Properties

Label 2175.2.c.a.349.1
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
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] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
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 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.a.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -4.00000 q^{21} +4.00000i q^{23} -3.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} -1.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +2.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} +8.00000i q^{38} -6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{42} +4.00000i q^{43} +4.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -12.0000 q^{56} +8.00000i q^{57} +1.00000i q^{58} +12.0000 q^{59} +6.00000 q^{61} -4.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} +4.00000 q^{69} +16.0000 q^{71} +3.00000i q^{72} +6.00000i q^{73} +6.00000 q^{74} -8.00000 q^{76} +6.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +16.0000i q^{83} -4.00000 q^{84} +4.00000 q^{86} +1.00000i q^{87} -2.00000 q^{89} -24.0000 q^{91} +4.00000i q^{92} -4.00000i q^{93} -5.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} - 2 q^{16} - 16 q^{19} - 8 q^{21} - 6 q^{24} - 12 q^{26} - 2 q^{29} + 8 q^{31} + 4 q^{34} - 2 q^{36} - 12 q^{39} + 4 q^{41} + 8 q^{46} - 18 q^{49} + 4 q^{51} + 2 q^{54} - 24 q^{56} + 24 q^{59} + 12 q^{61} - 14 q^{64} + 8 q^{69} + 32 q^{71} + 12 q^{74} - 16 q^{76} - 24 q^{79} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 4 q^{89} - 48 q^{91} - 10 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000i 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 8.00000i 1.29777i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) − 6.00000i − 0.832050i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 8.00000i 1.05963i
\(58\) 1.00000i 0.131306i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 4.00000i 0.503953i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 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
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 6.00000i 0.679366i
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) 4.00000i 0.417029i
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −18.0000 −1.76505
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 16.0000i − 1.54678i −0.633932 0.773389i \(-0.718560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 4.00000i 0.377964i
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 6.00000i 0.554700i
\(118\) − 12.0000i − 1.10469i
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 6.00000i − 0.543214i
\(123\) − 2.00000i − 0.180334i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 32.0000i 2.77475i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 16.0000i − 1.34269i
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 9.00000i 0.742307i
\(148\) 6.00000i 0.493197i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 24.0000i 1.94666i
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 12.0000i 0.954669i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) − 1.00000i − 0.0785674i
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) − 20.0000i − 1.54765i −0.633402 0.773823i \(-0.718342\pi\)
0.633402 0.773823i \(-0.281658\pi\)
\(168\) 12.0000i 0.925820i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 4.00000i 0.304997i
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 2.00000i 0.149906i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 24.0000i 1.77900i
\(183\) − 6.00000i − 0.443533i
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 7.00000i 0.505181i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 10.0000i 0.703598i
\(203\) 4.00000i 0.280745i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) − 4.00000i − 0.278019i
\(208\) 6.00000i 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 16.0000i − 1.09630i
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) − 16.0000i − 1.08615i
\(218\) − 2.00000i − 0.135457i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) − 6.00000i − 0.402694i
\(223\) − 12.0000i − 0.803579i −0.915732 0.401790i \(-0.868388\pi\)
0.915732 0.401790i \(-0.131612\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 8.00000i 0.529813i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000i 0.196960i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 12.0000i 0.779484i
\(238\) − 8.00000i − 0.518563i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 11.0000i 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 48.0000i 3.05417i
\(248\) − 12.0000i − 0.762001i
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 26.0000i 1.62184i 0.585160 + 0.810918i \(0.301032\pi\)
−0.585160 + 0.810918i \(0.698968\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 32.0000 1.96205
\(267\) 2.00000i 0.122398i
\(268\) − 8.00000i − 0.488678i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 24.0000i 1.45255i
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.00000i − 0.472225i
\(288\) 5.00000i 0.294628i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 6.00000i 0.351123i
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) − 10.0000i − 0.579284i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 16.0000i − 0.920697i
\(303\) 10.0000i 0.574485i
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 18.0000i 1.01905i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) −16.0000 −0.893033
\(322\) − 16.0000i − 0.891645i
\(323\) − 16.0000i − 0.890264i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 2.00000i − 0.110600i
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 16.0000i 0.878114i
\(333\) − 6.00000i − 0.328798i
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) − 30.0000i − 1.63420i −0.576493 0.817102i \(-0.695579\pi\)
0.576493 0.817102i \(-0.304421\pi\)
\(338\) 23.0000i 1.25104i
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) − 8.00000i − 0.432590i
\(343\) 8.00000i 0.431959i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 1.00000i 0.0536056i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) − 8.00000i − 0.423405i
\(358\) − 12.0000i − 0.634220i
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 10.0000i 0.525588i
\(363\) 11.0000i 0.577350i
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) − 4.00000i − 0.207390i
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000i 0.309016i
\(378\) − 4.00000i − 0.205738i
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 4.00000i 0.204658i
\(383\) − 28.0000i − 1.43073i −0.698749 0.715367i \(-0.746260\pi\)
0.698749 0.715367i \(-0.253740\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) − 14.0000i − 0.710742i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 27.0000i 1.36371i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 32.0000 1.60200
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 8.00000i 0.399004i
\(403\) − 24.0000i − 1.19553i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) − 6.00000i − 0.297044i
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 4.00000i − 0.197066i
\(413\) − 48.0000i − 2.36193i
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) − 8.00000i − 0.389434i
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) − 24.0000i − 1.16144i
\(428\) − 16.0000i − 0.773389i
\(429\) 0 0
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 32.0000i − 1.53077i
\(438\) − 6.00000i − 0.286691i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 12.0000i − 0.570782i
\(443\) − 20.0000i − 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) − 10.0000i − 0.472984i
\(448\) 28.0000i 1.32288i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 2.00000i − 0.0940721i
\(453\) − 16.0000i − 0.751746i
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 14.0000i 0.654177i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 6.00000i 0.277350i
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) − 36.0000i − 1.65703i
\(473\) 0 0
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000i 0.274721i
\(478\) 24.0000i 1.09773i
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) − 2.00000i − 0.0910975i
\(483\) − 16.0000i − 0.728025i
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) − 18.0000i − 0.814822i
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) − 2.00000i − 0.0900755i
\(494\) 48.0000 2.15962
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) − 64.0000i − 2.87079i
\(498\) − 16.0000i − 0.716977i
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 12.0000 0.534522
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) − 8.00000i − 0.354943i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 11.0000i 0.486136i
\(513\) − 8.00000i − 0.353209i
\(514\) 26.0000 1.14681
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) − 24.0000i − 1.05450i
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 32.0000i 1.38738i
\(533\) − 12.0000i − 0.519778i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) − 12.0000i − 0.517838i
\(538\) − 18.0000i − 0.776035i
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 20.0000i 0.859074i
\(543\) 10.0000i 0.429141i
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) − 32.0000i − 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) − 12.0000i − 0.510754i
\(553\) 48.0000i 2.04117i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 4.00000i 0.169334i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000i 0.253095i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) − 4.00000i − 0.167984i
\(568\) − 48.0000i − 2.01404i
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 4.00000i 0.167102i
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 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
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 64.0000 2.65517
\(582\) 14.0000i 0.580319i
\(583\) 0 0
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) − 16.0000i − 0.660391i −0.943913 0.330195i \(-0.892885\pi\)
0.943913 0.330195i \(-0.107115\pi\)
\(588\) 9.00000i 0.371154i
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) − 6.00000i − 0.246598i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) − 16.0000i − 0.654836i
\(598\) − 24.0000i − 0.981433i
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) 8.00000i 0.325785i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 40.0000i 1.62221i
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 28.0000i 1.12270i
\(623\) 8.00000i 0.320513i
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 14.0000i 0.558661i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 36.0000i 1.43200i
\(633\) − 8.00000i − 0.317971i
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 54.0000i 2.13956i
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 16.0000i 0.631470i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) − 3.00000i − 0.117851i
\(649\) 0 0
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) − 12.0000i − 0.469956i
\(653\) 50.0000i 1.95665i 0.207072 + 0.978326i \(0.433606\pi\)
−0.207072 + 0.978326i \(0.566394\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) − 6.00000i − 0.234082i
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 16.0000i 0.621858i
\(663\) − 12.0000i − 0.466041i
\(664\) 48.0000 1.86276
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) − 4.00000i − 0.154881i
\(668\) − 20.0000i − 0.773823i
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) 0 0
\(672\) 20.0000i 0.771517i
\(673\) 30.0000i 1.15642i 0.815890 + 0.578208i \(0.196248\pi\)
−0.815890 + 0.578208i \(0.803752\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) − 26.0000i − 0.999261i −0.866239 0.499631i \(-0.833469\pi\)
0.866239 0.499631i \(-0.166531\pi\)
\(678\) 2.00000i 0.0768095i
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 14.0000i 0.534133i
\(688\) − 4.00000i − 0.152499i
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 2.00000i 0.0760286i
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) 4.00000i 0.151511i
\(698\) − 2.00000i − 0.0757011i
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) − 48.0000i − 1.81035i
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 40.0000i 1.50435i
\(708\) − 12.0000i − 0.450988i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 6.00000i 0.224860i
\(713\) 16.0000i 0.599205i
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 24.0000i 0.896296i
\(718\) − 36.0000i − 1.34351i
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) − 45.0000i − 1.67473i
\(723\) − 2.00000i − 0.0743808i
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) 72.0000i 2.66850i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) − 6.00000i − 0.221766i
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) 2.00000i 0.0736210i
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 0 0
\(741\) 48.0000 1.76332
\(742\) 24.0000i 0.881068i
\(743\) 32.0000i 1.17397i 0.809599 + 0.586983i \(0.199684\pi\)
−0.809599 + 0.586983i \(0.800316\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) 34.0000 1.24483
\(747\) − 16.0000i − 0.585409i
\(748\) 0 0
\(749\) −64.0000 −2.33851
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 8.00000i 0.289809i
\(763\) − 8.00000i − 0.289619i
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −28.0000 −1.01168
\(767\) − 72.0000i − 2.59977i
\(768\) 17.0000i 0.613435i
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) − 2.00000i − 0.0719816i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −42.0000 −1.50771
\(777\) − 24.0000i − 0.860995i
\(778\) − 26.0000i − 0.932145i
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000i 0.286079i
\(783\) − 1.00000i − 0.0357371i
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) − 24.0000i − 0.855508i −0.903895 0.427754i \(-0.859305\pi\)
0.903895 0.427754i \(-0.140695\pi\)
\(788\) 6.00000i 0.213741i
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) − 36.0000i − 1.27840i
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) − 32.0000i − 1.13279i
\(799\) 0 0
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) − 18.0000i − 0.635602i
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) − 18.0000i − 0.633630i
\(808\) 30.0000i 1.05540i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 4.00000i 0.140372i
\(813\) 20.0000i 0.701431i
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) − 32.0000i − 1.11954i
\(818\) − 6.00000i − 0.209785i
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 8.00000i 0.278862i 0.990232 + 0.139431i \(0.0445274\pi\)
−0.990232 + 0.139431i \(0.955473\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 42.0000i 1.45609i
\(833\) − 18.0000i − 0.623663i
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000i 0.138260i
\(838\) 4.00000i 0.138178i
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 30.0000i − 1.03387i
\(843\) 6.00000i 0.206651i
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000i 1.51186i
\(848\) 6.00000i 0.206041i
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) − 16.0000i − 0.548151i
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) −48.0000 −1.64061
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) − 32.0000i − 1.08992i
\(863\) 52.0000i 1.77010i 0.465495 + 0.885050i \(0.345876\pi\)
−0.465495 + 0.885050i \(0.654124\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) − 13.0000i − 0.441503i
\(868\) − 16.0000i − 0.543075i
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) − 6.00000i − 0.203186i
\(873\) 14.0000i 0.473828i
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) − 32.0000i − 1.07445i −0.843437 0.537227i \(-0.819472\pi\)
0.843437 0.537227i \(-0.180528\pi\)
\(888\) − 18.0000i − 0.604040i
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) − 12.0000i − 0.401790i
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) − 24.0000i − 0.801337i
\(898\) − 6.00000i − 0.200223i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 8.00000i 0.265489i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) − 8.00000i − 0.264906i
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 2.00000i 0.0660098i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 18.0000i 0.592798i
\(923\) − 96.0000i − 3.15988i
\(924\) 0 0
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 4.00000i 0.131377i
\(928\) 5.00000i 0.164133i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) − 18.0000i − 0.589610i
\(933\) 28.0000i 0.916679i
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 32.0000i 1.04484i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) 8.00000i 0.260516i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 12.0000i 0.389742i
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) − 24.0000i − 0.777844i
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 12.0000i 0.387702i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 36.0000i − 1.16069i
\(963\) 16.0000i 0.515593i
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 56.0000i 1.80084i 0.435023 + 0.900419i \(0.356740\pi\)
−0.435023 + 0.900419i \(0.643260\pi\)
\(968\) 33.0000i 1.06066i
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 80.0000i 2.56468i
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) 12.0000i 0.383718i
\(979\) 0 0
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 24.0000i 0.765871i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 48.0000i 1.52708i
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) − 20.0000i − 0.635001i
\(993\) 16.0000i 0.507745i
\(994\) −64.0000 −2.02996
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) 38.0000i 1.20347i 0.798695 + 0.601736i \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) 36.0000i 1.13956i
\(999\) −6.00000 −0.189832
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 2175.2.c.a.349.1 2
5.2 odd 4 2175.2.a.h.1.1 1
5.3 odd 4 435.2.a.a.1.1 1
5.4 even 2 inner 2175.2.c.a.349.2 2
15.2 even 4 6525.2.a.e.1.1 1
15.8 even 4 1305.2.a.e.1.1 1
20.3 even 4 6960.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.a.1.1 1 5.3 odd 4
1305.2.a.e.1.1 1 15.8 even 4
2175.2.a.h.1.1 1 5.2 odd 4
2175.2.c.a.349.1 2 1.1 even 1 trivial
2175.2.c.a.349.2 2 5.4 even 2 inner
6525.2.a.e.1.1 1 15.2 even 4
6960.2.a.w.1.1 1 20.3 even 4