Properties

Label 1425.2.c.a.799.2
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.a.799.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} -2.00000i q^{12} -6.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} +3.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} +12.0000 q^{26} -1.00000i q^{27} +6.00000i q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -3.00000i q^{33} +6.00000 q^{34} +2.00000 q^{36} -8.00000i q^{37} +2.00000i q^{38} +6.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} +12.0000i q^{52} -6.00000i q^{53} +2.00000 q^{54} +1.00000i q^{57} +20.0000i q^{58} +7.00000 q^{61} +4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} +6.00000i q^{68} -4.00000 q^{69} +12.0000 q^{71} -11.0000i q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000i q^{77} +12.0000i q^{78} +1.00000 q^{81} -16.0000i q^{82} +4.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} +10.0000i q^{87} -10.0000 q^{89} -18.0000 q^{91} -8.00000i q^{92} +2.00000i q^{93} +6.00000 q^{94} +8.00000 q^{96} +2.00000i q^{97} -4.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} + 12 q^{14} - 8 q^{16} + 2 q^{19} + 6 q^{21} + 24 q^{26} + 20 q^{29} + 4 q^{31} + 12 q^{34} + 4 q^{36} + 12 q^{39} - 16 q^{41} + 12 q^{44} - 16 q^{46} - 4 q^{49} + 6 q^{51} + 4 q^{54} + 14 q^{61} + 16 q^{64} + 12 q^{66} - 8 q^{69} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 2 q^{81} - 12 q^{84} + 4 q^{86} - 20 q^{89} - 36 q^{91} + 12 q^{94} + 16 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) − 6.00000i − 1.27920i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 12.0000 2.35339
\(27\) − 1.00000i − 0.192450i
\(28\) 6.00000i 1.13389i
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) − 3.00000i − 0.522233i
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 2.00000i 0.324443i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 6.00000i 0.925820i
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 12.0000i 1.66410i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 20.0000i 2.62613i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 3.00000i 0.377964i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) − 11.0000i − 1.28745i −0.765256 0.643726i \(-0.777388\pi\)
0.765256 0.643726i \(-0.222612\pi\)
\(74\) 16.0000 1.85996
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 9.00000i 1.02565i
\(78\) 12.0000i 1.35873i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 16.0000i − 1.76690i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 10.0000i 1.07211i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) − 8.00000i − 0.834058i
\(93\) 2.00000i 0.207390i
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 4.00000i − 0.404061i
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 2.00000i 0.192450i
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 12.0000i 1.13389i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −20.0000 −1.85695
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 14.0000i 1.26750i
\(123\) − 8.00000i − 0.721336i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −13.0000 −1.13582 −0.567908 0.823092i \(-0.692247\pi\)
−0.567908 + 0.823092i \(0.692247\pi\)
\(132\) 6.00000i 0.522233i
\(133\) − 3.00000i − 0.260133i
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 24.0000i 2.01404i
\(143\) 18.0000i 1.50524i
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 22.0000 1.82073
\(147\) − 2.00000i − 0.164957i
\(148\) 16.0000i 1.31519i
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 3.00000i 0.242536i
\(154\) −18.0000 −1.45048
\(155\) 0 0
\(156\) −12.0000 −0.960769
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 2.00000i 0.157135i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 16.0000 1.24939
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 2.00000i 0.152499i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) −20.0000 −1.51620
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) − 20.0000i − 1.49906i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) − 36.0000i − 2.66850i
\(183\) 7.00000i 0.517455i
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 9.00000i 0.658145i
\(188\) 6.00000i 0.437595i
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 8.00000i 0.577350i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 6.00000i 0.426401i
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 4.00000i 0.281439i
\(203\) − 30.0000i − 2.10559i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −28.0000 −1.95085
\(207\) − 4.00000i − 0.278019i
\(208\) 24.0000i 1.66410i
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 12.0000i 0.822226i
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) − 6.00000i − 0.407307i
\(218\) − 40.0000i − 2.70914i
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) −18.0000 −1.21081
\(222\) 16.0000i 1.07385i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) −24.0000 −1.60357
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) −9.00000 −0.592157
\(232\) 0 0
\(233\) − 11.0000i − 0.720634i −0.932830 0.360317i \(-0.882669\pi\)
0.932830 0.360317i \(-0.117331\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) − 18.0000i − 1.16677i
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) − 4.00000i − 0.257130i
\(243\) 1.00000i 0.0641500i
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) − 6.00000i − 0.381771i
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) − 6.00000i − 0.377964i
\(253\) − 12.0000i − 0.754434i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 2.00000i 0.124515i
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) − 26.0000i − 1.60629i
\(263\) − 21.0000i − 1.29492i −0.762101 0.647458i \(-0.775832\pi\)
0.762101 0.647458i \(-0.224168\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) − 10.0000i − 0.611990i
\(268\) 16.0000i 0.977356i
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 12.0000i 0.727607i
\(273\) − 18.0000i − 1.08941i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) − 13.0000i − 0.781094i −0.920583 0.390547i \(-0.872286\pi\)
0.920583 0.390547i \(-0.127714\pi\)
\(278\) 10.0000i 0.599760i
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 19.0000i 1.12943i 0.825285 + 0.564716i \(0.191014\pi\)
−0.825285 + 0.564716i \(0.808986\pi\)
\(284\) −24.0000 −1.42414
\(285\) 0 0
\(286\) −36.0000 −2.12872
\(287\) 24.0000i 1.41668i
\(288\) 8.00000i 0.471405i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 22.0000i 1.28745i
\(293\) 4.00000i 0.233682i 0.993151 + 0.116841i \(0.0372769\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 4.00000 0.233285
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000i 0.174078i
\(298\) − 30.0000i − 1.73785i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) − 16.0000i − 0.920697i
\(303\) 2.00000i 0.114897i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) − 18.0000i − 1.02565i
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 12.0000i 0.672927i
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 24.0000i 1.33747i
\(323\) − 3.00000i − 0.166924i
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 32.0000 1.77232
\(327\) − 20.0000i − 1.10600i
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) 8.00000i 0.438397i
\(334\) 36.0000 1.96983
\(335\) 0 0
\(336\) −12.0000 −0.654654
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) − 46.0000i − 2.50207i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) − 2.00000i − 0.108148i
\(343\) − 15.0000i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) −28.0000 −1.50529
\(347\) − 3.00000i − 0.161048i −0.996753 0.0805242i \(-0.974341\pi\)
0.996753 0.0805242i \(-0.0256594\pi\)
\(348\) − 20.0000i − 1.07211i
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 24.0000i 1.27920i
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 20.0000 1.06000
\(357\) − 9.00000i − 0.476331i
\(358\) 20.0000i 1.05703i
\(359\) −25.0000 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.00000i 0.210235i
\(363\) − 2.00000i − 0.104973i
\(364\) 36.0000 1.88691
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) − 16.0000i − 0.834058i
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) − 4.00000i − 0.207390i
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) − 60.0000i − 3.09016i
\(378\) − 6.00000i − 0.308607i
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) − 6.00000i − 0.306987i
\(383\) 14.0000i 0.715367i 0.933843 + 0.357683i \(0.116433\pi\)
−0.933843 + 0.357683i \(0.883567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) 1.00000i 0.0508329i
\(388\) − 4.00000i − 0.203069i
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) − 13.0000i − 0.655763i
\(394\) −4.00000 −0.201517
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 7.00000i 0.351320i 0.984451 + 0.175660i \(0.0562059\pi\)
−0.984451 + 0.175660i \(0.943794\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 16.0000i 0.798007i
\(403\) − 12.0000i − 0.597763i
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 60.0000 2.97775
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) − 28.0000i − 1.37946i
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −48.0000 −2.35339
\(417\) 5.00000i 0.244851i
\(418\) − 6.00000i − 0.293470i
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) − 56.0000i − 2.72604i
\(423\) 3.00000i 0.145865i
\(424\) 0 0
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) − 21.0000i − 1.01626i
\(428\) − 4.00000i − 0.193347i
\(429\) −18.0000 −0.869048
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000i 0.192450i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) 40.0000 1.91565
\(437\) 4.00000i 0.191346i
\(438\) 22.0000i 1.05120i
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) − 36.0000i − 1.71235i
\(443\) 39.0000i 1.85295i 0.376361 + 0.926473i \(0.377175\pi\)
−0.376361 + 0.926473i \(0.622825\pi\)
\(444\) −16.0000 −0.759326
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) − 15.0000i − 0.709476i
\(448\) − 24.0000i − 1.13389i
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 12.0000i 0.564433i
\(453\) − 8.00000i − 0.375873i
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) − 3.00000i − 0.140334i −0.997535 0.0701670i \(-0.977647\pi\)
0.997535 0.0701670i \(-0.0223532\pi\)
\(458\) 30.0000i 1.40181i
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) − 18.0000i − 0.837436i
\(463\) − 31.0000i − 1.44069i −0.693615 0.720346i \(-0.743983\pi\)
0.693615 0.720346i \(-0.256017\pi\)
\(464\) −40.0000 −1.85695
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 17.0000i 0.786666i 0.919396 + 0.393333i \(0.128678\pi\)
−0.919396 + 0.393333i \(0.871322\pi\)
\(468\) − 12.0000i − 0.554700i
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 3.00000i 0.137940i
\(474\) 0 0
\(475\) 0 0
\(476\) 18.0000 0.825029
\(477\) 6.00000i 0.274721i
\(478\) 30.0000i 1.37217i
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 24.0000i 1.09317i
\(483\) 12.0000i 0.546019i
\(484\) 4.00000 0.181818
\(485\) 0 0
\(486\) −2.00000 −0.0907218
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 16.0000i 0.721336i
\(493\) − 30.0000i − 1.35113i
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 36.0000i − 1.61482i
\(498\) − 8.00000i − 0.358489i
\(499\) 35.0000 1.56682 0.783408 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 54.0000i 2.41014i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) − 23.0000i − 1.02147i
\(508\) − 4.00000i − 0.177471i
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −33.0000 −1.45983
\(512\) 32.0000i 1.41421i
\(513\) − 1.00000i − 0.0441511i
\(514\) 16.0000 0.705730
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 9.00000i 0.395820i
\(518\) − 48.0000i − 2.10900i
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) − 20.0000i − 0.875376i
\(523\) 14.0000i 0.612177i 0.952003 + 0.306089i \(0.0990204\pi\)
−0.952003 + 0.306089i \(0.900980\pi\)
\(524\) 26.0000 1.13582
\(525\) 0 0
\(526\) 42.0000 1.83129
\(527\) − 6.00000i − 0.261364i
\(528\) 12.0000i 0.522233i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000i 0.260133i
\(533\) 48.0000i 2.07911i
\(534\) 20.0000 0.865485
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000i 0.431532i
\(538\) 60.0000i 2.58678i
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 24.0000i 1.03089i
\(543\) 2.00000i 0.0858282i
\(544\) −24.0000 −1.02899
\(545\) 0 0
\(546\) 36.0000 1.54066
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) 10.0000 0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) − 3.00000i − 0.127114i −0.997978 0.0635570i \(-0.979756\pi\)
0.997978 0.0635570i \(-0.0202445\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 4.00000i 0.168730i
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −38.0000 −1.59726
\(567\) − 3.00000i − 0.125988i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) − 36.0000i − 1.50524i
\(573\) − 3.00000i − 0.125327i
\(574\) −48.0000 −2.00348
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) − 3.00000i − 0.124892i −0.998048 0.0624458i \(-0.980110\pi\)
0.998048 0.0624458i \(-0.0198901\pi\)
\(578\) 16.0000i 0.665512i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) − 4.00000i − 0.165805i
\(583\) 18.0000i 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) −8.00000 −0.330477
\(587\) 37.0000i 1.52715i 0.645717 + 0.763577i \(0.276559\pi\)
−0.645717 + 0.763577i \(0.723441\pi\)
\(588\) 4.00000i 0.164957i
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 32.0000i 1.31519i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 30.0000 1.22885
\(597\) 5.00000i 0.204636i
\(598\) 48.0000i 1.96287i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) − 6.00000i − 0.244542i
\(603\) 8.00000i 0.325785i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −4.00000 −0.162489
\(607\) − 18.0000i − 0.730597i −0.930890 0.365299i \(-0.880967\pi\)
0.930890 0.365299i \(-0.119033\pi\)
\(608\) − 8.00000i − 0.324443i
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) − 6.00000i − 0.242536i
\(613\) 9.00000i 0.363507i 0.983344 + 0.181753i \(0.0581772\pi\)
−0.983344 + 0.181753i \(0.941823\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) − 23.0000i − 0.925945i −0.886373 0.462973i \(-0.846783\pi\)
0.886373 0.462973i \(-0.153217\pi\)
\(618\) − 28.0000i − 1.12633i
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 14.0000i 0.561349i
\(623\) 30.0000i 1.20192i
\(624\) −24.0000 −0.960769
\(625\) 0 0
\(626\) −28.0000 −1.11911
\(627\) − 3.00000i − 0.119808i
\(628\) − 4.00000i − 0.159617i
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) − 28.0000i − 1.11290i
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 12.0000i 0.475457i
\(638\) − 60.0000i − 2.37542i
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) − 1.00000i − 0.0394362i −0.999806 0.0197181i \(-0.993723\pi\)
0.999806 0.0197181i \(-0.00627687\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 27.0000i 1.06148i 0.847535 + 0.530740i \(0.178086\pi\)
−0.847535 + 0.530740i \(0.821914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 32.0000i 1.25322i
\(653\) − 1.00000i − 0.0391330i −0.999809 0.0195665i \(-0.993771\pi\)
0.999809 0.0195665i \(-0.00622861\pi\)
\(654\) 40.0000 1.56412
\(655\) 0 0
\(656\) 32.0000 1.24939
\(657\) 11.0000i 0.429151i
\(658\) − 18.0000i − 0.701713i
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 24.0000i 0.932786i
\(663\) − 18.0000i − 0.699062i
\(664\) 0 0
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) 40.0000i 1.54881i
\(668\) 36.0000i 1.39288i
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) −21.0000 −0.810696
\(672\) − 24.0000i − 0.925820i
\(673\) − 16.0000i − 0.616755i −0.951264 0.308377i \(-0.900214\pi\)
0.951264 0.308377i \(-0.0997859\pi\)
\(674\) −44.0000 −1.69482
\(675\) 0 0
\(676\) 46.0000 1.76923
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) 12.0000i 0.460857i
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) − 12.0000i − 0.459504i
\(683\) − 6.00000i − 0.229584i −0.993390 0.114792i \(-0.963380\pi\)
0.993390 0.114792i \(-0.0366201\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 30.0000 1.14541
\(687\) 15.0000i 0.572286i
\(688\) 4.00000i 0.152499i
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) − 28.0000i − 1.06440i
\(693\) − 9.00000i − 0.341882i
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) − 50.0000i − 1.89253i
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) − 12.0000i − 0.452911i
\(703\) − 8.00000i − 0.301726i
\(704\) −24.0000 −0.904534
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) 18.0000 0.673633
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 15.0000i 0.560185i
\(718\) − 50.0000i − 1.86598i
\(719\) 35.0000 1.30528 0.652640 0.757668i \(-0.273661\pi\)
0.652640 + 0.757668i \(0.273661\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 2.00000i 0.0744323i
\(723\) 12.0000i 0.446285i
\(724\) −4.00000 −0.148659
\(725\) 0 0
\(726\) 4.00000 0.148454
\(727\) 7.00000i 0.259616i 0.991539 + 0.129808i \(0.0414360\pi\)
−0.991539 + 0.129808i \(0.958564\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) − 14.0000i − 0.517455i
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 32.0000 1.17954
\(737\) 24.0000i 0.884051i
\(738\) 16.0000i 0.588968i
\(739\) 45.0000 1.65535 0.827676 0.561206i \(-0.189663\pi\)
0.827676 + 0.561206i \(0.189663\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) − 36.0000i − 1.32160i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) − 4.00000i − 0.146352i
\(748\) − 18.0000i − 0.658145i
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 27.0000i 0.983935i
\(754\) 120.000 4.37014
\(755\) 0 0
\(756\) 6.00000 0.218218
\(757\) − 23.0000i − 0.835949i −0.908459 0.417975i \(-0.862740\pi\)
0.908459 0.417975i \(-0.137260\pi\)
\(758\) 60.0000i 2.17930i
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) − 4.00000i − 0.144905i
\(763\) 60.0000i 2.17215i
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −28.0000 −1.01168
\(767\) 0 0
\(768\) 16.0000i 0.577350i
\(769\) 45.0000 1.62274 0.811371 0.584532i \(-0.198722\pi\)
0.811371 + 0.584532i \(0.198722\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) − 8.00000i − 0.287926i
\(773\) − 36.0000i − 1.29483i −0.762138 0.647415i \(-0.775850\pi\)
0.762138 0.647415i \(-0.224150\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) − 24.0000i − 0.860995i
\(778\) 30.0000i 1.07555i
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 24.0000i 0.858238i
\(783\) − 10.0000i − 0.357371i
\(784\) 8.00000 0.285714
\(785\) 0 0
\(786\) 26.0000 0.927389
\(787\) − 8.00000i − 0.285169i −0.989783 0.142585i \(-0.954459\pi\)
0.989783 0.142585i \(-0.0455413\pi\)
\(788\) − 4.00000i − 0.142494i
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) − 42.0000i − 1.49146i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 6.00000i 0.212398i
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) − 56.0000i − 1.97743i
\(803\) 33.0000i 1.16454i
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) 30.0000i 1.05605i
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 60.0000i 2.10559i
\(813\) 12.0000i 0.420858i
\(814\) −48.0000 −1.68240
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) − 1.00000i − 0.0349856i
\(818\) − 20.0000i − 0.699284i
\(819\) 18.0000 0.628971
\(820\) 0 0
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 19.0000i 0.662298i 0.943578 + 0.331149i \(0.107436\pi\)
−0.943578 + 0.331149i \(0.892564\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 52.0000i 1.80822i 0.427303 + 0.904109i \(0.359464\pi\)
−0.427303 + 0.904109i \(0.640536\pi\)
\(828\) 8.00000i 0.278019i
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) − 48.0000i − 1.66410i
\(833\) 6.00000i 0.207888i
\(834\) −10.0000 −0.346272
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) − 2.00000i − 0.0691301i
\(838\) − 40.0000i − 1.38178i
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 4.00000i 0.137849i
\(843\) 2.00000i 0.0688837i
\(844\) 56.0000 1.92760
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 6.00000i 0.206162i
\(848\) 24.0000i 0.824163i
\(849\) −19.0000 −0.652078
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) − 24.0000i − 0.822226i
\(853\) 34.0000i 1.16414i 0.813139 + 0.582069i \(0.197757\pi\)
−0.813139 + 0.582069i \(0.802243\pi\)
\(854\) 42.0000 1.43721
\(855\) 0 0
\(856\) 0 0
\(857\) − 48.0000i − 1.63965i −0.572615 0.819824i \(-0.694071\pi\)
0.572615 0.819824i \(-0.305929\pi\)
\(858\) − 36.0000i − 1.22902i
\(859\) −35.0000 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) − 36.0000i − 1.22616i
\(863\) 44.0000i 1.49778i 0.662696 + 0.748889i \(0.269412\pi\)
−0.662696 + 0.748889i \(0.730588\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) 52.0000 1.76703
\(867\) 8.00000i 0.271694i
\(868\) 12.0000i 0.407307i
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −22.0000 −0.743311
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) −4.00000 −0.134917
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 4.00000i 0.134687i
\(883\) − 21.0000i − 0.706706i −0.935490 0.353353i \(-0.885041\pi\)
0.935490 0.353353i \(-0.114959\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) −78.0000 −2.62046
\(887\) 52.0000i 1.74599i 0.487730 + 0.872995i \(0.337825\pi\)
−0.487730 + 0.872995i \(0.662175\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) − 8.00000i − 0.267860i
\(893\) − 3.00000i − 0.100391i
\(894\) 30.0000 1.00335
\(895\) 0 0
\(896\) 0 0
\(897\) 24.0000i 0.801337i
\(898\) 40.0000i 1.33482i
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 48.0000i 1.59823i
\(903\) − 3.00000i − 0.0998337i
\(904\) 0 0
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 2.00000i 0.0664089i 0.999449 + 0.0332045i \(0.0105712\pi\)
−0.999449 + 0.0332045i \(0.989429\pi\)
\(908\) 36.0000i 1.19470i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 2.00000 0.0662630 0.0331315 0.999451i \(-0.489452\pi\)
0.0331315 + 0.999451i \(0.489452\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 12.0000i − 0.397142i
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −30.0000 −0.991228
\(917\) 39.0000i 1.28789i
\(918\) − 6.00000i − 0.198030i
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) − 66.0000i − 2.17359i
\(923\) − 72.0000i − 2.36991i
\(924\) 18.0000 0.592157
\(925\) 0 0
\(926\) 62.0000 2.03745
\(927\) − 14.0000i − 0.459820i
\(928\) − 80.0000i − 2.62613i
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 22.0000i 0.720634i
\(933\) 7.00000i 0.229170i
\(934\) −34.0000 −1.11251
\(935\) 0 0
\(936\) 0 0
\(937\) − 53.0000i − 1.73143i −0.500533 0.865717i \(-0.666863\pi\)
0.500533 0.865717i \(-0.333137\pi\)
\(938\) − 48.0000i − 1.56726i
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) − 4.00000i − 0.130327i
\(943\) − 32.0000i − 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) 0 0
\(949\) −66.0000 −2.14245
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) − 16.0000i − 0.518291i −0.965838 0.259145i \(-0.916559\pi\)
0.965838 0.259145i \(-0.0834409\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −30.0000 −0.970269
\(957\) − 30.0000i − 0.969762i
\(958\) 80.0000i 2.58468i
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 96.0000i − 3.09516i
\(963\) − 2.00000i − 0.0644491i
\(964\) −24.0000 −0.772988
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) − 15.0000i − 0.480878i
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −28.0000 −0.896258
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 32.0000i 1.02325i
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) − 16.0000i − 0.510581i
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 60.0000 1.91079
\(987\) − 9.00000i − 0.286473i
\(988\) 12.0000i 0.381771i
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) − 16.0000i − 0.508001i
\(993\) 12.0000i 0.380808i
\(994\) 72.0000 2.28370
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 7.00000i 0.221692i 0.993838 + 0.110846i \(0.0353561\pi\)
−0.993838 + 0.110846i \(0.964644\pi\)
\(998\) 70.0000i 2.21581i
\(999\) −8.00000 −0.253109
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 1425.2.c.a.799.2 2
5.2 odd 4 57.2.a.b.1.1 1
5.3 odd 4 1425.2.a.i.1.1 1
5.4 even 2 inner 1425.2.c.a.799.1 2
15.2 even 4 171.2.a.c.1.1 1
15.8 even 4 4275.2.a.a.1.1 1
20.7 even 4 912.2.a.d.1.1 1
35.27 even 4 2793.2.a.a.1.1 1
40.27 even 4 3648.2.a.y.1.1 1
40.37 odd 4 3648.2.a.h.1.1 1
55.32 even 4 6897.2.a.g.1.1 1
60.47 odd 4 2736.2.a.h.1.1 1
65.12 odd 4 9633.2.a.p.1.1 1
95.37 even 4 1083.2.a.d.1.1 1
105.62 odd 4 8379.2.a.q.1.1 1
285.227 odd 4 3249.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 1 5.2 odd 4
171.2.a.c.1.1 1 15.2 even 4
912.2.a.d.1.1 1 20.7 even 4
1083.2.a.d.1.1 1 95.37 even 4
1425.2.a.i.1.1 1 5.3 odd 4
1425.2.c.a.799.1 2 5.4 even 2 inner
1425.2.c.a.799.2 2 1.1 even 1 trivial
2736.2.a.h.1.1 1 60.47 odd 4
2793.2.a.a.1.1 1 35.27 even 4
3249.2.a.a.1.1 1 285.227 odd 4
3648.2.a.h.1.1 1 40.37 odd 4
3648.2.a.y.1.1 1 40.27 even 4
4275.2.a.a.1.1 1 15.8 even 4
6897.2.a.g.1.1 1 55.32 even 4
8379.2.a.q.1.1 1 105.62 odd 4
9633.2.a.p.1.1 1 65.12 odd 4