Properties

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

$q$-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 \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 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 \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 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 \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −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 \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 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 \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 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 \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 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 \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 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 \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 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 \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −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 \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 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 \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 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 \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 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 \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) −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 \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 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 \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 6.00000 + 8.00000i 0.489898 + 0.653197i
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 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 \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 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 \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 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 \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 2.00000i 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 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 \(0.297074\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(224\) 0 0
\(225\) −3.00000 4.00000i −0.200000 0.266667i
\(226\) 0 0
\(227\) 4.00000i 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 12.0000i 0.794719i
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 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 \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) −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 \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 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 \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 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 \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 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 \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 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 \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 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 \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) 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 \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.0000i 0.533002i
\(353\) 14.0000i 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 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 \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 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 \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −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 \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) 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 \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 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 \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 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 \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) −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 \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 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 \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 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 \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) −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 \(-0.752438\pi\)
0.712502 0.701670i \(-0.247562\pi\)
\(458\) 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 \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) −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 \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) −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 \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) −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 \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 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 \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 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 \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 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 \(0.122246\pi\)
−0.927156 + 0.374675i \(0.877754\pi\)
\(578\) 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 \(0.135430\pi\)
−0.910847 + 0.412744i \(0.864570\pi\)
\(588\) 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 \(0.149188\pi\)
−0.892162 + 0.451716i \(0.850812\pi\)
\(594\) 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 \(0.207165\pi\)
−0.795583 + 0.605844i \(0.792835\pi\)
\(614\) −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 \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) 12.0000i 0.482711i
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) 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 \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 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 \(-0.748762\pi\)
0.704352 0.709851i \(-0.251238\pi\)
\(644\) 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 \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 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 \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) −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 \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 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 \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) −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 \(-0.717421\pi\)
0.631160 0.775653i \(-0.282579\pi\)
\(734\) −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 \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) −6.00000 + 12.0000i −0.220564 + 0.441129i
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000i 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) −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 \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 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 \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) −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 \(-0.931391\pi\)
0.976861 0.213877i \(-0.0686091\pi\)
\(788\) 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 \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 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 \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 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 \(-0.977845\pi\)
0.997579 0.0695468i \(-0.0221553\pi\)
\(828\) 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 \(-0.207970\pi\)
0.794048 + 0.607855i \(0.207970\pi\)
\(840\) 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 \(-0.624514\pi\)
0.381273 0.924462i \(-0.375486\pi\)
\(854\) 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 \(-0.956523\pi\)
0.990687 0.136162i \(-0.0434766\pi\)
\(864\) 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 \(0.0323011\pi\)
−0.994856 + 0.101303i \(0.967699\pi\)
\(878\) 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 \(-0.584538\pi\)
−0.262471 + 0.964940i \(0.584538\pi\)
\(930\) 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 \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 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 \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) 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 \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) 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 \(0.183045\pi\)
−0.839164 + 0.543878i \(0.816955\pi\)
\(978\) 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 \(-0.917872\pi\)
0.966899 0.255160i \(-0.0821283\pi\)
\(984\) 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.a.339.1 2
5.2 odd 4 4225.2.a.m.1.1 1
5.3 odd 4 4225.2.a.e.1.1 1
5.4 even 2 inner 845.2.b.a.339.2 2
13.2 odd 12 845.2.l.b.654.1 4
13.3 even 3 845.2.n.a.529.2 4
13.4 even 6 845.2.n.b.484.2 4
13.5 odd 4 65.2.d.a.64.2 yes 2
13.6 odd 12 845.2.l.b.699.2 4
13.7 odd 12 845.2.l.a.699.2 4
13.8 odd 4 65.2.d.b.64.2 yes 2
13.9 even 3 845.2.n.a.484.1 4
13.10 even 6 845.2.n.b.529.1 4
13.11 odd 12 845.2.l.a.654.1 4
13.12 even 2 845.2.b.b.339.2 2
39.5 even 4 585.2.h.c.64.1 2
39.8 even 4 585.2.h.b.64.2 2
52.31 even 4 1040.2.f.a.129.1 2
52.47 even 4 1040.2.f.b.129.1 2
65.4 even 6 845.2.n.b.484.1 4
65.8 even 4 325.2.c.b.51.1 2
65.9 even 6 845.2.n.a.484.2 4
65.12 odd 4 4225.2.a.h.1.1 1
65.18 even 4 325.2.c.b.51.2 2
65.19 odd 12 845.2.l.a.699.1 4
65.24 odd 12 845.2.l.b.654.2 4
65.29 even 6 845.2.n.a.529.1 4
65.34 odd 4 65.2.d.a.64.1 2
65.38 odd 4 4225.2.a.k.1.1 1
65.44 odd 4 65.2.d.b.64.1 yes 2
65.47 even 4 325.2.c.e.51.2 2
65.49 even 6 845.2.n.b.529.2 4
65.54 odd 12 845.2.l.a.654.2 4
65.57 even 4 325.2.c.e.51.1 2
65.59 odd 12 845.2.l.b.699.1 4
65.64 even 2 845.2.b.b.339.1 2
195.44 even 4 585.2.h.b.64.1 2
195.164 even 4 585.2.h.c.64.2 2
260.99 even 4 1040.2.f.a.129.2 2
260.239 even 4 1040.2.f.b.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.d.a.64.1 2 65.34 odd 4
65.2.d.a.64.2 yes 2 13.5 odd 4
65.2.d.b.64.1 yes 2 65.44 odd 4
65.2.d.b.64.2 yes 2 13.8 odd 4
325.2.c.b.51.1 2 65.8 even 4
325.2.c.b.51.2 2 65.18 even 4
325.2.c.e.51.1 2 65.57 even 4
325.2.c.e.51.2 2 65.47 even 4
585.2.h.b.64.1 2 195.44 even 4
585.2.h.b.64.2 2 39.8 even 4
585.2.h.c.64.1 2 39.5 even 4
585.2.h.c.64.2 2 195.164 even 4
845.2.b.a.339.1 2 1.1 even 1 trivial
845.2.b.a.339.2 2 5.4 even 2 inner
845.2.b.b.339.1 2 65.64 even 2
845.2.b.b.339.2 2 13.12 even 2
845.2.l.a.654.1 4 13.11 odd 12
845.2.l.a.654.2 4 65.54 odd 12
845.2.l.a.699.1 4 65.19 odd 12
845.2.l.a.699.2 4 13.7 odd 12
845.2.l.b.654.1 4 13.2 odd 12
845.2.l.b.654.2 4 65.24 odd 12
845.2.l.b.699.1 4 65.59 odd 12
845.2.l.b.699.2 4 13.6 odd 12
845.2.n.a.484.1 4 13.9 even 3
845.2.n.a.484.2 4 65.9 even 6
845.2.n.a.529.1 4 65.29 even 6
845.2.n.a.529.2 4 13.3 even 3
845.2.n.b.484.1 4 65.4 even 6
845.2.n.b.484.2 4 13.4 even 6
845.2.n.b.529.1 4 13.10 even 6
845.2.n.b.529.2 4 65.49 even 6
1040.2.f.a.129.1 2 52.31 even 4
1040.2.f.a.129.2 2 260.99 even 4
1040.2.f.b.129.1 2 52.47 even 4
1040.2.f.b.129.2 2 260.239 even 4
4225.2.a.e.1.1 1 5.3 odd 4
4225.2.a.h.1.1 1 65.12 odd 4
4225.2.a.k.1.1 1 65.38 odd 4
4225.2.a.m.1.1 1 5.2 odd 4