Properties

Label 315.2.d.a.64.1
Level $315$
Weight $2$
Character 315.64
Analytic conductor $2.515$
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] = [315,2,Mod(64,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.51528766367\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 64.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 315.64
Dual form 315.2.d.a.64.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -2.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.00000i q^{2} -2.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +(2.00000 - 4.00000i) q^{10} +3.00000 q^{11} -1.00000i q^{13} -2.00000 q^{14} -4.00000 q^{16} -7.00000i q^{17} +(-4.00000 - 2.00000i) q^{20} -6.00000i q^{22} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -2.00000 q^{26} +2.00000i q^{28} -5.00000 q^{29} +2.00000 q^{31} +8.00000i q^{32} -14.0000 q^{34} +(1.00000 - 2.00000i) q^{35} +2.00000i q^{37} -2.00000 q^{41} +4.00000i q^{43} -6.00000 q^{44} +12.0000 q^{46} +3.00000i q^{47} -1.00000 q^{49} +(8.00000 - 6.00000i) q^{50} +2.00000i q^{52} +6.00000i q^{53} +(6.00000 + 3.00000i) q^{55} +10.0000i q^{58} +10.0000 q^{59} -8.00000 q^{61} -4.00000i q^{62} +8.00000 q^{64} +(1.00000 - 2.00000i) q^{65} +2.00000i q^{67} +14.0000i q^{68} +(-4.00000 - 2.00000i) q^{70} +8.00000 q^{71} -6.00000i q^{73} +4.00000 q^{74} -3.00000i q^{77} +5.00000 q^{79} +(-8.00000 - 4.00000i) q^{80} +4.00000i q^{82} -4.00000i q^{83} +(7.00000 - 14.0000i) q^{85} +8.00000 q^{86} -1.00000 q^{91} -12.0000i q^{92} +6.00000 q^{94} +7.00000i q^{97} +2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 4 q^{5} + 4 q^{10} + 6 q^{11} - 4 q^{14} - 8 q^{16} - 8 q^{20} + 6 q^{25} - 4 q^{26} - 10 q^{29} + 4 q^{31} - 28 q^{34} + 2 q^{35} - 4 q^{41} - 12 q^{44} + 24 q^{46} - 2 q^{49} + 16 q^{50} + 12 q^{55} + 20 q^{59} - 16 q^{61} + 16 q^{64} + 2 q^{65} - 8 q^{70} + 16 q^{71} + 8 q^{74} + 10 q^{79} - 16 q^{80} + 14 q^{85} + 16 q^{86} - 2 q^{91} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 4.00000i 0.632456 1.26491i
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 7.00000i 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −4.00000 2.00000i −0.894427 0.447214i
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −14.0000 −2.40098
\(35\) 1.00000 2.00000i 0.169031 0.338062i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 8.00000 6.00000i 1.13137 0.848528i
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 6.00000 + 3.00000i 0.809040 + 0.404520i
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000i 1.31306i
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 1.00000 2.00000i 0.124035 0.248069i
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 14.0000i 1.69775i
\(69\) 0 0
\(70\) −4.00000 2.00000i −0.478091 0.239046i
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −8.00000 4.00000i −0.894427 0.447214i
\(81\) 0 0
\(82\) 4.00000i 0.441726i
\(83\) 4.00000i 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 7.00000 14.0000i 0.759257 1.51851i
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 12.0000i 1.25109i
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 2.00000i 0.202031i
\(99\) 0 0
\(100\) −6.00000 8.00000i −0.600000 0.800000i
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 19.0000i 1.87213i 0.351833 + 0.936063i \(0.385559\pi\)
−0.351833 + 0.936063i \(0.614441\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 6.00000 12.0000i 0.572078 1.14416i
\(111\) 0 0
\(112\) 4.00000i 0.377964i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) −6.00000 + 12.0000i −0.559503 + 1.11901i
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 20.0000i 1.84115i
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 16.0000i 1.44857i
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −4.00000 2.00000i −0.350823 0.175412i
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) −2.00000 + 4.00000i −0.169031 + 0.338062i
\(141\) 0 0
\(142\) 16.0000i 1.34269i
\(143\) 3.00000i 0.250873i
\(144\) 0 0
\(145\) −10.0000 5.00000i −0.830455 0.415227i
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) 4.00000i 0.328798i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 4.00000 + 2.00000i 0.321288 + 0.160644i
\(156\) 0 0
\(157\) 18.0000i 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 0 0
\(160\) −8.00000 + 16.0000i −0.632456 + 1.26491i
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 3.00000i 0.232147i 0.993241 + 0.116073i \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) −28.0000 14.0000i −2.14750 1.07375i
\(171\) 0 0
\(172\) 8.00000i 0.609994i
\(173\) 9.00000i 0.684257i −0.939653 0.342129i \(-0.888852\pi\)
0.939653 0.342129i \(-0.111148\pi\)
\(174\) 0 0
\(175\) 4.00000 3.00000i 0.302372 0.226779i
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 + 4.00000i −0.147043 + 0.294086i
\(186\) 0 0
\(187\) 21.0000i 1.53567i
\(188\) 6.00000i 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i −0.817554 0.575853i \(-0.804670\pi\)
0.817554 0.575853i \(-0.195330\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 2.00000i 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 24.0000i 1.68863i
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) −4.00000 2.00000i −0.279372 0.139686i
\(206\) 38.0000 2.64759
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) −4.00000 + 8.00000i −0.272798 + 0.545595i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 10.0000i 0.677285i
\(219\) 0 0
\(220\) −12.0000 6.00000i −0.809040 0.404520i
\(221\) −7.00000 −0.470871
\(222\) 0 0
\(223\) 21.0000i 1.40626i −0.711059 0.703132i \(-0.751784\pi\)
0.711059 0.703132i \(-0.248216\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 17.0000i 1.12833i −0.825662 0.564165i \(-0.809198\pi\)
0.825662 0.564165i \(-0.190802\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 24.0000 + 12.0000i 1.58251 + 0.791257i
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −3.00000 + 6.00000i −0.195698 + 0.391397i
\(236\) −20.0000 −1.30189
\(237\) 0 0
\(238\) 14.0000i 0.907485i
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 4.00000i 0.257130i
\(243\) 0 0
\(244\) 16.0000 1.02430
\(245\) −2.00000 1.00000i −0.127775 0.0638877i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 22.0000 4.00000i 1.39140 0.252982i
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 18.0000i 1.13165i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 22.0000i 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −2.00000 + 4.00000i −0.124035 + 0.248069i
\(261\) 0 0
\(262\) 44.0000i 2.71833i
\(263\) 24.0000i 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) −6.00000 + 12.0000i −0.368577 + 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000i 0.244339i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 28.0000i 1.69775i
\(273\) 0 0
\(274\) −24.0000 −1.44989
\(275\) 9.00000 + 12.0000i 0.542720 + 0.723627i
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 0 0
\(280\) 0 0
\(281\) −7.00000 −0.417585 −0.208792 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) 11.0000i 0.653882i −0.945045 0.326941i \(-0.893982\pi\)
0.945045 0.326941i \(-0.106018\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 2.00000i 0.118056i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) −10.0000 + 20.0000i −0.587220 + 1.17444i
\(291\) 0 0
\(292\) 12.0000i 0.702247i
\(293\) 9.00000i 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 0 0
\(295\) 20.0000 + 10.0000i 1.16445 + 0.582223i
\(296\) 0 0
\(297\) 0 0
\(298\) 20.0000i 1.15857i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 26.0000i 1.49613i
\(303\) 0 0
\(304\) 0 0
\(305\) −16.0000 8.00000i −0.916157 0.458079i
\(306\) 0 0
\(307\) 7.00000i 0.399511i 0.979846 + 0.199756i \(0.0640148\pi\)
−0.979846 + 0.199756i \(0.935985\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 0 0
\(310\) 4.00000 8.00000i 0.227185 0.454369i
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 21.0000i 1.18699i −0.804838 0.593495i \(-0.797748\pi\)
0.804838 0.593495i \(-0.202252\pi\)
\(314\) −36.0000 −2.03160
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 12.0000i 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) −15.0000 −0.839839
\(320\) 16.0000 + 8.00000i 0.894427 + 0.447214i
\(321\) 0 0
\(322\) 12.0000i 0.668734i
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 3.00000i 0.221880 0.166410i
\(326\) 28.0000 1.55078
\(327\) 0 0
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 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
\(333\) 0 0
\(334\) 6.00000 0.328305
\(335\) −2.00000 + 4.00000i −0.109272 + 0.218543i
\(336\) 0 0
\(337\) 18.0000i 0.980522i −0.871576 0.490261i \(-0.836901\pi\)
0.871576 0.490261i \(-0.163099\pi\)
\(338\) 24.0000i 1.30543i
\(339\) 0 0
\(340\) −14.0000 + 28.0000i −0.759257 + 1.51851i
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) −6.00000 8.00000i −0.320713 0.427618i
\(351\) 0 0
\(352\) 24.0000i 1.27920i
\(353\) 11.0000i 0.585471i 0.956193 + 0.292735i \(0.0945655\pi\)
−0.956193 + 0.292735i \(0.905434\pi\)
\(354\) 0 0
\(355\) 16.0000 + 8.00000i 0.849192 + 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 40.0000i 2.11407i
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 36.0000i 1.89212i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 6.00000 12.0000i 0.314054 0.628109i
\(366\) 0 0
\(367\) 3.00000i 0.156599i −0.996930 0.0782994i \(-0.975051\pi\)
0.996930 0.0782994i \(-0.0249490\pi\)
\(368\) 24.0000i 1.25109i
\(369\) 0 0
\(370\) 8.00000 + 4.00000i 0.415900 + 0.207950i
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) −42.0000 −2.17177
\(375\) 0 0
\(376\) 0 0
\(377\) 5.00000i 0.257513i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.00000i 0.306987i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) 0 0
\(385\) 3.00000 6.00000i 0.152894 0.305788i
\(386\) −32.0000 −1.62876
\(387\) 0 0
\(388\) 14.0000i 0.710742i
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) 42.0000 2.12403
\(392\) 0 0
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) 10.0000 + 5.00000i 0.503155 + 0.251577i
\(396\) 0 0
\(397\) 7.00000i 0.351320i 0.984451 + 0.175660i \(0.0562059\pi\)
−0.984451 + 0.175660i \(0.943794\pi\)
\(398\) 20.0000i 1.00251i
\(399\) 0 0
\(400\) −12.0000 16.0000i −0.600000 0.800000i
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) 2.00000i 0.0996271i
\(404\) 24.0000 1.19404
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) 6.00000i 0.297409i
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) −4.00000 + 8.00000i −0.197546 + 0.395092i
\(411\) 0 0
\(412\) 38.0000i 1.87213i
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) 4.00000 8.00000i 0.196352 0.392705i
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −3.00000 −0.146211 −0.0731055 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) 26.0000i 1.26566i
\(423\) 0 0
\(424\) 0 0
\(425\) 28.0000 21.0000i 1.35820 1.01865i
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 16.0000i 0.773389i
\(429\) 0 0
\(430\) 16.0000 + 8.00000i 0.771589 + 0.385794i
\(431\) 23.0000 1.10787 0.553936 0.832560i \(-0.313125\pi\)
0.553936 + 0.832560i \(0.313125\pi\)
\(432\) 0 0
\(433\) 26.0000i 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) −30.0000 −1.43182 −0.715911 0.698192i \(-0.753988\pi\)
−0.715911 + 0.698192i \(0.753988\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 14.0000i 0.665912i
\(443\) 4.00000i 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −42.0000 −1.98876
\(447\) 0 0
\(448\) 8.00000i 0.377964i
\(449\) −5.00000 −0.235965 −0.117982 0.993016i \(-0.537643\pi\)
−0.117982 + 0.993016i \(0.537643\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 12.0000i 0.564433i
\(453\) 0 0
\(454\) −34.0000 −1.59570
\(455\) −2.00000 1.00000i −0.0937614 0.0468807i
\(456\) 0 0
\(457\) 38.0000i 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) 20.0000i 0.934539i
\(459\) 0 0
\(460\) 12.0000 24.0000i 0.559503 1.11901i
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 36.0000i 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 20.0000 0.928477
\(465\) 0 0
\(466\) 32.0000 1.48237
\(467\) 27.0000i 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 12.0000 + 6.00000i 0.553519 + 0.276759i
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 14.0000 0.641689
\(477\) 0 0
\(478\) 30.0000i 1.37217i
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 44.0000i 2.00415i
\(483\) 0 0
\(484\) 4.00000 0.181818
\(485\) −7.00000 + 14.0000i −0.317854 + 0.635707i
\(486\) 0 0
\(487\) 42.0000i 1.90320i 0.307337 + 0.951601i \(0.400562\pi\)
−0.307337 + 0.951601i \(0.599438\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −2.00000 + 4.00000i −0.0903508 + 0.180702i
\(491\) −7.00000 −0.315906 −0.157953 0.987447i \(-0.550489\pi\)
−0.157953 + 0.987447i \(0.550489\pi\)
\(492\) 0 0
\(493\) 35.0000i 1.57632i
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) 35.0000 1.56682 0.783408 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(500\) −4.00000 22.0000i −0.178885 0.983870i
\(501\) 0 0
\(502\) 36.0000i 1.60676i
\(503\) 9.00000i 0.401290i −0.979664 0.200645i \(-0.935696\pi\)
0.979664 0.200645i \(-0.0643038\pi\)
\(504\) 0 0
\(505\) −24.0000 12.0000i −1.06799 0.533993i
\(506\) 36.0000 1.60040
\(507\) 0 0
\(508\) 4.00000i 0.177471i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) −44.0000 −1.94076
\(515\) −19.0000 + 38.0000i −0.837240 + 1.67448i
\(516\) 0 0
\(517\) 9.00000i 0.395820i
\(518\) 4.00000i 0.175750i
\(519\) 0 0
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 44.0000 1.92215
\(525\) 0 0
\(526\) −48.0000 −2.09290
\(527\) 14.0000i 0.609850i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 24.0000 + 12.0000i 1.04249 + 0.521247i
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000i 0.0866296i
\(534\) 0 0
\(535\) −8.00000 + 16.0000i −0.345870 + 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 20.0000i 0.862261i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 27.0000 1.16082 0.580410 0.814324i \(-0.302892\pi\)
0.580410 + 0.814324i \(0.302892\pi\)
\(542\) 16.0000i 0.687259i
\(543\) 0 0
\(544\) 56.0000 2.40098
\(545\) −10.0000 5.00000i −0.428353 0.214176i
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 24.0000i 1.02523i
\(549\) 0 0
\(550\) 24.0000 18.0000i 1.02336 0.767523i
\(551\) 0 0
\(552\) 0 0
\(553\) 5.00000i 0.212622i
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 28.0000i 1.18640i 0.805056 + 0.593199i \(0.202135\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −4.00000 + 8.00000i −0.169031 + 0.338062i
\(561\) 0 0
\(562\) 14.0000i 0.590554i
\(563\) 24.0000i 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) −6.00000 + 12.0000i −0.252422 + 0.504844i
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 6.00000i 0.250873i
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) −24.0000 + 18.0000i −1.00087 + 0.750652i
\(576\) 0 0
\(577\) 43.0000i 1.79011i −0.445952 0.895057i \(-0.647135\pi\)
0.445952 0.895057i \(-0.352865\pi\)
\(578\) 64.0000i 2.66205i
\(579\) 0 0
\(580\) 20.0000 + 10.0000i 0.830455 + 0.415227i
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 18.0000i 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 12.0000i 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 20.0000 40.0000i 0.823387 1.64677i
\(591\) 0 0
\(592\) 8.00000i 0.328798i
\(593\) 41.0000i 1.68367i 0.539736 + 0.841834i \(0.318524\pi\)
−0.539736 + 0.841834i \(0.681476\pi\)
\(594\) 0 0
\(595\) −14.0000 7.00000i −0.573944 0.286972i
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) 12.0000i 0.490716i
\(599\) 25.0000 1.02147 0.510736 0.859738i \(-0.329373\pi\)
0.510736 + 0.859738i \(0.329373\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) 26.0000 1.05792
\(605\) −4.00000 2.00000i −0.162623 0.0813116i
\(606\) 0 0
\(607\) 27.0000i 1.09590i 0.836512 + 0.547948i \(0.184591\pi\)
−0.836512 + 0.547948i \(0.815409\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −16.0000 + 32.0000i −0.647821 + 1.29564i
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) 44.0000i 1.77714i 0.458738 + 0.888572i \(0.348302\pi\)
−0.458738 + 0.888572i \(0.651698\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000i 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −8.00000 4.00000i −0.321288 0.160644i
\(621\) 0 0
\(622\) 24.0000i 0.962312i
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) −42.0000 −1.67866
\(627\) 0 0
\(628\) 36.0000i 1.43656i
\(629\) 14.0000 0.558217
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) −2.00000 + 4.00000i −0.0793676 + 0.158735i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 30.0000i 1.18771i
\(639\) 0 0
\(640\) 0 0
\(641\) −22.0000 −0.868948 −0.434474 0.900684i \(-0.643066\pi\)
−0.434474 + 0.900684i \(0.643066\pi\)
\(642\) 0 0
\(643\) 1.00000i 0.0394362i −0.999806 0.0197181i \(-0.993723\pi\)
0.999806 0.0197181i \(-0.00627687\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000i 0.314512i 0.987558 + 0.157256i \(0.0502649\pi\)
−0.987558 + 0.157256i \(0.949735\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) −6.00000 8.00000i −0.235339 0.313786i
\(651\) 0 0
\(652\) 28.0000i 1.09656i
\(653\) 4.00000i 0.156532i −0.996933 0.0782660i \(-0.975062\pi\)
0.996933 0.0782660i \(-0.0249384\pi\)
\(654\) 0 0
\(655\) −44.0000 22.0000i −1.71922 0.859611i
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) 6.00000i 0.233904i
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 30.0000i 1.16160i
\(668\) 6.00000i 0.232147i
\(669\) 0 0
\(670\) 8.00000 + 4.00000i 0.309067 + 0.154533i
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 24.0000i 0.925132i 0.886585 + 0.462566i \(0.153071\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −36.0000 −1.38667
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 43.0000i 1.65262i 0.563212 + 0.826312i \(0.309565\pi\)
−0.563212 + 0.826312i \(0.690435\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) 16.0000i 0.612223i 0.951996 + 0.306111i \(0.0990280\pi\)
−0.951996 + 0.306111i \(0.900972\pi\)
\(684\) 0 0
\(685\) 12.0000 24.0000i 0.458496 0.916993i
\(686\) 2.00000 0.0763604
\(687\) 0 0
\(688\) 16.0000i 0.609994i
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) 20.0000 + 10.0000i 0.758643 + 0.379322i
\(696\) 0 0
\(697\) 14.0000i 0.530288i
\(698\) 40.0000i 1.51402i
\(699\) 0 0
\(700\) −8.00000 + 6.00000i −0.302372 + 0.226779i
\(701\) 13.0000 0.491003 0.245502 0.969396i \(-0.421047\pi\)
0.245502 + 0.969396i \(0.421047\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 22.0000 0.827981
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 16.0000 32.0000i 0.600469 1.20094i
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) 3.00000 6.00000i 0.112194 0.224387i
\(716\) 40.0000 1.49487
\(717\) 0 0
\(718\) 40.0000i 1.49279i
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 19.0000 0.707597
\(722\) 38.0000i 1.41421i
\(723\) 0 0
\(724\) 36.0000 1.33793
\(725\) −15.0000 20.0000i −0.557086 0.742781i
\(726\) 0 0
\(727\) 28.0000i 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −24.0000 12.0000i −0.888280 0.444140i
\(731\) 28.0000 1.03562
\(732\) 0 0
\(733\) 1.00000i 0.0369358i −0.999829 0.0184679i \(-0.994121\pi\)
0.999829 0.0184679i \(-0.00587886\pi\)
\(734\) −6.00000 −0.221464
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) 6.00000i 0.221013i
\(738\) 0 0
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) 4.00000 8.00000i 0.147043 0.294086i
\(741\) 0 0
\(742\) 12.0000i 0.440534i
\(743\) 14.0000i 0.513610i −0.966463 0.256805i \(-0.917330\pi\)
0.966463 0.256805i \(-0.0826698\pi\)
\(744\) 0 0
\(745\) 20.0000 + 10.0000i 0.732743 + 0.366372i
\(746\) 48.0000 1.75740
\(747\) 0 0
\(748\) 42.0000i 1.53567i
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −33.0000 −1.20419 −0.602094 0.798426i \(-0.705667\pi\)
−0.602094 + 0.798426i \(0.705667\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 0 0
\(754\) 10.0000 0.364179
\(755\) −26.0000 13.0000i −0.946237 0.473118i
\(756\) 0 0
\(757\) 32.0000i 1.16306i 0.813525 + 0.581530i \(0.197546\pi\)
−0.813525 + 0.581530i \(0.802454\pi\)
\(758\) 40.0000i 1.45287i
\(759\) 0 0
\(760\) 0 0
\(761\) −2.00000 −0.0724999 −0.0362500 0.999343i \(-0.511541\pi\)
−0.0362500 + 0.999343i \(0.511541\pi\)
\(762\) 0 0
\(763\) 5.00000i 0.181012i
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) 10.0000i 0.361079i
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) −12.0000 6.00000i −0.432450 0.216225i
\(771\) 0 0
\(772\) 32.0000i 1.15171i
\(773\) 21.0000i 0.755318i 0.925945 + 0.377659i \(0.123271\pi\)
−0.925945 + 0.377659i \(0.876729\pi\)
\(774\) 0 0
\(775\) 6.00000 + 8.00000i 0.215526 + 0.287368i
\(776\) 0 0
\(777\) 0 0
\(778\) 10.0000i 0.358517i
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 84.0000i 3.00383i
\(783\) 0 0
\(784\) 4.00000 0.142857
\(785\) 18.0000 36.0000i 0.642448 1.28490i
\(786\) 0 0
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) 4.00000i 0.142494i
\(789\) 0 0
\(790\) 10.0000 20.0000i 0.355784 0.711568i
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 8.00000i 0.284088i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 13.0000i 0.460484i 0.973133 + 0.230242i \(0.0739517\pi\)
−0.973133 + 0.230242i \(0.926048\pi\)
\(798\) 0 0
\(799\) 21.0000 0.742927
\(800\) −32.0000 + 24.0000i −1.13137 + 0.848528i
\(801\) 0 0
\(802\) 6.00000i 0.211867i
\(803\) 18.0000i 0.635206i
\(804\) 0 0
\(805\) 12.0000 + 6.00000i 0.422944 + 0.211472i
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 0 0
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 10.0000i 0.350931i
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) −14.0000 + 28.0000i −0.490399 + 0.980797i
\(816\) 0 0
\(817\) 0 0
\(818\) 40.0000i 1.39857i
\(819\) 0 0
\(820\) 8.00000 + 4.00000i 0.279372 + 0.139686i
\(821\) 23.0000 0.802706 0.401353 0.915924i \(-0.368540\pi\)
0.401353 + 0.915924i \(0.368540\pi\)
\(822\) 0 0
\(823\) 26.0000i 0.906303i −0.891434 0.453152i \(-0.850300\pi\)
0.891434 0.453152i \(-0.149700\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −20.0000 −0.695889
\(827\) 18.0000i 0.625921i 0.949766 + 0.312961i \(0.101321\pi\)
−0.949766 + 0.312961i \(0.898679\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −16.0000 8.00000i −0.555368 0.277684i
\(831\) 0 0
\(832\) 8.00000i 0.277350i
\(833\) 7.00000i 0.242536i
\(834\) 0 0
\(835\) −3.00000 + 6.00000i −0.103819 + 0.207639i
\(836\) 0 0
\(837\) 0 0
\(838\) 60.0000i 2.07267i
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 6.00000i 0.206774i
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) 24.0000 + 12.0000i 0.825625 + 0.412813i
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 24.0000i 0.824163i
\(849\) 0 0
\(850\) −42.0000 56.0000i −1.44059 1.92078i
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 54.0000i 1.84892i 0.381273 + 0.924462i \(0.375486\pi\)
−0.381273 + 0.924462i \(0.624514\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000i 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 8.00000 16.0000i 0.272798 0.545595i
\(861\) 0 0
\(862\) 46.0000i 1.56677i
\(863\) 54.0000i 1.83818i −0.394046 0.919091i \(-0.628925\pi\)
0.394046 0.919091i \(-0.371075\pi\)
\(864\) 0 0
\(865\) 9.00000 18.0000i 0.306009 0.612018i
\(866\) −52.0000 −1.76703
\(867\) 0 0
\(868\) 4.00000i 0.135769i
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.0000 2.00000i 0.371868 0.0676123i
\(876\) 0 0
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 60.0000i 2.02490i
\(879\) 0 0
\(880\) −24.0000 12.0000i −0.809040 0.404520i
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i −0.795656 0.605748i \(-0.792874\pi\)
0.795656 0.605748i \(-0.207126\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) 28.0000i 0.940148i 0.882627 + 0.470074i \(0.155773\pi\)
−0.882627 + 0.470074i \(0.844227\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 42.0000i 1.40626i
\(893\) 0 0
\(894\) 0 0
\(895\) −40.0000 20.0000i −1.33705 0.668526i
\(896\) 0 0
\(897\) 0 0
\(898\) 10.0000i 0.333704i
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) 42.0000 1.39922
\(902\) 12.0000i 0.399556i
\(903\) 0 0
\(904\) 0 0
\(905\) −36.0000 18.0000i −1.19668 0.598340i
\(906\) 0 0
\(907\) 38.0000i 1.26177i −0.775877 0.630885i \(-0.782692\pi\)
0.775877 0.630885i \(-0.217308\pi\)
\(908\) 34.0000i 1.12833i
\(909\) 0 0
\(910\) −2.00000 + 4.00000i −0.0662994 + 0.132599i
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) −76.0000 −2.51386
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) 22.0000i 0.726504i
\(918\) 0 0
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 24.0000i 0.790398i
\(923\) 8.00000i 0.263323i
\(924\) 0 0
\(925\) −8.00000 + 6.00000i −0.263038 + 0.197279i
\(926\) −72.0000 −2.36607
\(927\) 0 0
\(928\) 40.0000i 1.31306i
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 32.0000i 1.04819i
\(933\) 0 0
\(934\) −54.0000 −1.76693
\(935\) 21.0000 42.0000i 0.686773 1.37355i
\(936\) 0 0
\(937\) 13.0000i 0.424691i −0.977195 0.212346i \(-0.931890\pi\)
0.977195 0.212346i \(-0.0681103\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 0 0
\(940\) 6.00000 12.0000i 0.195698 0.391397i
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 12.0000i 0.390774i
\(944\) −40.0000 −1.30189
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) 42.0000i 1.36482i −0.730971 0.682408i \(-0.760933\pi\)
0.730971 0.682408i \(-0.239067\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) 0 0
\(955\) 6.00000 + 3.00000i 0.194155 + 0.0970777i
\(956\) −30.0000 −0.970269
\(957\) 0 0
\(958\) 60.0000i 1.93851i
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000i 0.128965i
\(963\) 0 0
\(964\) −44.0000 −1.41714
\(965\) 16.0000 32.0000i 0.515058 1.03012i
\(966\) 0 0
\(967\) 8.00000i 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 28.0000 + 14.0000i 0.899026 + 0.449513i
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 0 0
\(973\) 10.0000i 0.320585i
\(974\) 84.0000 2.69153
\(975\) 0 0
\(976\) 32.0000 1.02430
\(977\) 12.0000i 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 4.00000 + 2.00000i 0.127775 + 0.0638877i
\(981\) 0 0
\(982\) 14.0000i 0.446758i
\(983\) 51.0000i 1.62665i 0.581811 + 0.813324i \(0.302344\pi\)
−0.581811 + 0.813324i \(0.697656\pi\)
\(984\) 0 0
\(985\) 2.00000 4.00000i 0.0637253 0.127451i
\(986\) 70.0000 2.22925
\(987\) 0 0
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 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
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 20.0000 + 10.0000i 0.634043 + 0.317021i
\(996\) 0 0
\(997\) 13.0000i 0.411714i −0.978582 0.205857i \(-0.934002\pi\)
0.978582 0.205857i \(-0.0659982\pi\)
\(998\) 70.0000i 2.21581i
\(999\) 0 0
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 315.2.d.a.64.1 2
3.2 odd 2 35.2.b.a.29.2 yes 2
4.3 odd 2 5040.2.t.p.1009.2 2
5.2 odd 4 1575.2.a.k.1.1 1
5.3 odd 4 1575.2.a.a.1.1 1
5.4 even 2 inner 315.2.d.a.64.2 2
7.6 odd 2 2205.2.d.b.1324.1 2
12.11 even 2 560.2.g.b.449.2 2
15.2 even 4 175.2.a.a.1.1 1
15.8 even 4 175.2.a.c.1.1 1
15.14 odd 2 35.2.b.a.29.1 2
20.19 odd 2 5040.2.t.p.1009.1 2
21.2 odd 6 245.2.j.e.214.2 4
21.5 even 6 245.2.j.d.214.2 4
21.11 odd 6 245.2.j.e.79.1 4
21.17 even 6 245.2.j.d.79.1 4
21.20 even 2 245.2.b.a.99.2 2
24.5 odd 2 2240.2.g.h.449.2 2
24.11 even 2 2240.2.g.g.449.1 2
35.34 odd 2 2205.2.d.b.1324.2 2
60.23 odd 4 2800.2.a.l.1.1 1
60.47 odd 4 2800.2.a.w.1.1 1
60.59 even 2 560.2.g.b.449.1 2
105.44 odd 6 245.2.j.e.214.1 4
105.59 even 6 245.2.j.d.79.2 4
105.62 odd 4 1225.2.a.a.1.1 1
105.74 odd 6 245.2.j.e.79.2 4
105.83 odd 4 1225.2.a.i.1.1 1
105.89 even 6 245.2.j.d.214.1 4
105.104 even 2 245.2.b.a.99.1 2
120.29 odd 2 2240.2.g.h.449.1 2
120.59 even 2 2240.2.g.g.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.b.a.29.1 2 15.14 odd 2
35.2.b.a.29.2 yes 2 3.2 odd 2
175.2.a.a.1.1 1 15.2 even 4
175.2.a.c.1.1 1 15.8 even 4
245.2.b.a.99.1 2 105.104 even 2
245.2.b.a.99.2 2 21.20 even 2
245.2.j.d.79.1 4 21.17 even 6
245.2.j.d.79.2 4 105.59 even 6
245.2.j.d.214.1 4 105.89 even 6
245.2.j.d.214.2 4 21.5 even 6
245.2.j.e.79.1 4 21.11 odd 6
245.2.j.e.79.2 4 105.74 odd 6
245.2.j.e.214.1 4 105.44 odd 6
245.2.j.e.214.2 4 21.2 odd 6
315.2.d.a.64.1 2 1.1 even 1 trivial
315.2.d.a.64.2 2 5.4 even 2 inner
560.2.g.b.449.1 2 60.59 even 2
560.2.g.b.449.2 2 12.11 even 2
1225.2.a.a.1.1 1 105.62 odd 4
1225.2.a.i.1.1 1 105.83 odd 4
1575.2.a.a.1.1 1 5.3 odd 4
1575.2.a.k.1.1 1 5.2 odd 4
2205.2.d.b.1324.1 2 7.6 odd 2
2205.2.d.b.1324.2 2 35.34 odd 2
2240.2.g.g.449.1 2 24.11 even 2
2240.2.g.g.449.2 2 120.59 even 2
2240.2.g.h.449.1 2 120.29 odd 2
2240.2.g.h.449.2 2 24.5 odd 2
2800.2.a.l.1.1 1 60.23 odd 4
2800.2.a.w.1.1 1 60.47 odd 4
5040.2.t.p.1009.1 2 20.19 odd 2
5040.2.t.p.1009.2 2 4.3 odd 2