Properties

Label 845.2.b.b.339.2
Level $845$
Weight $2$
Character 845.339
Analytic conductor $6.747$
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] = [845,2,Mod(339,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("845.339");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.b (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: \(6.74735897080\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 339.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 845.339
Dual form 845.2.b.b.339.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -2.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -2.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +(-1.00000 + 2.00000i) q^{10} -2.00000 q^{11} +2.00000i q^{12} +(-2.00000 + 4.00000i) q^{15} -1.00000 q^{16} -1.00000i q^{18} -6.00000 q^{19} +(2.00000 + 1.00000i) q^{20} -2.00000i q^{22} -6.00000i q^{23} -6.00000 q^{24} +(3.00000 + 4.00000i) q^{25} +4.00000i q^{27} +6.00000 q^{29} +(-4.00000 - 2.00000i) q^{30} +6.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -1.00000 q^{36} +6.00000i q^{37} -6.00000i q^{38} +(-3.00000 + 6.00000i) q^{40} -8.00000 q^{41} +6.00000i q^{43} -2.00000 q^{44} +(-2.00000 - 1.00000i) q^{45} +6.00000 q^{46} -8.00000i q^{47} -2.00000i q^{48} +7.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} -12.0000i q^{53} -4.00000 q^{54} +(-4.00000 - 2.00000i) q^{55} -12.0000i q^{57} +6.00000i q^{58} -2.00000 q^{59} +(-2.00000 + 4.00000i) q^{60} +6.00000 q^{61} +6.00000i q^{62} -7.00000 q^{64} +4.00000 q^{66} -12.0000i q^{67} +12.0000 q^{69} -2.00000 q^{71} -3.00000i q^{72} -6.00000i q^{73} -6.00000 q^{74} +(-8.00000 + 6.00000i) q^{75} -6.00000 q^{76} +(-2.00000 - 1.00000i) q^{80} -11.0000 q^{81} -8.00000i q^{82} +4.00000i q^{83} -6.00000 q^{86} +12.0000i q^{87} -6.00000i q^{88} +8.00000 q^{89} +(1.00000 - 2.00000i) q^{90} -6.00000i q^{92} +12.0000i q^{93} +8.00000 q^{94} +(-12.0000 - 6.00000i) q^{95} -10.0000 q^{96} +6.00000i q^{97} +7.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{15} - 2 q^{16} - 12 q^{19} + 4 q^{20} - 12 q^{24} + 6 q^{25} + 12 q^{29} - 8 q^{30} + 12 q^{31} - 2 q^{36} - 6 q^{40} - 16 q^{41} - 4 q^{44} - 4 q^{45} + 12 q^{46} + 14 q^{49} - 8 q^{50} - 8 q^{54} - 8 q^{55} - 4 q^{59} - 4 q^{60} + 12 q^{61} - 14 q^{64} + 8 q^{66} + 24 q^{69} - 4 q^{71} - 12 q^{74} - 16 q^{75} - 12 q^{76} - 4 q^{80} - 22 q^{81} - 12 q^{86} + 16 q^{89} + 2 q^{90} + 16 q^{94} - 24 q^{95} - 20 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1\) \(-1\)

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 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) −2.00000 −0.816497
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) −1.00000 + 2.00000i −0.316228 + 0.632456i
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 0 0
\(14\) 0 0
\(15\) −2.00000 + 4.00000i −0.516398 + 1.03280i
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 2.00000 + 1.00000i 0.447214 + 0.223607i
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) −6.00000 −1.22474
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −4.00000 2.00000i −0.730297 0.365148i
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 6.00000i 0.973329i
\(39\) 0 0
\(40\) −3.00000 + 6.00000i −0.474342 + 0.948683i
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) −2.00000 −0.301511
\(45\) −2.00000 1.00000i −0.298142 0.149071i
\(46\) 6.00000 0.884652
\(47\) 8.00000i 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 2.00000i 0.288675i
\(49\) 7.00000 1.00000
\(50\) −4.00000 + 3.00000i −0.565685 + 0.424264i
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) −4.00000 −0.544331
\(55\) −4.00000 2.00000i −0.539360 0.269680i
\(56\) 0 0
\(57\) 12.0000i 1.58944i
\(58\) 6.00000i 0.787839i
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) −2.00000 + 4.00000i −0.258199 + 0.516398i
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 6.00000i 0.762001i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 12.0000i 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −6.00000 −0.697486
\(75\) −8.00000 + 6.00000i −0.923760 + 0.692820i
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 1.00000i −0.223607 0.111803i
\(81\) −11.0000 −1.22222
\(82\) 8.00000i 0.883452i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 12.0000i 1.28654i
\(88\) 6.00000i 0.639602i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 1.00000 2.00000i 0.105409 0.210819i
\(91\) 0 0
\(92\) 6.00000i 0.625543i
\(93\) 12.0000i 1.24434i
\(94\) 8.00000 0.825137
\(95\) −12.0000 6.00000i −1.23117 0.615587i
\(96\) −10.0000 −1.02062
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 2.00000 0.201008
\(100\) 3.00000 + 4.00000i 0.300000 + 0.400000i
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 6.00000i 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 2.00000 4.00000i 0.190693 0.381385i
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 12.0000 1.12390
\(115\) 6.00000 12.0000i 0.559503 1.11901i
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 2.00000i 0.184115i
\(119\) 0 0
\(120\) −12.0000 6.00000i −1.09545 0.547723i
\(121\) −7.00000 −0.636364
\(122\) 6.00000i 0.543214i
\(123\) 16.0000i 1.44267i
\(124\) 6.00000 0.538816
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 3.00000i 0.265165i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −4.00000 + 8.00000i −0.344265 + 0.688530i
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 12.0000i 1.02151i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 16.0000 1.34744
\(142\) 2.00000i 0.167836i
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 12.0000 + 6.00000i 0.996546 + 0.498273i
\(146\) 6.00000 0.496564
\(147\) 14.0000i 1.15470i
\(148\) 6.00000i 0.493197i
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) −6.00000 8.00000i −0.489898 0.653197i
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 18.0000i 1.45999i
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 + 6.00000i 0.963863 + 0.481932i
\(156\) 0 0
\(157\) 12.0000i 0.957704i 0.877896 + 0.478852i \(0.158947\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 24.0000 1.90332
\(160\) −5.00000 + 10.0000i −0.395285 + 0.790569i
\(161\) 0 0
\(162\) 11.0000i 0.864242i
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) −8.00000 −0.624695
\(165\) 4.00000 8.00000i 0.311400 0.622799i
\(166\) −4.00000 −0.310460
\(167\) 16.0000i 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 6.00000i 0.457496i
\(173\) 12.0000i 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000i 0.300658i
\(178\) 8.00000i 0.599625i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −2.00000 1.00000i −0.149071 0.0745356i
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 18.0000 1.32698
\(185\) −6.00000 + 12.0000i −0.441129 + 0.882258i
\(186\) −12.0000 −0.879883
\(187\) 0 0
\(188\) 8.00000i 0.583460i
\(189\) 0 0
\(190\) 6.00000 12.0000i 0.435286 0.870572i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 14.0000i 1.01036i
\(193\) 6.00000i 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 2.00000i 0.142134i
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −12.0000 + 9.00000i −0.848528 + 0.636396i
\(201\) 24.0000 1.69283
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) 0 0
\(205\) −16.0000 8.00000i −1.11749 0.558744i
\(206\) 6.00000 0.418040
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 4.00000i 0.274075i
\(214\) 6.00000 0.410152
\(215\) −6.00000 + 12.0000i −0.409197 + 0.818393i
\(216\) −12.0000 −0.816497
\(217\) 0 0
\(218\) 12.0000i 0.812743i
\(219\) 12.0000 0.810885
\(220\) −4.00000 2.00000i −0.269680 0.134840i
\(221\) 0 0
\(222\) 12.0000i 0.805387i
\(223\) 24.0000i 1.60716i −0.595198 0.803579i \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) 0 0
\(225\) −3.00000 4.00000i −0.200000 0.266667i
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 12.0000i 0.794719i
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 12.0000 + 6.00000i 0.791257 + 0.395628i
\(231\) 0 0
\(232\) 18.0000i 1.18176i
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 0 0
\(235\) 8.00000 16.0000i 0.521862 1.04372i
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 2.00000 4.00000i 0.129099 0.258199i
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 10.0000i 0.641500i
\(244\) 6.00000 0.384111
\(245\) 14.0000 + 7.00000i 0.894427 + 0.447214i
\(246\) 16.0000 1.02012
\(247\) 0 0
\(248\) 18.0000i 1.14300i
\(249\) −8.00000 −0.506979
\(250\) −11.0000 + 2.00000i −0.695701 + 0.126491i
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 12.0000i 0.741362i
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 12.0000 0.738549
\(265\) 12.0000 24.0000i 0.737154 1.47431i
\(266\) 0 0
\(267\) 16.0000i 0.979184i
\(268\) 12.0000i 0.733017i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −8.00000 4.00000i −0.486864 0.243432i
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −6.00000 8.00000i −0.361814 0.482418i
\(276\) 12.0000 0.722315
\(277\) 12.0000i 0.721010i 0.932757 + 0.360505i \(0.117396\pi\)
−0.932757 + 0.360505i \(0.882604\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 16.0000i 0.952786i
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) −2.00000 −0.118678
\(285\) 12.0000 24.0000i 0.710819 1.42164i
\(286\) 0 0
\(287\) 0 0
\(288\) 5.00000i 0.294628i
\(289\) 17.0000 1.00000
\(290\) −6.00000 + 12.0000i −0.352332 + 0.704664i
\(291\) −12.0000 −0.703452
\(292\) 6.00000i 0.351123i
\(293\) 26.0000i 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) −14.0000 −0.816497
\(295\) −4.00000 2.00000i −0.232889 0.116445i
\(296\) −18.0000 −1.04623
\(297\) 8.00000i 0.464207i
\(298\) 20.0000i 1.15857i
\(299\) 0 0
\(300\) −8.00000 + 6.00000i −0.461880 + 0.346410i
\(301\) 0 0
\(302\) 18.0000i 1.03578i
\(303\) 12.0000i 0.689382i
\(304\) 6.00000 0.344124
\(305\) 12.0000 + 6.00000i 0.687118 + 0.343559i
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) −6.00000 + 12.0000i −0.340777 + 0.681554i
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(314\) −12.0000 −0.677199
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 24.0000i 1.34585i
\(319\) −12.0000 −0.671871
\(320\) −14.0000 7.00000i −0.782624 0.391312i
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 24.0000i 1.32720i
\(328\) 24.0000i 1.32518i
\(329\) 0 0
\(330\) 8.00000 + 4.00000i 0.440386 + 0.220193i
\(331\) −30.0000 −1.64895 −0.824475 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 6.00000i 0.328798i
\(334\) 16.0000 0.875481
\(335\) 12.0000 24.0000i 0.655630 1.31126i
\(336\) 0 0
\(337\) 32.0000i 1.74315i −0.490261 0.871576i \(-0.663099\pi\)
0.490261 0.871576i \(-0.336901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 6.00000i 0.324443i
\(343\) 0 0
\(344\) −18.0000 −0.970495
\(345\) 24.0000 + 12.0000i 1.29212 + 0.646058i
\(346\) 12.0000 0.645124
\(347\) 6.00000i 0.322097i −0.986947 0.161048i \(-0.948512\pi\)
0.986947 0.161048i \(-0.0514875\pi\)
\(348\) 12.0000i 0.643268i
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.0000i 0.533002i
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 4.00000 0.212598
\(355\) −4.00000 2.00000i −0.212298 0.106149i
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 2.00000 0.105556 0.0527780 0.998606i \(-0.483192\pi\)
0.0527780 + 0.998606i \(0.483192\pi\)
\(360\) 3.00000 6.00000i 0.158114 0.316228i
\(361\) 17.0000 0.894737
\(362\) 2.00000i 0.105118i
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 6.00000 12.0000i 0.314054 0.628109i
\(366\) −12.0000 −0.627250
\(367\) 18.0000i 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 8.00000 0.416463
\(370\) −12.0000 6.00000i −0.623850 0.311925i
\(371\) 0 0
\(372\) 12.0000i 0.622171i
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) −22.0000 + 4.00000i −1.13608 + 0.206559i
\(376\) 24.0000 1.23771
\(377\) 0 0
\(378\) 0 0
\(379\) 18.0000 0.924598 0.462299 0.886724i \(-0.347025\pi\)
0.462299 + 0.886724i \(0.347025\pi\)
\(380\) −12.0000 6.00000i −0.615587 0.307794i
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 8.00000i 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) −6.00000 −0.306186
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 6.00000i 0.304997i
\(388\) 6.00000i 0.304604i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 21.0000i 1.06066i
\(393\) 24.0000i 1.21064i
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 18.0000i 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) −3.00000 4.00000i −0.150000 0.200000i
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 24.0000i 1.19701i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −22.0000 11.0000i −1.09319 0.546594i
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 8.00000 16.0000i 0.395092 0.790184i
\(411\) −4.00000 −0.197305
\(412\) 6.00000i 0.295599i
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −4.00000 + 8.00000i −0.196352 + 0.392705i
\(416\) 0 0
\(417\) 8.00000i 0.391762i
\(418\) 12.0000i 0.586939i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 36.0000 1.75453 0.877266 0.480004i \(-0.159365\pi\)
0.877266 + 0.480004i \(0.159365\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 8.00000i 0.388973i
\(424\) 36.0000 1.74831
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 0 0
\(428\) 6.00000i 0.290021i
\(429\) 0 0
\(430\) −12.0000 6.00000i −0.578691 0.289346i
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 16.0000i 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) 0 0
\(435\) −12.0000 + 24.0000i −0.575356 + 1.15071i
\(436\) 12.0000 0.574696
\(437\) 36.0000i 1.72211i
\(438\) 12.0000i 0.573382i
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 6.00000 12.0000i 0.286039 0.572078i
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 6.00000i 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) −12.0000 −0.569495
\(445\) 16.0000 + 8.00000i 0.758473 + 0.379236i
\(446\) 24.0000 1.13643
\(447\) 40.0000i 1.89194i
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 4.00000 3.00000i 0.188562 0.141421i
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) 36.0000i 1.69143i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 36.0000 1.68585
\(457\) 30.0000i 1.40334i 0.712502 + 0.701670i \(0.247562\pi\)
−0.712502 + 0.701670i \(0.752438\pi\)
\(458\) 12.0000i 0.560723i
\(459\) 0 0
\(460\) 6.00000 12.0000i 0.279751 0.559503i
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) −6.00000 −0.278543
\(465\) −12.0000 + 24.0000i −0.556487 + 1.11297i
\(466\) −24.0000 −1.11178
\(467\) 18.0000i 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 16.0000 + 8.00000i 0.738025 + 0.369012i
\(471\) −24.0000 −1.10586
\(472\) 6.00000i 0.276172i
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) −18.0000 24.0000i −0.825897 1.10120i
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 10.0000i 0.457389i
\(479\) −22.0000 −1.00521 −0.502603 0.864517i \(-0.667624\pi\)
−0.502603 + 0.864517i \(0.667624\pi\)
\(480\) −20.0000 10.0000i −0.912871 0.456435i
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −6.00000 + 12.0000i −0.272446 + 0.544892i
\(486\) 10.0000 0.453609
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 18.0000i 0.814822i
\(489\) −24.0000 −1.08532
\(490\) −7.00000 + 14.0000i −0.316228 + 0.632456i
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 16.0000i 0.721336i
\(493\) 0 0
\(494\) 0 0
\(495\) 4.00000 + 2.00000i 0.179787 + 0.0898933i
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 8.00000i 0.358489i
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 2.00000 + 11.0000i 0.0894427 + 0.491935i
\(501\) 32.0000 1.42965
\(502\) 12.0000i 0.535586i
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) −12.0000 6.00000i −0.533993 0.266996i
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 2.00000i 0.0887357i
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000i 0.486136i
\(513\) 24.0000i 1.05963i
\(514\) 0 0
\(515\) 6.00000 12.0000i 0.264392 0.528783i
\(516\) −12.0000 −0.528271
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 6.00000i 0.262613i
\(523\) 42.0000i 1.83653i 0.395964 + 0.918266i \(0.370410\pi\)
−0.395964 + 0.918266i \(0.629590\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) 4.00000i 0.174078i
\(529\) −13.0000 −0.565217
\(530\) 24.0000 + 12.0000i 1.04249 + 0.521247i
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 0 0
\(534\) −16.0000 −0.692388
\(535\) 6.00000 12.0000i 0.259403 0.518805i
\(536\) 36.0000 1.55496
\(537\) 24.0000i 1.03568i
\(538\) 18.0000i 0.776035i
\(539\) −14.0000 −0.603023
\(540\) −4.00000 + 8.00000i −0.172133 + 0.344265i
\(541\) −12.0000 −0.515920 −0.257960 0.966156i \(-0.583050\pi\)
−0.257960 + 0.966156i \(0.583050\pi\)
\(542\) 6.00000i 0.257722i
\(543\) 4.00000i 0.171656i
\(544\) 0 0
\(545\) 24.0000 + 12.0000i 1.02805 + 0.514024i
\(546\) 0 0
\(547\) 18.0000i 0.769624i 0.922995 + 0.384812i \(0.125734\pi\)
−0.922995 + 0.384812i \(0.874266\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) −6.00000 −0.256074
\(550\) 8.00000 6.00000i 0.341121 0.255841i
\(551\) −36.0000 −1.53365
\(552\) 36.0000i 1.53226i
\(553\) 0 0
\(554\) −12.0000 −0.509831
\(555\) −24.0000 12.0000i −1.01874 0.509372i
\(556\) −4.00000 −0.169638
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 6.00000i 0.254000i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 8.00000i 0.337460i
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) 16.0000 0.673722
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) 6.00000i 0.251754i
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 24.0000 + 12.0000i 1.00525 + 0.502625i
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.0000 18.0000i 1.00087 0.750652i
\(576\) 7.00000 0.291667
\(577\) 18.0000i 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) 17.0000i 0.707107i
\(579\) 12.0000 0.498703
\(580\) 12.0000 + 6.00000i 0.498273 + 0.249136i
\(581\) 0 0
\(582\) 12.0000i 0.497416i
\(583\) 24.0000i 0.993978i
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 20.0000i 0.825488i −0.910847 0.412744i \(-0.864570\pi\)
0.910847 0.412744i \(-0.135430\pi\)
\(588\) 14.0000i 0.577350i
\(589\) −36.0000 −1.48335
\(590\) 2.00000 4.00000i 0.0823387 0.164677i
\(591\) −4.00000 −0.164538
\(592\) 6.00000i 0.246598i
\(593\) 22.0000i 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 8.00000 0.328244
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 48.0000i 1.96451i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −18.0000 24.0000i −0.734847 0.979796i
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 18.0000 0.732410
\(605\) −14.0000 7.00000i −0.569181 0.284590i
\(606\) 12.0000 0.487467
\(607\) 18.0000i 0.730597i −0.930890 0.365299i \(-0.880967\pi\)
0.930890 0.365299i \(-0.119033\pi\)
\(608\) 30.0000i 1.21666i
\(609\) 0 0
\(610\) −6.00000 + 12.0000i −0.242933 + 0.485866i
\(611\) 0 0
\(612\) 0 0
\(613\) 30.0000i 1.21169i −0.795583 0.605844i \(-0.792835\pi\)
0.795583 0.605844i \(-0.207165\pi\)
\(614\) −12.0000 −0.484281
\(615\) 16.0000 32.0000i 0.645182 1.29036i
\(616\) 0 0
\(617\) 34.0000i 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 12.0000i 0.482711i
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 12.0000 + 6.00000i 0.481932 + 0.240966i
\(621\) 24.0000 0.963087
\(622\) 24.0000i 0.962312i
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) −8.00000 −0.319744
\(627\) 24.0000i 0.958468i
\(628\) 12.0000i 0.478852i
\(629\) 0 0
\(630\) 0 0
\(631\) −30.0000 −1.19428 −0.597141 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) 0 0
\(633\) 24.0000i 0.953914i
\(634\) −2.00000 −0.0794301
\(635\) −2.00000 + 4.00000i −0.0793676 + 0.158735i
\(636\) 24.0000 0.951662
\(637\) 0 0
\(638\) 12.0000i 0.475085i
\(639\) 2.00000 0.0791188
\(640\) −3.00000 + 6.00000i −0.118585 + 0.237171i
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) −24.0000 12.0000i −0.944999 0.472500i
\(646\) 0 0
\(647\) 6.00000i 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) 33.0000i 1.29636i
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000i 0.469956i
\(653\) 36.0000i 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) −24.0000 −0.938474
\(655\) −24.0000 12.0000i −0.937758 0.468879i
\(656\) 8.00000 0.312348
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 4.00000 8.00000i 0.155700 0.311400i
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 30.0000i 1.16598i
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 36.0000i 1.39393i
\(668\) 16.0000i 0.619059i
\(669\) 48.0000 1.85579
\(670\) 24.0000 + 12.0000i 0.927201 + 0.463600i
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 48.0000i 1.85026i −0.379646 0.925132i \(-0.623954\pi\)
0.379646 0.925132i \(-0.376046\pi\)
\(674\) 32.0000 1.23259
\(675\) −16.0000 + 12.0000i −0.615840 + 0.461880i
\(676\) 0 0
\(677\) 36.0000i 1.38359i 0.722093 + 0.691796i \(0.243180\pi\)
−0.722093 + 0.691796i \(0.756820\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 12.0000i 0.459504i
\(683\) 44.0000i 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) 6.00000 0.229416
\(685\) −2.00000 + 4.00000i −0.0764161 + 0.152832i
\(686\) 0 0
\(687\) 24.0000i 0.915657i
\(688\) 6.00000i 0.228748i
\(689\) 0 0
\(690\) −12.0000 + 24.0000i −0.456832 + 0.913664i
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) −8.00000 4.00000i −0.303457 0.151729i
\(696\) −36.0000 −1.36458
\(697\) 0 0
\(698\) 12.0000i 0.454207i
\(699\) −48.0000 −1.81553
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 36.0000i 1.35777i
\(704\) 14.0000 0.527645
\(705\) 32.0000 + 16.0000i 1.20519 + 0.602595i
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 4.00000i 0.150329i
\(709\) 12.0000 0.450669 0.225335 0.974281i \(-0.427652\pi\)
0.225335 + 0.974281i \(0.427652\pi\)
\(710\) 2.00000 4.00000i 0.0750587 0.150117i
\(711\) 0 0
\(712\) 24.0000i 0.899438i
\(713\) 36.0000i 1.34821i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 20.0000i 0.746914i
\(718\) 2.00000i 0.0746393i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 2.00000 + 1.00000i 0.0745356 + 0.0372678i
\(721\) 0 0
\(722\) 17.0000i 0.632674i
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 18.0000 + 24.0000i 0.668503 + 0.891338i
\(726\) 14.0000 0.519589
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 12.0000 + 6.00000i 0.444140 + 0.222070i
\(731\) 0 0
\(732\) 12.0000i 0.443533i
\(733\) 42.0000i 1.55131i 0.631160 + 0.775653i \(0.282579\pi\)
−0.631160 + 0.775653i \(0.717421\pi\)
\(734\) 18.0000 0.664392
\(735\) −14.0000 + 28.0000i −0.516398 + 1.03280i
\(736\) 30.0000 1.10581
\(737\) 24.0000i 0.884051i
\(738\) 8.00000i 0.294484i
\(739\) 6.00000 0.220714 0.110357 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) −6.00000 + 12.0000i −0.220564 + 0.441129i
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) −36.0000 −1.31982
\(745\) 40.0000 + 20.0000i 1.46549 + 0.732743i
\(746\) −4.00000 −0.146450
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 0 0
\(750\) −4.00000 22.0000i −0.146059 0.803326i
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 24.0000i 0.874609i
\(754\) 0 0
\(755\) 36.0000 + 18.0000i 1.31017 + 0.655087i
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 18.0000i 0.653789i
\(759\) −24.0000 −0.871145
\(760\) 18.0000 36.0000i 0.652929 1.30586i
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 0 0
\(768\) 34.0000i 1.22687i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.00000i 0.215945i
\(773\) 38.0000i 1.36677i −0.730061 0.683383i \(-0.760508\pi\)
0.730061 0.683383i \(-0.239492\pi\)
\(774\) 6.00000 0.215666
\(775\) 18.0000 + 24.0000i 0.646579 + 0.862105i
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) −7.00000 −0.250000
\(785\) −12.0000 + 24.0000i −0.428298 + 0.856597i
\(786\) 24.0000 0.856052
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 2.00000i 0.0712470i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000i 0.213201i
\(793\) 0 0
\(794\) 18.0000 0.638796
\(795\) 48.0000 + 24.0000i 1.70238 + 0.851192i
\(796\) 24.0000 0.850657
\(797\) 12.0000i 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −20.0000 + 15.0000i −0.707107 + 0.530330i
\(801\) −8.00000 −0.282666
\(802\) 16.0000i 0.564980i
\(803\) 12.0000i 0.423471i
\(804\) 24.0000 0.846415
\(805\) 0 0
\(806\) 0 0
\(807\) 36.0000i 1.26726i
\(808\) 18.0000i 0.633238i
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 11.0000 22.0000i 0.386501 0.773001i
\(811\) −30.0000 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) 0 0
\(813\) 12.0000i 0.420858i
\(814\) 12.0000 0.420600
\(815\) −12.0000 + 24.0000i −0.420342 + 0.840683i
\(816\) 0 0
\(817\) 36.0000i 1.25948i
\(818\) 24.0000i 0.839140i
\(819\) 0 0
\(820\) −16.0000 8.00000i −0.558744 0.279372i
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) 4.00000i 0.139516i
\(823\) 42.0000i 1.46403i 0.681290 + 0.732014i \(0.261419\pi\)
−0.681290 + 0.732014i \(0.738581\pi\)
\(824\) 18.0000 0.627060
\(825\) 16.0000 12.0000i 0.557048 0.417786i
\(826\) 0 0
\(827\) 4.00000i 0.139094i 0.997579 + 0.0695468i \(0.0221553\pi\)
−0.997579 + 0.0695468i \(0.977845\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −8.00000 4.00000i −0.277684 0.138842i
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 0 0
\(834\) 8.00000 0.277017
\(835\) 16.0000 32.0000i 0.553703 1.10741i
\(836\) 12.0000 0.415029
\(837\) 24.0000i 0.829561i
\(838\) 12.0000i 0.414533i
\(839\) −46.0000 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 36.0000i 1.24064i
\(843\) 16.0000i 0.551069i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 12.0000i 0.412082i
\(849\) −44.0000 −1.51008
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) 4.00000i 0.137038i
\(853\) 54.0000i 1.84892i 0.381273 + 0.924462i \(0.375486\pi\)
−0.381273 + 0.924462i \(0.624514\pi\)
\(854\) 0 0
\(855\) 12.0000 + 6.00000i 0.410391 + 0.205196i
\(856\) 18.0000 0.615227
\(857\) 24.0000i 0.819824i −0.912125 0.409912i \(-0.865559\pi\)
0.912125 0.409912i \(-0.134441\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −6.00000 + 12.0000i −0.204598 + 0.409197i
\(861\) 0 0
\(862\) 10.0000i 0.340601i
\(863\) 8.00000i 0.272323i 0.990687 + 0.136162i \(0.0434766\pi\)
−0.990687 + 0.136162i \(0.956523\pi\)
\(864\) −20.0000 −0.680414
\(865\) 12.0000 24.0000i 0.408012 0.816024i
\(866\) 16.0000 0.543702
\(867\) 34.0000i 1.15470i
\(868\) 0 0
\(869\) 0 0
\(870\) −24.0000 12.0000i −0.813676 0.406838i
\(871\) 0 0
\(872\) 36.0000i 1.21911i
\(873\) 6.00000i 0.203069i
\(874\) −36.0000 −1.21772
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 6.00000i 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 52.0000 1.75392
\(880\) 4.00000 + 2.00000i 0.134840 + 0.0674200i
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 2.00000i 0.0673054i −0.999434 0.0336527i \(-0.989286\pi\)
0.999434 0.0336527i \(-0.0107140\pi\)
\(884\) 0 0
\(885\) 4.00000 8.00000i 0.134459 0.268917i
\(886\) 6.00000 0.201574
\(887\) 42.0000i 1.41022i −0.709097 0.705111i \(-0.750897\pi\)
0.709097 0.705111i \(-0.249103\pi\)
\(888\) 36.0000i 1.20808i
\(889\) 0 0
\(890\) −8.00000 + 16.0000i −0.268161 + 0.536321i
\(891\) 22.0000 0.737028
\(892\) 24.0000i 0.803579i
\(893\) 48.0000i 1.60626i
\(894\) −40.0000 −1.33780
\(895\) 24.0000 + 12.0000i 0.802232 + 0.401116i
\(896\) 0 0
\(897\) 0 0
\(898\) 16.0000i 0.533927i
\(899\) 36.0000 1.20067
\(900\) −3.00000 4.00000i −0.100000 0.133333i
\(901\) 0 0
\(902\) 16.0000i 0.532742i
\(903\) 0 0
\(904\) 0 0
\(905\) 4.00000 + 2.00000i 0.132964 + 0.0664822i
\(906\) −36.0000 −1.19602
\(907\) 10.0000i 0.332045i −0.986122 0.166022i \(-0.946908\pi\)
0.986122 0.166022i \(-0.0530924\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 12.0000i 0.397360i
\(913\) 8.00000i 0.264761i
\(914\) −30.0000 −0.992312
\(915\) −12.0000 + 24.0000i −0.396708 + 0.793416i
\(916\) 12.0000 0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 36.0000 + 18.0000i 1.18688 + 0.593442i
\(921\) −24.0000 −0.790827
\(922\) 4.00000i 0.131733i
\(923\) 0 0
\(924\) 0 0
\(925\) −24.0000 + 18.0000i −0.789115 + 0.591836i
\(926\) 24.0000 0.788689
\(927\) 6.00000i 0.197066i
\(928\) 30.0000i 0.984798i
\(929\) 16.0000 0.524943 0.262471 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) −24.0000 12.0000i −0.786991 0.393496i
\(931\) −42.0000 −1.37649
\(932\) 24.0000i 0.786146i
\(933\) 48.0000i 1.57145i
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) 0 0
\(937\) 56.0000i 1.82944i −0.404088 0.914720i \(-0.632411\pi\)
0.404088 0.914720i \(-0.367589\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) 8.00000 16.0000i 0.260931 0.521862i
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 24.0000i 0.781962i
\(943\) 48.0000i 1.56310i
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 24.0000 18.0000i 0.778663 0.583997i
\(951\) −4.00000 −0.129709
\(952\) 0 0
\(953\) 24.0000i 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) 24.0000i 0.775810i
\(958\) 22.0000i 0.710788i
\(959\) 0 0
\(960\) 14.0000 28.0000i 0.451848 0.903696i
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 6.00000 12.0000i 0.193147 0.386294i
\(966\) 0 0
\(967\) 48.0000i 1.54358i −0.635880 0.771788i \(-0.719363\pi\)
0.635880 0.771788i \(-0.280637\pi\)
\(968\) 21.0000i 0.674966i
\(969\) 0 0
\(970\) −12.0000 6.00000i −0.385297 0.192648i
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 10.0000i 0.320750i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 34.0000i 1.08776i −0.839164 0.543878i \(-0.816955\pi\)
0.839164 0.543878i \(-0.183045\pi\)
\(978\) 24.0000i 0.767435i
\(979\) −16.0000 −0.511362
\(980\) 14.0000 + 7.00000i 0.447214 + 0.223607i
\(981\) −12.0000 −0.383131
\(982\) 12.0000i 0.382935i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 48.0000 1.53018
\(985\) −2.00000 + 4.00000i −0.0637253 + 0.127451i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 36.0000 1.14473
\(990\) −2.00000 + 4.00000i −0.0635642 + 0.127128i
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 30.0000i 0.952501i
\(993\) 60.0000i 1.90404i
\(994\) 0 0
\(995\) 48.0000 + 24.0000i 1.52170 + 0.760851i
\(996\) −8.00000 −0.253490
\(997\) 60.0000i 1.90022i −0.311916 0.950110i \(-0.600971\pi\)
0.311916 0.950110i \(-0.399029\pi\)
\(998\) 6.00000i 0.189927i
\(999\) −24.0000 −0.759326
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 845.2.b.b.339.2 2
5.2 odd 4 4225.2.a.h.1.1 1
5.3 odd 4 4225.2.a.k.1.1 1
5.4 even 2 inner 845.2.b.b.339.1 2
13.2 odd 12 845.2.l.a.654.1 4
13.3 even 3 845.2.n.b.529.1 4
13.4 even 6 845.2.n.a.484.1 4
13.5 odd 4 65.2.d.b.64.2 yes 2
13.6 odd 12 845.2.l.a.699.2 4
13.7 odd 12 845.2.l.b.699.2 4
13.8 odd 4 65.2.d.a.64.2 yes 2
13.9 even 3 845.2.n.b.484.2 4
13.10 even 6 845.2.n.a.529.2 4
13.11 odd 12 845.2.l.b.654.1 4
13.12 even 2 845.2.b.a.339.1 2
39.5 even 4 585.2.h.b.64.2 2
39.8 even 4 585.2.h.c.64.1 2
52.31 even 4 1040.2.f.b.129.1 2
52.47 even 4 1040.2.f.a.129.1 2
65.4 even 6 845.2.n.a.484.2 4
65.8 even 4 325.2.c.b.51.2 2
65.9 even 6 845.2.n.b.484.1 4
65.12 odd 4 4225.2.a.m.1.1 1
65.18 even 4 325.2.c.b.51.1 2
65.19 odd 12 845.2.l.b.699.1 4
65.24 odd 12 845.2.l.a.654.2 4
65.29 even 6 845.2.n.b.529.2 4
65.34 odd 4 65.2.d.b.64.1 yes 2
65.38 odd 4 4225.2.a.e.1.1 1
65.44 odd 4 65.2.d.a.64.1 2
65.47 even 4 325.2.c.e.51.1 2
65.49 even 6 845.2.n.a.529.1 4
65.54 odd 12 845.2.l.b.654.2 4
65.57 even 4 325.2.c.e.51.2 2
65.59 odd 12 845.2.l.a.699.1 4
65.64 even 2 845.2.b.a.339.2 2
195.44 even 4 585.2.h.c.64.2 2
195.164 even 4 585.2.h.b.64.1 2
260.99 even 4 1040.2.f.b.129.2 2
260.239 even 4 1040.2.f.a.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.d.a.64.1 2 65.44 odd 4
65.2.d.a.64.2 yes 2 13.8 odd 4
65.2.d.b.64.1 yes 2 65.34 odd 4
65.2.d.b.64.2 yes 2 13.5 odd 4
325.2.c.b.51.1 2 65.18 even 4
325.2.c.b.51.2 2 65.8 even 4
325.2.c.e.51.1 2 65.47 even 4
325.2.c.e.51.2 2 65.57 even 4
585.2.h.b.64.1 2 195.164 even 4
585.2.h.b.64.2 2 39.5 even 4
585.2.h.c.64.1 2 39.8 even 4
585.2.h.c.64.2 2 195.44 even 4
845.2.b.a.339.1 2 13.12 even 2
845.2.b.a.339.2 2 65.64 even 2
845.2.b.b.339.1 2 5.4 even 2 inner
845.2.b.b.339.2 2 1.1 even 1 trivial
845.2.l.a.654.1 4 13.2 odd 12
845.2.l.a.654.2 4 65.24 odd 12
845.2.l.a.699.1 4 65.59 odd 12
845.2.l.a.699.2 4 13.6 odd 12
845.2.l.b.654.1 4 13.11 odd 12
845.2.l.b.654.2 4 65.54 odd 12
845.2.l.b.699.1 4 65.19 odd 12
845.2.l.b.699.2 4 13.7 odd 12
845.2.n.a.484.1 4 13.4 even 6
845.2.n.a.484.2 4 65.4 even 6
845.2.n.a.529.1 4 65.49 even 6
845.2.n.a.529.2 4 13.10 even 6
845.2.n.b.484.1 4 65.9 even 6
845.2.n.b.484.2 4 13.9 even 3
845.2.n.b.529.1 4 13.3 even 3
845.2.n.b.529.2 4 65.29 even 6
1040.2.f.a.129.1 2 52.47 even 4
1040.2.f.a.129.2 2 260.239 even 4
1040.2.f.b.129.1 2 52.31 even 4
1040.2.f.b.129.2 2 260.99 even 4
4225.2.a.e.1.1 1 65.38 odd 4
4225.2.a.h.1.1 1 5.2 odd 4
4225.2.a.k.1.1 1 5.3 odd 4
4225.2.a.m.1.1 1 65.12 odd 4