Properties

Label 845.2.c.a.506.1
Level $845$
Weight $2$
Character 845.506
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(506,845)]
 
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([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("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), 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: \(6.74735897080\)
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 506.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.a.506.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000i q^{5} +2.00000i q^{6} +4.00000i q^{7} -3.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000i q^{5} +2.00000i q^{6} +4.00000i q^{7} -3.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{12} +4.00000 q^{14} +2.00000i q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} -8.00000i q^{21} -2.00000 q^{22} +6.00000 q^{23} +6.00000i q^{24} -1.00000 q^{25} +4.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} +2.00000 q^{30} -10.0000i q^{31} -5.00000i q^{32} +4.00000i q^{33} +2.00000i q^{34} +4.00000 q^{35} +1.00000 q^{36} +2.00000i q^{37} -6.00000 q^{38} -3.00000 q^{40} -6.00000i q^{41} -8.00000 q^{42} -10.0000 q^{43} -2.00000i q^{44} -1.00000i q^{45} -6.00000i q^{46} -4.00000i q^{47} +2.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +2.00000 q^{53} -4.00000i q^{54} -2.00000 q^{55} +12.0000 q^{56} +12.0000i q^{57} -2.00000i q^{58} -6.00000i q^{59} +2.00000i q^{60} +2.00000 q^{61} -10.0000 q^{62} +4.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} -2.00000 q^{68} -12.0000 q^{69} -4.00000i q^{70} +6.00000i q^{71} -3.00000i q^{72} +6.00000i q^{73} +2.00000 q^{74} +2.00000 q^{75} -6.00000i q^{76} +8.00000 q^{77} -12.0000 q^{79} +1.00000i q^{80} -11.0000 q^{81} -6.00000 q^{82} -16.0000i q^{83} -8.00000i q^{84} +2.00000i q^{85} +10.0000i q^{86} -4.00000 q^{87} -6.00000 q^{88} -2.00000i q^{89} -1.00000 q^{90} +6.00000 q^{92} +20.0000i q^{93} -4.00000 q^{94} -6.00000 q^{95} +10.0000i q^{96} -2.00000i q^{97} +9.00000i q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{10} - 4 q^{12} + 8 q^{14} - 2 q^{16} - 4 q^{17} - 4 q^{22} + 12 q^{23} - 2 q^{25} + 8 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{35} + 2 q^{36} - 12 q^{38} - 6 q^{40} - 16 q^{42} - 20 q^{43} + 4 q^{48} - 18 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 24 q^{56} + 4 q^{61} - 20 q^{62} - 14 q^{64} + 8 q^{66} - 4 q^{68} - 24 q^{69} + 4 q^{74} + 4 q^{75} + 16 q^{77} - 24 q^{79} - 22 q^{81} - 12 q^{82} - 8 q^{87} - 12 q^{88} - 2 q^{90} + 12 q^{92} - 8 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(-0.115027\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 2.00000i 0.816497i
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0
\(14\) 4.00000 1.06904
\(15\) 2.00000i 0.516398i
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 8.00000i − 1.74574i
\(22\) −2.00000 −0.426401
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 6.00000i 1.22474i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 4.00000i 0.755929i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) − 10.0000i − 1.79605i −0.439941 0.898027i \(-0.645001\pi\)
0.439941 0.898027i \(-0.354999\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 4.00000i 0.696311i
\(34\) 2.00000i 0.342997i
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) −8.00000 −1.23443
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) − 2.00000i − 0.301511i
\(45\) − 1.00000i − 0.149071i
\(46\) − 6.00000i − 0.884652i
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 2.00000 0.288675
\(49\) −9.00000 −1.28571
\(50\) 1.00000i 0.141421i
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) − 4.00000i − 0.544331i
\(55\) −2.00000 −0.269680
\(56\) 12.0000 1.60357
\(57\) 12.0000i 1.58944i
\(58\) − 2.00000i − 0.262613i
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 2.00000i 0.258199i
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.0000 −1.27000
\(63\) 4.00000i 0.503953i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) −2.00000 −0.242536
\(69\) −12.0000 −1.44463
\(70\) − 4.00000i − 0.478091i
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 2.00000 0.232495
\(75\) 2.00000 0.230940
\(76\) − 6.00000i − 0.688247i
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 1.00000i 0.111803i
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) − 8.00000i − 0.872872i
\(85\) 2.00000i 0.216930i
\(86\) 10.0000i 1.07833i
\(87\) −4.00000 −0.428845
\(88\) −6.00000 −0.639602
\(89\) − 2.00000i − 0.212000i −0.994366 0.106000i \(-0.966196\pi\)
0.994366 0.106000i \(-0.0338043\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 20.0000i 2.07390i
\(94\) −4.00000 −0.412568
\(95\) −6.00000 −0.615587
\(96\) 10.0000i 1.02062i
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 9.00000i 0.909137i
\(99\) − 2.00000i − 0.201008i
\(100\) −1.00000 −0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) − 2.00000i − 0.194257i
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 4.00000 0.384900
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 2.00000i 0.190693i
\(111\) − 4.00000i − 0.379663i
\(112\) − 4.00000i − 0.377964i
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 12.0000 1.12390
\(115\) − 6.00000i − 0.559503i
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) − 8.00000i − 0.733359i
\(120\) 6.00000 0.547723
\(121\) 7.00000 0.636364
\(122\) − 2.00000i − 0.181071i
\(123\) 12.0000i 1.08200i
\(124\) − 10.0000i − 0.898027i
\(125\) 1.00000i 0.0894427i
\(126\) 4.00000 0.356348
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) 20.0000 1.76090
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 24.0000 2.08106
\(134\) −4.00000 −0.345547
\(135\) − 4.00000i − 0.344265i
\(136\) 6.00000i 0.514496i
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 12.0000i 1.02151i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 4.00000 0.338062
\(141\) 8.00000i 0.673722i
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) − 2.00000i − 0.166091i
\(146\) 6.00000 0.496564
\(147\) 18.0000 1.48461
\(148\) 2.00000i 0.164399i
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) − 2.00000i − 0.163299i
\(151\) − 10.0000i − 0.813788i −0.913475 0.406894i \(-0.866612\pi\)
0.913475 0.406894i \(-0.133388\pi\)
\(152\) −18.0000 −1.45999
\(153\) −2.00000 −0.161690
\(154\) − 8.00000i − 0.644658i
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 12.0000i 0.954669i
\(159\) −4.00000 −0.317221
\(160\) −5.00000 −0.395285
\(161\) 24.0000i 1.89146i
\(162\) 11.0000i 0.864242i
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) 4.00000 0.311400
\(166\) −16.0000 −1.24184
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) −24.0000 −1.85164
\(169\) 0 0
\(170\) 2.00000 0.153393
\(171\) − 6.00000i − 0.458831i
\(172\) −10.0000 −0.762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 4.00000i 0.303239i
\(175\) − 4.00000i − 0.302372i
\(176\) 2.00000i 0.150756i
\(177\) 12.0000i 0.901975i
\(178\) −2.00000 −0.149906
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) − 18.0000i − 1.32698i
\(185\) 2.00000 0.147043
\(186\) 20.0000 1.46647
\(187\) 4.00000i 0.292509i
\(188\) − 4.00000i − 0.291730i
\(189\) 16.0000i 1.16383i
\(190\) 6.00000i 0.435286i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 14.0000 1.01036
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 3.00000i 0.212132i
\(201\) 8.00000i 0.564276i
\(202\) − 18.0000i − 1.26648i
\(203\) 8.00000i 0.561490i
\(204\) 4.00000 0.280056
\(205\) −6.00000 −0.419058
\(206\) 2.00000i 0.139347i
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 8.00000i 0.552052i
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 2.00000 0.137361
\(213\) − 12.0000i − 0.822226i
\(214\) − 10.0000i − 0.683586i
\(215\) 10.0000i 0.681994i
\(216\) − 12.0000i − 0.816497i
\(217\) 40.0000 2.71538
\(218\) 10.0000 0.677285
\(219\) − 12.0000i − 0.810885i
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 20.0000 1.33631
\(225\) −1.00000 −0.0666667
\(226\) 14.0000i 0.931266i
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 12.0000i 0.794719i
\(229\) 22.0000i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(230\) −6.00000 −0.395628
\(231\) −16.0000 −1.05272
\(232\) − 6.00000i − 0.393919i
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) − 6.00000i − 0.390567i
\(237\) 24.0000 1.55897
\(238\) −8.00000 −0.518563
\(239\) − 6.00000i − 0.388108i −0.980991 0.194054i \(-0.937836\pi\)
0.980991 0.194054i \(-0.0621637\pi\)
\(240\) − 2.00000i − 0.129099i
\(241\) − 10.0000i − 0.644157i −0.946713 0.322078i \(-0.895619\pi\)
0.946713 0.322078i \(-0.104381\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 10.0000 0.641500
\(244\) 2.00000 0.128037
\(245\) 9.00000i 0.574989i
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) −30.0000 −1.90500
\(249\) 32.0000i 2.02792i
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 4.00000i 0.251976i
\(253\) − 12.0000i − 0.754434i
\(254\) − 2.00000i − 0.125491i
\(255\) − 4.00000i − 0.250490i
\(256\) −17.0000 −1.06250
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) − 20.0000i − 1.24515i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 20.0000i − 1.23560i
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 12.0000 0.738549
\(265\) − 2.00000i − 0.122859i
\(266\) − 24.0000i − 1.47153i
\(267\) 4.00000i 0.244796i
\(268\) − 4.00000i − 0.244339i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −4.00000 −0.243432
\(271\) − 2.00000i − 0.121491i −0.998153 0.0607457i \(-0.980652\pi\)
0.998153 0.0607457i \(-0.0193479\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 2.00000i 0.120605i
\(276\) −12.0000 −0.722315
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) − 10.0000i − 0.598684i
\(280\) − 12.0000i − 0.717137i
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 8.00000 0.476393
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 6.00000i 0.356034i
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) − 5.00000i − 0.294628i
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 4.00000i 0.234484i
\(292\) 6.00000i 0.351123i
\(293\) − 22.0000i − 1.28525i −0.766179 0.642627i \(-0.777845\pi\)
0.766179 0.642627i \(-0.222155\pi\)
\(294\) − 18.0000i − 1.04978i
\(295\) −6.00000 −0.349334
\(296\) 6.00000 0.348743
\(297\) − 8.00000i − 0.464207i
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) − 40.0000i − 2.30556i
\(302\) −10.0000 −0.575435
\(303\) −36.0000 −2.06815
\(304\) 6.00000i 0.344124i
\(305\) − 2.00000i − 0.114520i
\(306\) 2.00000i 0.114332i
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) 8.00000 0.455842
\(309\) 4.00000 0.227552
\(310\) 10.0000i 0.567962i
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 6.00000i 0.338600i
\(315\) 4.00000 0.225374
\(316\) −12.0000 −0.675053
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 4.00000i 0.224309i
\(319\) − 4.00000i − 0.223957i
\(320\) 7.00000i 0.391312i
\(321\) −20.0000 −1.11629
\(322\) 24.0000 1.33747
\(323\) 12.0000i 0.667698i
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) − 20.0000i − 1.10600i
\(328\) −18.0000 −0.993884
\(329\) 16.0000 0.882109
\(330\) − 4.00000i − 0.220193i
\(331\) − 18.0000i − 0.989369i −0.869072 0.494685i \(-0.835284\pi\)
0.869072 0.494685i \(-0.164716\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) 2.00000i 0.109599i
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) 8.00000i 0.436436i
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 28.0000 1.52075
\(340\) 2.00000i 0.108465i
\(341\) −20.0000 −1.08306
\(342\) −6.00000 −0.324443
\(343\) − 8.00000i − 0.431959i
\(344\) 30.0000i 1.61749i
\(345\) 12.0000i 0.646058i
\(346\) − 6.00000i − 0.322562i
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) −4.00000 −0.214423
\(349\) 30.0000i 1.60586i 0.596071 + 0.802932i \(0.296728\pi\)
−0.596071 + 0.802932i \(0.703272\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −10.0000 −0.533002
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 12.0000 0.637793
\(355\) 6.00000 0.318447
\(356\) − 2.00000i − 0.106000i
\(357\) 16.0000i 0.846810i
\(358\) 12.0000i 0.634220i
\(359\) 10.0000i 0.527780i 0.964553 + 0.263890i \(0.0850056\pi\)
−0.964553 + 0.263890i \(0.914994\pi\)
\(360\) −3.00000 −0.158114
\(361\) −17.0000 −0.894737
\(362\) − 22.0000i − 1.15629i
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 4.00000i 0.209083i
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) −6.00000 −0.312772
\(369\) − 6.00000i − 0.312348i
\(370\) − 2.00000i − 0.103975i
\(371\) 8.00000i 0.415339i
\(372\) 20.0000i 1.03695i
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 4.00000 0.206835
\(375\) − 2.00000i − 0.103280i
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 16.0000 0.822951
\(379\) 10.0000i 0.513665i 0.966456 + 0.256833i \(0.0826790\pi\)
−0.966456 + 0.256833i \(0.917321\pi\)
\(380\) −6.00000 −0.307794
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 6.00000i 0.306186i
\(385\) − 8.00000i − 0.407718i
\(386\) 2.00000 0.101797
\(387\) −10.0000 −0.508329
\(388\) − 2.00000i − 0.101535i
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 27.0000i 1.36371i
\(393\) −40.0000 −2.01773
\(394\) −6.00000 −0.302276
\(395\) 12.0000i 0.603786i
\(396\) − 2.00000i − 0.100504i
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −48.0000 −2.40301
\(400\) 1.00000 0.0500000
\(401\) − 10.0000i − 0.499376i −0.968326 0.249688i \(-0.919672\pi\)
0.968326 0.249688i \(-0.0803281\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 11.0000i 0.546594i
\(406\) 8.00000 0.397033
\(407\) 4.00000 0.198273
\(408\) − 12.0000i − 0.594089i
\(409\) 18.0000i 0.890043i 0.895520 + 0.445021i \(0.146804\pi\)
−0.895520 + 0.445021i \(0.853196\pi\)
\(410\) 6.00000i 0.296319i
\(411\) − 4.00000i − 0.197305i
\(412\) −2.00000 −0.0985329
\(413\) 24.0000 1.18096
\(414\) − 6.00000i − 0.294884i
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) 0 0
\(418\) 12.0000i 0.586939i
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) −8.00000 −0.390360
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) − 12.0000i − 0.584151i
\(423\) − 4.00000i − 0.194487i
\(424\) − 6.00000i − 0.291386i
\(425\) 2.00000 0.0970143
\(426\) −12.0000 −0.581402
\(427\) 8.00000i 0.387147i
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) 14.0000i 0.674356i 0.941441 + 0.337178i \(0.109472\pi\)
−0.941441 + 0.337178i \(0.890528\pi\)
\(432\) −4.00000 −0.192450
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) − 40.0000i − 1.92006i
\(435\) 4.00000i 0.191785i
\(436\) 10.0000i 0.478913i
\(437\) − 36.0000i − 1.72211i
\(438\) −12.0000 −0.573382
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 6.00000i 0.286039i
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) − 4.00000i − 0.189832i
\(445\) −2.00000 −0.0948091
\(446\) 4.00000 0.189405
\(447\) − 36.0000i − 1.70274i
\(448\) − 28.0000i − 1.32288i
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −12.0000 −0.565058
\(452\) −14.0000 −0.658505
\(453\) 20.0000i 0.939682i
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 36.0000 1.68585
\(457\) 38.0000i 1.77757i 0.458329 + 0.888783i \(0.348448\pi\)
−0.458329 + 0.888783i \(0.651552\pi\)
\(458\) 22.0000 1.02799
\(459\) −8.00000 −0.373408
\(460\) − 6.00000i − 0.279751i
\(461\) 10.0000i 0.465746i 0.972507 + 0.232873i \(0.0748127\pi\)
−0.972507 + 0.232873i \(0.925187\pi\)
\(462\) 16.0000i 0.744387i
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 20.0000 0.927478
\(466\) 10.0000i 0.463241i
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 4.00000i 0.184506i
\(471\) 12.0000 0.552931
\(472\) −18.0000 −0.828517
\(473\) 20.0000i 0.919601i
\(474\) − 24.0000i − 1.10236i
\(475\) 6.00000i 0.275299i
\(476\) − 8.00000i − 0.366679i
\(477\) 2.00000 0.0915737
\(478\) −6.00000 −0.274434
\(479\) − 30.0000i − 1.37073i −0.728197 0.685367i \(-0.759642\pi\)
0.728197 0.685367i \(-0.240358\pi\)
\(480\) 10.0000 0.456435
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) − 48.0000i − 2.18408i
\(484\) 7.00000 0.318182
\(485\) −2.00000 −0.0908153
\(486\) − 10.0000i − 0.453609i
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) − 24.0000i − 1.08532i
\(490\) 9.00000 0.406579
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 12.0000i 0.541002i
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 10.0000i 0.449013i
\(497\) −24.0000 −1.07655
\(498\) 32.0000 1.43395
\(499\) 10.0000i 0.447661i 0.974628 + 0.223831i \(0.0718563\pi\)
−0.974628 + 0.223831i \(0.928144\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 24.0000i − 1.07224i
\(502\) 24.0000i 1.07117i
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 12.0000 0.534522
\(505\) − 18.0000i − 0.800989i
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) −4.00000 −0.177123
\(511\) −24.0000 −1.06170
\(512\) 11.0000i 0.486136i
\(513\) − 24.0000i − 1.05963i
\(514\) 2.00000i 0.0882162i
\(515\) 2.00000i 0.0881305i
\(516\) 20.0000 0.880451
\(517\) −8.00000 −0.351840
\(518\) 8.00000i 0.351500i
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 20.0000 0.873704
\(525\) 8.00000i 0.349149i
\(526\) − 14.0000i − 0.610429i
\(527\) 20.0000i 0.871214i
\(528\) − 4.00000i − 0.174078i
\(529\) 13.0000 0.565217
\(530\) −2.00000 −0.0868744
\(531\) − 6.00000i − 0.260378i
\(532\) 24.0000 1.04053
\(533\) 0 0
\(534\) 4.00000 0.173097
\(535\) − 10.0000i − 0.432338i
\(536\) −12.0000 −0.518321
\(537\) 24.0000 1.03568
\(538\) − 6.00000i − 0.258678i
\(539\) 18.0000i 0.775315i
\(540\) − 4.00000i − 0.172133i
\(541\) 22.0000i 0.945854i 0.881102 + 0.472927i \(0.156803\pi\)
−0.881102 + 0.472927i \(0.843197\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −44.0000 −1.88822
\(544\) 10.0000i 0.428746i
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 2.00000 0.0853579
\(550\) 2.00000 0.0852803
\(551\) − 12.0000i − 0.511217i
\(552\) 36.0000i 1.53226i
\(553\) − 48.0000i − 2.04117i
\(554\) − 14.0000i − 0.594803i
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 26.0000i 1.10166i 0.834619 + 0.550828i \(0.185688\pi\)
−0.834619 + 0.550828i \(0.814312\pi\)
\(558\) −10.0000 −0.423334
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) − 8.00000i − 0.337760i
\(562\) 6.00000 0.253095
\(563\) 22.0000 0.927189 0.463595 0.886047i \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) 8.00000i 0.336861i
\(565\) 14.0000i 0.588984i
\(566\) 2.00000i 0.0840663i
\(567\) − 44.0000i − 1.84783i
\(568\) 18.0000 0.755263
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) − 12.0000i − 0.502625i
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 24.0000i − 1.00174i
\(575\) −6.00000 −0.250217
\(576\) −7.00000 −0.291667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 13.0000i 0.540729i
\(579\) − 4.00000i − 0.166234i
\(580\) − 2.00000i − 0.0830455i
\(581\) 64.0000 2.65517
\(582\) 4.00000 0.165805
\(583\) − 4.00000i − 0.165663i
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) − 44.0000i − 1.81607i −0.418890 0.908037i \(-0.637581\pi\)
0.418890 0.908037i \(-0.362419\pi\)
\(588\) 18.0000 0.742307
\(589\) −60.0000 −2.47226
\(590\) 6.00000i 0.247016i
\(591\) 12.0000i 0.493614i
\(592\) − 2.00000i − 0.0821995i
\(593\) − 14.0000i − 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) −8.00000 −0.328244
\(595\) −8.00000 −0.327968
\(596\) 18.0000i 0.737309i
\(597\) −32.0000 −1.30967
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) − 6.00000i − 0.244949i
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −40.0000 −1.63028
\(603\) − 4.00000i − 0.162893i
\(604\) − 10.0000i − 0.406894i
\(605\) − 7.00000i − 0.284590i
\(606\) 36.0000i 1.46240i
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) −30.0000 −1.21666
\(609\) − 16.0000i − 0.648353i
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) −8.00000 −0.322854
\(615\) 12.0000 0.483887
\(616\) − 24.0000i − 0.966988i
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) 2.00000i 0.0803868i 0.999192 + 0.0401934i \(0.0127974\pi\)
−0.999192 + 0.0401934i \(0.987203\pi\)
\(620\) −10.0000 −0.401610
\(621\) 24.0000 0.963087
\(622\) − 4.00000i − 0.160385i
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 22.0000i 0.879297i
\(627\) 24.0000 0.958468
\(628\) −6.00000 −0.239426
\(629\) − 4.00000i − 0.159490i
\(630\) − 4.00000i − 0.159364i
\(631\) 14.0000i 0.557331i 0.960388 + 0.278666i \(0.0898921\pi\)
−0.960388 + 0.278666i \(0.910108\pi\)
\(632\) 36.0000i 1.43200i
\(633\) −24.0000 −0.953914
\(634\) −18.0000 −0.714871
\(635\) − 2.00000i − 0.0793676i
\(636\) −4.00000 −0.158610
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 6.00000i 0.237356i
\(640\) −3.00000 −0.118585
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 20.0000i 0.789337i
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 24.0000i 0.945732i
\(645\) − 20.0000i − 0.787499i
\(646\) 12.0000 0.472134
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 33.0000i 1.29636i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −80.0000 −3.13545
\(652\) 12.0000i 0.469956i
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −20.0000 −0.782062
\(655\) − 20.0000i − 0.781465i
\(656\) 6.00000i 0.234261i
\(657\) 6.00000i 0.234082i
\(658\) − 16.0000i − 0.623745i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 4.00000 0.155700
\(661\) − 2.00000i − 0.0777910i −0.999243 0.0388955i \(-0.987616\pi\)
0.999243 0.0388955i \(-0.0123839\pi\)
\(662\) −18.0000 −0.699590
\(663\) 0 0
\(664\) −48.0000 −1.86276
\(665\) − 24.0000i − 0.930680i
\(666\) 2.00000 0.0774984
\(667\) 12.0000 0.464642
\(668\) 12.0000i 0.464294i
\(669\) − 8.00000i − 0.309298i
\(670\) 4.00000i 0.154533i
\(671\) − 4.00000i − 0.154418i
\(672\) −40.0000 −1.54303
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 26.0000i 1.00148i
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) − 28.0000i − 1.07533i
\(679\) 8.00000 0.307012
\(680\) 6.00000 0.230089
\(681\) − 8.00000i − 0.306561i
\(682\) 20.0000i 0.765840i
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) − 6.00000i − 0.229416i
\(685\) 2.00000 0.0764161
\(686\) −8.00000 −0.305441
\(687\) − 44.0000i − 1.67870i
\(688\) 10.0000 0.381246
\(689\) 0 0
\(690\) 12.0000 0.456832
\(691\) 22.0000i 0.836919i 0.908235 + 0.418460i \(0.137430\pi\)
−0.908235 + 0.418460i \(0.862570\pi\)
\(692\) 6.00000 0.228086
\(693\) 8.00000 0.303895
\(694\) 22.0000i 0.835109i
\(695\) 0 0
\(696\) 12.0000i 0.454859i
\(697\) 12.0000i 0.454532i
\(698\) 30.0000 1.13552
\(699\) 20.0000 0.756469
\(700\) − 4.00000i − 0.151186i
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 14.0000i 0.527645i
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) 72.0000i 2.70784i
\(708\) 12.0000i 0.450988i
\(709\) − 10.0000i − 0.375558i −0.982211 0.187779i \(-0.939871\pi\)
0.982211 0.187779i \(-0.0601289\pi\)
\(710\) − 6.00000i − 0.225176i
\(711\) −12.0000 −0.450035
\(712\) −6.00000 −0.224860
\(713\) − 60.0000i − 2.24702i
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 12.0000i 0.448148i
\(718\) 10.0000 0.373197
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 8.00000i − 0.297936i
\(722\) 17.0000i 0.632674i
\(723\) 20.0000i 0.743808i
\(724\) 22.0000 0.817624
\(725\) −2.00000 −0.0742781
\(726\) 14.0000i 0.519589i
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) − 6.00000i − 0.222070i
\(731\) 20.0000 0.739727
\(732\) −4.00000 −0.147844
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 14.0000i 0.516749i
\(735\) − 18.0000i − 0.663940i
\(736\) − 30.0000i − 1.10581i
\(737\) −8.00000 −0.294684
\(738\) −6.00000 −0.220863
\(739\) − 6.00000i − 0.220714i −0.993892 0.110357i \(-0.964801\pi\)
0.993892 0.110357i \(-0.0351994\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 8.00000 0.293689
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) 60.0000 2.19971
\(745\) 18.0000 0.659469
\(746\) − 34.0000i − 1.24483i
\(747\) − 16.0000i − 0.585409i
\(748\) 4.00000i 0.146254i
\(749\) 40.0000i 1.46157i
\(750\) −2.00000 −0.0730297
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 48.0000 1.74922
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 16.0000i 0.581914i
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 10.0000 0.363216
\(759\) 24.0000i 0.871145i
\(760\) 18.0000i 0.652929i
\(761\) − 18.0000i − 0.652499i −0.945284 0.326250i \(-0.894215\pi\)
0.945284 0.326250i \(-0.105785\pi\)
\(762\) 4.00000i 0.144905i
\(763\) −40.0000 −1.44810
\(764\) 0 0
\(765\) 2.00000i 0.0723102i
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 34.0000 1.22687
\(769\) 26.0000i 0.937584i 0.883309 + 0.468792i \(0.155311\pi\)
−0.883309 + 0.468792i \(0.844689\pi\)
\(770\) −8.00000 −0.288300
\(771\) 4.00000 0.144056
\(772\) 2.00000i 0.0719816i
\(773\) − 2.00000i − 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 10.0000i 0.359443i
\(775\) 10.0000i 0.359211i
\(776\) −6.00000 −0.215387
\(777\) 16.0000 0.573997
\(778\) − 10.0000i − 0.358517i
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 12.0000i 0.429119i
\(783\) 8.00000 0.285897
\(784\) 9.00000 0.321429
\(785\) 6.00000i 0.214149i
\(786\) 40.0000i 1.42675i
\(787\) − 8.00000i − 0.285169i −0.989783 0.142585i \(-0.954459\pi\)
0.989783 0.142585i \(-0.0455413\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −28.0000 −0.996826
\(790\) 12.0000 0.426941
\(791\) − 56.0000i − 1.99113i
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) 4.00000i 0.141865i
\(796\) 16.0000 0.567105
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 48.0000i 1.69918i
\(799\) 8.00000i 0.283020i
\(800\) 5.00000i 0.176777i
\(801\) − 2.00000i − 0.0706665i
\(802\) −10.0000 −0.353112
\(803\) 12.0000 0.423471
\(804\) 8.00000i 0.282138i
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) − 54.0000i − 1.89971i
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 11.0000 0.386501
\(811\) − 42.0000i − 1.47482i −0.675446 0.737410i \(-0.736049\pi\)
0.675446 0.737410i \(-0.263951\pi\)
\(812\) 8.00000i 0.280745i
\(813\) 4.00000i 0.140286i
\(814\) − 4.00000i − 0.140200i
\(815\) 12.0000 0.420342
\(816\) −4.00000 −0.140028
\(817\) 60.0000i 2.09913i
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 50.0000i 1.74501i 0.488603 + 0.872506i \(0.337507\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) −4.00000 −0.139516
\(823\) 46.0000 1.60346 0.801730 0.597687i \(-0.203913\pi\)
0.801730 + 0.597687i \(0.203913\pi\)
\(824\) 6.00000i 0.209020i
\(825\) − 4.00000i − 0.139262i
\(826\) − 24.0000i − 0.835067i
\(827\) 32.0000i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(828\) 6.00000 0.208514
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 16.0000i 0.555368i
\(831\) −28.0000 −0.971309
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) −12.0000 −0.415029
\(837\) − 40.0000i − 1.38260i
\(838\) 40.0000i 1.38178i
\(839\) − 38.0000i − 1.31191i −0.754802 0.655953i \(-0.772267\pi\)
0.754802 0.655953i \(-0.227733\pi\)
\(840\) 24.0000i 0.828079i
\(841\) −25.0000 −0.862069
\(842\) 10.0000 0.344623
\(843\) − 12.0000i − 0.413302i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 28.0000i 0.962091i
\(848\) −2.00000 −0.0686803
\(849\) 4.00000 0.137280
\(850\) − 2.00000i − 0.0685994i
\(851\) 12.0000i 0.411355i
\(852\) − 12.0000i − 0.411113i
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) 8.00000 0.273754
\(855\) −6.00000 −0.205196
\(856\) − 30.0000i − 1.02538i
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 10.0000i 0.340997i
\(861\) −48.0000 −1.63584
\(862\) 14.0000 0.476842
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) − 20.0000i − 0.680414i
\(865\) − 6.00000i − 0.204006i
\(866\) 10.0000i 0.339814i
\(867\) 26.0000 0.883006
\(868\) 40.0000 1.35769
\(869\) 24.0000i 0.814144i
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) 30.0000 1.01593
\(873\) − 2.00000i − 0.0676897i
\(874\) −36.0000 −1.21772
\(875\) −4.00000 −0.135225
\(876\) − 12.0000i − 0.405442i
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 0 0
\(879\) 44.0000i 1.48408i
\(880\) 2.00000 0.0674200
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 9.00000i 0.303046i
\(883\) 22.0000 0.740359 0.370179 0.928960i \(-0.379296\pi\)
0.370179 + 0.928960i \(0.379296\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) − 14.0000i − 0.470339i
\(887\) 58.0000 1.94745 0.973725 0.227728i \(-0.0731298\pi\)
0.973725 + 0.227728i \(0.0731298\pi\)
\(888\) −12.0000 −0.402694
\(889\) 8.00000i 0.268311i
\(890\) 2.00000i 0.0670402i
\(891\) 22.0000i 0.737028i
\(892\) 4.00000i 0.133930i
\(893\) −24.0000 −0.803129
\(894\) −36.0000 −1.20402
\(895\) 12.0000i 0.401116i
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) − 20.0000i − 0.667037i
\(900\) −1.00000 −0.0333333
\(901\) −4.00000 −0.133259
\(902\) 12.0000i 0.399556i
\(903\) 80.0000i 2.66223i
\(904\) 42.0000i 1.39690i
\(905\) − 22.0000i − 0.731305i
\(906\) 20.0000 0.664455
\(907\) 34.0000 1.12895 0.564476 0.825450i \(-0.309078\pi\)
0.564476 + 0.825450i \(0.309078\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 18.0000 0.597022
\(910\) 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
\(913\) −32.0000 −1.05905
\(914\) 38.0000 1.25693
\(915\) 4.00000i 0.132236i
\(916\) 22.0000i 0.726900i
\(917\) 80.0000i 2.64183i
\(918\) 8.00000i 0.264039i
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) −18.0000 −0.593442
\(921\) 16.0000i 0.527218i
\(922\) 10.0000 0.329332
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) − 2.00000i − 0.0657596i
\(926\) 24.0000 0.788689
\(927\) −2.00000 −0.0656886
\(928\) − 10.0000i − 0.328266i
\(929\) − 38.0000i − 1.24674i −0.781927 0.623370i \(-0.785763\pi\)
0.781927 0.623370i \(-0.214237\pi\)
\(930\) − 20.0000i − 0.655826i
\(931\) 54.0000i 1.76978i
\(932\) −10.0000 −0.327561
\(933\) −8.00000 −0.261908
\(934\) − 10.0000i − 0.327210i
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 44.0000 1.43589
\(940\) −4.00000 −0.130466
\(941\) 18.0000i 0.586783i 0.955992 + 0.293392i \(0.0947840\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(942\) − 12.0000i − 0.390981i
\(943\) − 36.0000i − 1.17232i
\(944\) 6.00000i 0.195283i
\(945\) 16.0000 0.520480
\(946\) 20.0000 0.650256
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 24.0000 0.779484
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) 36.0000i 1.16738i
\(952\) −24.0000 −0.777844
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) − 2.00000i − 0.0647524i
\(955\) 0 0
\(956\) − 6.00000i − 0.194054i
\(957\) 8.00000i 0.258603i
\(958\) −30.0000 −0.969256
\(959\) −8.00000 −0.258333
\(960\) − 14.0000i − 0.451848i
\(961\) −69.0000 −2.22581
\(962\) 0 0
\(963\) 10.0000 0.322245
\(964\) − 10.0000i − 0.322078i
\(965\) 2.00000 0.0643823
\(966\) −48.0000 −1.54437
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) − 21.0000i − 0.674966i
\(969\) − 24.0000i − 0.770991i
\(970\) 2.00000i 0.0642161i
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) −24.0000 −0.767435
\(979\) −4.00000 −0.127841
\(980\) 9.00000i 0.287494i
\(981\) 10.0000i 0.319275i
\(982\) − 24.0000i − 0.765871i
\(983\) − 56.0000i − 1.78612i −0.449935 0.893061i \(-0.648553\pi\)
0.449935 0.893061i \(-0.351447\pi\)
\(984\) 36.0000 1.14764
\(985\) −6.00000 −0.191176
\(986\) 4.00000i 0.127386i
\(987\) −32.0000 −1.01857
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 2.00000i 0.0635642i
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) −50.0000 −1.58750
\(993\) 36.0000i 1.14243i
\(994\) 24.0000i 0.761234i
\(995\) − 16.0000i − 0.507234i
\(996\) 32.0000i 1.01396i
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 10.0000 0.316544
\(999\) 8.00000i 0.253109i
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.c.a.506.1 2
13.2 odd 12 845.2.e.b.191.1 2
13.3 even 3 845.2.m.b.316.2 4
13.4 even 6 845.2.m.b.361.2 4
13.5 odd 4 65.2.a.a.1.1 1
13.6 odd 12 845.2.e.b.146.1 2
13.7 odd 12 845.2.e.a.146.1 2
13.8 odd 4 845.2.a.a.1.1 1
13.9 even 3 845.2.m.b.361.1 4
13.10 even 6 845.2.m.b.316.1 4
13.11 odd 12 845.2.e.a.191.1 2
13.12 even 2 inner 845.2.c.a.506.2 2
39.5 even 4 585.2.a.h.1.1 1
39.8 even 4 7605.2.a.f.1.1 1
52.31 even 4 1040.2.a.f.1.1 1
65.18 even 4 325.2.b.b.274.2 2
65.34 odd 4 4225.2.a.g.1.1 1
65.44 odd 4 325.2.a.d.1.1 1
65.57 even 4 325.2.b.b.274.1 2
91.83 even 4 3185.2.a.e.1.1 1
104.5 odd 4 4160.2.a.q.1.1 1
104.83 even 4 4160.2.a.f.1.1 1
143.109 even 4 7865.2.a.c.1.1 1
156.83 odd 4 9360.2.a.ca.1.1 1
195.44 even 4 2925.2.a.f.1.1 1
195.83 odd 4 2925.2.c.h.2224.1 2
195.122 odd 4 2925.2.c.h.2224.2 2
260.239 even 4 5200.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.a.1.1 1 13.5 odd 4
325.2.a.d.1.1 1 65.44 odd 4
325.2.b.b.274.1 2 65.57 even 4
325.2.b.b.274.2 2 65.18 even 4
585.2.a.h.1.1 1 39.5 even 4
845.2.a.a.1.1 1 13.8 odd 4
845.2.c.a.506.1 2 1.1 even 1 trivial
845.2.c.a.506.2 2 13.12 even 2 inner
845.2.e.a.146.1 2 13.7 odd 12
845.2.e.a.191.1 2 13.11 odd 12
845.2.e.b.146.1 2 13.6 odd 12
845.2.e.b.191.1 2 13.2 odd 12
845.2.m.b.316.1 4 13.10 even 6
845.2.m.b.316.2 4 13.3 even 3
845.2.m.b.361.1 4 13.9 even 3
845.2.m.b.361.2 4 13.4 even 6
1040.2.a.f.1.1 1 52.31 even 4
2925.2.a.f.1.1 1 195.44 even 4
2925.2.c.h.2224.1 2 195.83 odd 4
2925.2.c.h.2224.2 2 195.122 odd 4
3185.2.a.e.1.1 1 91.83 even 4
4160.2.a.f.1.1 1 104.83 even 4
4160.2.a.q.1.1 1 104.5 odd 4
4225.2.a.g.1.1 1 65.34 odd 4
5200.2.a.d.1.1 1 260.239 even 4
7605.2.a.f.1.1 1 39.8 even 4
7865.2.a.c.1.1 1 143.109 even 4
9360.2.a.ca.1.1 1 156.83 odd 4