Properties

Label 2070.2.d.a.829.3
Level $2070$
Weight $2$
Character 2070.829
Analytic conductor $16.529$
Analytic rank $0$
Dimension $4$
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] = [2070,2,Mod(829,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2070.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.d (of order \(2\), degree \(1\), 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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 829.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2070.829
Dual form 2070.2.d.a.829.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +2.00000i q^{7} -1.00000i q^{8} +(-2.12132 - 0.707107i) q^{10} -4.24264 q^{11} +4.82843i q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.17157i q^{17} +2.24264 q^{19} +(0.707107 - 2.12132i) q^{20} -4.24264i q^{22} +1.00000i q^{23} +(-4.00000 - 3.00000i) q^{25} -4.82843 q^{26} -2.00000i q^{28} -7.65685 q^{29} +6.00000 q^{31} +1.00000i q^{32} -1.17157 q^{34} +(-4.24264 - 1.41421i) q^{35} -3.41421i q^{37} +2.24264i q^{38} +(2.12132 + 0.707107i) q^{40} +1.17157 q^{41} +1.75736i q^{43} +4.24264 q^{44} -1.00000 q^{46} +4.82843i q^{47} +3.00000 q^{49} +(3.00000 - 4.00000i) q^{50} -4.82843i q^{52} -13.4142i q^{53} +(3.00000 - 9.00000i) q^{55} +2.00000 q^{56} -7.65685i q^{58} -8.48528 q^{59} -3.41421 q^{61} +6.00000i q^{62} -1.00000 q^{64} +(-10.2426 - 3.41421i) q^{65} +0.585786i q^{67} -1.17157i q^{68} +(1.41421 - 4.24264i) q^{70} -5.65685 q^{71} -3.65685i q^{73} +3.41421 q^{74} -2.24264 q^{76} -8.48528i q^{77} -7.65685 q^{79} +(-0.707107 + 2.12132i) q^{80} +1.17157i q^{82} +1.41421i q^{83} +(-2.48528 - 0.828427i) q^{85} -1.75736 q^{86} +4.24264i q^{88} +14.8284 q^{89} -9.65685 q^{91} -1.00000i q^{92} -4.82843 q^{94} +(-1.58579 + 4.75736i) q^{95} +6.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 8 q^{14} + 4 q^{16} - 8 q^{19} - 16 q^{25} - 8 q^{26} - 8 q^{29} + 24 q^{31} - 16 q^{34} + 16 q^{41} - 4 q^{46} + 12 q^{49} + 12 q^{50} + 12 q^{55} + 8 q^{56} - 8 q^{61} - 4 q^{64} - 24 q^{65} + 8 q^{74} + 8 q^{76} - 8 q^{79} + 24 q^{85} - 24 q^{86} + 48 q^{89} - 16 q^{91} - 8 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −0.707107 + 2.12132i −0.316228 + 0.948683i
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −2.12132 0.707107i −0.670820 0.223607i
\(11\) −4.24264 −1.27920 −0.639602 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) 4.82843i 1.33916i 0.742738 + 0.669582i \(0.233527\pi\)
−0.742738 + 0.669582i \(0.766473\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.17157i 0.284148i 0.989856 + 0.142074i \(0.0453771\pi\)
−0.989856 + 0.142074i \(0.954623\pi\)
\(18\) 0 0
\(19\) 2.24264 0.514497 0.257249 0.966345i \(-0.417184\pi\)
0.257249 + 0.966345i \(0.417184\pi\)
\(20\) 0.707107 2.12132i 0.158114 0.474342i
\(21\) 0 0
\(22\) 4.24264i 0.904534i
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) −4.00000 3.00000i −0.800000 0.600000i
\(26\) −4.82843 −0.946932
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −1.17157 −0.200923
\(35\) −4.24264 1.41421i −0.717137 0.239046i
\(36\) 0 0
\(37\) 3.41421i 0.561293i −0.959811 0.280647i \(-0.909451\pi\)
0.959811 0.280647i \(-0.0905489\pi\)
\(38\) 2.24264i 0.363804i
\(39\) 0 0
\(40\) 2.12132 + 0.707107i 0.335410 + 0.111803i
\(41\) 1.17157 0.182969 0.0914845 0.995807i \(-0.470839\pi\)
0.0914845 + 0.995807i \(0.470839\pi\)
\(42\) 0 0
\(43\) 1.75736i 0.267995i 0.990982 + 0.133997i \(0.0427814\pi\)
−0.990982 + 0.133997i \(0.957219\pi\)
\(44\) 4.24264 0.639602
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.82843i 0.704298i 0.935944 + 0.352149i \(0.114549\pi\)
−0.935944 + 0.352149i \(0.885451\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 3.00000 4.00000i 0.424264 0.565685i
\(51\) 0 0
\(52\) 4.82843i 0.669582i
\(53\) 13.4142i 1.84258i −0.388872 0.921292i \(-0.627135\pi\)
0.388872 0.921292i \(-0.372865\pi\)
\(54\) 0 0
\(55\) 3.00000 9.00000i 0.404520 1.21356i
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 7.65685i 1.00539i
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) −3.41421 −0.437145 −0.218573 0.975821i \(-0.570140\pi\)
−0.218573 + 0.975821i \(0.570140\pi\)
\(62\) 6.00000i 0.762001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −10.2426 3.41421i −1.27044 0.423481i
\(66\) 0 0
\(67\) 0.585786i 0.0715652i 0.999360 + 0.0357826i \(0.0113924\pi\)
−0.999360 + 0.0357826i \(0.988608\pi\)
\(68\) 1.17157i 0.142074i
\(69\) 0 0
\(70\) 1.41421 4.24264i 0.169031 0.507093i
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 3.65685i 0.428002i −0.976833 0.214001i \(-0.931350\pi\)
0.976833 0.214001i \(-0.0686496\pi\)
\(74\) 3.41421 0.396894
\(75\) 0 0
\(76\) −2.24264 −0.257249
\(77\) 8.48528i 0.966988i
\(78\) 0 0
\(79\) −7.65685 −0.861463 −0.430732 0.902480i \(-0.641744\pi\)
−0.430732 + 0.902480i \(0.641744\pi\)
\(80\) −0.707107 + 2.12132i −0.0790569 + 0.237171i
\(81\) 0 0
\(82\) 1.17157i 0.129379i
\(83\) 1.41421i 0.155230i 0.996983 + 0.0776151i \(0.0247305\pi\)
−0.996983 + 0.0776151i \(0.975269\pi\)
\(84\) 0 0
\(85\) −2.48528 0.828427i −0.269567 0.0898555i
\(86\) −1.75736 −0.189501
\(87\) 0 0
\(88\) 4.24264i 0.452267i
\(89\) 14.8284 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(90\) 0 0
\(91\) −9.65685 −1.01231
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) −4.82843 −0.498014
\(95\) −1.58579 + 4.75736i −0.162698 + 0.488095i
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
\(100\) 4.00000 + 3.00000i 0.400000 + 0.300000i
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) 0 0
\(103\) 12.1421i 1.19640i −0.801347 0.598200i \(-0.795883\pi\)
0.801347 0.598200i \(-0.204117\pi\)
\(104\) 4.82843 0.473466
\(105\) 0 0
\(106\) 13.4142 1.30290
\(107\) 15.0711i 1.45698i 0.685059 + 0.728488i \(0.259776\pi\)
−0.685059 + 0.728488i \(0.740224\pi\)
\(108\) 0 0
\(109\) −15.4142 −1.47641 −0.738207 0.674574i \(-0.764327\pi\)
−0.738207 + 0.674574i \(0.764327\pi\)
\(110\) 9.00000 + 3.00000i 0.858116 + 0.286039i
\(111\) 0 0
\(112\) 2.00000i 0.188982i
\(113\) 16.9706i 1.59646i −0.602355 0.798228i \(-0.705771\pi\)
0.602355 0.798228i \(-0.294229\pi\)
\(114\) 0 0
\(115\) −2.12132 0.707107i −0.197814 0.0659380i
\(116\) 7.65685 0.710921
\(117\) 0 0
\(118\) 8.48528i 0.781133i
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 3.41421i 0.309108i
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 9.19239 6.36396i 0.822192 0.569210i
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 3.41421 10.2426i 0.299446 0.898339i
\(131\) 10.3431 0.903685 0.451842 0.892098i \(-0.350767\pi\)
0.451842 + 0.892098i \(0.350767\pi\)
\(132\) 0 0
\(133\) 4.48528i 0.388923i
\(134\) −0.585786 −0.0506042
\(135\) 0 0
\(136\) 1.17157 0.100462
\(137\) 8.48528i 0.724947i 0.931994 + 0.362473i \(0.118068\pi\)
−0.931994 + 0.362473i \(0.881932\pi\)
\(138\) 0 0
\(139\) −16.9706 −1.43942 −0.719712 0.694273i \(-0.755726\pi\)
−0.719712 + 0.694273i \(0.755726\pi\)
\(140\) 4.24264 + 1.41421i 0.358569 + 0.119523i
\(141\) 0 0
\(142\) 5.65685i 0.474713i
\(143\) 20.4853i 1.71307i
\(144\) 0 0
\(145\) 5.41421 16.2426i 0.449626 1.34888i
\(146\) 3.65685 0.302643
\(147\) 0 0
\(148\) 3.41421i 0.280647i
\(149\) −12.7279 −1.04271 −0.521356 0.853339i \(-0.674574\pi\)
−0.521356 + 0.853339i \(0.674574\pi\)
\(150\) 0 0
\(151\) −15.6569 −1.27414 −0.637068 0.770807i \(-0.719853\pi\)
−0.637068 + 0.770807i \(0.719853\pi\)
\(152\) 2.24264i 0.181902i
\(153\) 0 0
\(154\) 8.48528 0.683763
\(155\) −4.24264 + 12.7279i −0.340777 + 1.02233i
\(156\) 0 0
\(157\) 6.92893i 0.552989i −0.961016 0.276494i \(-0.910827\pi\)
0.961016 0.276494i \(-0.0891728\pi\)
\(158\) 7.65685i 0.609147i
\(159\) 0 0
\(160\) −2.12132 0.707107i −0.167705 0.0559017i
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 1.17157i 0.0917647i 0.998947 + 0.0458823i \(0.0146099\pi\)
−0.998947 + 0.0458823i \(0.985390\pi\)
\(164\) −1.17157 −0.0914845
\(165\) 0 0
\(166\) −1.41421 −0.109764
\(167\) 8.82843i 0.683164i −0.939852 0.341582i \(-0.889037\pi\)
0.939852 0.341582i \(-0.110963\pi\)
\(168\) 0 0
\(169\) −10.3137 −0.793362
\(170\) 0.828427 2.48528i 0.0635375 0.190612i
\(171\) 0 0
\(172\) 1.75736i 0.133997i
\(173\) 3.17157i 0.241130i −0.992705 0.120565i \(-0.961529\pi\)
0.992705 0.120565i \(-0.0384707\pi\)
\(174\) 0 0
\(175\) 6.00000 8.00000i 0.453557 0.604743i
\(176\) −4.24264 −0.319801
\(177\) 0 0
\(178\) 14.8284i 1.11144i
\(179\) −10.3431 −0.773083 −0.386542 0.922272i \(-0.626330\pi\)
−0.386542 + 0.922272i \(0.626330\pi\)
\(180\) 0 0
\(181\) 16.3848 1.21787 0.608935 0.793220i \(-0.291597\pi\)
0.608935 + 0.793220i \(0.291597\pi\)
\(182\) 9.65685i 0.715814i
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 7.24264 + 2.41421i 0.532490 + 0.177497i
\(186\) 0 0
\(187\) 4.97056i 0.363484i
\(188\) 4.82843i 0.352149i
\(189\) 0 0
\(190\) −4.75736 1.58579i −0.345135 0.115045i
\(191\) 5.17157 0.374202 0.187101 0.982341i \(-0.440091\pi\)
0.187101 + 0.982341i \(0.440091\pi\)
\(192\) 0 0
\(193\) 12.9706i 0.933642i 0.884352 + 0.466821i \(0.154601\pi\)
−0.884352 + 0.466821i \(0.845399\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 24.6274i 1.75463i −0.479914 0.877315i \(-0.659332\pi\)
0.479914 0.877315i \(-0.340668\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) −3.00000 + 4.00000i −0.212132 + 0.282843i
\(201\) 0 0
\(202\) 9.31371i 0.655310i
\(203\) 15.3137i 1.07481i
\(204\) 0 0
\(205\) −0.828427 + 2.48528i −0.0578599 + 0.173580i
\(206\) 12.1421 0.845983
\(207\) 0 0
\(208\) 4.82843i 0.334791i
\(209\) −9.51472 −0.658147
\(210\) 0 0
\(211\) −20.4853 −1.41026 −0.705132 0.709076i \(-0.749112\pi\)
−0.705132 + 0.709076i \(0.749112\pi\)
\(212\) 13.4142i 0.921292i
\(213\) 0 0
\(214\) −15.0711 −1.03024
\(215\) −3.72792 1.24264i −0.254242 0.0847474i
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 15.4142i 1.04398i
\(219\) 0 0
\(220\) −3.00000 + 9.00000i −0.202260 + 0.606780i
\(221\) −5.65685 −0.380521
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 16.9706 1.12887
\(227\) 14.3848i 0.954751i 0.878699 + 0.477376i \(0.158412\pi\)
−0.878699 + 0.477376i \(0.841588\pi\)
\(228\) 0 0
\(229\) −1.75736 −0.116130 −0.0580648 0.998313i \(-0.518493\pi\)
−0.0580648 + 0.998313i \(0.518493\pi\)
\(230\) 0.707107 2.12132i 0.0466252 0.139876i
\(231\) 0 0
\(232\) 7.65685i 0.502697i
\(233\) 26.1421i 1.71263i 0.516455 + 0.856314i \(0.327251\pi\)
−0.516455 + 0.856314i \(0.672749\pi\)
\(234\) 0 0
\(235\) −10.2426 3.41421i −0.668156 0.222719i
\(236\) 8.48528 0.552345
\(237\) 0 0
\(238\) 2.34315i 0.151884i
\(239\) −11.1716 −0.722629 −0.361314 0.932444i \(-0.617672\pi\)
−0.361314 + 0.932444i \(0.617672\pi\)
\(240\) 0 0
\(241\) −10.4853 −0.675416 −0.337708 0.941251i \(-0.609652\pi\)
−0.337708 + 0.941251i \(0.609652\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 0 0
\(244\) 3.41421 0.218573
\(245\) −2.12132 + 6.36396i −0.135526 + 0.406579i
\(246\) 0 0
\(247\) 10.8284i 0.688996i
\(248\) 6.00000i 0.381000i
\(249\) 0 0
\(250\) 6.36396 + 9.19239i 0.402492 + 0.581378i
\(251\) 24.7279 1.56081 0.780406 0.625273i \(-0.215012\pi\)
0.780406 + 0.625273i \(0.215012\pi\)
\(252\) 0 0
\(253\) 4.24264i 0.266733i
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 6.82843 0.424298
\(260\) 10.2426 + 3.41421i 0.635222 + 0.211741i
\(261\) 0 0
\(262\) 10.3431i 0.639002i
\(263\) 17.6569i 1.08877i 0.838836 + 0.544384i \(0.183237\pi\)
−0.838836 + 0.544384i \(0.816763\pi\)
\(264\) 0 0
\(265\) 28.4558 + 9.48528i 1.74803 + 0.582676i
\(266\) −4.48528 −0.275010
\(267\) 0 0
\(268\) 0.585786i 0.0357826i
\(269\) −24.8284 −1.51382 −0.756908 0.653521i \(-0.773291\pi\)
−0.756908 + 0.653521i \(0.773291\pi\)
\(270\) 0 0
\(271\) −1.65685 −0.100647 −0.0503234 0.998733i \(-0.516025\pi\)
−0.0503234 + 0.998733i \(0.516025\pi\)
\(272\) 1.17157i 0.0710370i
\(273\) 0 0
\(274\) −8.48528 −0.512615
\(275\) 16.9706 + 12.7279i 1.02336 + 0.767523i
\(276\) 0 0
\(277\) 30.4853i 1.83168i −0.401540 0.915842i \(-0.631525\pi\)
0.401540 0.915842i \(-0.368475\pi\)
\(278\) 16.9706i 1.01783i
\(279\) 0 0
\(280\) −1.41421 + 4.24264i −0.0845154 + 0.253546i
\(281\) 30.1421 1.79813 0.899065 0.437816i \(-0.144248\pi\)
0.899065 + 0.437816i \(0.144248\pi\)
\(282\) 0 0
\(283\) 27.2132i 1.61766i −0.588045 0.808829i \(-0.700102\pi\)
0.588045 0.808829i \(-0.299898\pi\)
\(284\) 5.65685 0.335673
\(285\) 0 0
\(286\) 20.4853 1.21132
\(287\) 2.34315i 0.138312i
\(288\) 0 0
\(289\) 15.6274 0.919260
\(290\) 16.2426 + 5.41421i 0.953801 + 0.317934i
\(291\) 0 0
\(292\) 3.65685i 0.214001i
\(293\) 28.0416i 1.63821i 0.573644 + 0.819105i \(0.305529\pi\)
−0.573644 + 0.819105i \(0.694471\pi\)
\(294\) 0 0
\(295\) 6.00000 18.0000i 0.349334 1.04800i
\(296\) −3.41421 −0.198447
\(297\) 0 0
\(298\) 12.7279i 0.737309i
\(299\) −4.82843 −0.279235
\(300\) 0 0
\(301\) −3.51472 −0.202585
\(302\) 15.6569i 0.900951i
\(303\) 0 0
\(304\) 2.24264 0.128624
\(305\) 2.41421 7.24264i 0.138237 0.414712i
\(306\) 0 0
\(307\) 25.4558i 1.45284i 0.687250 + 0.726421i \(0.258818\pi\)
−0.687250 + 0.726421i \(0.741182\pi\)
\(308\) 8.48528i 0.483494i
\(309\) 0 0
\(310\) −12.7279 4.24264i −0.722897 0.240966i
\(311\) −17.7990 −1.00929 −0.504644 0.863328i \(-0.668376\pi\)
−0.504644 + 0.863328i \(0.668376\pi\)
\(312\) 0 0
\(313\) 18.0000i 1.01742i 0.860938 + 0.508710i \(0.169877\pi\)
−0.860938 + 0.508710i \(0.830123\pi\)
\(314\) 6.92893 0.391022
\(315\) 0 0
\(316\) 7.65685 0.430732
\(317\) 2.48528i 0.139587i 0.997561 + 0.0697937i \(0.0222341\pi\)
−0.997561 + 0.0697937i \(0.977766\pi\)
\(318\) 0 0
\(319\) 32.4853 1.81883
\(320\) 0.707107 2.12132i 0.0395285 0.118585i
\(321\) 0 0
\(322\) 2.00000i 0.111456i
\(323\) 2.62742i 0.146193i
\(324\) 0 0
\(325\) 14.4853 19.3137i 0.803499 1.07133i
\(326\) −1.17157 −0.0648874
\(327\) 0 0
\(328\) 1.17157i 0.0646893i
\(329\) −9.65685 −0.532400
\(330\) 0 0
\(331\) −3.51472 −0.193186 −0.0965932 0.995324i \(-0.530795\pi\)
−0.0965932 + 0.995324i \(0.530795\pi\)
\(332\) 1.41421i 0.0776151i
\(333\) 0 0
\(334\) 8.82843 0.483070
\(335\) −1.24264 0.414214i −0.0678927 0.0226309i
\(336\) 0 0
\(337\) 9.51472i 0.518300i 0.965837 + 0.259150i \(0.0834424\pi\)
−0.965837 + 0.259150i \(0.916558\pi\)
\(338\) 10.3137i 0.560992i
\(339\) 0 0
\(340\) 2.48528 + 0.828427i 0.134783 + 0.0449278i
\(341\) −25.4558 −1.37851
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 1.75736 0.0947505
\(345\) 0 0
\(346\) 3.17157 0.170505
\(347\) 7.79899i 0.418672i 0.977844 + 0.209336i \(0.0671302\pi\)
−0.977844 + 0.209336i \(0.932870\pi\)
\(348\) 0 0
\(349\) −25.3137 −1.35501 −0.677506 0.735517i \(-0.736939\pi\)
−0.677506 + 0.735517i \(0.736939\pi\)
\(350\) 8.00000 + 6.00000i 0.427618 + 0.320713i
\(351\) 0 0
\(352\) 4.24264i 0.226134i
\(353\) 26.1421i 1.39141i 0.718330 + 0.695703i \(0.244907\pi\)
−0.718330 + 0.695703i \(0.755093\pi\)
\(354\) 0 0
\(355\) 4.00000 12.0000i 0.212298 0.636894i
\(356\) −14.8284 −0.785905
\(357\) 0 0
\(358\) 10.3431i 0.546652i
\(359\) −19.7990 −1.04495 −0.522475 0.852654i \(-0.674991\pi\)
−0.522475 + 0.852654i \(0.674991\pi\)
\(360\) 0 0
\(361\) −13.9706 −0.735293
\(362\) 16.3848i 0.861165i
\(363\) 0 0
\(364\) 9.65685 0.506157
\(365\) 7.75736 + 2.58579i 0.406039 + 0.135346i
\(366\) 0 0
\(367\) 34.9706i 1.82545i −0.408576 0.912724i \(-0.633975\pi\)
0.408576 0.912724i \(-0.366025\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0 0
\(370\) −2.41421 + 7.24264i −0.125509 + 0.376527i
\(371\) 26.8284 1.39286
\(372\) 0 0
\(373\) 19.2132i 0.994822i 0.867515 + 0.497411i \(0.165716\pi\)
−0.867515 + 0.497411i \(0.834284\pi\)
\(374\) 4.97056 0.257022
\(375\) 0 0
\(376\) 4.82843 0.249007
\(377\) 36.9706i 1.90408i
\(378\) 0 0
\(379\) 19.2132 0.986916 0.493458 0.869770i \(-0.335733\pi\)
0.493458 + 0.869770i \(0.335733\pi\)
\(380\) 1.58579 4.75736i 0.0813491 0.244047i
\(381\) 0 0
\(382\) 5.17157i 0.264601i
\(383\) 18.6274i 0.951817i 0.879495 + 0.475908i \(0.157881\pi\)
−0.879495 + 0.475908i \(0.842119\pi\)
\(384\) 0 0
\(385\) 18.0000 + 6.00000i 0.917365 + 0.305788i
\(386\) −12.9706 −0.660184
\(387\) 0 0
\(388\) 6.00000i 0.304604i
\(389\) 6.38478 0.323721 0.161861 0.986814i \(-0.448251\pi\)
0.161861 + 0.986814i \(0.448251\pi\)
\(390\) 0 0
\(391\) −1.17157 −0.0592490
\(392\) 3.00000i 0.151523i
\(393\) 0 0
\(394\) 24.6274 1.24071
\(395\) 5.41421 16.2426i 0.272419 0.817256i
\(396\) 0 0
\(397\) 8.34315i 0.418730i 0.977838 + 0.209365i \(0.0671398\pi\)
−0.977838 + 0.209365i \(0.932860\pi\)
\(398\) 14.0000i 0.701757i
\(399\) 0 0
\(400\) −4.00000 3.00000i −0.200000 0.150000i
\(401\) 28.9706 1.44672 0.723360 0.690471i \(-0.242597\pi\)
0.723360 + 0.690471i \(0.242597\pi\)
\(402\) 0 0
\(403\) 28.9706i 1.44313i
\(404\) −9.31371 −0.463374
\(405\) 0 0
\(406\) 15.3137 0.760007
\(407\) 14.4853i 0.718009i
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) −2.48528 0.828427i −0.122739 0.0409131i
\(411\) 0 0
\(412\) 12.1421i 0.598200i
\(413\) 16.9706i 0.835067i
\(414\) 0 0
\(415\) −3.00000 1.00000i −0.147264 0.0490881i
\(416\) −4.82843 −0.236733
\(417\) 0 0
\(418\) 9.51472i 0.465380i
\(419\) −24.9289 −1.21786 −0.608929 0.793225i \(-0.708401\pi\)
−0.608929 + 0.793225i \(0.708401\pi\)
\(420\) 0 0
\(421\) 10.2426 0.499196 0.249598 0.968350i \(-0.419702\pi\)
0.249598 + 0.968350i \(0.419702\pi\)
\(422\) 20.4853i 0.997208i
\(423\) 0 0
\(424\) −13.4142 −0.651452
\(425\) 3.51472 4.68629i 0.170489 0.227319i
\(426\) 0 0
\(427\) 6.82843i 0.330451i
\(428\) 15.0711i 0.728488i
\(429\) 0 0
\(430\) 1.24264 3.72792i 0.0599255 0.179776i
\(431\) −8.48528 −0.408722 −0.204361 0.978896i \(-0.565512\pi\)
−0.204361 + 0.978896i \(0.565512\pi\)
\(432\) 0 0
\(433\) 26.9706i 1.29612i 0.761588 + 0.648061i \(0.224420\pi\)
−0.761588 + 0.648061i \(0.775580\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 15.4142 0.738207
\(437\) 2.24264i 0.107280i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −9.00000 3.00000i −0.429058 0.143019i
\(441\) 0 0
\(442\) 5.65685i 0.269069i
\(443\) 23.3137i 1.10767i 0.832627 + 0.553834i \(0.186836\pi\)
−0.832627 + 0.553834i \(0.813164\pi\)
\(444\) 0 0
\(445\) −10.4853 + 31.4558i −0.497050 + 1.49115i
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 2.00000i 0.0944911i
\(449\) −22.8284 −1.07734 −0.538670 0.842517i \(-0.681073\pi\)
−0.538670 + 0.842517i \(0.681073\pi\)
\(450\) 0 0
\(451\) −4.97056 −0.234055
\(452\) 16.9706i 0.798228i
\(453\) 0 0
\(454\) −14.3848 −0.675111
\(455\) 6.82843 20.4853i 0.320122 0.960365i
\(456\) 0 0
\(457\) 1.02944i 0.0481550i −0.999710 0.0240775i \(-0.992335\pi\)
0.999710 0.0240775i \(-0.00766485\pi\)
\(458\) 1.75736i 0.0821160i
\(459\) 0 0
\(460\) 2.12132 + 0.707107i 0.0989071 + 0.0329690i
\(461\) 12.8284 0.597479 0.298740 0.954335i \(-0.403434\pi\)
0.298740 + 0.954335i \(0.403434\pi\)
\(462\) 0 0
\(463\) 2.97056i 0.138054i 0.997615 + 0.0690269i \(0.0219894\pi\)
−0.997615 + 0.0690269i \(0.978011\pi\)
\(464\) −7.65685 −0.355461
\(465\) 0 0
\(466\) −26.1421 −1.21101
\(467\) 39.3553i 1.82115i 0.413346 + 0.910574i \(0.364360\pi\)
−0.413346 + 0.910574i \(0.635640\pi\)
\(468\) 0 0
\(469\) −1.17157 −0.0540982
\(470\) 3.41421 10.2426i 0.157486 0.472458i
\(471\) 0 0
\(472\) 8.48528i 0.390567i
\(473\) 7.45584i 0.342820i
\(474\) 0 0
\(475\) −8.97056 6.72792i −0.411598 0.308698i
\(476\) 2.34315 0.107398
\(477\) 0 0
\(478\) 11.1716i 0.510976i
\(479\) 4.68629 0.214122 0.107061 0.994252i \(-0.465856\pi\)
0.107061 + 0.994252i \(0.465856\pi\)
\(480\) 0 0
\(481\) 16.4853 0.751664
\(482\) 10.4853i 0.477591i
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −12.7279 4.24264i −0.577945 0.192648i
\(486\) 0 0
\(487\) 27.9411i 1.26613i 0.774097 + 0.633067i \(0.218204\pi\)
−0.774097 + 0.633067i \(0.781796\pi\)
\(488\) 3.41421i 0.154554i
\(489\) 0 0
\(490\) −6.36396 2.12132i −0.287494 0.0958315i
\(491\) −26.6274 −1.20168 −0.600839 0.799370i \(-0.705167\pi\)
−0.600839 + 0.799370i \(0.705167\pi\)
\(492\) 0 0
\(493\) 8.97056i 0.404014i
\(494\) −10.8284 −0.487194
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 11.3137i 0.507489i
\(498\) 0 0
\(499\) −13.4558 −0.602366 −0.301183 0.953566i \(-0.597382\pi\)
−0.301183 + 0.953566i \(0.597382\pi\)
\(500\) −9.19239 + 6.36396i −0.411096 + 0.284605i
\(501\) 0 0
\(502\) 24.7279i 1.10366i
\(503\) 22.1421i 0.987269i −0.869669 0.493635i \(-0.835668\pi\)
0.869669 0.493635i \(-0.164332\pi\)
\(504\) 0 0
\(505\) −6.58579 + 19.7574i −0.293064 + 0.879191i
\(506\) 4.24264 0.188608
\(507\) 0 0
\(508\) 2.00000i 0.0887357i
\(509\) 22.9706 1.01815 0.509076 0.860721i \(-0.329987\pi\)
0.509076 + 0.860721i \(0.329987\pi\)
\(510\) 0 0
\(511\) 7.31371 0.323539
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 25.7574 + 8.58579i 1.13500 + 0.378335i
\(516\) 0 0
\(517\) 20.4853i 0.900942i
\(518\) 6.82843i 0.300024i
\(519\) 0 0
\(520\) −3.41421 + 10.2426i −0.149723 + 0.449170i
\(521\) 16.2843 0.713427 0.356713 0.934214i \(-0.383897\pi\)
0.356713 + 0.934214i \(0.383897\pi\)
\(522\) 0 0
\(523\) 2.92893i 0.128073i −0.997948 0.0640366i \(-0.979603\pi\)
0.997948 0.0640366i \(-0.0203974\pi\)
\(524\) −10.3431 −0.451842
\(525\) 0 0
\(526\) −17.6569 −0.769875
\(527\) 7.02944i 0.306207i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) −9.48528 + 28.4558i −0.412014 + 1.23604i
\(531\) 0 0
\(532\) 4.48528i 0.194462i
\(533\) 5.65685i 0.245026i
\(534\) 0 0
\(535\) −31.9706 10.6569i −1.38221 0.460736i
\(536\) 0.585786 0.0253021
\(537\) 0 0
\(538\) 24.8284i 1.07043i
\(539\) −12.7279 −0.548230
\(540\) 0 0
\(541\) 35.9411 1.54523 0.772615 0.634875i \(-0.218948\pi\)
0.772615 + 0.634875i \(0.218948\pi\)
\(542\) 1.65685i 0.0711680i
\(543\) 0 0
\(544\) −1.17157 −0.0502308
\(545\) 10.8995 32.6985i 0.466883 1.40065i
\(546\) 0 0
\(547\) 4.48528i 0.191777i 0.995392 + 0.0958884i \(0.0305692\pi\)
−0.995392 + 0.0958884i \(0.969431\pi\)
\(548\) 8.48528i 0.362473i
\(549\) 0 0
\(550\) −12.7279 + 16.9706i −0.542720 + 0.723627i
\(551\) −17.1716 −0.731534
\(552\) 0 0
\(553\) 15.3137i 0.651205i
\(554\) 30.4853 1.29520
\(555\) 0 0
\(556\) 16.9706 0.719712
\(557\) 14.8701i 0.630065i −0.949081 0.315032i \(-0.897985\pi\)
0.949081 0.315032i \(-0.102015\pi\)
\(558\) 0 0
\(559\) −8.48528 −0.358889
\(560\) −4.24264 1.41421i −0.179284 0.0597614i
\(561\) 0 0
\(562\) 30.1421i 1.27147i
\(563\) 21.8995i 0.922954i 0.887152 + 0.461477i \(0.152680\pi\)
−0.887152 + 0.461477i \(0.847320\pi\)
\(564\) 0 0
\(565\) 36.0000 + 12.0000i 1.51453 + 0.504844i
\(566\) 27.2132 1.14386
\(567\) 0 0
\(568\) 5.65685i 0.237356i
\(569\) 25.6569 1.07559 0.537796 0.843075i \(-0.319257\pi\)
0.537796 + 0.843075i \(0.319257\pi\)
\(570\) 0 0
\(571\) −42.7279 −1.78811 −0.894054 0.447959i \(-0.852151\pi\)
−0.894054 + 0.447959i \(0.852151\pi\)
\(572\) 20.4853i 0.856533i
\(573\) 0 0
\(574\) −2.34315 −0.0978010
\(575\) 3.00000 4.00000i 0.125109 0.166812i
\(576\) 0 0
\(577\) 20.6274i 0.858731i 0.903131 + 0.429365i \(0.141263\pi\)
−0.903131 + 0.429365i \(0.858737\pi\)
\(578\) 15.6274i 0.650015i
\(579\) 0 0
\(580\) −5.41421 + 16.2426i −0.224813 + 0.674439i
\(581\) −2.82843 −0.117343
\(582\) 0 0
\(583\) 56.9117i 2.35704i
\(584\) −3.65685 −0.151322
\(585\) 0 0
\(586\) −28.0416 −1.15839
\(587\) 26.8284i 1.10733i 0.832740 + 0.553664i \(0.186771\pi\)
−0.832740 + 0.553664i \(0.813229\pi\)
\(588\) 0 0
\(589\) 13.4558 0.554438
\(590\) 18.0000 + 6.00000i 0.741048 + 0.247016i
\(591\) 0 0
\(592\) 3.41421i 0.140323i
\(593\) 12.3431i 0.506872i 0.967352 + 0.253436i \(0.0815608\pi\)
−0.967352 + 0.253436i \(0.918439\pi\)
\(594\) 0 0
\(595\) 1.65685 4.97056i 0.0679244 0.203773i
\(596\) 12.7279 0.521356
\(597\) 0 0
\(598\) 4.82843i 0.197449i
\(599\) −40.1421 −1.64016 −0.820082 0.572247i \(-0.806072\pi\)
−0.820082 + 0.572247i \(0.806072\pi\)
\(600\) 0 0
\(601\) −20.9706 −0.855407 −0.427704 0.903919i \(-0.640677\pi\)
−0.427704 + 0.903919i \(0.640677\pi\)
\(602\) 3.51472i 0.143249i
\(603\) 0 0
\(604\) 15.6569 0.637068
\(605\) −4.94975 + 14.8492i −0.201236 + 0.603708i
\(606\) 0 0
\(607\) 16.9706i 0.688814i 0.938820 + 0.344407i \(0.111920\pi\)
−0.938820 + 0.344407i \(0.888080\pi\)
\(608\) 2.24264i 0.0909511i
\(609\) 0 0
\(610\) 7.24264 + 2.41421i 0.293246 + 0.0977486i
\(611\) −23.3137 −0.943172
\(612\) 0 0
\(613\) 14.2426i 0.575255i −0.957742 0.287627i \(-0.907134\pi\)
0.957742 0.287627i \(-0.0928665\pi\)
\(614\) −25.4558 −1.02731
\(615\) 0 0
\(616\) −8.48528 −0.341882
\(617\) 31.3137i 1.26064i −0.776334 0.630321i \(-0.782923\pi\)
0.776334 0.630321i \(-0.217077\pi\)
\(618\) 0 0
\(619\) 19.2132 0.772244 0.386122 0.922448i \(-0.373814\pi\)
0.386122 + 0.922448i \(0.373814\pi\)
\(620\) 4.24264 12.7279i 0.170389 0.511166i
\(621\) 0 0
\(622\) 17.7990i 0.713674i
\(623\) 29.6569i 1.18818i
\(624\) 0 0
\(625\) 7.00000 + 24.0000i 0.280000 + 0.960000i
\(626\) −18.0000 −0.719425
\(627\) 0 0
\(628\) 6.92893i 0.276494i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 35.1716 1.40016 0.700079 0.714065i \(-0.253148\pi\)
0.700079 + 0.714065i \(0.253148\pi\)
\(632\) 7.65685i 0.304573i
\(633\) 0 0
\(634\) −2.48528 −0.0987031
\(635\) −4.24264 1.41421i −0.168364 0.0561214i
\(636\) 0 0
\(637\) 14.4853i 0.573928i
\(638\) 32.4853i 1.28610i
\(639\) 0 0
\(640\) 2.12132 + 0.707107i 0.0838525 + 0.0279508i
\(641\) 8.28427 0.327209 0.163605 0.986526i \(-0.447688\pi\)
0.163605 + 0.986526i \(0.447688\pi\)
\(642\) 0 0
\(643\) 32.3848i 1.27713i −0.769568 0.638565i \(-0.779528\pi\)
0.769568 0.638565i \(-0.220472\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) −2.62742 −0.103374
\(647\) 29.6569i 1.16593i −0.812497 0.582966i \(-0.801892\pi\)
0.812497 0.582966i \(-0.198108\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 19.3137 + 14.4853i 0.757546 + 0.568159i
\(651\) 0 0
\(652\) 1.17157i 0.0458823i
\(653\) 43.4558i 1.70056i −0.526332 0.850279i \(-0.676433\pi\)
0.526332 0.850279i \(-0.323567\pi\)
\(654\) 0 0
\(655\) −7.31371 + 21.9411i −0.285770 + 0.857311i
\(656\) 1.17157 0.0457422
\(657\) 0 0
\(658\) 9.65685i 0.376463i
\(659\) −26.1005 −1.01673 −0.508366 0.861141i \(-0.669750\pi\)
−0.508366 + 0.861141i \(0.669750\pi\)
\(660\) 0 0
\(661\) 9.27208 0.360642 0.180321 0.983608i \(-0.442286\pi\)
0.180321 + 0.983608i \(0.442286\pi\)
\(662\) 3.51472i 0.136603i
\(663\) 0 0
\(664\) 1.41421 0.0548821
\(665\) −9.51472 3.17157i −0.368965 0.122988i
\(666\) 0 0
\(667\) 7.65685i 0.296475i
\(668\) 8.82843i 0.341582i
\(669\) 0 0
\(670\) 0.414214 1.24264i 0.0160025 0.0480074i
\(671\) 14.4853 0.559198
\(672\) 0 0
\(673\) 39.9411i 1.53962i −0.638275 0.769809i \(-0.720352\pi\)
0.638275 0.769809i \(-0.279648\pi\)
\(674\) −9.51472 −0.366493
\(675\) 0 0
\(676\) 10.3137 0.396681
\(677\) 24.5269i 0.942646i 0.881961 + 0.471323i \(0.156223\pi\)
−0.881961 + 0.471323i \(0.843777\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) −0.828427 + 2.48528i −0.0317687 + 0.0953062i
\(681\) 0 0
\(682\) 25.4558i 0.974755i
\(683\) 8.48528i 0.324680i −0.986735 0.162340i \(-0.948096\pi\)
0.986735 0.162340i \(-0.0519042\pi\)
\(684\) 0 0
\(685\) −18.0000 6.00000i −0.687745 0.229248i
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 1.75736i 0.0669987i
\(689\) 64.7696 2.46752
\(690\) 0 0
\(691\) −18.8284 −0.716267 −0.358134 0.933670i \(-0.616587\pi\)
−0.358134 + 0.933670i \(0.616587\pi\)
\(692\) 3.17157i 0.120565i
\(693\) 0 0
\(694\) −7.79899 −0.296046
\(695\) 12.0000 36.0000i 0.455186 1.36556i
\(696\) 0 0
\(697\) 1.37258i 0.0519903i
\(698\) 25.3137i 0.958138i
\(699\) 0 0
\(700\) −6.00000 + 8.00000i −0.226779 + 0.302372i
\(701\) −38.1838 −1.44218 −0.721090 0.692841i \(-0.756359\pi\)
−0.721090 + 0.692841i \(0.756359\pi\)
\(702\) 0 0
\(703\) 7.65685i 0.288784i
\(704\) 4.24264 0.159901
\(705\) 0 0
\(706\) −26.1421 −0.983872
\(707\) 18.6274i 0.700556i
\(708\) 0 0
\(709\) −35.4142 −1.33001 −0.665004 0.746839i \(-0.731570\pi\)
−0.665004 + 0.746839i \(0.731570\pi\)
\(710\) 12.0000 + 4.00000i 0.450352 + 0.150117i
\(711\) 0 0
\(712\) 14.8284i 0.555719i
\(713\) 6.00000i 0.224702i
\(714\) 0 0
\(715\) 43.4558 + 14.4853i 1.62516 + 0.541719i
\(716\) 10.3431 0.386542
\(717\) 0 0
\(718\) 19.7990i 0.738892i
\(719\) 28.1421 1.04952 0.524762 0.851249i \(-0.324154\pi\)
0.524762 + 0.851249i \(0.324154\pi\)
\(720\) 0 0
\(721\) 24.2843 0.904394
\(722\) 13.9706i 0.519931i
\(723\) 0 0
\(724\) −16.3848 −0.608935
\(725\) 30.6274 + 22.9706i 1.13747 + 0.853105i
\(726\) 0 0
\(727\) 31.6569i 1.17409i 0.809555 + 0.587044i \(0.199708\pi\)
−0.809555 + 0.587044i \(0.800292\pi\)
\(728\) 9.65685i 0.357907i
\(729\) 0 0
\(730\) −2.58579 + 7.75736i −0.0957042 + 0.287113i
\(731\) −2.05887 −0.0761502
\(732\) 0 0
\(733\) 8.58579i 0.317123i 0.987349 + 0.158562i \(0.0506857\pi\)
−0.987349 + 0.158562i \(0.949314\pi\)
\(734\) 34.9706 1.29079
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 2.48528i 0.0915465i
\(738\) 0 0
\(739\) −42.6274 −1.56807 −0.784037 0.620714i \(-0.786843\pi\)
−0.784037 + 0.620714i \(0.786843\pi\)
\(740\) −7.24264 2.41421i −0.266245 0.0887483i
\(741\) 0 0
\(742\) 26.8284i 0.984903i
\(743\) 2.34315i 0.0859617i −0.999076 0.0429808i \(-0.986315\pi\)
0.999076 0.0429808i \(-0.0136854\pi\)
\(744\) 0 0
\(745\) 9.00000 27.0000i 0.329734 0.989203i
\(746\) −19.2132 −0.703445
\(747\) 0 0
\(748\) 4.97056i 0.181742i
\(749\) −30.1421 −1.10137
\(750\) 0 0
\(751\) −48.8284 −1.78177 −0.890887 0.454224i \(-0.849916\pi\)
−0.890887 + 0.454224i \(0.849916\pi\)
\(752\) 4.82843i 0.176075i
\(753\) 0 0
\(754\) 36.9706 1.34639
\(755\) 11.0711 33.2132i 0.402917 1.20875i
\(756\) 0 0
\(757\) 42.2426i 1.53533i −0.640848 0.767667i \(-0.721417\pi\)
0.640848 0.767667i \(-0.278583\pi\)
\(758\) 19.2132i 0.697855i
\(759\) 0 0
\(760\) 4.75736 + 1.58579i 0.172568 + 0.0575225i
\(761\) 24.3431 0.882438 0.441219 0.897399i \(-0.354546\pi\)
0.441219 + 0.897399i \(0.354546\pi\)
\(762\) 0 0
\(763\) 30.8284i 1.11606i
\(764\) −5.17157 −0.187101
\(765\) 0 0
\(766\) −18.6274 −0.673036
\(767\) 40.9706i 1.47936i
\(768\) 0 0
\(769\) −45.7990 −1.65155 −0.825777 0.563997i \(-0.809263\pi\)
−0.825777 + 0.563997i \(0.809263\pi\)
\(770\) −6.00000 + 18.0000i −0.216225 + 0.648675i
\(771\) 0 0
\(772\) 12.9706i 0.466821i
\(773\) 16.9289i 0.608891i 0.952530 + 0.304446i \(0.0984712\pi\)
−0.952530 + 0.304446i \(0.901529\pi\)
\(774\) 0 0
\(775\) −24.0000 18.0000i −0.862105 0.646579i
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 6.38478i 0.228905i
\(779\) 2.62742 0.0941370
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 1.17157i 0.0418954i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 14.6985 + 4.89949i 0.524611 + 0.174870i
\(786\) 0 0
\(787\) 2.72792i 0.0972399i −0.998817 0.0486200i \(-0.984518\pi\)
0.998817 0.0486200i \(-0.0154823\pi\)
\(788\) 24.6274i 0.877315i
\(789\) 0 0
\(790\) 16.2426 + 5.41421i 0.577887 + 0.192629i
\(791\) 33.9411 1.20681
\(792\) 0 0
\(793\) 16.4853i 0.585410i
\(794\) −8.34315 −0.296087
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) 21.4142i 0.758530i −0.925288 0.379265i \(-0.876177\pi\)
0.925288 0.379265i \(-0.123823\pi\)
\(798\) 0 0
\(799\) −5.65685 −0.200125
\(800\) 3.00000 4.00000i 0.106066 0.141421i
\(801\) 0 0
\(802\) 28.9706i 1.02299i
\(803\) 15.5147i 0.547503i
\(804\) 0 0
\(805\) 1.41421 4.24264i 0.0498445 0.149533i
\(806\) −28.9706 −1.02044
\(807\) 0 0
\(808\) 9.31371i 0.327655i
\(809\) 13.3137 0.468085 0.234043 0.972226i \(-0.424804\pi\)
0.234043 + 0.972226i \(0.424804\pi\)
\(810\) 0 0
\(811\) 6.82843 0.239779 0.119889 0.992787i \(-0.461746\pi\)
0.119889 + 0.992787i \(0.461746\pi\)
\(812\) 15.3137i 0.537406i
\(813\) 0 0
\(814\) −14.4853 −0.507709
\(815\) −2.48528 0.828427i −0.0870556 0.0290185i
\(816\) 0 0
\(817\) 3.94113i 0.137883i
\(818\) 18.0000i 0.629355i
\(819\) 0 0
\(820\) 0.828427 2.48528i 0.0289299 0.0867898i
\(821\) 6.68629 0.233353 0.116677 0.993170i \(-0.462776\pi\)
0.116677 + 0.993170i \(0.462776\pi\)
\(822\) 0 0
\(823\) 32.2843i 1.12536i 0.826675 + 0.562679i \(0.190229\pi\)
−0.826675 + 0.562679i \(0.809771\pi\)
\(824\) −12.1421 −0.422991
\(825\) 0 0
\(826\) 16.9706 0.590481
\(827\) 18.1005i 0.629416i 0.949188 + 0.314708i \(0.101907\pi\)
−0.949188 + 0.314708i \(0.898093\pi\)
\(828\) 0 0
\(829\) −50.2843 −1.74644 −0.873222 0.487322i \(-0.837974\pi\)
−0.873222 + 0.487322i \(0.837974\pi\)
\(830\) 1.00000 3.00000i 0.0347105 0.104132i
\(831\) 0 0
\(832\) 4.82843i 0.167396i
\(833\) 3.51472i 0.121778i
\(834\) 0 0
\(835\) 18.7279 + 6.24264i 0.648106 + 0.216035i
\(836\) 9.51472 0.329073
\(837\) 0 0
\(838\) 24.9289i 0.861156i
\(839\) −33.9411 −1.17178 −0.585889 0.810391i \(-0.699255\pi\)
−0.585889 + 0.810391i \(0.699255\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 10.2426i 0.352985i
\(843\) 0 0
\(844\) 20.4853 0.705132
\(845\) 7.29289 21.8787i 0.250883 0.752649i
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 13.4142i 0.460646i
\(849\) 0 0
\(850\) 4.68629 + 3.51472i 0.160738 + 0.120554i
\(851\) 3.41421 0.117038
\(852\) 0 0
\(853\) 1.31371i 0.0449805i 0.999747 + 0.0224903i \(0.00715948\pi\)
−0.999747 + 0.0224903i \(0.992841\pi\)
\(854\) 6.82843 0.233664
\(855\) 0 0
\(856\) 15.0711 0.515118
\(857\) 45.1716i 1.54303i −0.636210 0.771516i \(-0.719499\pi\)
0.636210 0.771516i \(-0.280501\pi\)
\(858\) 0 0
\(859\) 32.2843 1.10153 0.550763 0.834662i \(-0.314337\pi\)
0.550763 + 0.834662i \(0.314337\pi\)
\(860\) 3.72792 + 1.24264i 0.127121 + 0.0423737i
\(861\) 0 0
\(862\) 8.48528i 0.289010i
\(863\) 22.6274i 0.770246i −0.922865 0.385123i \(-0.874159\pi\)
0.922865 0.385123i \(-0.125841\pi\)
\(864\) 0 0
\(865\) 6.72792 + 2.24264i 0.228756 + 0.0762521i
\(866\) −26.9706 −0.916497
\(867\) 0 0
\(868\) 12.0000i 0.407307i
\(869\) 32.4853 1.10199
\(870\) 0 0
\(871\) −2.82843 −0.0958376
\(872\) 15.4142i 0.521991i
\(873\) 0 0
\(874\) −2.24264 −0.0758585
\(875\) 12.7279 + 18.3848i 0.430282 + 0.621519i
\(876\) 0 0
\(877\) 11.1716i 0.377237i 0.982050 + 0.188619i \(0.0604010\pi\)
−0.982050 + 0.188619i \(0.939599\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 3.00000 9.00000i 0.101130 0.303390i
\(881\) 2.14214 0.0721704 0.0360852 0.999349i \(-0.488511\pi\)
0.0360852 + 0.999349i \(0.488511\pi\)
\(882\) 0 0
\(883\) 13.1716i 0.443259i 0.975131 + 0.221629i \(0.0711375\pi\)
−0.975131 + 0.221629i \(0.928862\pi\)
\(884\) 5.65685 0.190261
\(885\) 0 0
\(886\) −23.3137 −0.783239
\(887\) 48.4264i 1.62600i −0.582264 0.813000i \(-0.697833\pi\)
0.582264 0.813000i \(-0.302167\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) −31.4558 10.4853i −1.05440 0.351467i
\(891\) 0 0
\(892\) 16.0000i 0.535720i
\(893\) 10.8284i 0.362359i
\(894\) 0 0
\(895\) 7.31371 21.9411i 0.244470 0.733411i
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 22.8284i 0.761794i
\(899\) −45.9411 −1.53222
\(900\) 0 0
\(901\) 15.7157 0.523567
\(902\) 4.97056i 0.165502i
\(903\) 0 0
\(904\) −16.9706 −0.564433
\(905\) −11.5858 + 34.7574i −0.385125 + 1.15537i
\(906\) 0 0
\(907\) 27.2132i 0.903600i −0.892119 0.451800i \(-0.850782\pi\)
0.892119 0.451800i \(-0.149218\pi\)
\(908\) 14.3848i 0.477376i
\(909\) 0 0
\(910\) 20.4853 + 6.82843i 0.679080 + 0.226360i
\(911\) 0.201010 0.00665976 0.00332988 0.999994i \(-0.498940\pi\)
0.00332988 + 0.999994i \(0.498940\pi\)
\(912\) 0 0
\(913\) 6.00000i 0.198571i
\(914\) 1.02944 0.0340508
\(915\) 0 0
\(916\) 1.75736 0.0580648
\(917\) 20.6863i 0.683122i
\(918\) 0 0
\(919\) −5.02944 −0.165906 −0.0829529 0.996553i \(-0.526435\pi\)
−0.0829529 + 0.996553i \(0.526435\pi\)
\(920\) −0.707107 + 2.12132i −0.0233126 + 0.0699379i
\(921\) 0 0
\(922\) 12.8284i 0.422482i
\(923\) 27.3137i 0.899042i
\(924\) 0 0
\(925\) −10.2426 + 13.6569i −0.336776 + 0.449035i
\(926\) −2.97056 −0.0976187
\(927\) 0 0
\(928\) 7.65685i 0.251349i
\(929\) 42.1421 1.38264 0.691319 0.722549i \(-0.257030\pi\)
0.691319 + 0.722549i \(0.257030\pi\)
\(930\) 0 0
\(931\) 6.72792 0.220499
\(932\) 26.1421i 0.856314i
\(933\) 0 0
\(934\) −39.3553 −1.28775
\(935\) 10.5442 + 3.51472i 0.344831 + 0.114944i
\(936\) 0 0
\(937\) 44.4264i 1.45135i −0.688039 0.725674i \(-0.741528\pi\)
0.688039 0.725674i \(-0.258472\pi\)
\(938\) 1.17157i 0.0382532i
\(939\) 0 0
\(940\) 10.2426 + 3.41421i 0.334078 + 0.111359i
\(941\) 31.5563 1.02871 0.514354 0.857578i \(-0.328032\pi\)
0.514354 + 0.857578i \(0.328032\pi\)
\(942\) 0 0
\(943\) 1.17157i 0.0381517i
\(944\) −8.48528 −0.276172
\(945\) 0 0
\(946\) 7.45584 0.242410
\(947\) 1.45584i 0.0473086i 0.999720 + 0.0236543i \(0.00753010\pi\)
−0.999720 + 0.0236543i \(0.992470\pi\)
\(948\) 0 0
\(949\) 17.6569 0.573166
\(950\) 6.72792 8.97056i 0.218283 0.291043i
\(951\) 0 0
\(952\) 2.34315i 0.0759418i
\(953\) 40.0833i 1.29842i −0.760607 0.649212i \(-0.775099\pi\)
0.760607 0.649212i \(-0.224901\pi\)
\(954\) 0 0
\(955\) −3.65685 + 10.9706i −0.118333 + 0.354999i
\(956\) 11.1716 0.361314
\(957\) 0 0
\(958\) 4.68629i 0.151407i
\(959\) −16.9706 −0.548008
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 16.4853i 0.531507i
\(963\) 0 0
\(964\) 10.4853 0.337708
\(965\) −27.5147 9.17157i −0.885730 0.295243i
\(966\) 0 0
\(967\) 4.97056i 0.159843i −0.996801 0.0799213i \(-0.974533\pi\)
0.996801 0.0799213i \(-0.0254669\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 4.24264 12.7279i 0.136223 0.408669i
\(971\) 17.6152 0.565299 0.282650 0.959223i \(-0.408787\pi\)
0.282650 + 0.959223i \(0.408787\pi\)
\(972\) 0 0
\(973\) 33.9411i 1.08810i
\(974\) −27.9411 −0.895291
\(975\) 0 0
\(976\) −3.41421 −0.109286
\(977\) 19.7990i 0.633426i 0.948521 + 0.316713i \(0.102579\pi\)
−0.948521 + 0.316713i \(0.897421\pi\)
\(978\) 0 0
\(979\) −62.9117 −2.01067
\(980\) 2.12132 6.36396i 0.0677631 0.203289i
\(981\) 0 0
\(982\) 26.6274i 0.849715i
\(983\) 20.4853i 0.653379i 0.945132 + 0.326690i \(0.105933\pi\)
−0.945132 + 0.326690i \(0.894067\pi\)
\(984\) 0 0
\(985\) 52.2426 + 17.4142i 1.66459 + 0.554863i
\(986\) 8.97056 0.285681
\(987\) 0 0
\(988\) 10.8284i 0.344498i
\(989\) −1.75736 −0.0558808
\(990\) 0 0
\(991\) 51.9411 1.64996 0.824982 0.565159i \(-0.191185\pi\)
0.824982 + 0.565159i \(0.191185\pi\)
\(992\) 6.00000i 0.190500i
\(993\) 0 0
\(994\) 11.3137 0.358849
\(995\) −9.89949 + 29.6985i −0.313835 + 0.941505i
\(996\) 0 0
\(997\) 46.9706i 1.48757i 0.668417 + 0.743786i \(0.266972\pi\)
−0.668417 + 0.743786i \(0.733028\pi\)
\(998\) 13.4558i 0.425937i
\(999\) 0 0
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 2070.2.d.a.829.3 4
3.2 odd 2 690.2.d.b.139.2 4
5.4 even 2 inner 2070.2.d.a.829.1 4
15.2 even 4 3450.2.a.bk.1.2 2
15.8 even 4 3450.2.a.bg.1.2 2
15.14 odd 2 690.2.d.b.139.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.b.139.2 4 3.2 odd 2
690.2.d.b.139.4 yes 4 15.14 odd 2
2070.2.d.a.829.1 4 5.4 even 2 inner
2070.2.d.a.829.3 4 1.1 even 1 trivial
3450.2.a.bg.1.2 2 15.8 even 4
3450.2.a.bk.1.2 2 15.2 even 4