Properties

Label 276.2.g.a.137.8
Level $276$
Weight $2$
Character 276.137
Analytic conductor $2.204$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [276,2,Mod(137,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.g (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: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 15x^{4} - 30x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 137.8
Root \(1.06789 - 0.545948i\) of defining polynomial
Character \(\chi\) \(=\) 276.137
Dual form 276.2.g.a.137.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.28078 + 1.16602i) q^{3} +0.936426 q^{5} -3.88884i q^{7} +(0.280776 + 2.98683i) q^{9} +O(q^{10})\) \(q+(1.28078 + 1.16602i) q^{3} +0.936426 q^{5} -3.88884i q^{7} +(0.280776 + 2.98683i) q^{9} +4.27156 q^{11} +2.56155 q^{13} +(1.19935 + 1.09190i) q^{15} -7.60669 q^{17} +6.07263i q^{19} +(4.53448 - 4.98074i) q^{21} +(-2.39871 + 4.15286i) q^{23} -4.12311 q^{25} +(-3.12311 + 4.15286i) q^{27} -5.97366i q^{29} +3.68466 q^{31} +(5.47091 + 4.98074i) q^{33} -3.64162i q^{35} -2.18379i q^{37} +(3.28078 + 2.98683i) q^{39} -10.6378i q^{41} -6.07263i q^{43} +(0.262926 + 2.79695i) q^{45} +1.30957i q^{47} -8.12311 q^{49} +(-9.74247 - 8.86958i) q^{51} -8.13254 q^{53} +4.00000 q^{55} +(-7.08084 + 7.77769i) q^{57} -4.66410i q^{59} +7.77769i q^{61} +(11.6153 - 1.09190i) q^{63} +2.39871 q^{65} +11.6665i q^{67} +(-7.91453 + 2.52193i) q^{69} +2.33205i q^{71} -5.68466 q^{73} +(-5.28078 - 4.80764i) q^{75} -16.6114i q^{77} -1.70505i q^{79} +(-8.84233 + 1.67726i) q^{81} +0.525853 q^{83} -7.12311 q^{85} +(6.96543 - 7.65093i) q^{87} +1.46228 q^{89} -9.96148i q^{91} +(4.71922 + 4.29640i) q^{93} +5.68658i q^{95} +9.96148i q^{97} +(1.19935 + 12.7584i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 6 q^{9} + 4 q^{13} + 8 q^{27} - 20 q^{31} + 18 q^{39} - 32 q^{49} + 32 q^{55} - 14 q^{69} + 4 q^{73} - 34 q^{75} - 46 q^{81} - 24 q^{85} - 2 q^{87} + 46 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.28078 + 1.16602i 0.739457 + 0.673204i
\(4\) 0 0
\(5\) 0.936426 0.418783 0.209391 0.977832i \(-0.432852\pi\)
0.209391 + 0.977832i \(0.432852\pi\)
\(6\) 0 0
\(7\) 3.88884i 1.46984i −0.678151 0.734922i \(-0.737219\pi\)
0.678151 0.734922i \(-0.262781\pi\)
\(8\) 0 0
\(9\) 0.280776 + 2.98683i 0.0935921 + 0.995611i
\(10\) 0 0
\(11\) 4.27156 1.28792 0.643962 0.765058i \(-0.277290\pi\)
0.643962 + 0.765058i \(0.277290\pi\)
\(12\) 0 0
\(13\) 2.56155 0.710447 0.355223 0.934781i \(-0.384405\pi\)
0.355223 + 0.934781i \(0.384405\pi\)
\(14\) 0 0
\(15\) 1.19935 + 1.09190i 0.309672 + 0.281926i
\(16\) 0 0
\(17\) −7.60669 −1.84489 −0.922447 0.386124i \(-0.873814\pi\)
−0.922447 + 0.386124i \(0.873814\pi\)
\(18\) 0 0
\(19\) 6.07263i 1.39316i 0.717480 + 0.696579i \(0.245295\pi\)
−0.717480 + 0.696579i \(0.754705\pi\)
\(20\) 0 0
\(21\) 4.53448 4.98074i 0.989506 1.08689i
\(22\) 0 0
\(23\) −2.39871 + 4.15286i −0.500165 + 0.865930i
\(24\) 0 0
\(25\) −4.12311 −0.824621
\(26\) 0 0
\(27\) −3.12311 + 4.15286i −0.601042 + 0.799217i
\(28\) 0 0
\(29\) 5.97366i 1.10928i −0.832090 0.554641i \(-0.812856\pi\)
0.832090 0.554641i \(-0.187144\pi\)
\(30\) 0 0
\(31\) 3.68466 0.661784 0.330892 0.943669i \(-0.392650\pi\)
0.330892 + 0.943669i \(0.392650\pi\)
\(32\) 0 0
\(33\) 5.47091 + 4.98074i 0.952363 + 0.867035i
\(34\) 0 0
\(35\) 3.64162i 0.615545i
\(36\) 0 0
\(37\) 2.18379i 0.359013i −0.983757 0.179507i \(-0.942550\pi\)
0.983757 0.179507i \(-0.0574501\pi\)
\(38\) 0 0
\(39\) 3.28078 + 2.98683i 0.525345 + 0.478276i
\(40\) 0 0
\(41\) 10.6378i 1.66134i −0.556766 0.830669i \(-0.687958\pi\)
0.556766 0.830669i \(-0.312042\pi\)
\(42\) 0 0
\(43\) 6.07263i 0.926068i −0.886340 0.463034i \(-0.846761\pi\)
0.886340 0.463034i \(-0.153239\pi\)
\(44\) 0 0
\(45\) 0.262926 + 2.79695i 0.0391948 + 0.416944i
\(46\) 0 0
\(47\) 1.30957i 0.191020i 0.995428 + 0.0955101i \(0.0304482\pi\)
−0.995428 + 0.0955101i \(0.969552\pi\)
\(48\) 0 0
\(49\) −8.12311 −1.16044
\(50\) 0 0
\(51\) −9.74247 8.86958i −1.36422 1.24199i
\(52\) 0 0
\(53\) −8.13254 −1.11709 −0.558545 0.829474i \(-0.688640\pi\)
−0.558545 + 0.829474i \(0.688640\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −7.08084 + 7.77769i −0.937880 + 1.03018i
\(58\) 0 0
\(59\) 4.66410i 0.607214i −0.952797 0.303607i \(-0.901809\pi\)
0.952797 0.303607i \(-0.0981909\pi\)
\(60\) 0 0
\(61\) 7.77769i 0.995831i 0.867226 + 0.497915i \(0.165901\pi\)
−0.867226 + 0.497915i \(0.834099\pi\)
\(62\) 0 0
\(63\) 11.6153 1.09190i 1.46339 0.137566i
\(64\) 0 0
\(65\) 2.39871 0.297523
\(66\) 0 0
\(67\) 11.6665i 1.42529i 0.701523 + 0.712647i \(0.252504\pi\)
−0.701523 + 0.712647i \(0.747496\pi\)
\(68\) 0 0
\(69\) −7.91453 + 2.52193i −0.952798 + 0.303605i
\(70\) 0 0
\(71\) 2.33205i 0.276763i 0.990379 + 0.138382i \(0.0441900\pi\)
−0.990379 + 0.138382i \(0.955810\pi\)
\(72\) 0 0
\(73\) −5.68466 −0.665339 −0.332669 0.943043i \(-0.607949\pi\)
−0.332669 + 0.943043i \(0.607949\pi\)
\(74\) 0 0
\(75\) −5.28078 4.80764i −0.609772 0.555138i
\(76\) 0 0
\(77\) 16.6114i 1.89305i
\(78\) 0 0
\(79\) 1.70505i 0.191833i −0.995389 0.0959167i \(-0.969422\pi\)
0.995389 0.0959167i \(-0.0305782\pi\)
\(80\) 0 0
\(81\) −8.84233 + 1.67726i −0.982481 + 0.186363i
\(82\) 0 0
\(83\) 0.525853 0.0577199 0.0288599 0.999583i \(-0.490812\pi\)
0.0288599 + 0.999583i \(0.490812\pi\)
\(84\) 0 0
\(85\) −7.12311 −0.772609
\(86\) 0 0
\(87\) 6.96543 7.65093i 0.746773 0.820266i
\(88\) 0 0
\(89\) 1.46228 0.155001 0.0775006 0.996992i \(-0.475306\pi\)
0.0775006 + 0.996992i \(0.475306\pi\)
\(90\) 0 0
\(91\) 9.96148i 1.04425i
\(92\) 0 0
\(93\) 4.71922 + 4.29640i 0.489361 + 0.445516i
\(94\) 0 0
\(95\) 5.68658i 0.583430i
\(96\) 0 0
\(97\) 9.96148i 1.01143i 0.862699 + 0.505717i \(0.168772\pi\)
−0.862699 + 0.505717i \(0.831228\pi\)
\(98\) 0 0
\(99\) 1.19935 + 12.7584i 0.120539 + 1.28227i
\(100\) 0 0
\(101\) 7.28323i 0.724709i 0.932040 + 0.362354i \(0.118027\pi\)
−0.932040 + 0.362354i \(0.881973\pi\)
\(102\) 0 0
\(103\) 11.6665i 1.14954i −0.818316 0.574769i \(-0.805092\pi\)
0.818316 0.574769i \(-0.194908\pi\)
\(104\) 0 0
\(105\) 4.24621 4.66410i 0.414388 0.455169i
\(106\) 0 0
\(107\) 1.87285 0.181056 0.0905278 0.995894i \(-0.471145\pi\)
0.0905278 + 0.995894i \(0.471145\pi\)
\(108\) 0 0
\(109\) 7.77769i 0.744967i −0.928039 0.372484i \(-0.878506\pi\)
0.928039 0.372484i \(-0.121494\pi\)
\(110\) 0 0
\(111\) 2.54635 2.79695i 0.241689 0.265475i
\(112\) 0 0
\(113\) −5.73384 −0.539394 −0.269697 0.962945i \(-0.586924\pi\)
−0.269697 + 0.962945i \(0.586924\pi\)
\(114\) 0 0
\(115\) −2.24621 + 3.88884i −0.209460 + 0.362637i
\(116\) 0 0
\(117\) 0.719224 + 7.65093i 0.0664922 + 0.707329i
\(118\) 0 0
\(119\) 29.5812i 2.71171i
\(120\) 0 0
\(121\) 7.24621 0.658746
\(122\) 0 0
\(123\) 12.4039 13.6246i 1.11842 1.22849i
\(124\) 0 0
\(125\) −8.54312 −0.764120
\(126\) 0 0
\(127\) −7.68466 −0.681903 −0.340952 0.940081i \(-0.610749\pi\)
−0.340952 + 0.940081i \(0.610749\pi\)
\(128\) 0 0
\(129\) 7.08084 7.77769i 0.623433 0.684787i
\(130\) 0 0
\(131\) 9.61528i 0.840091i 0.907503 + 0.420045i \(0.137986\pi\)
−0.907503 + 0.420045i \(0.862014\pi\)
\(132\) 0 0
\(133\) 23.6155 2.04773
\(134\) 0 0
\(135\) −2.92456 + 3.88884i −0.251706 + 0.334698i
\(136\) 0 0
\(137\) 11.8782 1.01483 0.507414 0.861703i \(-0.330602\pi\)
0.507414 + 0.861703i \(0.330602\pi\)
\(138\) 0 0
\(139\) 13.9309 1.18160 0.590800 0.806818i \(-0.298812\pi\)
0.590800 + 0.806818i \(0.298812\pi\)
\(140\) 0 0
\(141\) −1.52699 + 1.67726i −0.128596 + 0.141251i
\(142\) 0 0
\(143\) 10.9418 0.915001
\(144\) 0 0
\(145\) 5.59390i 0.464548i
\(146\) 0 0
\(147\) −10.4039 9.47174i −0.858098 0.781216i
\(148\) 0 0
\(149\) 8.95369 0.733515 0.366757 0.930317i \(-0.380468\pi\)
0.366757 + 0.930317i \(0.380468\pi\)
\(150\) 0 0
\(151\) 14.5616 1.18500 0.592501 0.805570i \(-0.298141\pi\)
0.592501 + 0.805570i \(0.298141\pi\)
\(152\) 0 0
\(153\) −2.13578 22.7199i −0.172668 1.83680i
\(154\) 0 0
\(155\) 3.45041 0.277144
\(156\) 0 0
\(157\) 12.1453i 0.969298i −0.874709 0.484649i \(-0.838947\pi\)
0.874709 0.484649i \(-0.161053\pi\)
\(158\) 0 0
\(159\) −10.4160 9.48274i −0.826040 0.752030i
\(160\) 0 0
\(161\) 16.1498 + 9.32819i 1.27278 + 0.735164i
\(162\) 0 0
\(163\) −1.43845 −0.112668 −0.0563339 0.998412i \(-0.517941\pi\)
−0.0563339 + 0.998412i \(0.517941\pi\)
\(164\) 0 0
\(165\) 5.12311 + 4.66410i 0.398833 + 0.363099i
\(166\) 0 0
\(167\) 1.02248i 0.0791219i 0.999217 + 0.0395609i \(0.0125959\pi\)
−0.999217 + 0.0395609i \(0.987404\pi\)
\(168\) 0 0
\(169\) −6.43845 −0.495265
\(170\) 0 0
\(171\) −18.1379 + 1.70505i −1.38704 + 0.130389i
\(172\) 0 0
\(173\) 18.6564i 1.41842i 0.704998 + 0.709209i \(0.250948\pi\)
−0.704998 + 0.709209i \(0.749052\pi\)
\(174\) 0 0
\(175\) 16.0341i 1.21207i
\(176\) 0 0
\(177\) 5.43845 5.97366i 0.408779 0.449008i
\(178\) 0 0
\(179\) 8.01862i 0.599340i −0.954043 0.299670i \(-0.903123\pi\)
0.954043 0.299670i \(-0.0968766\pi\)
\(180\) 0 0
\(181\) 22.1067i 1.64318i 0.570078 + 0.821591i \(0.306913\pi\)
−0.570078 + 0.821591i \(0.693087\pi\)
\(182\) 0 0
\(183\) −9.06897 + 9.96148i −0.670398 + 0.736374i
\(184\) 0 0
\(185\) 2.04496i 0.150348i
\(186\) 0 0
\(187\) −32.4924 −2.37608
\(188\) 0 0
\(189\) 16.1498 + 12.1453i 1.17473 + 0.883438i
\(190\) 0 0
\(191\) 20.5366 1.48598 0.742990 0.669303i \(-0.233407\pi\)
0.742990 + 0.669303i \(0.233407\pi\)
\(192\) 0 0
\(193\) 18.5616 1.33609 0.668045 0.744121i \(-0.267131\pi\)
0.668045 + 0.744121i \(0.267131\pi\)
\(194\) 0 0
\(195\) 3.07221 + 2.79695i 0.220005 + 0.200294i
\(196\) 0 0
\(197\) 19.9660i 1.42252i −0.702932 0.711258i \(-0.748126\pi\)
0.702932 0.711258i \(-0.251874\pi\)
\(198\) 0 0
\(199\) 3.88884i 0.275673i −0.990455 0.137836i \(-0.955985\pi\)
0.990455 0.137836i \(-0.0440148\pi\)
\(200\) 0 0
\(201\) −13.6035 + 14.9422i −0.959514 + 1.05394i
\(202\) 0 0
\(203\) −23.2306 −1.63047
\(204\) 0 0
\(205\) 9.96148i 0.695740i
\(206\) 0 0
\(207\) −13.0774 5.99851i −0.908941 0.416925i
\(208\) 0 0
\(209\) 25.9396i 1.79428i
\(210\) 0 0
\(211\) −24.4924 −1.68613 −0.843064 0.537813i \(-0.819251\pi\)
−0.843064 + 0.537813i \(0.819251\pi\)
\(212\) 0 0
\(213\) −2.71922 + 2.98683i −0.186318 + 0.204654i
\(214\) 0 0
\(215\) 5.68658i 0.387821i
\(216\) 0 0
\(217\) 14.3291i 0.972720i
\(218\) 0 0
\(219\) −7.28078 6.62845i −0.491989 0.447909i
\(220\) 0 0
\(221\) −19.4849 −1.31070
\(222\) 0 0
\(223\) 2.24621 0.150417 0.0752087 0.997168i \(-0.476038\pi\)
0.0752087 + 0.997168i \(0.476038\pi\)
\(224\) 0 0
\(225\) −1.15767 12.3150i −0.0771781 0.821002i
\(226\) 0 0
\(227\) −3.21985 −0.213709 −0.106855 0.994275i \(-0.534078\pi\)
−0.106855 + 0.994275i \(0.534078\pi\)
\(228\) 0 0
\(229\) 3.41011i 0.225346i −0.993632 0.112673i \(-0.964059\pi\)
0.993632 0.112673i \(-0.0359413\pi\)
\(230\) 0 0
\(231\) 19.3693 21.2755i 1.27441 1.39983i
\(232\) 0 0
\(233\) 15.3019i 1.00246i 0.865315 + 0.501229i \(0.167119\pi\)
−0.865315 + 0.501229i \(0.832881\pi\)
\(234\) 0 0
\(235\) 1.22631i 0.0799959i
\(236\) 0 0
\(237\) 1.98813 2.18379i 0.129143 0.141852i
\(238\) 0 0
\(239\) 14.2794i 0.923656i −0.886970 0.461828i \(-0.847194\pi\)
0.886970 0.461828i \(-0.152806\pi\)
\(240\) 0 0
\(241\) 9.96148i 0.641675i −0.947134 0.320838i \(-0.896036\pi\)
0.947134 0.320838i \(-0.103964\pi\)
\(242\) 0 0
\(243\) −13.2808 8.16217i −0.851962 0.523603i
\(244\) 0 0
\(245\) −7.60669 −0.485974
\(246\) 0 0
\(247\) 15.5554i 0.989765i
\(248\) 0 0
\(249\) 0.673500 + 0.613157i 0.0426813 + 0.0388572i
\(250\) 0 0
\(251\) 25.1035 1.58452 0.792259 0.610184i \(-0.208905\pi\)
0.792259 + 0.610184i \(0.208905\pi\)
\(252\) 0 0
\(253\) −10.2462 + 17.7392i −0.644174 + 1.11525i
\(254\) 0 0
\(255\) −9.12311 8.30571i −0.571311 0.520124i
\(256\) 0 0
\(257\) 13.2569i 0.826942i −0.910517 0.413471i \(-0.864316\pi\)
0.910517 0.413471i \(-0.135684\pi\)
\(258\) 0 0
\(259\) −8.49242 −0.527693
\(260\) 0 0
\(261\) 17.8423 1.67726i 1.10441 0.103820i
\(262\) 0 0
\(263\) 21.8836 1.34940 0.674702 0.738091i \(-0.264272\pi\)
0.674702 + 0.738091i \(0.264272\pi\)
\(264\) 0 0
\(265\) −7.61553 −0.467818
\(266\) 0 0
\(267\) 1.87285 + 1.70505i 0.114617 + 0.104348i
\(268\) 0 0
\(269\) 13.2569i 0.808287i −0.914696 0.404144i \(-0.867570\pi\)
0.914696 0.404144i \(-0.132430\pi\)
\(270\) 0 0
\(271\) 14.2462 0.865396 0.432698 0.901539i \(-0.357562\pi\)
0.432698 + 0.901539i \(0.357562\pi\)
\(272\) 0 0
\(273\) 11.6153 12.7584i 0.702991 0.772175i
\(274\) 0 0
\(275\) −17.6121 −1.06205
\(276\) 0 0
\(277\) −3.43845 −0.206596 −0.103298 0.994650i \(-0.532940\pi\)
−0.103298 + 0.994650i \(0.532940\pi\)
\(278\) 0 0
\(279\) 1.03457 + 11.0055i 0.0619378 + 0.658879i
\(280\) 0 0
\(281\) −23.6412 −1.41032 −0.705158 0.709050i \(-0.749124\pi\)
−0.705158 + 0.709050i \(0.749124\pi\)
\(282\) 0 0
\(283\) 7.29895i 0.433877i 0.976185 + 0.216939i \(0.0696072\pi\)
−0.976185 + 0.216939i \(0.930393\pi\)
\(284\) 0 0
\(285\) −6.63068 + 7.28323i −0.392768 + 0.431421i
\(286\) 0 0
\(287\) −41.3686 −2.44191
\(288\) 0 0
\(289\) 40.8617 2.40363
\(290\) 0 0
\(291\) −11.6153 + 12.7584i −0.680902 + 0.747912i
\(292\) 0 0
\(293\) 24.1671 1.41186 0.705928 0.708284i \(-0.250530\pi\)
0.705928 + 0.708284i \(0.250530\pi\)
\(294\) 0 0
\(295\) 4.36758i 0.254290i
\(296\) 0 0
\(297\) −13.3405 + 17.7392i −0.774096 + 1.02933i
\(298\) 0 0
\(299\) −6.14441 + 10.6378i −0.355340 + 0.615198i
\(300\) 0 0
\(301\) −23.6155 −1.36118
\(302\) 0 0
\(303\) −8.49242 + 9.32819i −0.487877 + 0.535891i
\(304\) 0 0
\(305\) 7.28323i 0.417037i
\(306\) 0 0
\(307\) −5.75379 −0.328386 −0.164193 0.986428i \(-0.552502\pi\)
−0.164193 + 0.986428i \(0.552502\pi\)
\(308\) 0 0
\(309\) 13.6035 14.9422i 0.773873 0.850033i
\(310\) 0 0
\(311\) 10.6378i 0.603212i 0.953433 + 0.301606i \(0.0975227\pi\)
−0.953433 + 0.301606i \(0.902477\pi\)
\(312\) 0 0
\(313\) 4.36758i 0.246870i 0.992353 + 0.123435i \(0.0393911\pi\)
−0.992353 + 0.123435i \(0.960609\pi\)
\(314\) 0 0
\(315\) 10.8769 1.02248i 0.612844 0.0576102i
\(316\) 0 0
\(317\) 18.6564i 1.04785i −0.851765 0.523923i \(-0.824468\pi\)
0.851765 0.523923i \(-0.175532\pi\)
\(318\) 0 0
\(319\) 25.5169i 1.42867i
\(320\) 0 0
\(321\) 2.39871 + 2.18379i 0.133883 + 0.121887i
\(322\) 0 0
\(323\) 46.1927i 2.57023i
\(324\) 0 0
\(325\) −10.5616 −0.585850
\(326\) 0 0
\(327\) 9.06897 9.96148i 0.501515 0.550871i
\(328\) 0 0
\(329\) 5.09271 0.280770
\(330\) 0 0
\(331\) 12.8078 0.703978 0.351989 0.936004i \(-0.385505\pi\)
0.351989 + 0.936004i \(0.385505\pi\)
\(332\) 0 0
\(333\) 6.52262 0.613157i 0.357437 0.0336008i
\(334\) 0 0
\(335\) 10.9248i 0.596888i
\(336\) 0 0
\(337\) 15.5554i 0.847355i 0.905813 + 0.423678i \(0.139261\pi\)
−0.905813 + 0.423678i \(0.860739\pi\)
\(338\) 0 0
\(339\) −7.34376 6.68579i −0.398859 0.363122i
\(340\) 0 0
\(341\) 15.7392 0.852327
\(342\) 0 0
\(343\) 4.36758i 0.235827i
\(344\) 0 0
\(345\) −7.41138 + 2.36160i −0.399015 + 0.127144i
\(346\) 0 0
\(347\) 28.5588i 1.53311i −0.642176 0.766557i \(-0.721968\pi\)
0.642176 0.766557i \(-0.278032\pi\)
\(348\) 0 0
\(349\) −3.19224 −0.170876 −0.0854382 0.996343i \(-0.527229\pi\)
−0.0854382 + 0.996343i \(0.527229\pi\)
\(350\) 0 0
\(351\) −8.00000 + 10.6378i −0.427008 + 0.567802i
\(352\) 0 0
\(353\) 17.9210i 0.953838i 0.878947 + 0.476919i \(0.158247\pi\)
−0.878947 + 0.476919i \(0.841753\pi\)
\(354\) 0 0
\(355\) 2.18379i 0.115904i
\(356\) 0 0
\(357\) −34.4924 + 37.8869i −1.82553 + 2.00519i
\(358\) 0 0
\(359\) 20.8319 1.09947 0.549734 0.835340i \(-0.314729\pi\)
0.549734 + 0.835340i \(0.314729\pi\)
\(360\) 0 0
\(361\) −17.8769 −0.940889
\(362\) 0 0
\(363\) 9.28078 + 8.44926i 0.487114 + 0.443471i
\(364\) 0 0
\(365\) −5.32326 −0.278632
\(366\) 0 0
\(367\) 23.8118i 1.24297i 0.783428 + 0.621483i \(0.213469\pi\)
−0.783428 + 0.621483i \(0.786531\pi\)
\(368\) 0 0
\(369\) 31.7732 2.98683i 1.65405 0.155488i
\(370\) 0 0
\(371\) 31.6262i 1.64195i
\(372\) 0 0
\(373\) 28.9270i 1.49778i −0.662694 0.748891i \(-0.730587\pi\)
0.662694 0.748891i \(-0.269413\pi\)
\(374\) 0 0
\(375\) −10.9418 9.96148i −0.565033 0.514409i
\(376\) 0 0
\(377\) 15.3019i 0.788086i
\(378\) 0 0
\(379\) 33.7733i 1.73482i 0.497597 + 0.867408i \(0.334216\pi\)
−0.497597 + 0.867408i \(0.665784\pi\)
\(380\) 0 0
\(381\) −9.84233 8.96050i −0.504238 0.459060i
\(382\) 0 0
\(383\) 12.2888 0.627929 0.313965 0.949435i \(-0.398343\pi\)
0.313965 + 0.949435i \(0.398343\pi\)
\(384\) 0 0
\(385\) 15.5554i 0.792775i
\(386\) 0 0
\(387\) 18.1379 1.70505i 0.922003 0.0866727i
\(388\) 0 0
\(389\) −5.20798 −0.264055 −0.132028 0.991246i \(-0.542149\pi\)
−0.132028 + 0.991246i \(0.542149\pi\)
\(390\) 0 0
\(391\) 18.2462 31.5895i 0.922751 1.59755i
\(392\) 0 0
\(393\) −11.2116 + 12.3150i −0.565553 + 0.621211i
\(394\) 0 0
\(395\) 1.59666i 0.0803365i
\(396\) 0 0
\(397\) 0.0691303 0.00346955 0.00173478 0.999998i \(-0.499448\pi\)
0.00173478 + 0.999998i \(0.499448\pi\)
\(398\) 0 0
\(399\) 30.2462 + 27.5363i 1.51420 + 1.37854i
\(400\) 0 0
\(401\) −2.80928 −0.140289 −0.0701444 0.997537i \(-0.522346\pi\)
−0.0701444 + 0.997537i \(0.522346\pi\)
\(402\) 0 0
\(403\) 9.43845 0.470163
\(404\) 0 0
\(405\) −8.28019 + 1.57063i −0.411446 + 0.0780454i
\(406\) 0 0
\(407\) 9.32819i 0.462381i
\(408\) 0 0
\(409\) 8.31534 0.411167 0.205584 0.978640i \(-0.434091\pi\)
0.205584 + 0.978640i \(0.434091\pi\)
\(410\) 0 0
\(411\) 15.2134 + 13.8503i 0.750421 + 0.683186i
\(412\) 0 0
\(413\) −18.1379 −0.892510
\(414\) 0 0
\(415\) 0.492423 0.0241721
\(416\) 0 0
\(417\) 17.8423 + 16.2437i 0.873743 + 0.795459i
\(418\) 0 0
\(419\) −3.97626 −0.194253 −0.0971266 0.995272i \(-0.530965\pi\)
−0.0971266 + 0.995272i \(0.530965\pi\)
\(420\) 0 0
\(421\) 2.18379i 0.106431i −0.998583 0.0532157i \(-0.983053\pi\)
0.998583 0.0532157i \(-0.0169471\pi\)
\(422\) 0 0
\(423\) −3.91146 + 0.367696i −0.190182 + 0.0178780i
\(424\) 0 0
\(425\) 31.3632 1.52134
\(426\) 0 0
\(427\) 30.2462 1.46372
\(428\) 0 0
\(429\) 14.0140 + 12.7584i 0.676604 + 0.615983i
\(430\) 0 0
\(431\) −26.6811 −1.28518 −0.642591 0.766210i \(-0.722140\pi\)
−0.642591 + 0.766210i \(0.722140\pi\)
\(432\) 0 0
\(433\) 14.3291i 0.688611i 0.938858 + 0.344305i \(0.111886\pi\)
−0.938858 + 0.344305i \(0.888114\pi\)
\(434\) 0 0
\(435\) 6.52262 7.16453i 0.312736 0.343513i
\(436\) 0 0
\(437\) −25.2188 14.5665i −1.20638 0.696808i
\(438\) 0 0
\(439\) 25.9309 1.23761 0.618806 0.785543i \(-0.287617\pi\)
0.618806 + 0.785543i \(0.287617\pi\)
\(440\) 0 0
\(441\) −2.28078 24.2624i −0.108608 1.15535i
\(442\) 0 0
\(443\) 35.5549i 1.68926i 0.535347 + 0.844632i \(0.320181\pi\)
−0.535347 + 0.844632i \(0.679819\pi\)
\(444\) 0 0
\(445\) 1.36932 0.0649118
\(446\) 0 0
\(447\) 11.4677 + 10.4402i 0.542402 + 0.493805i
\(448\) 0 0
\(449\) 25.9396i 1.22417i 0.790793 + 0.612083i \(0.209668\pi\)
−0.790793 + 0.612083i \(0.790332\pi\)
\(450\) 0 0
\(451\) 45.4398i 2.13968i
\(452\) 0 0
\(453\) 18.6501 + 16.9791i 0.876258 + 0.797749i
\(454\) 0 0
\(455\) 9.32819i 0.437312i
\(456\) 0 0
\(457\) 1.22631i 0.0573646i 0.999589 + 0.0286823i \(0.00913110\pi\)
−0.999589 + 0.0286823i \(0.990869\pi\)
\(458\) 0 0
\(459\) 23.7565 31.5895i 1.10886 1.47447i
\(460\) 0 0
\(461\) 10.6378i 0.495450i −0.968830 0.247725i \(-0.920317\pi\)
0.968830 0.247725i \(-0.0796829\pi\)
\(462\) 0 0
\(463\) −40.4924 −1.88184 −0.940921 0.338626i \(-0.890038\pi\)
−0.940921 + 0.338626i \(0.890038\pi\)
\(464\) 0 0
\(465\) 4.41921 + 4.02326i 0.204936 + 0.186574i
\(466\) 0 0
\(467\) −20.0108 −0.925989 −0.462995 0.886361i \(-0.653225\pi\)
−0.462995 + 0.886361i \(0.653225\pi\)
\(468\) 0 0
\(469\) 45.3693 2.09496
\(470\) 0 0
\(471\) 14.1617 15.5554i 0.652536 0.716754i
\(472\) 0 0
\(473\) 25.9396i 1.19270i
\(474\) 0 0
\(475\) 25.0381i 1.14883i
\(476\) 0 0
\(477\) −2.28343 24.2905i −0.104551 1.11219i
\(478\) 0 0
\(479\) 26.9764 1.23258 0.616290 0.787519i \(-0.288635\pi\)
0.616290 + 0.787519i \(0.288635\pi\)
\(480\) 0 0
\(481\) 5.59390i 0.255060i
\(482\) 0 0
\(483\) 9.80740 + 30.7784i 0.446252 + 1.40047i
\(484\) 0 0
\(485\) 9.32819i 0.423571i
\(486\) 0 0
\(487\) −23.0540 −1.04468 −0.522338 0.852739i \(-0.674940\pi\)
−0.522338 + 0.852739i \(0.674940\pi\)
\(488\) 0 0
\(489\) −1.84233 1.67726i −0.0833130 0.0758485i
\(490\) 0 0
\(491\) 8.01862i 0.361875i −0.983495 0.180938i \(-0.942087\pi\)
0.983495 0.180938i \(-0.0579132\pi\)
\(492\) 0 0
\(493\) 45.4398i 2.04651i
\(494\) 0 0
\(495\) 1.12311 + 11.9473i 0.0504798 + 0.536992i
\(496\) 0 0
\(497\) 9.06897 0.406799
\(498\) 0 0
\(499\) −14.5616 −0.651865 −0.325932 0.945393i \(-0.605678\pi\)
−0.325932 + 0.945393i \(0.605678\pi\)
\(500\) 0 0
\(501\) −1.19224 + 1.30957i −0.0532652 + 0.0585072i
\(502\) 0 0
\(503\) −31.4785 −1.40356 −0.701778 0.712396i \(-0.747610\pi\)
−0.701778 + 0.712396i \(0.747610\pi\)
\(504\) 0 0
\(505\) 6.82021i 0.303495i
\(506\) 0 0
\(507\) −8.24621 7.50738i −0.366227 0.333415i
\(508\) 0 0
\(509\) 3.35453i 0.148687i −0.997233 0.0743434i \(-0.976314\pi\)
0.997233 0.0743434i \(-0.0236861\pi\)
\(510\) 0 0
\(511\) 22.1067i 0.977945i
\(512\) 0 0
\(513\) −25.2188 18.9655i −1.11344 0.837346i
\(514\) 0 0
\(515\) 10.9248i 0.481406i
\(516\) 0 0
\(517\) 5.59390i 0.246019i
\(518\) 0 0
\(519\) −21.7538 + 23.8947i −0.954885 + 1.04886i
\(520\) 0 0
\(521\) 15.8545 0.694599 0.347299 0.937754i \(-0.387099\pi\)
0.347299 + 0.937754i \(0.387099\pi\)
\(522\) 0 0
\(523\) 3.88884i 0.170047i −0.996379 0.0850236i \(-0.972903\pi\)
0.996379 0.0850236i \(-0.0270966\pi\)
\(524\) 0 0
\(525\) −18.6962 + 20.5361i −0.815967 + 0.896270i
\(526\) 0 0
\(527\) −28.0281 −1.22092
\(528\) 0 0
\(529\) −11.4924 19.9230i −0.499671 0.866216i
\(530\) 0 0
\(531\) 13.9309 1.30957i 0.604548 0.0568304i
\(532\) 0 0
\(533\) 27.2492i 1.18029i
\(534\) 0 0
\(535\) 1.75379 0.0758229
\(536\) 0 0
\(537\) 9.34991 10.2701i 0.403478 0.443186i
\(538\) 0 0
\(539\) −34.6983 −1.49456
\(540\) 0 0
\(541\) −12.1771 −0.523534 −0.261767 0.965131i \(-0.584305\pi\)
−0.261767 + 0.965131i \(0.584305\pi\)
\(542\) 0 0
\(543\) −25.7770 + 28.3138i −1.10620 + 1.21506i
\(544\) 0 0
\(545\) 7.28323i 0.311979i
\(546\) 0 0
\(547\) 12.8078 0.547620 0.273810 0.961784i \(-0.411716\pi\)
0.273810 + 0.961784i \(0.411716\pi\)
\(548\) 0 0
\(549\) −23.2306 + 2.18379i −0.991460 + 0.0932019i
\(550\) 0 0
\(551\) 36.2759 1.54540
\(552\) 0 0
\(553\) −6.63068 −0.281965
\(554\) 0 0
\(555\) 2.38447 2.61914i 0.101215 0.111176i
\(556\) 0 0
\(557\) −6.25969 −0.265232 −0.132616 0.991168i \(-0.542338\pi\)
−0.132616 + 0.991168i \(0.542338\pi\)
\(558\) 0 0
\(559\) 15.5554i 0.657922i
\(560\) 0 0
\(561\) −41.6155 37.8869i −1.75701 1.59959i
\(562\) 0 0
\(563\) 34.4030 1.44991 0.724957 0.688794i \(-0.241859\pi\)
0.724957 + 0.688794i \(0.241859\pi\)
\(564\) 0 0
\(565\) −5.36932 −0.225889
\(566\) 0 0
\(567\) 6.52262 + 34.3864i 0.273924 + 1.44409i
\(568\) 0 0
\(569\) −9.47954 −0.397403 −0.198702 0.980060i \(-0.563672\pi\)
−0.198702 + 0.980060i \(0.563672\pi\)
\(570\) 0 0
\(571\) 28.1794i 1.17927i 0.807670 + 0.589635i \(0.200728\pi\)
−0.807670 + 0.589635i \(0.799272\pi\)
\(572\) 0 0
\(573\) 26.3029 + 23.9462i 1.09882 + 1.00037i
\(574\) 0 0
\(575\) 9.89012 17.1227i 0.412446 0.714064i
\(576\) 0 0
\(577\) −31.6847 −1.31905 −0.659525 0.751683i \(-0.729242\pi\)
−0.659525 + 0.751683i \(0.729242\pi\)
\(578\) 0 0
\(579\) 23.7732 + 21.6432i 0.987980 + 0.899461i
\(580\) 0 0
\(581\) 2.04496i 0.0848392i
\(582\) 0 0
\(583\) −34.7386 −1.43873
\(584\) 0 0
\(585\) 0.673500 + 7.16453i 0.0278458 + 0.296217i
\(586\) 0 0
\(587\) 0.287088i 0.0118494i 0.999982 + 0.00592470i \(0.00188590\pi\)
−0.999982 + 0.00592470i \(0.998114\pi\)
\(588\) 0 0
\(589\) 22.3756i 0.921970i
\(590\) 0 0
\(591\) 23.2808 25.5719i 0.957643 1.05189i
\(592\) 0 0
\(593\) 16.6114i 0.682149i 0.940036 + 0.341075i \(0.110791\pi\)
−0.940036 + 0.341075i \(0.889209\pi\)
\(594\) 0 0
\(595\) 27.7006i 1.13562i
\(596\) 0 0
\(597\) 4.53448 4.98074i 0.185584 0.203848i
\(598\) 0 0
\(599\) 32.2004i 1.31567i −0.753162 0.657836i \(-0.771472\pi\)
0.753162 0.657836i \(-0.228528\pi\)
\(600\) 0 0
\(601\) −21.4384 −0.874493 −0.437246 0.899342i \(-0.644046\pi\)
−0.437246 + 0.899342i \(0.644046\pi\)
\(602\) 0 0
\(603\) −34.8460 + 3.27569i −1.41904 + 0.133396i
\(604\) 0 0
\(605\) 6.78554 0.275872
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −29.7533 27.0875i −1.20566 1.09764i
\(610\) 0 0
\(611\) 3.35453i 0.135710i
\(612\) 0 0
\(613\) 37.6621i 1.52116i −0.649245 0.760579i \(-0.724915\pi\)
0.649245 0.760579i \(-0.275085\pi\)
\(614\) 0 0
\(615\) 11.6153 12.7584i 0.468375 0.514469i
\(616\) 0 0
\(617\) −20.4214 −0.822133 −0.411067 0.911605i \(-0.634844\pi\)
−0.411067 + 0.911605i \(0.634844\pi\)
\(618\) 0 0
\(619\) 6.07263i 0.244080i −0.992525 0.122040i \(-0.961056\pi\)
0.992525 0.122040i \(-0.0389436\pi\)
\(620\) 0 0
\(621\) −9.75480 22.9313i −0.391447 0.920201i
\(622\) 0 0
\(623\) 5.68658i 0.227828i
\(624\) 0 0
\(625\) 12.6155 0.504621
\(626\) 0 0
\(627\) −30.2462 + 33.2228i −1.20792 + 1.32679i
\(628\) 0 0
\(629\) 16.6114i 0.662341i
\(630\) 0 0
\(631\) 1.70505i 0.0678771i 0.999424 + 0.0339385i \(0.0108050\pi\)
−0.999424 + 0.0339385i \(0.989195\pi\)
\(632\) 0 0
\(633\) −31.3693 28.5588i −1.24682 1.13511i
\(634\) 0 0
\(635\) −7.19612 −0.285569
\(636\) 0 0
\(637\) −20.8078 −0.824434
\(638\) 0 0
\(639\) −6.96543 + 0.654784i −0.275548 + 0.0259029i
\(640\) 0 0
\(641\) 12.9300 0.510703 0.255351 0.966848i \(-0.417809\pi\)
0.255351 + 0.966848i \(0.417809\pi\)
\(642\) 0 0
\(643\) 16.0341i 0.632324i −0.948705 0.316162i \(-0.897606\pi\)
0.948705 0.316162i \(-0.102394\pi\)
\(644\) 0 0
\(645\) 6.63068 7.28323i 0.261083 0.286777i
\(646\) 0 0
\(647\) 13.2569i 0.521182i −0.965449 0.260591i \(-0.916083\pi\)
0.965449 0.260591i \(-0.0839175\pi\)
\(648\) 0 0
\(649\) 19.9230i 0.782044i
\(650\) 0 0
\(651\) 16.7080 18.3523i 0.654839 0.719284i
\(652\) 0 0
\(653\) 8.01862i 0.313793i −0.987615 0.156896i \(-0.949851\pi\)
0.987615 0.156896i \(-0.0501489\pi\)
\(654\) 0 0
\(655\) 9.00400i 0.351815i
\(656\) 0 0
\(657\) −1.59612 16.9791i −0.0622705 0.662419i
\(658\) 0 0
\(659\) −13.8664 −0.540158 −0.270079 0.962838i \(-0.587050\pi\)
−0.270079 + 0.962838i \(0.587050\pi\)
\(660\) 0 0
\(661\) 38.8884i 1.51258i 0.654234 + 0.756292i \(0.272991\pi\)
−0.654234 + 0.756292i \(0.727009\pi\)
\(662\) 0 0
\(663\) −24.9559 22.7199i −0.969205 0.882368i
\(664\) 0 0
\(665\) 22.1142 0.857552
\(666\) 0 0
\(667\) 24.8078 + 14.3291i 0.960560 + 0.554823i
\(668\) 0 0
\(669\) 2.87689 + 2.61914i 0.111227 + 0.101262i
\(670\) 0 0
\(671\) 33.2228i 1.28255i
\(672\) 0 0
\(673\) 35.5464 1.37021 0.685106 0.728443i \(-0.259756\pi\)
0.685106 + 0.728443i \(0.259756\pi\)
\(674\) 0 0
\(675\) 12.8769 17.1227i 0.495632 0.659052i
\(676\) 0 0
\(677\) 11.6477 0.447657 0.223829 0.974629i \(-0.428144\pi\)
0.223829 + 0.974629i \(0.428144\pi\)
\(678\) 0 0
\(679\) 38.7386 1.48665
\(680\) 0 0
\(681\) −4.12391 3.75443i −0.158029 0.143870i
\(682\) 0 0
\(683\) 0.287088i 0.0109851i 0.999985 + 0.00549256i \(0.00174835\pi\)
−0.999985 + 0.00549256i \(0.998252\pi\)
\(684\) 0 0
\(685\) 11.1231 0.424992
\(686\) 0 0
\(687\) 3.97626 4.36758i 0.151704 0.166634i
\(688\) 0 0
\(689\) −20.8319 −0.793634
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) 49.6155 4.66410i 1.88474 0.177174i
\(694\) 0 0
\(695\) 13.0452 0.494834
\(696\) 0 0
\(697\) 80.9181i 3.06499i
\(698\) 0 0
\(699\) −17.8423 + 19.5983i −0.674859 + 0.741274i
\(700\) 0 0
\(701\) −41.7792 −1.57798 −0.788989 0.614408i \(-0.789395\pi\)
−0.788989 + 0.614408i \(0.789395\pi\)
\(702\) 0 0
\(703\) 13.2614 0.500162
\(704\) 0 0
\(705\) −1.42991 + 1.57063i −0.0538536 + 0.0591535i
\(706\) 0 0
\(707\) 28.3234 1.06521
\(708\) 0 0
\(709\) 28.9270i 1.08637i −0.839612 0.543187i \(-0.817217\pi\)
0.839612 0.543187i \(-0.182783\pi\)
\(710\) 0 0
\(711\) 5.09271 0.478739i 0.190991 0.0179541i
\(712\) 0 0
\(713\) −8.83841 + 15.3019i −0.331001 + 0.573059i
\(714\) 0 0
\(715\) 10.2462 0.383187
\(716\) 0 0
\(717\) 16.6501 18.2887i 0.621809 0.683004i
\(718\) 0 0
\(719\) 9.90237i 0.369296i −0.982805 0.184648i \(-0.940885\pi\)
0.982805 0.184648i \(-0.0591145\pi\)
\(720\) 0 0
\(721\) −45.3693 −1.68964
\(722\) 0 0
\(723\) 11.6153 12.7584i 0.431978 0.474491i
\(724\) 0 0
\(725\) 24.6300i 0.914737i
\(726\) 0 0
\(727\) 31.5895i 1.17159i −0.810460 0.585795i \(-0.800783\pi\)
0.810460 0.585795i \(-0.199217\pi\)
\(728\) 0 0
\(729\) −7.49242 25.9396i −0.277497 0.960726i
\(730\) 0 0
\(731\) 46.1927i 1.70850i
\(732\) 0 0
\(733\) 53.2175i 1.96563i −0.184585 0.982817i \(-0.559094\pi\)
0.184585 0.982817i \(-0.440906\pi\)
\(734\) 0 0
\(735\) −9.74247 8.86958i −0.359356 0.327159i
\(736\) 0 0
\(737\) 49.8343i 1.83567i
\(738\) 0 0
\(739\) −12.8078 −0.471141 −0.235571 0.971857i \(-0.575696\pi\)
−0.235571 + 0.971857i \(0.575696\pi\)
\(740\) 0 0
\(741\) −18.1379 + 19.9230i −0.666314 + 0.731888i
\(742\) 0 0
\(743\) −19.4849 −0.714833 −0.357417 0.933945i \(-0.616342\pi\)
−0.357417 + 0.933945i \(0.616342\pi\)
\(744\) 0 0
\(745\) 8.38447 0.307183
\(746\) 0 0
\(747\) 0.147647 + 1.57063i 0.00540212 + 0.0574665i
\(748\) 0 0
\(749\) 7.28323i 0.266123i
\(750\) 0 0
\(751\) 6.07263i 0.221594i 0.993843 + 0.110797i \(0.0353403\pi\)
−0.993843 + 0.110797i \(0.964660\pi\)
\(752\) 0 0
\(753\) 32.1520 + 29.2713i 1.17168 + 1.06670i
\(754\) 0 0
\(755\) 13.6358 0.496258
\(756\) 0 0
\(757\) 33.2945i 1.21011i −0.796183 0.605055i \(-0.793151\pi\)
0.796183 0.605055i \(-0.206849\pi\)
\(758\) 0 0
\(759\) −33.8074 + 10.7726i −1.22713 + 0.391020i
\(760\) 0 0
\(761\) 45.9056i 1.66408i 0.554719 + 0.832038i \(0.312826\pi\)
−0.554719 + 0.832038i \(0.687174\pi\)
\(762\) 0 0
\(763\) −30.2462 −1.09499
\(764\) 0 0
\(765\) −2.00000 21.2755i −0.0723102 0.769218i
\(766\) 0 0
\(767\) 11.9473i 0.431393i
\(768\) 0 0
\(769\) 21.1493i 0.762662i −0.924439 0.381331i \(-0.875466\pi\)
0.924439 0.381331i \(-0.124534\pi\)
\(770\) 0 0
\(771\) 15.4579 16.9791i 0.556701 0.611488i
\(772\) 0 0
\(773\) 50.0270 1.79935 0.899673 0.436565i \(-0.143805\pi\)
0.899673 + 0.436565i \(0.143805\pi\)
\(774\) 0 0
\(775\) −15.1922 −0.545721
\(776\) 0 0
\(777\) −10.8769 9.90237i −0.390206 0.355245i
\(778\) 0 0
\(779\) 64.5992 2.31451
\(780\) 0 0
\(781\) 9.96148i 0.356450i
\(782\) 0 0
\(783\) 24.8078 + 18.6564i 0.886557 + 0.666725i
\(784\) 0 0
\(785\) 11.3732i 0.405925i
\(786\) 0 0
\(787\) 12.8928i 0.459580i −0.973240 0.229790i \(-0.926196\pi\)
0.973240 0.229790i \(-0.0738040\pi\)
\(788\) 0 0
\(789\) 28.0281 + 25.5169i 0.997825 + 0.908424i
\(790\) 0 0
\(791\) 22.2980i 0.792826i
\(792\) 0 0
\(793\) 19.9230i 0.707485i
\(794\) 0 0
\(795\) −9.75379 8.87989i −0.345931 0.314937i
\(796\) 0 0
\(797\) −19.3697 −0.686109 −0.343054 0.939316i \(-0.611461\pi\)
−0.343054 + 0.939316i \(0.611461\pi\)
\(798\) 0 0
\(799\) 9.96148i 0.352412i
\(800\) 0 0
\(801\) 0.410574 + 4.36758i 0.0145069 + 0.154321i
\(802\) 0 0
\(803\) −24.2824 −0.856906
\(804\) 0 0
\(805\) 15.1231 + 8.73516i 0.533019 + 0.307874i
\(806\) 0 0
\(807\) 15.4579 16.9791i 0.544142 0.597693i
\(808\) 0 0
\(809\) 5.23827i 0.184168i 0.995751 + 0.0920839i \(0.0293528\pi\)
−0.995751 + 0.0920839i \(0.970647\pi\)
\(810\) 0 0
\(811\) 39.6847 1.39352 0.696758 0.717306i \(-0.254625\pi\)
0.696758 + 0.717306i \(0.254625\pi\)
\(812\) 0 0
\(813\) 18.2462 + 16.6114i 0.639923 + 0.582588i
\(814\) 0 0
\(815\) −1.34700 −0.0471833
\(816\) 0 0
\(817\) 36.8769 1.29016
\(818\) 0 0
\(819\) 29.7533 2.79695i 1.03966 0.0977333i
\(820\) 0 0
\(821\) 44.5960i 1.55641i 0.628010 + 0.778206i \(0.283870\pi\)
−0.628010 + 0.778206i \(0.716130\pi\)
\(822\) 0 0
\(823\) −22.5616 −0.786446 −0.393223 0.919443i \(-0.628640\pi\)
−0.393223 + 0.919443i \(0.628640\pi\)
\(824\) 0 0
\(825\) −22.5571 20.5361i −0.785339 0.714976i
\(826\) 0 0
\(827\) 6.67026 0.231948 0.115974 0.993252i \(-0.463001\pi\)
0.115974 + 0.993252i \(0.463001\pi\)
\(828\) 0 0
\(829\) −47.8617 −1.66231 −0.831153 0.556043i \(-0.812319\pi\)
−0.831153 + 0.556043i \(0.812319\pi\)
\(830\) 0 0
\(831\) −4.40388 4.00931i −0.152769 0.139081i
\(832\) 0 0
\(833\) 61.7899 2.14089
\(834\) 0 0
\(835\) 0.957477i 0.0331349i
\(836\) 0 0
\(837\) −11.5076 + 15.3019i −0.397760 + 0.528909i
\(838\) 0 0
\(839\) −6.14441 −0.212129 −0.106064 0.994359i \(-0.533825\pi\)
−0.106064 + 0.994359i \(0.533825\pi\)
\(840\) 0 0
\(841\) −6.68466 −0.230505
\(842\) 0 0
\(843\) −30.2791 27.5662i −1.04287 0.949431i
\(844\) 0 0
\(845\) −6.02913 −0.207408
\(846\) 0 0
\(847\) 28.1794i 0.968255i
\(848\) 0 0
\(849\) −8.51075 + 9.34832i −0.292088 + 0.320834i
\(850\) 0 0
\(851\) 9.06897 + 5.23827i 0.310880 + 0.179566i
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) −16.9848 + 1.59666i −0.580869 + 0.0546045i
\(856\) 0 0
\(857\) 15.3019i 0.522701i −0.965244 0.261351i \(-0.915832\pi\)
0.965244 0.261351i \(-0.0841679\pi\)
\(858\) 0 0
\(859\) −0.946025 −0.0322779 −0.0161390 0.999870i \(-0.505137\pi\)
−0.0161390 + 0.999870i \(0.505137\pi\)
\(860\) 0 0
\(861\) −52.9839 48.2368i −1.80569 1.64390i
\(862\) 0 0
\(863\) 50.5697i 1.72141i −0.509103 0.860706i \(-0.670023\pi\)
0.509103 0.860706i \(-0.329977\pi\)
\(864\) 0 0
\(865\) 17.4703i 0.594009i
\(866\) 0 0
\(867\) 52.3348 + 47.6458i 1.77738 + 1.61814i
\(868\) 0 0
\(869\) 7.28323i 0.247067i
\(870\) 0 0
\(871\) 29.8844i 1.01260i
\(872\) 0 0
\(873\) −29.7533 + 2.79695i −1.00700 + 0.0946624i
\(874\) 0 0
\(875\) 33.2228i 1.12314i
\(876\) 0 0
\(877\) 10.9848 0.370932 0.185466 0.982651i \(-0.440621\pi\)
0.185466 + 0.982651i \(0.440621\pi\)
\(878\) 0 0
\(879\) 30.9526 + 28.1794i 1.04401 + 0.950467i
\(880\) 0 0
\(881\) 28.6692 0.965889 0.482945 0.875651i \(-0.339567\pi\)
0.482945 + 0.875651i \(0.339567\pi\)
\(882\) 0 0
\(883\) 7.50758 0.252650 0.126325 0.991989i \(-0.459682\pi\)
0.126325 + 0.991989i \(0.459682\pi\)
\(884\) 0 0
\(885\) 5.09271 5.59390i 0.171189 0.188037i
\(886\) 0 0
\(887\) 43.8606i 1.47270i 0.676604 + 0.736348i \(0.263451\pi\)
−0.676604 + 0.736348i \(0.736549\pi\)
\(888\) 0 0
\(889\) 29.8844i 1.00229i
\(890\) 0 0
\(891\) −37.7705 + 7.16453i −1.26536 + 0.240021i
\(892\) 0 0
\(893\) −7.95253 −0.266121
\(894\) 0 0
\(895\) 7.50885i 0.250993i
\(896\) 0 0
\(897\) −20.2735 + 6.46006i −0.676912 + 0.215695i
\(898\) 0 0
\(899\) 22.0109i 0.734105i
\(900\) 0 0
\(901\) 61.8617 2.06091
\(902\) 0 0
\(903\) −30.2462 27.5363i −1.00653 0.916350i
\(904\) 0 0
\(905\) 20.7013i 0.688136i
\(906\) 0 0
\(907\) 38.1409i 1.26645i −0.773969 0.633223i \(-0.781731\pi\)
0.773969 0.633223i \(-0.218269\pi\)
\(908\) 0 0
\(909\) −21.7538 + 2.04496i −0.721528 + 0.0678270i
\(910\) 0 0
\(911\) −35.2242 −1.16703 −0.583514 0.812103i \(-0.698323\pi\)
−0.583514 + 0.812103i \(0.698323\pi\)
\(912\) 0 0
\(913\) 2.24621 0.0743387
\(914\) 0 0
\(915\) −8.49242 + 9.32819i −0.280751 + 0.308380i
\(916\) 0 0
\(917\) 37.3923 1.23480
\(918\) 0 0
\(919\) 47.1449i 1.55517i 0.628781 + 0.777583i \(0.283554\pi\)
−0.628781 + 0.777583i \(0.716446\pi\)
\(920\) 0 0
\(921\) −7.36932 6.70906i −0.242827 0.221071i
\(922\) 0 0
\(923\) 5.97366i 0.196626i
\(924\) 0 0
\(925\) 9.00400i 0.296050i
\(926\) 0 0
\(927\) 34.8460 3.27569i 1.14449 0.107588i
\(928\) 0 0
\(929\) 22.5851i 0.740993i 0.928834 + 0.370496i \(0.120812\pi\)
−0.928834 + 0.370496i \(0.879188\pi\)
\(930\) 0 0
\(931\) 49.3287i 1.61668i
\(932\) 0 0
\(933\) −12.4039 + 13.6246i −0.406085 + 0.446049i
\(934\) 0 0
\(935\) −30.4268 −0.995062
\(936\) 0 0
\(937\) 16.7817i 0.548234i −0.961696 0.274117i \(-0.911614\pi\)
0.961696 0.274117i \(-0.0883855\pi\)
\(938\) 0 0
\(939\) −5.09271 + 5.59390i −0.166194 + 0.182550i
\(940\) 0 0
\(941\) −5.50328 −0.179402 −0.0897009 0.995969i \(-0.528591\pi\)
−0.0897009 + 0.995969i \(0.528591\pi\)
\(942\) 0 0
\(943\) 44.1771 + 25.5169i 1.43860 + 0.830943i
\(944\) 0 0
\(945\) 15.1231 + 11.3732i 0.491955 + 0.369969i
\(946\) 0 0
\(947\) 13.2569i 0.430791i −0.976527 0.215396i \(-0.930896\pi\)
0.976527 0.215396i \(-0.0691041\pi\)
\(948\) 0 0
\(949\) −14.5616 −0.472688
\(950\) 0 0
\(951\) 21.7538 23.8947i 0.705415 0.774837i
\(952\) 0 0
\(953\) 23.3459 0.756249 0.378124 0.925755i \(-0.376569\pi\)
0.378124 + 0.925755i \(0.376569\pi\)
\(954\) 0 0
\(955\) 19.2311 0.622302
\(956\) 0 0
\(957\) 29.7533 32.6814i 0.961786 1.05644i
\(958\) 0 0
\(959\) 46.1927i 1.49164i
\(960\) 0 0
\(961\) −17.4233 −0.562042
\(962\) 0 0
\(963\) 0.525853 + 5.59390i 0.0169454 + 0.180261i
\(964\) 0 0
\(965\) 17.3815 0.559531
\(966\) 0 0
\(967\) −57.1619 −1.83820 −0.919102 0.394020i \(-0.871084\pi\)
−0.919102 + 0.394020i \(0.871084\pi\)
\(968\) 0 0
\(969\) 53.8617 59.1625i 1.73029 1.90057i
\(970\) 0 0
\(971\) 9.36426 0.300514 0.150257 0.988647i \(-0.451990\pi\)
0.150257 + 0.988647i \(0.451990\pi\)
\(972\) 0 0
\(973\) 54.1750i 1.73677i
\(974\) 0 0
\(975\) −13.5270 12.3150i −0.433210 0.394396i
\(976\) 0 0
\(977\) −5.73384 −0.183442 −0.0917209 0.995785i \(-0.529237\pi\)
−0.0917209 + 0.995785i \(0.529237\pi\)
\(978\) 0 0
\(979\) 6.24621 0.199630
\(980\) 0 0
\(981\) 23.2306 2.18379i 0.741697 0.0697231i
\(982\) 0 0
\(983\) −24.5776 −0.783905 −0.391953 0.919985i \(-0.628200\pi\)
−0.391953 + 0.919985i \(0.628200\pi\)
\(984\) 0 0
\(985\) 18.6966i 0.595725i
\(986\) 0 0
\(987\) 6.52262 + 5.93822i 0.207617 + 0.189016i
\(988\) 0 0
\(989\) 25.2188 + 14.5665i 0.801910 + 0.463187i
\(990\) 0 0
\(991\) −12.4924 −0.396835 −0.198417 0.980118i \(-0.563580\pi\)
−0.198417 + 0.980118i \(0.563580\pi\)
\(992\) 0 0
\(993\) 16.4039 + 14.9342i 0.520561 + 0.473921i
\(994\) 0 0
\(995\) 3.64162i 0.115447i
\(996\) 0 0
\(997\) −22.8769 −0.724519 −0.362259 0.932077i \(-0.617994\pi\)
−0.362259 + 0.932077i \(0.617994\pi\)
\(998\) 0 0
\(999\) 9.06897 + 6.82021i 0.286930 + 0.215782i
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 276.2.g.a.137.8 yes 8
3.2 odd 2 inner 276.2.g.a.137.5 8
4.3 odd 2 1104.2.m.b.689.2 8
12.11 even 2 1104.2.m.b.689.3 8
23.22 odd 2 inner 276.2.g.a.137.7 yes 8
69.68 even 2 inner 276.2.g.a.137.6 yes 8
92.91 even 2 1104.2.m.b.689.1 8
276.275 odd 2 1104.2.m.b.689.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
276.2.g.a.137.5 8 3.2 odd 2 inner
276.2.g.a.137.6 yes 8 69.68 even 2 inner
276.2.g.a.137.7 yes 8 23.22 odd 2 inner
276.2.g.a.137.8 yes 8 1.1 even 1 trivial
1104.2.m.b.689.1 8 92.91 even 2
1104.2.m.b.689.2 8 4.3 odd 2
1104.2.m.b.689.3 8 12.11 even 2
1104.2.m.b.689.4 8 276.275 odd 2