} + 20 q^{41} - 4 q^{43} + 2 q^{45} - 7 q^{47} - q^{51} + 2 q^{53} - 2 q^{55} - 12 q^{57} + 14 q^{59} - 8 q^{61} - 5 q^{65} - 14 q^{67} + 8 q^{69} - 10 q^{73} + q^{75} + 11 q^{79} - q^{81} + 8 q^{83} - 2 q^{85} + 3 q^{87} + 4 q^{89} - 2 q^{93} - 6 q^{95} + 6 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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 −0.684819 0.728714i \(-0.740119\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i −0.780750 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.500000 0.866025i 0.121268 0.210042i −0.799000 0.601331i \(-0.794637\pi\)
0.920268 + 0.391289i \(0.127971\pi\)
\(18\) 0 0
\(19\) −3.00000 5.19615i −0.688247 1.19208i −0.972404 0.233301i \(-0.925047\pi\)
0.284157 0.958778i \(-0.408286\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 1.00000 1.73205i 0.179605 0.311086i −0.762140 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) 0 0
\(33\) −0.500000 0.866025i −0.0870388 0.150756i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 6.92820i −0.657596 1.13899i −0.981236 0.192809i \(-0.938240\pi\)
0.323640 0.946180i \(-0.395093\pi\)
\(38\) 0 0
\(39\) 2.50000 4.33013i 0.400320 0.693375i
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 1.00000 1.73205i 0.149071 0.258199i
\(46\) 0 0
\(47\) −3.50000 6.06218i −0.510527 0.884260i −0.999926 0.0121990i \(-0.996117\pi\)
0.489398 0.872060i \(-0.337217\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.500000 0.866025i −0.0700140 0.121268i
\(52\) 0 0
\(53\) 1.00000 1.73205i 0.137361 0.237915i −0.789136 0.614218i \(-0.789471\pi\)
0.926497 + 0.376303i \(0.122805\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 7.00000 12.1244i 0.911322 1.57846i 0.0991242 0.995075i \(-0.468396\pi\)
0.812198 0.583382i \(-0.198271\pi\)
\(60\) 0 0
\(61\) −4.00000 6.92820i −0.512148 0.887066i −0.999901 0.0140840i \(-0.995517\pi\)
0.487753 0.872982i \(-0.337817\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.50000 4.33013i −0.310087 0.537086i
\(66\) 0 0
\(67\) −7.00000 + 12.1244i −0.855186 + 1.48123i 0.0212861 + 0.999773i \(0.493224\pi\)
−0.876472 + 0.481452i \(0.840109\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −5.00000 + 8.66025i −0.585206 + 1.01361i 0.409644 + 0.912245i \(0.365653\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 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
\(85\) −1.00000 −0.108465
\(86\) 0 0
\(87\) 1.50000 2.59808i 0.160817 0.278543i
\(88\) 0 0
\(89\) 2.00000 + 3.46410i 0.212000 + 0.367194i 0.952340 0.305038i \(-0.0986691\pi\)
−0.740341 + 0.672232i \(0.765336\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 1.73205i −0.103695 0.179605i
\(94\) 0 0
\(95\) −3.00000 + 5.19615i −0.307794 + 0.533114i
\(96\) 0 0
\(97\) 3.00000 0.304604 0.152302 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −5.00000 + 8.66025i −0.497519 + 0.861727i −0.999996 0.00286291i \(-0.999089\pi\)
0.502477 + 0.864590i \(0.332422\pi\)
\(102\) 0 0
\(103\) −3.50000 6.06218i −0.344865 0.597324i 0.640464 0.767988i \(-0.278742\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.00000 15.5885i −0.870063 1.50699i −0.861931 0.507026i \(-0.830745\pi\)
−0.00813215 0.999967i \(-0.502589\pi\)
\(108\) 0 0
\(109\) 5.50000 9.52628i 0.526804 0.912452i −0.472708 0.881219i \(-0.656723\pi\)
0.999512 0.0312328i \(-0.00994332\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 2.00000 3.46410i 0.186501 0.323029i
\(116\) 0 0
\(117\) 5.00000 + 8.66025i 0.462250 + 0.800641i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 + 8.66025i 0.454545 + 0.787296i
\(122\) 0 0
\(123\) 5.00000 8.66025i 0.450835 0.780869i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) −1.00000 + 1.73205i −0.0880451 + 0.152499i
\(130\) 0 0
\(131\) 6.00000 + 10.3923i 0.524222 + 0.907980i 0.999602 + 0.0281993i \(0.00897729\pi\)
−0.475380 + 0.879781i \(0.657689\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.50000 4.33013i −0.215166 0.372678i
\(136\) 0 0
\(137\) −8.00000 + 13.8564i −0.683486 + 1.18383i 0.290424 + 0.956898i \(0.406204\pi\)
−0.973910 + 0.226935i \(0.927130\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −7.00000 −0.589506
\(142\) 0 0
\(143\) 2.50000 4.33013i 0.209061 0.362103i
\(144\) 0 0
\(145\) −1.50000 2.59808i −0.124568 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.0000 + 19.0526i 0.901155 + 1.56085i 0.825997 + 0.563675i \(0.190613\pi\)
0.0751583 + 0.997172i \(0.476054\pi\)
\(150\) 0 0
\(151\) 1.50000 2.59808i 0.122068 0.211428i −0.798515 0.601975i \(-0.794381\pi\)
0.920583 + 0.390547i \(0.127714\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −7.00000 + 12.1244i −0.558661 + 0.967629i 0.438948 + 0.898513i \(0.355351\pi\)
−0.997609 + 0.0691164i \(0.977982\pi\)
\(158\) 0 0
\(159\) −1.00000 1.73205i −0.0793052 0.137361i
\(160\) 0 0
\(161\) 0 0
\(162\) 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
\(165\) −0.500000 + 0.866025i −0.0389249 + 0.0674200i
\(166\) 0 0
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 6.00000 10.3923i 0.458831 0.794719i
\(172\) 0 0
\(173\) 10.5000 + 18.1865i 0.798300 + 1.38270i 0.920722 + 0.390218i \(0.127601\pi\)
−0.122422 + 0.992478i \(0.539066\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.00000 12.1244i −0.526152 0.911322i
\(178\) 0 0
\(179\) 10.0000 17.3205i 0.747435 1.29460i −0.201613 0.979465i \(-0.564618\pi\)
0.949048 0.315130i \(-0.102048\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) −0.500000 0.866025i −0.0365636 0.0633300i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.50000 2.59808i −0.108536 0.187990i 0.806641 0.591041i \(-0.201283\pi\)
−0.915177 + 0.403051i \(0.867950\pi\)
\(192\) 0 0
\(193\) 6.00000 10.3923i 0.431889 0.748054i −0.565147 0.824991i \(-0.691180\pi\)
0.997036 + 0.0769360i \(0.0245137\pi\)
\(194\) 0 0
\(195\) −5.00000 −0.358057
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −11.0000 + 19.0526i −0.779769 + 1.35060i 0.152305 + 0.988334i \(0.451330\pi\)
−0.932075 + 0.362267i \(0.882003\pi\)
\(200\) 0 0
\(201\) 7.00000 + 12.1244i 0.493742 + 0.855186i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5.00000 8.66025i −0.349215 0.604858i
\(206\) 0 0
\(207\) −4.00000 + 6.92820i −0.278019 + 0.481543i
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 + 1.73205i 0.0681994 + 0.118125i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 5.00000 + 8.66025i 0.337869 + 0.585206i
\(220\) 0 0
\(221\) 2.50000 4.33013i 0.168168 0.291276i
\(222\) 0 0
\(223\) −29.0000 −1.94198 −0.970992 0.239113i \(-0.923143\pi\)
−0.970992 + 0.239113i \(0.923143\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 4.50000 7.79423i 0.298675 0.517321i −0.677158 0.735838i \(-0.736789\pi\)
0.975833 + 0.218517i \(0.0701218\pi\)
\(228\) 0 0
\(229\) 1.00000 + 1.73205i 0.0660819 + 0.114457i 0.897173 0.441679i \(-0.145617\pi\)
−0.831092 + 0.556136i \(0.812283\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 + 6.92820i 0.262049 + 0.453882i 0.966786 0.255586i \(-0.0822686\pi\)
−0.704737 + 0.709468i \(0.748935\pi\)
\(234\) 0 0
\(235\) −3.50000 + 6.06218i −0.228315 + 0.395453i
\(236\) 0 0
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) −8.00000 + 13.8564i −0.515325 + 0.892570i 0.484516 + 0.874782i \(0.338996\pi\)
−0.999842 + 0.0177875i \(0.994338\pi\)
\(242\) 0 0
\(243\) 8.00000 + 13.8564i 0.513200 + 0.888889i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −15.0000 25.9808i −0.954427 1.65312i
\(248\) 0 0
\(249\) 2.00000 3.46410i 0.126745 0.219529i
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −0.500000 + 0.866025i −0.0313112 + 0.0542326i
\(256\) 0 0
\(257\) 9.00000 + 15.5885i 0.561405 + 0.972381i 0.997374 + 0.0724199i \(0.0230722\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 + 5.19615i 0.185695 + 0.321634i
\(262\) 0 0
\(263\) −9.00000 + 15.5885i −0.554964 + 0.961225i 0.442943 + 0.896550i \(0.353935\pi\)
−0.997906 + 0.0646755i \(0.979399\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 0 0
\(269\) −4.00000 + 6.92820i −0.243884 + 0.422420i −0.961817 0.273692i \(-0.911755\pi\)
0.717933 + 0.696112i \(0.245088\pi\)
\(270\) 0 0
\(271\) −8.00000 13.8564i −0.485965 0.841717i 0.513905 0.857847i \(-0.328199\pi\)
−0.999870 + 0.0161307i \(0.994865\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.500000 + 0.866025i 0.0301511 + 0.0522233i
\(276\) 0 0
\(277\) −10.0000 + 17.3205i −0.600842 + 1.04069i 0.391852 + 0.920028i \(0.371834\pi\)
−0.992694 + 0.120660i \(0.961499\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) −0.500000 + 0.866025i −0.0297219 + 0.0514799i −0.880504 0.474039i \(-0.842796\pi\)
0.850782 + 0.525519i \(0.176129\pi\)
\(284\) 0 0
\(285\) 3.00000 + 5.19615i 0.177705 + 0.307794i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 + 13.8564i 0.470588 + 0.815083i
\(290\) 0 0
\(291\) 1.50000 2.59808i 0.0879316 0.152302i
\(292\) 0 0
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 0 0
\(297\) 2.50000 4.33013i 0.145065 0.251259i
\(298\) 0 0
\(299\) 10.0000 + 17.3205i 0.578315 + 1.00167i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 5.00000 + 8.66025i 0.287242 + 0.497519i
\(304\) 0 0
\(305\) −4.00000 + 6.92820i −0.229039 + 0.396708i
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 7.00000 12.1244i 0.396934 0.687509i −0.596412 0.802678i \(-0.703408\pi\)
0.993346 + 0.115169i \(0.0367410\pi\)
\(312\) 0 0
\(313\) −12.5000 21.6506i −0.706542 1.22377i −0.966132 0.258047i \(-0.916921\pi\)
0.259590 0.965719i \(-0.416412\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.00000 15.5885i −0.505490 0.875535i −0.999980 0.00635137i \(-0.997978\pi\)
0.494489 0.869184i \(-0.335355\pi\)
\(318\) 0 0
\(319\) 1.50000 2.59808i 0.0839839 0.145464i
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −2.50000 + 4.33013i −0.138675 + 0.240192i
\(326\) 0 0
\(327\) −5.50000 9.52628i −0.304151 0.526804i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 17.3205i −0.549650 0.952021i −0.998298 0.0583130i \(-0.981428\pi\)
0.448649 0.893708i \(-0.351905\pi\)
\(332\) 0 0
\(333\) 8.00000 13.8564i 0.438397 0.759326i
\(334\) 0 0
\(335\) 14.0000 0.764902
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 8.00000 13.8564i 0.434500 0.752577i
\(340\) 0 0
\(341\) −1.00000 1.73205i −0.0541530 0.0937958i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.00000 3.46410i −0.107676 0.186501i
\(346\) 0 0
\(347\) −4.00000 + 6.92820i −0.214731 + 0.371925i −0.953189 0.302374i \(-0.902221\pi\)
0.738458 + 0.674299i \(0.235554\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 25.0000 1.33440
\(352\) 0 0
\(353\) 4.50000 7.79423i 0.239511 0.414845i −0.721063 0.692869i \(-0.756346\pi\)
0.960574 + 0.278024i \(0.0896796\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.00000 + 6.92820i 0.211112 + 0.365657i 0.952063 0.305903i \(-0.0989582\pi\)
−0.740951 + 0.671559i \(0.765625\pi\)
\(360\) 0 0
\(361\) −8.50000 + 14.7224i −0.447368 + 0.774865i
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −11.5000 + 19.9186i −0.600295 + 1.03974i 0.392481 + 0.919760i \(0.371617\pi\)
−0.992776 + 0.119982i \(0.961716\pi\)
\(368\) 0 0
\(369\) 10.0000 + 17.3205i 0.520579 + 0.901670i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 10.3923i −0.310668 0.538093i 0.667839 0.744306i \(-0.267219\pi\)
−0.978507 + 0.206213i \(0.933886\pi\)
\(374\) 0 0
\(375\) 0.500000 0.866025i 0.0258199 0.0447214i
\(376\) 0 0
\(377\) 15.0000 0.772539
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −3.00000 + 5.19615i −0.153695 + 0.266207i
\(382\) 0 0
\(383\) −16.0000 27.7128i −0.817562 1.41606i −0.907474 0.420109i \(-0.861992\pi\)
0.0899119 0.995950i \(-0.471341\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.00000 3.46410i −0.101666 0.176090i
\(388\) 0 0
\(389\) 7.50000 12.9904i 0.380265 0.658638i −0.610835 0.791758i \(-0.709166\pi\)
0.991100 + 0.133120i \(0.0424994\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 5.50000 9.52628i 0.276735 0.479319i
\(396\) 0 0
\(397\) −11.5000 19.9186i −0.577168 0.999685i −0.995802 0.0915300i \(-0.970824\pi\)
0.418634 0.908155i \(-0.362509\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.50000 + 9.52628i 0.274657 + 0.475720i 0.970049 0.242911i \(-0.0781024\pi\)
−0.695392 + 0.718631i \(0.744769\pi\)
\(402\) 0 0
\(403\) 5.00000 8.66025i 0.249068 0.431398i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −5.00000 + 8.66025i −0.247234 + 0.428222i −0.962757 0.270367i \(-0.912855\pi\)
0.715523 + 0.698589i \(0.246188\pi\)
\(410\) 0 0
\(411\) 8.00000 + 13.8564i 0.394611 + 0.683486i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2.00000 3.46410i −0.0981761 0.170046i
\(416\) 0 0
\(417\) −8.00000 + 13.8564i −0.391762 + 0.678551i
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 7.00000 12.1244i 0.340352 0.589506i
\(424\) 0 0
\(425\) 0.500000 + 0.866025i 0.0242536 + 0.0420084i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.50000 4.33013i −0.120701 0.209061i
\(430\) 0 0
\(431\) 16.5000 28.5788i 0.794777 1.37659i −0.128204 0.991748i \(-0.540921\pi\)
0.922981 0.384846i \(-0.125746\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 0 0
\(437\) 12.0000 20.7846i 0.574038 0.994263i
\(438\) 0 0
\(439\) −14.0000 24.2487i −0.668184 1.15733i −0.978412 0.206666i \(-0.933739\pi\)
0.310228 0.950662i \(-0.399595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.0000 + 22.5167i 0.617649 + 1.06980i 0.989914 + 0.141672i \(0.0452479\pi\)
−0.372265 + 0.928126i \(0.621419\pi\)
\(444\) 0 0
\(445\) 2.00000 3.46410i 0.0948091 0.164214i
\(446\) 0 0
\(447\) 22.0000 1.04056
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 5.00000 8.66025i 0.235441 0.407795i
\(452\) 0 0
\(453\) −1.50000 2.59808i −0.0704761 0.122068i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.00000 + 15.5885i 0.421002 + 0.729197i 0.996038 0.0889312i \(-0.0283451\pi\)
−0.575036 + 0.818128i \(0.695012\pi\)
\(458\) 0 0
\(459\) 2.50000 4.33013i 0.116690 0.202113i
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) −1.00000 + 1.73205i −0.0463739 + 0.0803219i
\(466\) 0 0
\(467\) 9.50000 + 16.4545i 0.439608 + 0.761423i 0.997659 0.0683836i \(-0.0217842\pi\)
−0.558052 + 0.829806i \(0.688451\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 7.00000 + 12.1244i 0.322543 + 0.558661i
\(472\) 0 0
\(473\) −1.00000 + 1.73205i −0.0459800 + 0.0796398i
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −8.00000 + 13.8564i −0.365529 + 0.633115i −0.988861 0.148842i \(-0.952445\pi\)
0.623332 + 0.781958i \(0.285779\pi\)
\(480\) 0 0
\(481\) −20.0000 34.6410i −0.911922 1.57949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.50000 2.59808i −0.0681115 0.117973i
\(486\) 0 0
\(487\) 10.0000 17.3205i 0.453143 0.784867i −0.545436 0.838152i \(-0.683636\pi\)
0.998579 + 0.0532853i \(0.0169693\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 1.50000 2.59808i 0.0675566 0.117011i
\(494\) 0 0
\(495\) −1.00000 1.73205i −0.0449467 0.0778499i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.50000 + 4.33013i 0.111915 + 0.193843i 0.916542 0.399937i \(-0.130968\pi\)
−0.804627 + 0.593780i \(0.797635\pi\)
\(500\) 0 0
\(501\) −10.5000 + 18.1865i −0.469105 + 0.812514i
\(502\) 0 0
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 6.00000 10.3923i 0.266469 0.461538i
\(508\) 0 0
\(509\) −12.0000 20.7846i −0.531891 0.921262i −0.999307 0.0372243i \(-0.988148\pi\)
0.467416 0.884037i \(-0.345185\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −15.0000 25.9808i −0.662266 1.14708i
\(514\) 0 0
\(515\) −3.50000 + 6.06218i −0.154228 + 0.267131i
\(516\) 0 0
\(517\) −7.00000 −0.307860
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) 15.0000 25.9808i 0.657162 1.13824i −0.324185 0.945994i \(-0.605090\pi\)
0.981347 0.192244i \(-0.0615766\pi\)
\(522\) 0 0
\(523\) 8.00000 + 13.8564i 0.349816 + 0.605898i 0.986216 0.165460i \(-0.0529109\pi\)
−0.636401 + 0.771358i \(0.719578\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.00000 1.73205i −0.0435607 0.0754493i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 28.0000 1.21510
\(532\) 0 0
\(533\) 50.0000 2.16574
\(534\) 0 0
\(535\) −9.00000 + 15.5885i −0.389104 + 0.673948i
\(536\) 0 0
\(537\) −10.0000 17.3205i −0.431532 0.747435i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.5000 + 21.6506i 0.537417 + 0.930834i 0.999042 + 0.0437584i \(0.0139332\pi\)
−0.461625 + 0.887075i \(0.652733\pi\)
\(542\) 0 0
\(543\) −5.00000 + 8.66025i −0.214571 + 0.371647i
\(544\) 0 0
\(545\) −11.0000 −0.471188
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 8.00000 13.8564i 0.341432 0.591377i
\(550\) 0 0
\(551\) −9.00000 15.5885i −0.383413 0.664091i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4.00000 + 6.92820i 0.169791 + 0.294086i
\(556\) 0 0
\(557\) 6.00000 10.3923i 0.254228 0.440336i −0.710457 0.703740i \(-0.751512\pi\)
0.964686 + 0.263404i \(0.0848453\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) 6.00000 10.3923i 0.252870 0.437983i −0.711445 0.702742i \(-0.751959\pi\)
0.964315 + 0.264758i \(0.0852922\pi\)
\(564\) 0 0
\(565\) −8.00000 13.8564i −0.336563 0.582943i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.00000 8.66025i −0.209611 0.363057i 0.741981 0.670421i \(-0.233886\pi\)
−0.951592 + 0.307364i \(0.900553\pi\)
\(570\) 0 0
\(571\) −6.00000 + 10.3923i −0.251092 + 0.434904i −0.963827 0.266529i \(-0.914123\pi\)
0.712735 + 0.701434i \(0.247456\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 5.50000 9.52628i 0.228968 0.396584i −0.728535 0.685009i \(-0.759798\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) 0 0
\(579\) −6.00000 10.3923i −0.249351 0.431889i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.00000 1.73205i −0.0414158 0.0717342i
\(584\) 0 0
\(585\) 5.00000 8.66025i 0.206725 0.358057i
\(586\) 0 0
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −9.00000 + 15.5885i −0.370211 + 0.641223i
\(592\) 0 0
\(593\) 11.5000 + 19.9186i 0.472248 + 0.817958i 0.999496 0.0317536i \(-0.0101092\pi\)
−0.527247 + 0.849712i \(0.676776\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0000 + 19.0526i 0.450200 + 0.779769i
\(598\) 0 0
\(599\) −17.5000 + 30.3109i −0.715031 + 1.23847i 0.247917 + 0.968781i \(0.420254\pi\)
−0.962948 + 0.269688i \(0.913079\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −28.0000 −1.14025
\(604\) 0 0
\(605\) 5.00000 8.66025i 0.203279 0.352089i
\(606\) 0 0
\(607\) −14.5000 25.1147i −0.588537 1.01938i −0.994424 0.105453i \(-0.966371\pi\)
0.405887 0.913923i \(-0.366962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17.5000 30.3109i −0.707974 1.22625i
\(612\) 0 0
\(613\) 13.0000 22.5167i 0.525065 0.909439i −0.474509 0.880251i \(-0.657374\pi\)
0.999574 0.0291886i \(-0.00929235\pi\)
\(614\) 0 0
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 10.0000 + 17.3205i 0.401286 + 0.695048i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) −3.00000 + 5.19615i −0.119808 + 0.207514i
\(628\) 0 0
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 31.0000 1.23409 0.617045 0.786928i \(-0.288330\pi\)
0.617045 + 0.786928i \(0.288330\pi\)
\(632\) 0 0
\(633\) 8.50000 14.7224i 0.337845 0.585164i
\(634\) 0 0
\(635\) 3.00000 + 5.19615i 0.119051 + 0.206203i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 15.5885i 0.355479 0.615707i −0.631721 0.775196i \(-0.717651\pi\)
0.987200 + 0.159489i \(0.0509845\pi\)
\(642\) 0 0
\(643\) 15.0000 0.591542 0.295771 0.955259i \(-0.404423\pi\)
0.295771 + 0.955259i \(0.404423\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −14.0000 + 24.2487i −0.550397 + 0.953315i 0.447849 + 0.894109i \(0.352190\pi\)
−0.998246 + 0.0592060i \(0.981143\pi\)
\(648\) 0 0
\(649\) −7.00000 12.1244i −0.274774 0.475923i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.00000 12.1244i −0.273931 0.474463i 0.695934 0.718106i \(-0.254991\pi\)
−0.969865 + 0.243643i \(0.921657\pi\)
\(654\) 0 0
\(655\) 6.00000 10.3923i 0.234439 0.406061i
\(656\) 0 0
\(657\) −20.0000 −0.780274
\(658\) 0 0
\(659\) −37.0000 −1.44132 −0.720658 0.693291i \(-0.756160\pi\)
−0.720658 + 0.693291i \(0.756160\pi\)
\(660\) 0 0
\(661\) −2.00000 + 3.46410i −0.0777910 + 0.134738i −0.902297 0.431116i \(-0.858120\pi\)
0.824506 + 0.565854i \(0.191453\pi\)
\(662\) 0 0
\(663\) −2.50000 4.33013i −0.0970920 0.168168i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000 + 10.3923i 0.232321 + 0.402392i
\(668\) 0 0
\(669\) −14.5000 + 25.1147i −0.560602 + 0.970992i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) −2.50000 + 4.33013i −0.0962250 + 0.166667i
\(676\) 0 0
\(677\) 15.5000 + 26.8468i 0.595713 + 1.03181i 0.993446 + 0.114304i \(0.0364639\pi\)
−0.397732 + 0.917501i \(0.630203\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −4.50000 7.79423i −0.172440 0.298675i
\(682\) 0 0
\(683\) −24.0000 + 41.5692i −0.918334 + 1.59060i −0.116390 + 0.993204i \(0.537132\pi\)
−0.801945 + 0.597398i \(0.796201\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 0 0
\(689\) 5.00000 8.66025i 0.190485 0.329929i
\(690\) 0 0
\(691\) 14.0000 + 24.2487i 0.532585 + 0.922464i 0.999276 + 0.0380440i \(0.0121127\pi\)
−0.466691 + 0.884420i \(0.654554\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.00000 + 13.8564i 0.303457 + 0.525603i
\(696\) 0 0
\(697\) 5.00000 8.66025i 0.189389 0.328031i
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 0 0
\(703\) −24.0000 + 41.5692i −0.905177 + 1.56781i
\(704\) 0 0
\(705\) 3.50000 + 6.06218i 0.131818 + 0.228315i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −13.5000 23.3827i −0.507003 0.878155i −0.999967 0.00810550i \(-0.997420\pi\)
0.492964 0.870050i \(-0.335913\pi\)
\(710\) 0 0
\(711\) −11.0000 + 19.0526i −0.412532 + 0.714527i
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −5.00000 −0.186989
\(716\) 0 0
\(717\) 6.50000 11.2583i 0.242747 0.420450i
\(718\) 0 0
\(719\) 6.00000 + 10.3923i 0.223762 + 0.387568i 0.955947 0.293538i \(-0.0948328\pi\)
−0.732185 + 0.681106i \(0.761499\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 8.00000 + 13.8564i 0.297523 + 0.515325i
\(724\) 0 0
\(725\) −1.50000 + 2.59808i −0.0557086 + 0.0964901i
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −1.00000 + 1.73205i −0.0369863 + 0.0640622i
\(732\) 0 0
\(733\) −16.5000 28.5788i −0.609441 1.05558i −0.991333 0.131376i \(-0.958060\pi\)
0.381891 0.924207i \(-0.375273\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.00000 + 12.1244i 0.257848 + 0.446606i
\(738\) 0 0
\(739\) −1.50000 + 2.59808i −0.0551784 + 0.0955718i −0.892295 0.451452i \(-0.850906\pi\)
0.837117 + 0.547024i \(0.184239\pi\)
\(740\) 0 0
\(741\) −30.0000 −1.10208
\(742\) 0 0
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 0 0
\(745\) 11.0000 19.0526i 0.403009 0.698032i
\(746\) 0 0
\(747\) 4.00000 + 6.92820i 0.146352 + 0.253490i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −6.50000 11.2583i −0.237188 0.410822i 0.722718 0.691143i \(-0.242893\pi\)
−0.959906 + 0.280321i \(0.909559\pi\)
\(752\) 0 0
\(753\) 2.00000 3.46410i 0.0728841 0.126239i
\(754\) 0 0
\(755\) −3.00000 −0.109181
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 0 0
\(759\) 2.00000 3.46410i 0.0725954 0.125739i
\(760\) 0 0
\(761\) −6.00000 10.3923i −0.217500 0.376721i 0.736543 0.676391i \(-0.236457\pi\)
−0.954043 + 0.299670i \(0.903123\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.00000 1.73205i −0.0361551 0.0626224i
\(766\) 0 0
\(767\) 35.0000 60.6218i 1.26378 2.18893i
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) −10.5000 + 18.1865i −0.377659 + 0.654124i −0.990721 0.135910i \(-0.956604\pi\)
0.613062 + 0.790034i \(0.289937\pi\)
\(774\) 0 0
\(775\) 1.00000 + 1.73205i 0.0359211 + 0.0622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −30.0000 51.9615i −1.07486 1.86171i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −1.50000 + 2.59808i −0.0534692 + 0.0926114i −0.891521 0.452979i \(-0.850361\pi\)
0.838052 + 0.545590i \(0.183695\pi\)
\(788\) 0 0
\(789\) 9.00000 + 15.5885i 0.320408 + 0.554964i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 34.6410i −0.710221 1.23014i
\(794\) 0 0
\(795\) −1.00000 + 1.73205i −0.0354663 + 0.0614295i
\(796\) 0 0
\(797\) −37.0000 −1.31061 −0.655304 0.755366i \(-0.727459\pi\)
−0.655304 + 0.755366i \(0.727459\pi\)
\(798\) 0 0
\(799\) −7.00000 −0.247642
\(800\) 0 0
\(801\) −4.00000 + 6.92820i −0.141333 + 0.244796i
\(802\) 0 0
\(803\) 5.00000 + 8.66025i 0.176446 + 0.305614i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.00000 + 6.92820i 0.140807 + 0.243884i
\(808\) 0 0
\(809\) −2.50000 + 4.33013i −0.0878953 + 0.152239i −0.906621 0.421945i \(-0.861347\pi\)
0.818726 + 0.574184i \(0.194681\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 8.00000 13.8564i 0.280228 0.485369i
\(816\) 0 0
\(817\) 6.00000 + 10.3923i 0.209913 + 0.363581i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.50000 + 2.59808i 0.0523504 + 0.0906735i 0.891013 0.453978i \(-0.149995\pi\)
−0.838663 + 0.544651i \(0.816662\pi\)
\(822\) 0 0
\(823\) −7.00000 + 12.1244i −0.244005 + 0.422628i −0.961851 0.273573i \(-0.911795\pi\)
0.717847 + 0.696201i \(0.245128\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) 23.0000 39.8372i 0.798823 1.38360i −0.121560 0.992584i \(-0.538790\pi\)
0.920383 0.391018i \(-0.127877\pi\)
\(830\) 0 0
\(831\) 10.0000 + 17.3205i 0.346896 + 0.600842i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 10.5000 + 18.1865i 0.363367 + 0.629371i
\(836\) 0 0
\(837\) 5.00000 8.66025i 0.172825 0.299342i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −6.50000 + 11.2583i −0.223872 + 0.387757i
\(844\) 0 0
\(845\) −6.00000 10.3923i −0.206406 0.357506i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.500000 + 0.866025i 0.0171600 + 0.0297219i
\(850\) 0 0
\(851\) 16.0000 27.7128i 0.548473 0.949983i
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −21.0000 + 36.3731i −0.717346 + 1.24248i 0.244701 + 0.969599i \(0.421310\pi\)
−0.962048 + 0.272882i \(0.912023\pi\)
\(858\) 0 0
\(859\) 26.0000 + 45.0333i 0.887109 + 1.53652i 0.843278 + 0.537478i \(0.180623\pi\)
0.0438309 + 0.999039i \(0.486044\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15.0000 + 25.9808i 0.510606 + 0.884395i 0.999924 + 0.0122903i \(0.00391222\pi\)
−0.489319 + 0.872105i \(0.662754\pi\)
\(864\) 0 0
\(865\) 10.5000 18.1865i 0.357011 0.618361i
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 11.0000 0.373149
\(870\) 0 0
\(871\) −35.0000 + 60.6218i −1.18593 + 2.05409i
\(872\) 0 0
\(873\) 3.00000 + 5.19615i 0.101535 + 0.175863i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.0000 + 46.7654i 0.911725 + 1.57915i 0.811626 + 0.584177i \(0.198583\pi\)
0.100099 + 0.994977i \(0.468084\pi\)
\(878\) 0 0
\(879\) −4.50000 + 7.79423i −0.151781 + 0.262893i
\(880\) 0 0
\(881\) −16.0000 −0.539054 −0.269527 0.962993i \(-0.586867\pi\)
−0.269527 + 0.962993i \(0.586867\pi\)
\(882\) 0 0
\(883\) −56.0000 −1.88455 −0.942275 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 0 0
\(885\) −7.00000 + 12.1244i −0.235302 + 0.407556i
\(886\) 0 0
\(887\) 4.00000 + 6.92820i 0.134307 + 0.232626i 0.925332 0.379157i \(-0.123786\pi\)
−0.791026 + 0.611783i \(0.790453\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.500000 + 0.866025i 0.0167506 + 0.0290129i
\(892\) 0 0
\(893\) −21.0000 + 36.3731i −0.702738 + 1.21718i
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) 3.00000 5.19615i 0.100056 0.173301i
\(900\) 0 0
\(901\) −1.00000 1.73205i −0.0333148 0.0577030i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 + 8.66025i 0.166206 + 0.287877i
\(906\) 0 0
\(907\) −8.00000 + 13.8564i −0.265636 + 0.460094i −0.967730 0.251990i \(-0.918915\pi\)
0.702094 + 0.712084i \(0.252248\pi\)
\(908\) 0 0
\(909\) −20.0000 −0.663358
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 2.00000 3.46410i 0.0661903 0.114645i
\(914\) 0 0
\(915\) 4.00000 + 6.92820i 0.132236 + 0.229039i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −14.5000 25.1147i −0.478311 0.828459i 0.521380 0.853325i \(-0.325417\pi\)
−0.999691 + 0.0248659i \(0.992084\pi\)
\(920\) 0 0
\(921\) −11.5000 + 19.9186i −0.378938 + 0.656340i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 7.00000 12.1244i 0.229910 0.398216i
\(928\) 0 0
\(929\) 9.00000 + 15.5885i 0.295280 + 0.511441i 0.975050 0.221985i \(-0.0712536\pi\)
−0.679770 + 0.733426i \(0.737920\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −7.00000 12.1244i −0.229170 0.396934i
\(934\) 0 0
\(935\) −0.500000 + 0.866025i −0.0163517 + 0.0283221i
\(936\) 0 0
\(937\) −9.00000 −0.294017 −0.147009 0.989135i \(-0.546964\pi\)
−0.147009 + 0.989135i \(0.546964\pi\)
\(938\) 0 0
\(939\) −25.0000 −0.815844
\(940\) 0 0
\(941\) 20.0000 34.6410i 0.651981 1.12926i −0.330660 0.943750i \(-0.607271\pi\)
0.982641 0.185515i \(-0.0593953\pi\)
\(942\) 0 0
\(943\) 20.0000 + 34.6410i 0.651290 + 1.12807i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000 + 46.7654i 0.877382 + 1.51967i 0.854203 + 0.519939i \(0.174045\pi\)
0.0231788 + 0.999731i \(0.492621\pi\)
\(948\) 0 0
\(949\) −25.0000 + 43.3013i −0.811534 + 1.40562i
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 0 0
\(955\) −1.50000 + 2.59808i −0.0485389 + 0.0840718i
\(956\) 0 0
\(957\) −1.50000 2.59808i −0.0484881 0.0839839i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 13.5000 + 23.3827i 0.435484 + 0.754280i
\(962\) 0 0
\(963\) 18.0000 31.1769i 0.580042 1.00466i
\(964\) 0 0
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 0 0
\(969\) −3.00000 + 5.19615i −0.0963739 + 0.166924i
\(970\) 0 0
\(971\) 1.00000 + 1.73205i 0.0320915 + 0.0555842i 0.881625 0.471950i \(-0.156450\pi\)
−0.849534 + 0.527535i \(0.823117\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2.50000 + 4.33013i 0.0800641 + 0.138675i
\(976\) 0 0
\(977\) 27.0000 46.7654i 0.863807 1.49616i −0.00442082 0.999990i \(-0.501407\pi\)
0.868227 0.496167i \(-0.165259\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) 22.0000 0.702406
\(982\) 0 0
\(983\) −26.5000 + 45.8993i −0.845219 + 1.46396i 0.0402124 + 0.999191i \(0.487197\pi\)
−0.885431 + 0.464771i \(0.846137\pi\)
\(984\) 0 0
\(985\) 9.00000 + 15.5885i 0.286764 + 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 6.92820i −0.127193 0.220304i
\(990\) 0 0
\(991\) −6.00000 + 10.3923i −0.190596 + 0.330122i −0.945448 0.325773i \(-0.894375\pi\)
0.754852 + 0.655895i \(0.227709\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 22.0000 0.697447
\(996\) 0 0
\(997\) −1.50000 + 2.59808i −0.0475055 + 0.0822819i −0.888800 0.458295i \(-0.848460\pi\)
0.841295 + 0.540576i \(0.181794\pi\)
\(998\) 0 0
\(999\) −20.0000 34.6410i −0.632772 1.09599i
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 980.2.i.g.361.1 2
7.2 even 3 inner 980.2.i.g.961.1 2
7.3 odd 6 980.2.a.f.1.1 yes 1
7.4 even 3 980.2.a.d.1.1 1
7.5 odd 6 980.2.i.e.961.1 2
7.6 odd 2 980.2.i.e.361.1 2
21.11 odd 6 8820.2.a.i.1.1 1
21.17 even 6 8820.2.a.v.1.1 1
28.3 even 6 3920.2.a.n.1.1 1
28.11 odd 6 3920.2.a.z.1.1 1
35.3 even 12 4900.2.e.k.2549.2 2
35.4 even 6 4900.2.a.o.1.1 1
35.17 even 12 4900.2.e.k.2549.1 2
35.18 odd 12 4900.2.e.j.2549.1 2
35.24 odd 6 4900.2.a.h.1.1 1
35.32 odd 12 4900.2.e.j.2549.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.d.1.1 1 7.4 even 3
980.2.a.f.1.1 yes 1 7.3 odd 6
980.2.i.e.361.1 2 7.6 odd 2
980.2.i.e.961.1 2 7.5 odd 6
980.2.i.g.361.1 2 1.1 even 1 trivial
980.2.i.g.961.1 2 7.2 even 3 inner
3920.2.a.n.1.1 1 28.3 even 6
3920.2.a.z.1.1 1 28.11 odd 6
4900.2.a.h.1.1 1 35.24 odd 6
4900.2.a.o.1.1 1 35.4 even 6
4900.2.e.j.2549.1 2 35.18 odd 12
4900.2.e.j.2549.2 2 35.32 odd 12
4900.2.e.k.2549.1 2 35.17 even 12
4900.2.e.k.2549.2 2 35.3 even 12
8820.2.a.i.1.1 1 21.11 odd 6
8820.2.a.v.1.1 1 21.17 even 6