Properties

Label 2448.2.be.n.1585.1
Level $2448$
Weight $2$
Character 2448.1585
Analytic conductor $19.547$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2448,2,Mod(1441,2448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2448, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2448.1441");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2448.be (of order \(4\), 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: \(19.5473784148\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1585.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2448.1585
Dual form 2448.2.be.n.1441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.00000 - 3.00000i) q^{5} +(3.00000 + 3.00000i) q^{7} +(1.00000 + 1.00000i) q^{11} +2.00000 q^{13} +(-1.00000 - 4.00000i) q^{17} -6.00000i q^{19} +(1.00000 + 1.00000i) q^{23} -13.0000i q^{25} +(-1.00000 + 1.00000i) q^{29} +(1.00000 - 1.00000i) q^{31} +18.0000 q^{35} +(-3.00000 + 3.00000i) q^{37} +(-1.00000 - 1.00000i) q^{41} +2.00000i q^{43} +11.0000i q^{49} +4.00000i q^{53} +6.00000 q^{55} +6.00000i q^{59} +(1.00000 + 1.00000i) q^{61} +(6.00000 - 6.00000i) q^{65} +4.00000 q^{67} +(-5.00000 + 5.00000i) q^{71} +(1.00000 - 1.00000i) q^{73} +6.00000i q^{77} +(-9.00000 - 9.00000i) q^{79} +14.0000i q^{83} +(-15.0000 - 9.00000i) q^{85} -6.00000 q^{89} +(6.00000 + 6.00000i) q^{91} +(-18.0000 - 18.0000i) q^{95} +(13.0000 - 13.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 6 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} + 2 q^{23} - 2 q^{29} + 2 q^{31} + 36 q^{35} - 6 q^{37} - 2 q^{41} + 12 q^{55} + 2 q^{61} + 12 q^{65} + 8 q^{67} - 10 q^{71} + 2 q^{73} - 18 q^{79}+ \cdots + 26 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(1361\) \(1873\) \(2143\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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 0
\(4\) 0 0
\(5\) 3.00000 3.00000i 1.34164 1.34164i 0.447214 0.894427i \(-0.352416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 3.00000 + 3.00000i 1.13389 + 1.13389i 0.989524 + 0.144370i \(0.0461154\pi\)
0.144370 + 0.989524i \(0.453885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 + 1.00000i 0.301511 + 0.301511i 0.841605 0.540094i \(-0.181611\pi\)
−0.540094 + 0.841605i \(0.681611\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 4.00000i −0.242536 0.970143i
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 + 1.00000i 0.208514 + 0.208514i 0.803636 0.595121i \(-0.202896\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) 0 0
\(25\) 13.0000i 2.60000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 + 1.00000i −0.185695 + 0.185695i −0.793832 0.608137i \(-0.791917\pi\)
0.608137 + 0.793832i \(0.291917\pi\)
\(30\) 0 0
\(31\) 1.00000 1.00000i 0.179605 0.179605i −0.611578 0.791184i \(-0.709465\pi\)
0.791184 + 0.611578i \(0.209465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 18.0000 3.04256
\(36\) 0 0
\(37\) −3.00000 + 3.00000i −0.493197 + 0.493197i −0.909312 0.416115i \(-0.863391\pi\)
0.416115 + 0.909312i \(0.363391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 1.00000i −0.156174 0.156174i 0.624695 0.780869i \(-0.285223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 11.0000i 1.57143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.00000i 0.128037 + 0.128037i 0.768221 0.640184i \(-0.221142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 6.00000i 0.744208 0.744208i
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 + 5.00000i −0.593391 + 0.593391i −0.938546 0.345155i \(-0.887826\pi\)
0.345155 + 0.938546i \(0.387826\pi\)
\(72\) 0 0
\(73\) 1.00000 1.00000i 0.117041 0.117041i −0.646160 0.763202i \(-0.723626\pi\)
0.763202 + 0.646160i \(0.223626\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) −9.00000 9.00000i −1.01258 1.01258i −0.999920 0.0126592i \(-0.995970\pi\)
−0.0126592 0.999920i \(-0.504030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) 0 0
\(85\) −15.0000 9.00000i −1.62698 0.976187i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.00000 + 6.00000i 0.628971 + 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −18.0000 18.0000i −1.84676 1.84676i
\(96\) 0 0
\(97\) 13.0000 13.0000i 1.31995 1.31995i 0.406138 0.913812i \(-0.366875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.00000 7.00000i 0.676716 0.676716i −0.282540 0.959256i \(-0.591177\pi\)
0.959256 + 0.282540i \(0.0911770\pi\)
\(108\) 0 0
\(109\) 9.00000 + 9.00000i 0.862044 + 0.862044i 0.991575 0.129532i \(-0.0413474\pi\)
−0.129532 + 0.991575i \(0.541347\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.00000 5.00000i −0.470360 0.470360i 0.431671 0.902031i \(-0.357924\pi\)
−0.902031 + 0.431671i \(0.857924\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.00000 15.0000i 0.825029 1.37505i
\(120\) 0 0
\(121\) 9.00000i 0.818182i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −24.0000 24.0000i −2.14663 2.14663i
\(126\) 0 0
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.00000 + 9.00000i −0.786334 + 0.786334i −0.980891 0.194557i \(-0.937673\pi\)
0.194557 + 0.980891i \(0.437673\pi\)
\(132\) 0 0
\(133\) 18.0000 18.0000i 1.56080 1.56080i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −11.0000 + 11.0000i −0.933008 + 0.933008i −0.997893 0.0648849i \(-0.979332\pi\)
0.0648849 + 0.997893i \(0.479332\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 + 2.00000i 0.167248 + 0.167248i
\(144\) 0 0
\(145\) 6.00000i 0.498273i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 2.00000i 0.162758i −0.996683 0.0813788i \(-0.974068\pi\)
0.996683 0.0813788i \(-0.0259324\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000i 0.481932i
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000i 0.472866i
\(162\) 0 0
\(163\) −1.00000 1.00000i −0.0783260 0.0783260i 0.666858 0.745184i \(-0.267639\pi\)
−0.745184 + 0.666858i \(0.767639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.0000 + 13.0000i −1.00597 + 1.00597i −0.00598813 + 0.999982i \(0.501906\pi\)
−0.999982 + 0.00598813i \(0.998094\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.00000 7.00000i 0.532200 0.532200i −0.389026 0.921227i \(-0.627189\pi\)
0.921227 + 0.389026i \(0.127189\pi\)
\(174\) 0 0
\(175\) 39.0000 39.0000i 2.94812 2.94812i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.0000i 1.34538i −0.739923 0.672692i \(-0.765138\pi\)
0.739923 0.672692i \(-0.234862\pi\)
\(180\) 0 0
\(181\) 9.00000 + 9.00000i 0.668965 + 0.668965i 0.957476 0.288512i \(-0.0931604\pi\)
−0.288512 + 0.957476i \(0.593160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 18.0000i 1.32339i
\(186\) 0 0
\(187\) 3.00000 5.00000i 0.219382 0.365636i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) 9.00000 + 9.00000i 0.647834 + 0.647834i 0.952469 0.304635i \(-0.0985345\pi\)
−0.304635 + 0.952469i \(0.598534\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0000 + 11.0000i 0.783718 + 0.783718i 0.980456 0.196738i \(-0.0630350\pi\)
−0.196738 + 0.980456i \(0.563035\pi\)
\(198\) 0 0
\(199\) −7.00000 + 7.00000i −0.496217 + 0.496217i −0.910258 0.414041i \(-0.864117\pi\)
0.414041 + 0.910258i \(0.364117\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.00000 6.00000i 0.415029 0.415029i
\(210\) 0 0
\(211\) −5.00000 5.00000i −0.344214 0.344214i 0.513735 0.857949i \(-0.328262\pi\)
−0.857949 + 0.513735i \(0.828262\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.00000 + 6.00000i 0.409197 + 0.409197i
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 8.00000i −0.134535 0.538138i
\(222\) 0 0
\(223\) 26.0000i 1.74109i −0.492090 0.870544i \(-0.663767\pi\)
0.492090 0.870544i \(-0.336233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.0000 15.0000i −0.995585 0.995585i 0.00440533 0.999990i \(-0.498598\pi\)
−0.999990 + 0.00440533i \(0.998598\pi\)
\(228\) 0 0
\(229\) 16.0000i 1.05731i 0.848837 + 0.528655i \(0.177303\pi\)
−0.848837 + 0.528655i \(0.822697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.0000 19.0000i 1.24473 1.24473i 0.286716 0.958016i \(-0.407437\pi\)
0.958016 0.286716i \(-0.0925635\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 9.00000 9.00000i 0.579741 0.579741i −0.355091 0.934832i \(-0.615550\pi\)
0.934832 + 0.355091i \(0.115550\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 33.0000 + 33.0000i 2.10829 + 2.10829i
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.0000i 1.24757i 0.781598 + 0.623783i \(0.214405\pi\)
−0.781598 + 0.623783i \(0.785595\pi\)
\(258\) 0 0
\(259\) −18.0000 −1.11847
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 12.0000 + 12.0000i 0.737154 + 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.0000 + 13.0000i −0.792624 + 0.792624i −0.981920 0.189296i \(-0.939379\pi\)
0.189296 + 0.981920i \(0.439379\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.0000 13.0000i 0.783929 0.783929i
\(276\) 0 0
\(277\) 9.00000 9.00000i 0.540758 0.540758i −0.382993 0.923751i \(-0.625107\pi\)
0.923751 + 0.382993i \(0.125107\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000i 0.477240i 0.971113 + 0.238620i \(0.0766950\pi\)
−0.971113 + 0.238620i \(0.923305\pi\)
\(282\) 0 0
\(283\) 15.0000 + 15.0000i 0.891657 + 0.891657i 0.994679 0.103022i \(-0.0328511\pi\)
−0.103022 + 0.994679i \(0.532851\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) −15.0000 + 8.00000i −0.882353 + 0.470588i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 0 0
\(295\) 18.0000 + 18.0000i 1.04800 + 1.04800i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000 + 2.00000i 0.115663 + 0.115663i
\(300\) 0 0
\(301\) −6.00000 + 6.00000i −0.345834 + 0.345834i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.0000 19.0000i 1.07739 1.07739i 0.0806486 0.996743i \(-0.474301\pi\)
0.996743 0.0806486i \(-0.0256991\pi\)
\(312\) 0 0
\(313\) 9.00000 + 9.00000i 0.508710 + 0.508710i 0.914130 0.405420i \(-0.132875\pi\)
−0.405420 + 0.914130i \(0.632875\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.00000 5.00000i −0.280828 0.280828i 0.552611 0.833439i \(-0.313631\pi\)
−0.833439 + 0.552611i \(0.813631\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 + 6.00000i −1.33540 + 0.333849i
\(324\) 0 0
\(325\) 26.0000i 1.44222i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000i 0.109930i 0.998488 + 0.0549650i \(0.0175047\pi\)
−0.998488 + 0.0549650i \(0.982495\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 12.0000i 0.655630 0.655630i
\(336\) 0 0
\(337\) −15.0000 + 15.0000i −0.817102 + 0.817102i −0.985687 0.168585i \(-0.946080\pi\)
0.168585 + 0.985687i \(0.446080\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −12.0000 + 12.0000i −0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.0000 + 17.0000i 0.912608 + 0.912608i 0.996477 0.0838690i \(-0.0267277\pi\)
−0.0838690 + 0.996477i \(0.526728\pi\)
\(348\) 0 0
\(349\) 12.0000i 0.642345i −0.947021 0.321173i \(-0.895923\pi\)
0.947021 0.321173i \(-0.104077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 30.0000i 1.59223i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.0000i 0.738892i −0.929252 0.369446i \(-0.879548\pi\)
0.929252 0.369446i \(-0.120452\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000i 0.314054i
\(366\) 0 0
\(367\) −21.0000 21.0000i −1.09619 1.09619i −0.994852 0.101339i \(-0.967687\pi\)
−0.101339 0.994852i \(-0.532313\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 + 12.0000i −0.623009 + 0.623009i
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 + 2.00000i −0.103005 + 0.103005i
\(378\) 0 0
\(379\) −15.0000 + 15.0000i −0.770498 + 0.770498i −0.978194 0.207695i \(-0.933404\pi\)
0.207695 + 0.978194i \(0.433404\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.0000i 1.32854i 0.747494 + 0.664269i \(0.231257\pi\)
−0.747494 + 0.664269i \(0.768743\pi\)
\(384\) 0 0
\(385\) 18.0000 + 18.0000i 0.917365 + 0.917365i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.0000i 0.811232i −0.914044 0.405616i \(-0.867057\pi\)
0.914044 0.405616i \(-0.132943\pi\)
\(390\) 0 0
\(391\) 3.00000 5.00000i 0.151717 0.252861i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −54.0000 −2.71703
\(396\) 0 0
\(397\) −23.0000 23.0000i −1.15434 1.15434i −0.985674 0.168663i \(-0.946055\pi\)
−0.168663 0.985674i \(-0.553945\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 9.00000i −0.449439 0.449439i 0.445729 0.895168i \(-0.352944\pi\)
−0.895168 + 0.445729i \(0.852944\pi\)
\(402\) 0 0
\(403\) 2.00000 2.00000i 0.0996271 0.0996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.0000 + 18.0000i −0.885722 + 0.885722i
\(414\) 0 0
\(415\) 42.0000 + 42.0000i 2.06170 + 2.06170i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.00000 + 5.00000i 0.244266 + 0.244266i 0.818612 0.574346i \(-0.194744\pi\)
−0.574346 + 0.818612i \(0.694744\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −52.0000 + 13.0000i −2.52237 + 0.630593i
\(426\) 0 0
\(427\) 6.00000i 0.290360i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.00000 7.00000i −0.337178 0.337178i 0.518126 0.855304i \(-0.326630\pi\)
−0.855304 + 0.518126i \(0.826630\pi\)
\(432\) 0 0
\(433\) 12.0000i 0.576683i 0.957528 + 0.288342i \(0.0931039\pi\)
−0.957528 + 0.288342i \(0.906896\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.00000 6.00000i 0.287019 0.287019i
\(438\) 0 0
\(439\) −3.00000 + 3.00000i −0.143182 + 0.143182i −0.775064 0.631882i \(-0.782283\pi\)
0.631882 + 0.775064i \(0.282283\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −18.0000 + 18.0000i −0.853282 + 0.853282i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.00000 + 3.00000i 0.141579 + 0.141579i 0.774344 0.632765i \(-0.218080\pi\)
−0.632765 + 0.774344i \(0.718080\pi\)
\(450\) 0 0
\(451\) 2.00000i 0.0941763i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 36.0000 1.68771
\(456\) 0 0
\(457\) 20.0000i 0.935561i 0.883845 + 0.467780i \(0.154946\pi\)
−0.883845 + 0.467780i \(0.845054\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 28.0000i 1.30409i 0.758180 + 0.652045i \(0.226089\pi\)
−0.758180 + 0.652045i \(0.773911\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000i 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) 0 0
\(469\) 12.0000 + 12.0000i 0.554109 + 0.554109i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.00000 + 2.00000i −0.0919601 + 0.0919601i
\(474\) 0 0
\(475\) −78.0000 −3.57889
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.00000 + 9.00000i −0.411220 + 0.411220i −0.882164 0.470943i \(-0.843914\pi\)
0.470943 + 0.882164i \(0.343914\pi\)
\(480\) 0 0
\(481\) −6.00000 + 6.00000i −0.273576 + 0.273576i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 78.0000i 3.54180i
\(486\) 0 0
\(487\) −13.0000 13.0000i −0.589086 0.589086i 0.348298 0.937384i \(-0.386760\pi\)
−0.937384 + 0.348298i \(0.886760\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000i 0.0902587i −0.998981 0.0451294i \(-0.985630\pi\)
0.998981 0.0451294i \(-0.0143700\pi\)
\(492\) 0 0
\(493\) 5.00000 + 3.00000i 0.225189 + 0.135113i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −30.0000 −1.34568
\(498\) 0 0
\(499\) −13.0000 13.0000i −0.581960 0.581960i 0.353482 0.935441i \(-0.384998\pi\)
−0.935441 + 0.353482i \(0.884998\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19.0000 19.0000i −0.847168 0.847168i 0.142611 0.989779i \(-0.454450\pi\)
−0.989779 + 0.142611i \(0.954450\pi\)
\(504\) 0 0
\(505\) 18.0000 18.0000i 0.800989 0.800989i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.0000 + 23.0000i 1.00765 + 1.00765i 0.999971 + 0.00767777i \(0.00244393\pi\)
0.00767777 + 0.999971i \(0.497556\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.00000 3.00000i −0.217803 0.130682i
\(528\) 0 0
\(529\) 21.0000i 0.913043i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.00000 2.00000i −0.0866296 0.0866296i
\(534\) 0 0
\(535\) 42.0000i 1.81582i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11.0000 + 11.0000i −0.473804 + 0.473804i
\(540\) 0 0
\(541\) −19.0000 + 19.0000i −0.816874 + 0.816874i −0.985654 0.168780i \(-0.946017\pi\)
0.168780 + 0.985654i \(0.446017\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 54.0000 2.31311
\(546\) 0 0
\(547\) −15.0000 + 15.0000i −0.641354 + 0.641354i −0.950888 0.309535i \(-0.899827\pi\)
0.309535 + 0.950888i \(0.399827\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 + 6.00000i 0.255609 + 0.255609i
\(552\) 0 0
\(553\) 54.0000i 2.29631i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 4.00000i 0.169182i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.0000i 0.927189i 0.886047 + 0.463595i \(0.153441\pi\)
−0.886047 + 0.463595i \(0.846559\pi\)
\(564\) 0 0
\(565\) −30.0000 −1.26211
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.0000i 0.670755i −0.942084 0.335377i \(-0.891136\pi\)
0.942084 0.335377i \(-0.108864\pi\)
\(570\) 0 0
\(571\) −13.0000 13.0000i −0.544033 0.544033i 0.380676 0.924709i \(-0.375691\pi\)
−0.924709 + 0.380676i \(0.875691\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.0000 13.0000i 0.542137 0.542137i
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −42.0000 + 42.0000i −1.74245 + 1.74245i
\(582\) 0 0
\(583\) −4.00000 + 4.00000i −0.165663 + 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 38.0000i 1.56843i 0.620491 + 0.784214i \(0.286934\pi\)
−0.620491 + 0.784214i \(0.713066\pi\)
\(588\) 0 0
\(589\) −6.00000 6.00000i −0.247226 0.247226i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.0000i 0.985562i 0.870153 + 0.492781i \(0.164020\pi\)
−0.870153 + 0.492781i \(0.835980\pi\)
\(594\) 0 0
\(595\) −18.0000 72.0000i −0.737928 2.95171i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) 1.00000 + 1.00000i 0.0407909 + 0.0407909i 0.727208 0.686417i \(-0.240818\pi\)
−0.686417 + 0.727208i \(0.740818\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.0000 27.0000i −1.09771 1.09771i
\(606\) 0 0
\(607\) 9.00000 9.00000i 0.365299 0.365299i −0.500461 0.865759i \(-0.666836\pi\)
0.865759 + 0.500461i \(0.166836\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.00000 + 5.00000i −0.201292 + 0.201292i −0.800554 0.599261i \(-0.795461\pi\)
0.599261 + 0.800554i \(0.295461\pi\)
\(618\) 0 0
\(619\) 15.0000 + 15.0000i 0.602901 + 0.602901i 0.941081 0.338180i \(-0.109811\pi\)
−0.338180 + 0.941081i \(0.609811\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −18.0000 18.0000i −0.721155 0.721155i
\(624\) 0 0
\(625\) −79.0000 −3.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.0000 + 9.00000i 0.598089 + 0.358854i
\(630\) 0 0
\(631\) 22.0000i 0.875806i 0.899022 + 0.437903i \(0.144279\pi\)
−0.899022 + 0.437903i \(0.855721\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 42.0000 + 42.0000i 1.66672 + 1.66672i
\(636\) 0 0
\(637\) 22.0000i 0.871672i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.0000 11.0000i 0.434474 0.434474i −0.455673 0.890147i \(-0.650601\pi\)
0.890147 + 0.455673i \(0.150601\pi\)
\(642\) 0 0
\(643\) −31.0000 + 31.0000i −1.22252 + 1.22252i −0.255788 + 0.966733i \(0.582335\pi\)
−0.966733 + 0.255788i \(0.917665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) −6.00000 + 6.00000i −0.235521 + 0.235521i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.0000 + 11.0000i 0.430463 + 0.430463i 0.888786 0.458323i \(-0.151550\pi\)
−0.458323 + 0.888786i \(0.651550\pi\)
\(654\) 0 0
\(655\) 54.0000i 2.10995i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 36.0000i 1.40024i 0.714026 + 0.700119i \(0.246870\pi\)
−0.714026 + 0.700119i \(0.753130\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 108.000i 4.18806i
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.00000i 0.0772091i
\(672\) 0 0
\(673\) −11.0000 11.0000i −0.424019 0.424019i 0.462566 0.886585i \(-0.346929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.00000 + 1.00000i −0.0384331 + 0.0384331i −0.726062 0.687629i \(-0.758652\pi\)
0.687629 + 0.726062i \(0.258652\pi\)
\(678\) 0 0
\(679\) 78.0000 2.99337
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.00000 + 1.00000i −0.0382639 + 0.0382639i −0.725980 0.687716i \(-0.758613\pi\)
0.687716 + 0.725980i \(0.258613\pi\)
\(684\) 0 0
\(685\) −18.0000 + 18.0000i −0.687745 + 0.687745i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.00000i 0.304776i
\(690\) 0 0
\(691\) 31.0000 + 31.0000i 1.17930 + 1.17930i 0.979924 + 0.199372i \(0.0638901\pi\)
0.199372 + 0.979924i \(0.436110\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 66.0000i 2.50352i
\(696\) 0 0
\(697\) −3.00000 + 5.00000i −0.113633 + 0.189389i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 18.0000 + 18.0000i 0.678883 + 0.678883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.0000 + 18.0000i 0.676960 + 0.676960i
\(708\) 0 0
\(709\) 33.0000 33.0000i 1.23934 1.23934i 0.279070 0.960271i \(-0.409974\pi\)
0.960271 0.279070i \(-0.0900263\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.0000 + 13.0000i −0.484818 + 0.484818i −0.906666 0.421848i \(-0.861381\pi\)
0.421848 + 0.906666i \(0.361381\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.0000 + 13.0000i 0.482808 + 0.482808i
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 2.00000i 0.295891 0.0739727i
\(732\) 0 0
\(733\) 28.0000i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 + 4.00000i 0.147342 + 0.147342i
\(738\) 0 0
\(739\) 14.0000i 0.514998i −0.966279 0.257499i \(-0.917102\pi\)
0.966279 0.257499i \(-0.0828985\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.0000 15.0000i 0.550297 0.550297i −0.376230 0.926526i \(-0.622780\pi\)
0.926526 + 0.376230i \(0.122780\pi\)
\(744\) 0 0
\(745\) 30.0000 30.0000i 1.09911 1.09911i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 42.0000 1.53465
\(750\) 0 0
\(751\) −19.0000 + 19.0000i −0.693320 + 0.693320i −0.962961 0.269641i \(-0.913095\pi\)
0.269641 + 0.962961i \(0.413095\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.00000 6.00000i −0.218362 0.218362i
\(756\) 0 0
\(757\) 16.0000i 0.581530i −0.956795 0.290765i \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 54.0000i 1.95493i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.0000i 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) −13.0000 13.0000i −0.466974 0.466974i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 + 6.00000i −0.214972 + 0.214972i
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −30.0000 + 30.0000i −1.07075 + 1.07075i
\(786\) 0 0
\(787\) 5.00000 5.00000i 0.178231 0.178231i −0.612353 0.790584i \(-0.709777\pi\)
0.790584 + 0.612353i \(0.209777\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 30.0000i 1.06668i
\(792\) 0 0
\(793\) 2.00000 + 2.00000i 0.0710221 + 0.0710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.0000i 0.708436i 0.935163 + 0.354218i \(0.115253\pi\)
−0.935163 + 0.354218i \(0.884747\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 18.0000 + 18.0000i 0.634417 + 0.634417i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.00000 + 3.00000i 0.105474 + 0.105474i 0.757875 0.652400i \(-0.226238\pi\)
−0.652400 + 0.757875i \(0.726238\pi\)
\(810\) 0 0
\(811\) −3.00000 + 3.00000i −0.105344 + 0.105344i −0.757814 0.652470i \(-0.773733\pi\)
0.652470 + 0.757814i \(0.273733\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 + 1.00000i −0.0349002 + 0.0349002i −0.724342 0.689441i \(-0.757856\pi\)
0.689441 + 0.724342i \(0.257856\pi\)
\(822\) 0 0
\(823\) −25.0000 25.0000i −0.871445 0.871445i 0.121185 0.992630i \(-0.461331\pi\)
−0.992630 + 0.121185i \(0.961331\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.00000 + 5.00000i 0.173867 + 0.173867i 0.788676 0.614809i \(-0.210767\pi\)
−0.614809 + 0.788676i \(0.710767\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 44.0000 11.0000i 1.52451 0.381127i
\(834\) 0 0
\(835\) 78.0000i 2.69930i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27.0000 27.0000i −0.932144 0.932144i 0.0656962 0.997840i \(-0.479073\pi\)
−0.997840 + 0.0656962i \(0.979073\pi\)
\(840\) 0 0
\(841\) 27.0000i 0.931034i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −27.0000 + 27.0000i −0.928828 + 0.928828i
\(846\) 0 0
\(847\) 27.0000 27.0000i 0.927731 0.927731i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 5.00000 5.00000i 0.171197 0.171197i −0.616308 0.787505i \(-0.711372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −33.0000 33.0000i −1.12726 1.12726i −0.990621 0.136637i \(-0.956370\pi\)
−0.136637 0.990621i \(-0.543630\pi\)
\(858\) 0 0
\(859\) 42.0000i 1.43302i 0.697576 + 0.716511i \(0.254262\pi\)
−0.697576 + 0.716511i \(0.745738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 0 0
\(865\) 42.0000i 1.42804i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 18.0000i 0.610608i
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 144.000i 4.86809i
\(876\) 0 0
\(877\) −23.0000 23.0000i −0.776655 0.776655i 0.202606 0.979260i \(-0.435059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17.0000 + 17.0000i −0.572745 + 0.572745i −0.932894 0.360150i \(-0.882726\pi\)
0.360150 + 0.932894i \(0.382726\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −41.0000 + 41.0000i −1.37665 + 1.37665i −0.526422 + 0.850224i \(0.676467\pi\)
−0.850224 + 0.526422i \(0.823533\pi\)
\(888\) 0 0
\(889\) −42.0000 + 42.0000i −1.40863 + 1.40863i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −54.0000 54.0000i −1.80502 1.80502i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.00000i 0.0667037i
\(900\) 0 0
\(901\) 16.0000 4.00000i 0.533037 0.133259i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 54.0000 1.79502
\(906\) 0 0
\(907\) −33.0000 33.0000i −1.09575 1.09575i −0.994902 0.100845i \(-0.967845\pi\)
−0.100845 0.994902i \(-0.532155\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21.0000 + 21.0000i 0.695761 + 0.695761i 0.963493 0.267732i \(-0.0862743\pi\)
−0.267732 + 0.963493i \(0.586274\pi\)
\(912\) 0 0
\(913\) −14.0000 + 14.0000i −0.463332 + 0.463332i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −54.0000 −1.78324
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.0000 + 10.0000i −0.329154 + 0.329154i
\(924\) 0 0
\(925\) 39.0000 + 39.0000i 1.28231 + 1.28231i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.00000 + 7.00000i 0.229663 + 0.229663i 0.812552 0.582889i \(-0.198078\pi\)
−0.582889 + 0.812552i \(0.698078\pi\)
\(930\) 0 0
\(931\) 66.0000 2.16306
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.00000 24.0000i −0.196221 0.784884i
\(936\) 0 0
\(937\) 32.0000i 1.04539i −0.852518 0.522697i \(-0.824926\pi\)
0.852518 0.522697i \(-0.175074\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.0000 + 27.0000i 0.880175 + 0.880175i 0.993552 0.113377i \(-0.0361668\pi\)
−0.113377 + 0.993552i \(0.536167\pi\)
\(942\) 0 0
\(943\) 2.00000i 0.0651290i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000 27.0000i 0.877382 0.877382i −0.115881 0.993263i \(-0.536969\pi\)
0.993263 + 0.115881i \(0.0369691\pi\)
\(948\) 0 0
\(949\) 2.00000 2.00000i 0.0649227 0.0649227i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 0 0
\(955\) 72.0000 72.0000i 2.32987 2.32987i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 18.0000i −0.581250 0.581250i
\(960\) 0 0
\(961\) 29.0000i 0.935484i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 54.0000 1.73832
\(966\) 0 0
\(967\) 58.0000i 1.86515i −0.360971 0.932577i \(-0.617555\pi\)
0.360971 0.932577i \(-0.382445\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 14.0000i 0.449281i 0.974442 + 0.224641i \(0.0721208\pi\)
−0.974442 + 0.224641i \(0.927879\pi\)
\(972\) 0 0
\(973\) −66.0000 −2.11586
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.00000i 0.255943i −0.991778 0.127971i \(-0.959153\pi\)
0.991778 0.127971i \(-0.0408466\pi\)
\(978\) 0 0
\(979\) −6.00000 6.00000i −0.191761 0.191761i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.00000 3.00000i 0.0956851 0.0956851i −0.657644 0.753329i \(-0.728447\pi\)
0.753329 + 0.657644i \(0.228447\pi\)
\(984\) 0 0
\(985\) 66.0000 2.10293
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.00000 + 2.00000i −0.0635963 + 0.0635963i
\(990\) 0 0
\(991\) −11.0000 + 11.0000i −0.349427 + 0.349427i −0.859896 0.510469i \(-0.829472\pi\)
0.510469 + 0.859896i \(0.329472\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 42.0000i 1.33149i
\(996\) 0 0
\(997\) −3.00000 3.00000i −0.0950110 0.0950110i 0.658004 0.753015i \(-0.271401\pi\)
−0.753015 + 0.658004i \(0.771401\pi\)
\(998\) 0 0
\(999\) 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 2448.2.be.n.1585.1 2
3.2 odd 2 272.2.o.d.225.1 2
4.3 odd 2 1224.2.w.f.361.1 2
12.11 even 2 136.2.k.a.89.1 yes 2
17.13 even 4 inner 2448.2.be.n.1441.1 2
24.5 odd 2 1088.2.o.g.769.1 2
24.11 even 2 1088.2.o.p.769.1 2
51.8 odd 8 4624.2.a.q.1.1 2
51.26 odd 8 4624.2.a.q.1.2 2
51.47 odd 4 272.2.o.d.81.1 2
68.47 odd 4 1224.2.w.f.217.1 2
204.47 even 4 136.2.k.a.81.1 2
204.59 even 8 2312.2.a.h.1.2 2
204.83 even 8 2312.2.b.f.577.1 2
204.155 even 8 2312.2.b.f.577.2 2
204.179 even 8 2312.2.a.h.1.1 2
408.149 odd 4 1088.2.o.g.897.1 2
408.251 even 4 1088.2.o.p.897.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.k.a.81.1 2 204.47 even 4
136.2.k.a.89.1 yes 2 12.11 even 2
272.2.o.d.81.1 2 51.47 odd 4
272.2.o.d.225.1 2 3.2 odd 2
1088.2.o.g.769.1 2 24.5 odd 2
1088.2.o.g.897.1 2 408.149 odd 4
1088.2.o.p.769.1 2 24.11 even 2
1088.2.o.p.897.1 2 408.251 even 4
1224.2.w.f.217.1 2 68.47 odd 4
1224.2.w.f.361.1 2 4.3 odd 2
2312.2.a.h.1.1 2 204.179 even 8
2312.2.a.h.1.2 2 204.59 even 8
2312.2.b.f.577.1 2 204.83 even 8
2312.2.b.f.577.2 2 204.155 even 8
2448.2.be.n.1441.1 2 17.13 even 4 inner
2448.2.be.n.1585.1 2 1.1 even 1 trivial
4624.2.a.q.1.1 2 51.8 odd 8
4624.2.a.q.1.2 2 51.26 odd 8