Properties

Label 756.2.l.a.289.1
Level $756$
Weight $2$
Character 756.289
Analytic conductor $6.037$
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] = [756,2,Mod(289,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.l (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 756.289
Dual form 756.2.l.a.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q-2.00000 q^{5} +(2.00000 + 1.73205i) q^{7} -4.00000 q^{11} +(-1.50000 - 2.59808i) q^{13} +(3.50000 + 6.06218i) q^{17} +(-2.50000 + 4.33013i) q^{19} -4.00000 q^{23} -1.00000 q^{25} +(-0.500000 + 0.866025i) q^{29} +(1.50000 - 2.59808i) q^{31} +(-4.00000 - 3.46410i) q^{35} +(-5.50000 + 9.52628i) q^{37} +(-4.50000 - 7.79423i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(1.50000 + 2.59808i) q^{47} +(1.00000 + 6.92820i) q^{49} +(1.50000 + 2.59808i) q^{53} +8.00000 q^{55} +(-3.50000 + 6.06218i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-6.50000 + 11.2583i) q^{67} +8.00000 q^{71} +(-3.50000 - 6.06218i) q^{73} +(-8.00000 - 6.92820i) q^{77} +(4.50000 + 7.79423i) q^{79} +(0.500000 - 0.866025i) q^{83} +(-7.00000 - 12.1244i) q^{85} +(7.50000 - 12.9904i) q^{89} +(1.50000 - 7.79423i) q^{91} +(5.00000 - 8.66025i) q^{95} +(8.50000 - 14.7224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} + 4 q^{7} - 8 q^{11} - 3 q^{13} + 7 q^{17} - 5 q^{19} - 8 q^{23} - 2 q^{25} - q^{29} + 3 q^{31} - 8 q^{35} - 11 q^{37} - 9 q^{41} - 5 q^{43} + 3 q^{47} + 2 q^{49} + 3 q^{53} + 16 q^{55} - 7 q^{59} - 3 q^{61} + 6 q^{65} - 13 q^{67} + 16 q^{71} - 7 q^{73} - 16 q^{77} + 9 q^{79} + q^{83} - 14 q^{85} + 15 q^{89} + 3 q^{91} + 10 q^{95} + 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.50000 2.59808i −0.416025 0.720577i 0.579510 0.814965i \(-0.303244\pi\)
−0.995535 + 0.0943882i \(0.969911\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.50000 + 6.06218i 0.848875 + 1.47029i 0.882213 + 0.470850i \(0.156053\pi\)
−0.0333386 + 0.999444i \(0.510614\pi\)
\(18\) 0 0
\(19\) −2.50000 + 4.33013i −0.573539 + 0.993399i 0.422659 + 0.906289i \(0.361097\pi\)
−0.996199 + 0.0871106i \(0.972237\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.500000 + 0.866025i −0.0928477 + 0.160817i −0.908708 0.417432i \(-0.862930\pi\)
0.815861 + 0.578249i \(0.196264\pi\)
\(30\) 0 0
\(31\) 1.50000 2.59808i 0.269408 0.466628i −0.699301 0.714827i \(-0.746505\pi\)
0.968709 + 0.248199i \(0.0798387\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 3.46410i −0.676123 0.585540i
\(36\) 0 0
\(37\) −5.50000 + 9.52628i −0.904194 + 1.56611i −0.0821995 + 0.996616i \(0.526194\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.50000 7.79423i −0.702782 1.21725i −0.967486 0.252924i \(-0.918608\pi\)
0.264704 0.964330i \(-0.414726\pi\)
\(42\) 0 0
\(43\) −2.50000 + 4.33013i −0.381246 + 0.660338i −0.991241 0.132068i \(-0.957838\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.50000 + 2.59808i 0.218797 + 0.378968i 0.954441 0.298401i \(-0.0964533\pi\)
−0.735643 + 0.677369i \(0.763120\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 + 2.59808i 0.206041 + 0.356873i 0.950464 0.310835i \(-0.100609\pi\)
−0.744423 + 0.667708i \(0.767275\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.50000 + 6.06218i −0.455661 + 0.789228i −0.998726 0.0504625i \(-0.983930\pi\)
0.543065 + 0.839691i \(0.317264\pi\)
\(60\) 0 0
\(61\) −1.50000 2.59808i −0.192055 0.332650i 0.753876 0.657017i \(-0.228182\pi\)
−0.945931 + 0.324367i \(0.894849\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) −6.50000 + 11.2583i −0.794101 + 1.37542i 0.129307 + 0.991605i \(0.458725\pi\)
−0.923408 + 0.383819i \(0.874609\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −3.50000 6.06218i −0.409644 0.709524i 0.585206 0.810885i \(-0.301014\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 6.92820i −0.911685 0.789542i
\(78\) 0 0
\(79\) 4.50000 + 7.79423i 0.506290 + 0.876919i 0.999974 + 0.00727784i \(0.00231663\pi\)
−0.493684 + 0.869641i \(0.664350\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.500000 0.866025i 0.0548821 0.0950586i −0.837279 0.546776i \(-0.815855\pi\)
0.892161 + 0.451717i \(0.149188\pi\)
\(84\) 0 0
\(85\) −7.00000 12.1244i −0.759257 1.31507i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.50000 12.9904i 0.794998 1.37698i −0.127842 0.991795i \(-0.540805\pi\)
0.922840 0.385183i \(-0.125862\pi\)
\(90\) 0 0
\(91\) 1.50000 7.79423i 0.157243 0.817057i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.00000 8.66025i 0.512989 0.888523i
\(96\) 0 0
\(97\) 8.50000 14.7224i 0.863044 1.49484i −0.00593185 0.999982i \(-0.501888\pi\)
0.868976 0.494854i \(-0.164778\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.50000 + 2.59808i −0.145010 + 0.251166i −0.929377 0.369132i \(-0.879655\pi\)
0.784366 + 0.620298i \(0.212988\pi\)
\(108\) 0 0
\(109\) −3.50000 6.06218i −0.335239 0.580651i 0.648292 0.761392i \(-0.275484\pi\)
−0.983531 + 0.180741i \(0.942150\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.500000 0.866025i −0.0470360 0.0814688i 0.841549 0.540181i \(-0.181644\pi\)
−0.888585 + 0.458712i \(0.848311\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.50000 + 18.1865i −0.320844 + 1.66716i
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −12.5000 + 4.33013i −1.08389 + 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 2.50000 + 4.33013i 0.212047 + 0.367277i 0.952355 0.304991i \(-0.0986536\pi\)
−0.740308 + 0.672268i \(0.765320\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 + 10.3923i 0.501745 + 0.869048i
\(144\) 0 0
\(145\) 1.00000 1.73205i 0.0830455 0.143839i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.00000 + 5.19615i −0.240966 + 0.417365i
\(156\) 0 0
\(157\) 6.50000 11.2583i 0.518756 0.898513i −0.481006 0.876717i \(-0.659728\pi\)
0.999762 0.0217953i \(-0.00693820\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 6.92820i −0.630488 0.546019i
\(162\) 0 0
\(163\) 9.50000 16.4545i 0.744097 1.28881i −0.206518 0.978443i \(-0.566213\pi\)
0.950615 0.310372i \(-0.100454\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.5000 + 19.9186i 0.889897 + 1.54135i 0.839996 + 0.542592i \(0.182557\pi\)
0.0499004 + 0.998754i \(0.484110\pi\)
\(168\) 0 0
\(169\) 2.00000 3.46410i 0.153846 0.266469i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.500000 0.866025i −0.0380143 0.0658427i 0.846392 0.532560i \(-0.178770\pi\)
−0.884407 + 0.466717i \(0.845437\pi\)
\(174\) 0 0
\(175\) −2.00000 1.73205i −0.151186 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.5000 18.1865i −0.784807 1.35933i −0.929114 0.369792i \(-0.879429\pi\)
0.144308 0.989533i \(-0.453905\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.0000 19.0526i 0.808736 1.40077i
\(186\) 0 0
\(187\) −14.0000 24.2487i −1.02378 1.77324i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.50000 + 12.9904i 0.542681 + 0.939951i 0.998749 + 0.0500060i \(0.0159241\pi\)
−0.456068 + 0.889945i \(0.650743\pi\)
\(192\) 0 0
\(193\) 0.500000 0.866025i 0.0359908 0.0623379i −0.847469 0.530845i \(-0.821875\pi\)
0.883460 + 0.468507i \(0.155208\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 0 0
\(199\) 6.50000 + 11.2583i 0.460773 + 0.798082i 0.999000 0.0447181i \(-0.0142390\pi\)
−0.538227 + 0.842800i \(0.680906\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.50000 + 0.866025i −0.175466 + 0.0607831i
\(204\) 0 0
\(205\) 9.00000 + 15.5885i 0.628587 + 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 17.3205i 0.691714 1.19808i
\(210\) 0 0
\(211\) 6.50000 + 11.2583i 0.447478 + 0.775055i 0.998221 0.0596196i \(-0.0189888\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.00000 8.66025i 0.340997 0.590624i
\(216\) 0 0
\(217\) 7.50000 2.59808i 0.509133 0.176369i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.5000 18.1865i 0.706306 1.22336i
\(222\) 0 0
\(223\) 3.50000 6.06218i 0.234377 0.405953i −0.724714 0.689050i \(-0.758028\pi\)
0.959092 + 0.283096i \(0.0913615\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5000 + 25.1147i −0.949927 + 1.64532i −0.204354 + 0.978897i \(0.565509\pi\)
−0.745573 + 0.666424i \(0.767824\pi\)
\(234\) 0 0
\(235\) −3.00000 5.19615i −0.195698 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.5000 18.1865i −0.679189 1.17639i −0.975226 0.221213i \(-0.928999\pi\)
0.296037 0.955176i \(-0.404335\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.00000 13.8564i −0.127775 0.885253i
\(246\) 0 0
\(247\) 15.0000 0.954427
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −27.5000 + 9.52628i −1.70877 + 0.591934i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −3.00000 5.19615i −0.184289 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.500000 0.866025i −0.0304855 0.0528025i 0.850380 0.526169i \(-0.176372\pi\)
−0.880866 + 0.473366i \(0.843039\pi\)
\(270\) 0 0
\(271\) 1.50000 2.59808i 0.0911185 0.157822i −0.816864 0.576831i \(-0.804289\pi\)
0.907982 + 0.419009i \(0.137622\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.50000 + 14.7224i −0.507067 + 0.878267i 0.492899 + 0.870087i \(0.335937\pi\)
−0.999967 + 0.00818015i \(0.997396\pi\)
\(282\) 0 0
\(283\) −0.500000 + 0.866025i −0.0297219 + 0.0514799i −0.880504 0.474039i \(-0.842796\pi\)
0.850782 + 0.525519i \(0.176129\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.50000 23.3827i 0.265627 1.38024i
\(288\) 0 0
\(289\) −16.0000 + 27.7128i −0.941176 + 1.63017i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.5000 + 23.3827i 0.788678 + 1.36603i 0.926777 + 0.375613i \(0.122568\pi\)
−0.138098 + 0.990419i \(0.544099\pi\)
\(294\) 0 0
\(295\) 7.00000 12.1244i 0.407556 0.705907i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) −12.5000 + 4.33013i −0.720488 + 0.249584i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.00000 + 5.19615i 0.171780 + 0.297531i
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.5000 + 26.8468i −0.878924 + 1.52234i −0.0264017 + 0.999651i \(0.508405\pi\)
−0.852523 + 0.522690i \(0.824928\pi\)
\(312\) 0 0
\(313\) 8.50000 + 14.7224i 0.480448 + 0.832161i 0.999748 0.0224310i \(-0.00714060\pi\)
−0.519300 + 0.854592i \(0.673807\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.50000 7.79423i −0.252745 0.437767i 0.711535 0.702650i \(-0.248000\pi\)
−0.964281 + 0.264883i \(0.914667\pi\)
\(318\) 0 0
\(319\) 2.00000 3.46410i 0.111979 0.193952i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −35.0000 −1.94745
\(324\) 0 0
\(325\) 1.50000 + 2.59808i 0.0832050 + 0.144115i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.50000 + 7.79423i −0.0826977 + 0.429710i
\(330\) 0 0
\(331\) 12.5000 + 21.6506i 0.687062 + 1.19003i 0.972784 + 0.231714i \(0.0744333\pi\)
−0.285722 + 0.958313i \(0.592233\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 13.0000 22.5167i 0.710266 1.23022i
\(336\) 0 0
\(337\) −1.50000 2.59808i −0.0817102 0.141526i 0.822274 0.569091i \(-0.192705\pi\)
−0.903985 + 0.427565i \(0.859372\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 + 10.3923i −0.324918 + 0.562775i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.50000 + 12.9904i −0.402621 + 0.697360i −0.994041 0.109003i \(-0.965234\pi\)
0.591420 + 0.806363i \(0.298567\pi\)
\(348\) 0 0
\(349\) 2.50000 4.33013i 0.133822 0.231786i −0.791325 0.611396i \(-0.790608\pi\)
0.925147 + 0.379610i \(0.123942\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.5000 + 26.8468i −0.818059 + 1.41692i 0.0890519 + 0.996027i \(0.471616\pi\)
−0.907111 + 0.420892i \(0.861717\pi\)
\(360\) 0 0
\(361\) −3.00000 5.19615i −0.157895 0.273482i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.00000 + 12.1244i 0.366397 + 0.634618i
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.50000 + 7.79423i −0.0778761 + 0.404656i
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 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
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 16.0000 + 13.8564i 0.815436 + 0.706188i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −14.0000 24.2487i −0.708010 1.22631i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.00000 15.5885i −0.452839 0.784340i
\(396\) 0 0
\(397\) 10.5000 18.1865i 0.526980 0.912756i −0.472526 0.881317i \(-0.656658\pi\)
0.999506 0.0314391i \(-0.0100090\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.0000 38.1051i 1.09050 1.88880i
\(408\) 0 0
\(409\) −11.5000 + 19.9186i −0.568638 + 0.984911i 0.428063 + 0.903749i \(0.359196\pi\)
−0.996701 + 0.0811615i \(0.974137\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −17.5000 + 6.06218i −0.861119 + 0.298300i
\(414\) 0 0
\(415\) −1.00000 + 1.73205i −0.0490881 + 0.0850230i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.50000 + 12.9904i 0.366399 + 0.634622i 0.989000 0.147918i \(-0.0472572\pi\)
−0.622601 + 0.782540i \(0.713924\pi\)
\(420\) 0 0
\(421\) 4.50000 7.79423i 0.219317 0.379867i −0.735283 0.677761i \(-0.762951\pi\)
0.954599 + 0.297893i \(0.0962839\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.50000 6.06218i −0.169775 0.294059i
\(426\) 0 0
\(427\) 1.50000 7.79423i 0.0725901 0.377189i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.5000 + 33.7750i 0.939282 + 1.62688i 0.766814 + 0.641869i \(0.221841\pi\)
0.172468 + 0.985015i \(0.444826\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.0000 17.3205i 0.478365 0.828552i
\(438\) 0 0
\(439\) 10.5000 + 18.1865i 0.501138 + 0.867996i 0.999999 + 0.00131415i \(0.000418308\pi\)
−0.498861 + 0.866682i \(0.666248\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.5000 25.1147i −0.688916 1.19324i −0.972189 0.234198i \(-0.924754\pi\)
0.283273 0.959039i \(-0.408580\pi\)
\(444\) 0 0
\(445\) −15.0000 + 25.9808i −0.711068 + 1.23161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 18.0000 + 31.1769i 0.847587 + 1.46806i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.00000 + 15.5885i −0.140642 + 0.730798i
\(456\) 0 0
\(457\) −15.5000 26.8468i −0.725059 1.25584i −0.958950 0.283577i \(-0.908479\pi\)
0.233890 0.972263i \(-0.424854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.5000 19.9186i 0.535608 0.927701i −0.463525 0.886084i \(-0.653416\pi\)
0.999134 0.0416172i \(-0.0132510\pi\)
\(462\) 0 0
\(463\) 2.50000 + 4.33013i 0.116185 + 0.201238i 0.918253 0.395995i \(-0.129600\pi\)
−0.802068 + 0.597233i \(0.796267\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.50000 14.7224i 0.393333 0.681273i −0.599554 0.800334i \(-0.704655\pi\)
0.992887 + 0.119062i \(0.0379886\pi\)
\(468\) 0 0
\(469\) −32.5000 + 11.2583i −1.50071 + 0.519861i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000 17.3205i 0.459800 0.796398i
\(474\) 0 0
\(475\) 2.50000 4.33013i 0.114708 0.198680i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 33.0000 1.50467
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17.0000 + 29.4449i −0.771930 + 1.33702i
\(486\) 0 0
\(487\) 0.500000 + 0.866025i 0.0226572 + 0.0392434i 0.877132 0.480250i \(-0.159454\pi\)
−0.854475 + 0.519493i \(0.826121\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.50000 7.79423i −0.203082 0.351749i 0.746438 0.665455i \(-0.231763\pi\)
−0.949520 + 0.313707i \(0.898429\pi\)
\(492\) 0 0
\(493\) −7.00000 −0.315264
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 + 13.8564i 0.717698 + 0.621545i
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 3.50000 18.1865i 0.154831 0.804525i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −6.00000 10.3923i −0.263880 0.457053i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.50000 + 2.59808i 0.0657162 + 0.113824i 0.897011 0.442007i \(-0.145733\pi\)
−0.831295 + 0.555831i \(0.812400\pi\)
\(522\) 0 0
\(523\) −14.5000 + 25.1147i −0.634041 + 1.09819i 0.352677 + 0.935745i \(0.385272\pi\)
−0.986718 + 0.162446i \(0.948062\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.0000 0.914774
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −13.5000 + 23.3827i −0.584750 + 1.01282i
\(534\) 0 0
\(535\) 3.00000 5.19615i 0.129701 0.224649i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 27.7128i −0.172292 1.19368i
\(540\) 0 0
\(541\) 16.5000 28.5788i 0.709390 1.22870i −0.255693 0.966758i \(-0.582304\pi\)
0.965084 0.261942i \(-0.0843630\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.00000 + 12.1244i 0.299847 + 0.519350i
\(546\) 0 0
\(547\) −16.5000 + 28.5788i −0.705489 + 1.22194i 0.261026 + 0.965332i \(0.415939\pi\)
−0.966515 + 0.256611i \(0.917394\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.50000 4.33013i −0.106504 0.184470i
\(552\) 0 0
\(553\) −4.50000 + 23.3827i −0.191359 + 0.994333i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.50000 + 9.52628i 0.233042 + 0.403641i 0.958702 0.284413i \(-0.0917985\pi\)
−0.725660 + 0.688054i \(0.758465\pi\)
\(558\) 0 0
\(559\) 15.0000 0.634432
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.50000 11.2583i 0.273942 0.474482i −0.695925 0.718114i \(-0.745006\pi\)
0.969868 + 0.243632i \(0.0783389\pi\)
\(564\) 0 0
\(565\) 1.00000 + 1.73205i 0.0420703 + 0.0728679i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.5000 + 23.3827i 0.565949 + 0.980253i 0.996961 + 0.0779066i \(0.0248236\pi\)
−0.431011 + 0.902347i \(0.641843\pi\)
\(570\) 0 0
\(571\) 1.50000 2.59808i 0.0627730 0.108726i −0.832931 0.553377i \(-0.813339\pi\)
0.895704 + 0.444651i \(0.146672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −3.50000 6.06218i −0.145707 0.252372i 0.783930 0.620850i \(-0.213212\pi\)
−0.929636 + 0.368478i \(0.879879\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.50000 0.866025i 0.103717 0.0359288i
\(582\) 0 0
\(583\) −6.00000 10.3923i −0.248495 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.5000 32.0429i 0.763577 1.32255i −0.177419 0.984135i \(-0.556775\pi\)
0.940996 0.338418i \(-0.109892\pi\)
\(588\) 0 0
\(589\) 7.50000 + 12.9904i 0.309032 + 0.535259i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.5000 23.3827i 0.554379 0.960212i −0.443573 0.896238i \(-0.646289\pi\)
0.997952 0.0639736i \(-0.0203773\pi\)
\(594\) 0 0
\(595\) 7.00000 36.3731i 0.286972 1.49115i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.50000 + 2.59808i −0.0612883 + 0.106155i −0.895042 0.445983i \(-0.852854\pi\)
0.833753 + 0.552137i \(0.186188\pi\)
\(600\) 0 0
\(601\) −17.5000 + 30.3109i −0.713840 + 1.23641i 0.249565 + 0.968358i \(0.419712\pi\)
−0.963405 + 0.268049i \(0.913621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −48.0000 −1.94826 −0.974130 0.225989i \(-0.927439\pi\)
−0.974130 + 0.225989i \(0.927439\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.50000 7.79423i 0.182051 0.315321i
\(612\) 0 0
\(613\) 4.50000 + 7.79423i 0.181753 + 0.314806i 0.942478 0.334269i \(-0.108489\pi\)
−0.760724 + 0.649075i \(0.775156\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.5000 + 37.2391i 0.865557 + 1.49919i 0.866493 + 0.499190i \(0.166369\pi\)
−0.000935233 1.00000i \(0.500298\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 37.5000 12.9904i 1.50241 0.520449i
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −77.0000 −3.07019
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 16.5000 12.9904i 0.653754 0.514698i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 6.50000 + 11.2583i 0.256335 + 0.443985i 0.965257 0.261301i \(-0.0841516\pi\)
−0.708922 + 0.705287i \(0.750818\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.50000 + 2.59808i 0.0589711 + 0.102141i 0.894004 0.448059i \(-0.147885\pi\)
−0.835033 + 0.550200i \(0.814551\pi\)
\(648\) 0 0
\(649\) 14.0000 24.2487i 0.549548 0.951845i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.5000 21.6506i 0.486931 0.843389i −0.512956 0.858415i \(-0.671450\pi\)
0.999887 + 0.0150258i \(0.00478303\pi\)
\(660\) 0 0
\(661\) 8.50000 14.7224i 0.330612 0.572636i −0.652020 0.758202i \(-0.726078\pi\)
0.982632 + 0.185565i \(0.0594116\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 25.0000 8.66025i 0.969458 0.335830i
\(666\) 0 0
\(667\) 2.00000 3.46410i 0.0774403 0.134131i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.00000 + 10.3923i 0.231627 + 0.401190i
\(672\) 0 0
\(673\) 12.5000 21.6506i 0.481840 0.834571i −0.517943 0.855415i \(-0.673302\pi\)
0.999783 + 0.0208444i \(0.00663546\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.5000 + 23.3827i 0.518847 + 0.898670i 0.999760 + 0.0219013i \(0.00697196\pi\)
−0.480913 + 0.876768i \(0.659695\pi\)
\(678\) 0 0
\(679\) 42.5000 14.7224i 1.63100 0.564995i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −10.5000 18.1865i −0.401771 0.695888i 0.592168 0.805814i \(-0.298272\pi\)
−0.993940 + 0.109926i \(0.964939\pi\)
\(684\) 0 0
\(685\) 28.0000 1.06983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.50000 7.79423i 0.171436 0.296936i
\(690\) 0 0
\(691\) −5.50000 9.52628i −0.209230 0.362397i 0.742242 0.670132i \(-0.233762\pi\)
−0.951472 + 0.307735i \(0.900429\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.00000 8.66025i −0.189661 0.328502i
\(696\) 0 0
\(697\) 31.5000 54.5596i 1.19315 2.06659i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) −27.5000 47.6314i −1.03718 1.79645i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.00000 3.46410i −0.150435 0.130281i
\(708\) 0 0
\(709\) 12.5000 + 21.6506i 0.469447 + 0.813107i 0.999390 0.0349269i \(-0.0111198\pi\)
−0.529943 + 0.848034i \(0.677787\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.00000 + 10.3923i −0.224702 + 0.389195i
\(714\) 0 0
\(715\) −12.0000 20.7846i −0.448775 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.50000 + 12.9904i −0.279703 + 0.484459i −0.971311 0.237814i \(-0.923569\pi\)
0.691608 + 0.722273i \(0.256903\pi\)
\(720\) 0 0
\(721\) 16.0000 + 13.8564i 0.595871 + 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.500000 0.866025i 0.0185695 0.0321634i
\(726\) 0 0
\(727\) 23.5000 40.7032i 0.871567 1.50960i 0.0111912 0.999937i \(-0.496438\pi\)
0.860376 0.509661i \(-0.170229\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −35.0000 −1.29452
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.0000 45.0333i 0.957722 1.65882i
\(738\) 0 0
\(739\) 10.5000 + 18.1865i 0.386249 + 0.669002i 0.991942 0.126696i \(-0.0404373\pi\)
−0.605693 + 0.795699i \(0.707104\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.50000 7.79423i −0.165089 0.285943i 0.771598 0.636111i \(-0.219458\pi\)
−0.936687 + 0.350168i \(0.886124\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7.50000 + 2.59808i −0.274044 + 0.0949316i
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 3.50000 18.1865i 0.126709 0.658397i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.0000 0.758266
\(768\) 0 0
\(769\) 2.50000 + 4.33013i 0.0901523 + 0.156148i 0.907575 0.419890i \(-0.137931\pi\)
−0.817423 + 0.576038i \(0.804598\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.5000 + 19.9186i 0.413626 + 0.716422i 0.995283 0.0970125i \(-0.0309287\pi\)
−0.581657 + 0.813434i \(0.697595\pi\)
\(774\) 0 0
\(775\) −1.50000 + 2.59808i −0.0538816 + 0.0933257i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 45.0000 1.61229
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −13.0000 + 22.5167i −0.463990 + 0.803654i
\(786\) 0 0
\(787\) −10.5000 + 18.1865i −0.374285 + 0.648280i −0.990220 0.139517i \(-0.955445\pi\)
0.615935 + 0.787797i \(0.288778\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.500000 2.59808i 0.0177780 0.0923770i
\(792\) 0 0
\(793\) −4.50000 + 7.79423i −0.159800 + 0.276781i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.50000 + 12.9904i 0.265664 + 0.460143i 0.967737 0.251961i \(-0.0810756\pi\)
−0.702074 + 0.712104i \(0.747742\pi\)
\(798\) 0 0
\(799\) −10.5000 + 18.1865i −0.371463 + 0.643393i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.0000 + 24.2487i 0.494049 + 0.855718i
\(804\) 0 0
\(805\) 16.0000 + 13.8564i 0.563926 + 0.488374i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.500000 0.866025i −0.0175791 0.0304478i 0.857102 0.515147i \(-0.172263\pi\)
−0.874681 + 0.484699i \(0.838929\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19.0000 + 32.9090i −0.665541 + 1.15275i
\(816\) 0 0
\(817\) −12.5000 21.6506i −0.437320 0.757460i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26.5000 45.8993i −0.924856 1.60190i −0.791792 0.610791i \(-0.790852\pi\)
−0.133064 0.991107i \(-0.542482\pi\)
\(822\) 0 0
\(823\) 11.5000 19.9186i 0.400865 0.694318i −0.592966 0.805228i \(-0.702043\pi\)
0.993831 + 0.110910i \(0.0353764\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −9.50000 16.4545i −0.329949 0.571488i 0.652553 0.757743i \(-0.273698\pi\)
−0.982501 + 0.186256i \(0.940365\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −38.5000 + 30.3109i −1.33395 + 1.05021i
\(834\) 0 0
\(835\) −23.0000 39.8372i −0.795948 1.37862i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.5000 + 19.9186i −0.397024 + 0.687666i −0.993357 0.115071i \(-0.963290\pi\)
0.596333 + 0.802737i \(0.296624\pi\)
\(840\) 0 0
\(841\) 14.0000 + 24.2487i 0.482759 + 0.836162i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.00000 + 6.92820i −0.137604 + 0.238337i
\(846\) 0 0
\(847\) 10.0000 + 8.66025i 0.343604 + 0.297570i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 22.0000 38.1051i 0.754150 1.30623i
\(852\) 0 0
\(853\) −17.5000 + 30.3109i −0.599189 + 1.03783i 0.393753 + 0.919216i \(0.371177\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.5000 35.5070i 0.697828 1.20867i −0.271390 0.962470i \(-0.587483\pi\)
0.969218 0.246204i \(-0.0791834\pi\)
\(864\) 0 0
\(865\) 1.00000 + 1.73205i 0.0340010 + 0.0588915i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.0000 31.1769i −0.610608 1.05760i
\(870\) 0 0
\(871\) 39.0000 1.32146
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.0000 + 20.7846i 0.811348 + 0.702648i
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −15.0000 −0.501956
\(894\) 0 0
\(895\) 21.0000 + 36.3731i 0.701953 + 1.21582i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.50000 + 2.59808i 0.0500278 + 0.0866507i
\(900\) 0 0
\(901\) −10.5000 + 18.1865i −0.349806 + 0.605881i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −19.5000 + 33.7750i −0.646064 + 1.11902i 0.337991 + 0.941149i \(0.390253\pi\)
−0.984055 + 0.177866i \(0.943081\pi\)
\(912\) 0 0
\(913\) −2.00000 + 3.46410i −0.0661903 + 0.114645i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.00000 + 6.92820i 0.264183 + 0.228789i
\(918\) 0 0
\(919\) 17.5000 30.3109i 0.577272 0.999864i −0.418519 0.908208i \(-0.637451\pi\)
0.995791 0.0916559i \(-0.0292160\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 20.7846i −0.394985 0.684134i
\(924\) 0 0
\(925\) 5.50000 9.52628i 0.180839 0.313222i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.50000 + 2.59808i 0.0492134 + 0.0852401i 0.889583 0.456774i \(-0.150995\pi\)
−0.840369 + 0.542014i \(0.817662\pi\)
\(930\) 0 0
\(931\) −32.5000 12.9904i −1.06514 0.425743i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 28.0000 + 48.4974i 0.915698 + 1.58604i
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.5000 51.0955i 0.961673 1.66567i 0.243372 0.969933i \(-0.421747\pi\)
0.718301 0.695733i \(-0.244920\pi\)
\(942\) 0 0
\(943\) 18.0000 + 31.1769i 0.586161 + 1.01526i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12.5000 21.6506i −0.406195 0.703551i 0.588264 0.808669i \(-0.299811\pi\)
−0.994460 + 0.105118i \(0.966478\pi\)
\(948\) 0 0
\(949\) −10.5000 + 18.1865i −0.340844 + 0.590360i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) −15.0000 25.9808i −0.485389 0.840718i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −28.0000 24.2487i −0.904167 0.783032i
\(960\) 0 0
\(961\) 11.0000 + 19.0526i 0.354839 + 0.614599i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.00000 + 1.73205i −0.0321911 + 0.0557567i
\(966\) 0 0
\(967\) −23.5000 40.7032i −0.755709 1.30893i −0.945021 0.327009i \(-0.893959\pi\)
0.189312 0.981917i \(-0.439374\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.50000 + 12.9904i −0.240686 + 0.416881i −0.960910 0.276861i \(-0.910706\pi\)
0.720224 + 0.693742i \(0.244039\pi\)
\(972\) 0 0
\(973\) −2.50000 + 12.9904i −0.0801463 + 0.416452i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.50000 2.59808i 0.0479893 0.0831198i −0.841033 0.540984i \(-0.818052\pi\)
0.889022 + 0.457864i \(0.151385\pi\)
\(978\) 0 0
\(979\) −30.0000 + 51.9615i −0.958804 + 1.66070i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −52.0000 −1.65686
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.0000 17.3205i 0.317982 0.550760i
\(990\) 0 0
\(991\) 18.5000 + 32.0429i 0.587672 + 1.01788i 0.994537 + 0.104389i \(0.0332887\pi\)
−0.406865 + 0.913488i \(0.633378\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13.0000 22.5167i −0.412128 0.713826i
\(996\) 0 0
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 0 0
\(999\) 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 756.2.l.a.289.1 2
3.2 odd 2 252.2.l.a.205.1 yes 2
4.3 odd 2 3024.2.t.b.289.1 2
7.2 even 3 5292.2.j.c.3529.1 2
7.3 odd 6 5292.2.i.b.2125.1 2
7.4 even 3 756.2.i.a.613.1 2
7.5 odd 6 5292.2.j.b.3529.1 2
7.6 odd 2 5292.2.l.b.3313.1 2
9.2 odd 6 2268.2.k.a.1297.1 2
9.4 even 3 756.2.i.a.37.1 2
9.5 odd 6 252.2.i.a.121.1 yes 2
9.7 even 3 2268.2.k.b.1297.1 2
12.11 even 2 1008.2.t.b.961.1 2
21.2 odd 6 1764.2.j.c.1177.1 2
21.5 even 6 1764.2.j.a.1177.1 2
21.11 odd 6 252.2.i.a.25.1 2
21.17 even 6 1764.2.i.b.1537.1 2
21.20 even 2 1764.2.l.b.961.1 2
28.11 odd 6 3024.2.q.e.2881.1 2
36.23 even 6 1008.2.q.f.625.1 2
36.31 odd 6 3024.2.q.e.2305.1 2
63.4 even 3 inner 756.2.l.a.361.1 2
63.5 even 6 1764.2.j.a.589.1 2
63.11 odd 6 2268.2.k.a.1621.1 2
63.13 odd 6 5292.2.i.b.1549.1 2
63.23 odd 6 1764.2.j.c.589.1 2
63.25 even 3 2268.2.k.b.1621.1 2
63.31 odd 6 5292.2.l.b.361.1 2
63.32 odd 6 252.2.l.a.193.1 yes 2
63.40 odd 6 5292.2.j.b.1765.1 2
63.41 even 6 1764.2.i.b.373.1 2
63.58 even 3 5292.2.j.c.1765.1 2
63.59 even 6 1764.2.l.b.949.1 2
84.11 even 6 1008.2.q.f.529.1 2
252.67 odd 6 3024.2.t.b.1873.1 2
252.95 even 6 1008.2.t.b.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.i.a.25.1 2 21.11 odd 6
252.2.i.a.121.1 yes 2 9.5 odd 6
252.2.l.a.193.1 yes 2 63.32 odd 6
252.2.l.a.205.1 yes 2 3.2 odd 2
756.2.i.a.37.1 2 9.4 even 3
756.2.i.a.613.1 2 7.4 even 3
756.2.l.a.289.1 2 1.1 even 1 trivial
756.2.l.a.361.1 2 63.4 even 3 inner
1008.2.q.f.529.1 2 84.11 even 6
1008.2.q.f.625.1 2 36.23 even 6
1008.2.t.b.193.1 2 252.95 even 6
1008.2.t.b.961.1 2 12.11 even 2
1764.2.i.b.373.1 2 63.41 even 6
1764.2.i.b.1537.1 2 21.17 even 6
1764.2.j.a.589.1 2 63.5 even 6
1764.2.j.a.1177.1 2 21.5 even 6
1764.2.j.c.589.1 2 63.23 odd 6
1764.2.j.c.1177.1 2 21.2 odd 6
1764.2.l.b.949.1 2 63.59 even 6
1764.2.l.b.961.1 2 21.20 even 2
2268.2.k.a.1297.1 2 9.2 odd 6
2268.2.k.a.1621.1 2 63.11 odd 6
2268.2.k.b.1297.1 2 9.7 even 3
2268.2.k.b.1621.1 2 63.25 even 3
3024.2.q.e.2305.1 2 36.31 odd 6
3024.2.q.e.2881.1 2 28.11 odd 6
3024.2.t.b.289.1 2 4.3 odd 2
3024.2.t.b.1873.1 2 252.67 odd 6
5292.2.i.b.1549.1 2 63.13 odd 6
5292.2.i.b.2125.1 2 7.3 odd 6
5292.2.j.b.1765.1 2 63.40 odd 6
5292.2.j.b.3529.1 2 7.5 odd 6
5292.2.j.c.1765.1 2 63.58 even 3
5292.2.j.c.3529.1 2 7.2 even 3
5292.2.l.b.361.1 2 63.31 odd 6
5292.2.l.b.3313.1 2 7.6 odd 2