Properties

Label 768.2.c.a.767.1
Level $768$
Weight $2$
Character 768.767
Analytic conductor $6.133$
Analytic rank $1$
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] = [768,2,Mod(767,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.767");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 767.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 768.767
Dual form 768.2.c.a.767.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.41421i) q^{3} -2.82843i q^{5} +2.82843i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.41421i) q^{3} -2.82843i q^{5} +2.82843i q^{7} +(-1.00000 + 2.82843i) q^{9} -2.00000 q^{11} -4.00000 q^{13} +(-4.00000 + 2.82843i) q^{15} -5.65685i q^{17} +2.82843i q^{19} +(4.00000 - 2.82843i) q^{21} -8.00000 q^{23} -3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} +2.82843i q^{29} +8.48528i q^{31} +(2.00000 + 2.82843i) q^{33} +8.00000 q^{35} -4.00000 q^{37} +(4.00000 + 5.65685i) q^{39} +2.82843i q^{43} +(8.00000 + 2.82843i) q^{45} -1.00000 q^{49} +(-8.00000 + 5.65685i) q^{51} +8.48528i q^{53} +5.65685i q^{55} +(4.00000 - 2.82843i) q^{57} -6.00000 q^{59} -4.00000 q^{61} +(-8.00000 - 2.82843i) q^{63} +11.3137i q^{65} -14.1421i q^{67} +(8.00000 + 11.3137i) q^{69} -8.00000 q^{71} +10.0000 q^{73} +(3.00000 + 4.24264i) q^{75} -5.65685i q^{77} -2.82843i q^{79} +(-7.00000 - 5.65685i) q^{81} -6.00000 q^{83} -16.0000 q^{85} +(4.00000 - 2.82843i) q^{87} -5.65685i q^{89} -11.3137i q^{91} +(12.0000 - 8.48528i) q^{93} +8.00000 q^{95} -6.00000 q^{97} +(2.00000 - 5.65685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{9} - 4 q^{11} - 8 q^{13} - 8 q^{15} + 8 q^{21} - 16 q^{23} - 6 q^{25} + 10 q^{27} + 4 q^{33} + 16 q^{35} - 8 q^{37} + 8 q^{39} + 16 q^{45} - 2 q^{49} - 16 q^{51} + 8 q^{57} - 12 q^{59} - 8 q^{61} - 16 q^{63} + 16 q^{69} - 16 q^{71} + 20 q^{73} + 6 q^{75} - 14 q^{81} - 12 q^{83} - 32 q^{85} + 8 q^{87} + 24 q^{93} + 16 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.00000 1.41421i −0.577350 0.816497i
\(4\) 0 0
\(5\) 2.82843i 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −4.00000 + 2.82843i −1.03280 + 0.730297i
\(16\) 0 0
\(17\) 5.65685i 1.37199i −0.727607 0.685994i \(-0.759367\pi\)
0.727607 0.685994i \(-0.240633\pi\)
\(18\) 0 0
\(19\) 2.82843i 0.648886i 0.945905 + 0.324443i \(0.105177\pi\)
−0.945905 + 0.324443i \(0.894823\pi\)
\(20\) 0 0
\(21\) 4.00000 2.82843i 0.872872 0.617213i
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 5.00000 1.41421i 0.962250 0.272166i
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 0 0
\(33\) 2.00000 + 2.82843i 0.348155 + 0.492366i
\(34\) 0 0
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 4.00000 + 5.65685i 0.640513 + 0.905822i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 2.82843i 0.431331i 0.976467 + 0.215666i \(0.0691921\pi\)
−0.976467 + 0.215666i \(0.930808\pi\)
\(44\) 0 0
\(45\) 8.00000 + 2.82843i 1.19257 + 0.421637i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −8.00000 + 5.65685i −1.12022 + 0.792118i
\(52\) 0 0
\(53\) 8.48528i 1.16554i 0.812636 + 0.582772i \(0.198032\pi\)
−0.812636 + 0.582772i \(0.801968\pi\)
\(54\) 0 0
\(55\) 5.65685i 0.762770i
\(56\) 0 0
\(57\) 4.00000 2.82843i 0.529813 0.374634i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) −8.00000 2.82843i −1.00791 0.356348i
\(64\) 0 0
\(65\) 11.3137i 1.40329i
\(66\) 0 0
\(67\) 14.1421i 1.72774i −0.503718 0.863868i \(-0.668035\pi\)
0.503718 0.863868i \(-0.331965\pi\)
\(68\) 0 0
\(69\) 8.00000 + 11.3137i 0.963087 + 1.36201i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 3.00000 + 4.24264i 0.346410 + 0.489898i
\(76\) 0 0
\(77\) 5.65685i 0.644658i
\(78\) 0 0
\(79\) 2.82843i 0.318223i −0.987261 0.159111i \(-0.949137\pi\)
0.987261 0.159111i \(-0.0508629\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) 4.00000 2.82843i 0.428845 0.303239i
\(88\) 0 0
\(89\) 5.65685i 0.599625i −0.953998 0.299813i \(-0.903076\pi\)
0.953998 0.299813i \(-0.0969242\pi\)
\(90\) 0 0
\(91\) 11.3137i 1.18600i
\(92\) 0 0
\(93\) 12.0000 8.48528i 1.24434 0.879883i
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 2.00000 5.65685i 0.201008 0.568535i
\(100\) 0 0
\(101\) 2.82843i 0.281439i −0.990050 0.140720i \(-0.955058\pi\)
0.990050 0.140720i \(-0.0449416\pi\)
\(102\) 0 0
\(103\) 8.48528i 0.836080i −0.908429 0.418040i \(-0.862717\pi\)
0.908429 0.418040i \(-0.137283\pi\)
\(104\) 0 0
\(105\) −8.00000 11.3137i −0.780720 1.10410i
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 4.00000 + 5.65685i 0.379663 + 0.536925i
\(112\) 0 0
\(113\) 11.3137i 1.06430i 0.846649 + 0.532152i \(0.178617\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 22.6274i 2.11002i
\(116\) 0 0
\(117\) 4.00000 11.3137i 0.369800 1.04595i
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) 14.1421i 1.25491i −0.778652 0.627456i \(-0.784096\pi\)
0.778652 0.627456i \(-0.215904\pi\)
\(128\) 0 0
\(129\) 4.00000 2.82843i 0.352180 0.249029i
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) −4.00000 14.1421i −0.344265 1.21716i
\(136\) 0 0
\(137\) 11.3137i 0.966595i 0.875456 + 0.483298i \(0.160561\pi\)
−0.875456 + 0.483298i \(0.839439\pi\)
\(138\) 0 0
\(139\) 8.48528i 0.719712i 0.933008 + 0.359856i \(0.117174\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 1.00000 + 1.41421i 0.0824786 + 0.116642i
\(148\) 0 0
\(149\) 2.82843i 0.231714i −0.993266 0.115857i \(-0.963039\pi\)
0.993266 0.115857i \(-0.0369614\pi\)
\(150\) 0 0
\(151\) 8.48528i 0.690522i −0.938507 0.345261i \(-0.887790\pi\)
0.938507 0.345261i \(-0.112210\pi\)
\(152\) 0 0
\(153\) 16.0000 + 5.65685i 1.29352 + 0.457330i
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 12.0000 8.48528i 0.951662 0.672927i
\(160\) 0 0
\(161\) 22.6274i 1.78329i
\(162\) 0 0
\(163\) 8.48528i 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) 8.00000 5.65685i 0.622799 0.440386i
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −8.00000 2.82843i −0.611775 0.216295i
\(172\) 0 0
\(173\) 2.82843i 0.215041i 0.994203 + 0.107521i \(0.0342912\pi\)
−0.994203 + 0.107521i \(0.965709\pi\)
\(174\) 0 0
\(175\) 8.48528i 0.641427i
\(176\) 0 0
\(177\) 6.00000 + 8.48528i 0.450988 + 0.637793i
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 4.00000 + 5.65685i 0.295689 + 0.418167i
\(184\) 0 0
\(185\) 11.3137i 0.831800i
\(186\) 0 0
\(187\) 11.3137i 0.827340i
\(188\) 0 0
\(189\) 4.00000 + 14.1421i 0.290957 + 1.02869i
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 16.0000 11.3137i 1.14578 0.810191i
\(196\) 0 0
\(197\) 14.1421i 1.00759i −0.863825 0.503793i \(-0.831938\pi\)
0.863825 0.503793i \(-0.168062\pi\)
\(198\) 0 0
\(199\) 8.48528i 0.601506i −0.953702 0.300753i \(-0.902762\pi\)
0.953702 0.300753i \(-0.0972379\pi\)
\(200\) 0 0
\(201\) −20.0000 + 14.1421i −1.41069 + 0.997509i
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.00000 22.6274i 0.556038 1.57271i
\(208\) 0 0
\(209\) 5.65685i 0.391293i
\(210\) 0 0
\(211\) 2.82843i 0.194717i −0.995249 0.0973585i \(-0.968961\pi\)
0.995249 0.0973585i \(-0.0310393\pi\)
\(212\) 0 0
\(213\) 8.00000 + 11.3137i 0.548151 + 0.775203i
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) −10.0000 14.1421i −0.675737 0.955637i
\(220\) 0 0
\(221\) 22.6274i 1.52208i
\(222\) 0 0
\(223\) 2.82843i 0.189405i −0.995506 0.0947027i \(-0.969810\pi\)
0.995506 0.0947027i \(-0.0301901\pi\)
\(224\) 0 0
\(225\) 3.00000 8.48528i 0.200000 0.565685i
\(226\) 0 0
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −8.00000 + 5.65685i −0.526361 + 0.372194i
\(232\) 0 0
\(233\) 28.2843i 1.85296i −0.376339 0.926482i \(-0.622817\pi\)
0.376339 0.926482i \(-0.377183\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 + 2.82843i −0.259828 + 0.183726i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −1.00000 + 15.5563i −0.0641500 + 0.997940i
\(244\) 0 0
\(245\) 2.82843i 0.180702i
\(246\) 0 0
\(247\) 11.3137i 0.719874i
\(248\) 0 0
\(249\) 6.00000 + 8.48528i 0.380235 + 0.537733i
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 16.0000 + 22.6274i 1.00196 + 1.41698i
\(256\) 0 0
\(257\) 11.3137i 0.705730i −0.935674 0.352865i \(-0.885208\pi\)
0.935674 0.352865i \(-0.114792\pi\)
\(258\) 0 0
\(259\) 11.3137i 0.703000i
\(260\) 0 0
\(261\) −8.00000 2.82843i −0.495188 0.175075i
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) −8.00000 + 5.65685i −0.489592 + 0.346194i
\(268\) 0 0
\(269\) 14.1421i 0.862261i 0.902290 + 0.431131i \(0.141885\pi\)
−0.902290 + 0.431131i \(0.858115\pi\)
\(270\) 0 0
\(271\) 19.7990i 1.20270i 0.798985 + 0.601351i \(0.205371\pi\)
−0.798985 + 0.601351i \(0.794629\pi\)
\(272\) 0 0
\(273\) −16.0000 + 11.3137i −0.968364 + 0.684737i
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) −24.0000 8.48528i −1.43684 0.508001i
\(280\) 0 0
\(281\) 5.65685i 0.337460i 0.985662 + 0.168730i \(0.0539665\pi\)
−0.985662 + 0.168730i \(0.946033\pi\)
\(282\) 0 0
\(283\) 25.4558i 1.51319i −0.653882 0.756596i \(-0.726861\pi\)
0.653882 0.756596i \(-0.273139\pi\)
\(284\) 0 0
\(285\) −8.00000 11.3137i −0.473879 0.670166i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 6.00000 + 8.48528i 0.351726 + 0.497416i
\(292\) 0 0
\(293\) 31.1127i 1.81762i 0.417207 + 0.908812i \(0.363009\pi\)
−0.417207 + 0.908812i \(0.636991\pi\)
\(294\) 0 0
\(295\) 16.9706i 0.988064i
\(296\) 0 0
\(297\) −10.0000 + 2.82843i −0.580259 + 0.164122i
\(298\) 0 0
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −4.00000 + 2.82843i −0.229794 + 0.162489i
\(304\) 0 0
\(305\) 11.3137i 0.647821i
\(306\) 0 0
\(307\) 8.48528i 0.484281i 0.970241 + 0.242140i \(0.0778494\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) −12.0000 + 8.48528i −0.682656 + 0.482711i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) −8.00000 + 22.6274i −0.450749 + 1.27491i
\(316\) 0 0
\(317\) 31.1127i 1.74746i −0.486408 0.873732i \(-0.661693\pi\)
0.486408 0.873732i \(-0.338307\pi\)
\(318\) 0 0
\(319\) 5.65685i 0.316723i
\(320\) 0 0
\(321\) 6.00000 + 8.48528i 0.334887 + 0.473602i
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) 12.0000 0.665640
\(326\) 0 0
\(327\) 4.00000 + 5.65685i 0.221201 + 0.312825i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.82843i 0.155464i −0.996974 0.0777322i \(-0.975232\pi\)
0.996974 0.0777322i \(-0.0247679\pi\)
\(332\) 0 0
\(333\) 4.00000 11.3137i 0.219199 0.619987i
\(334\) 0 0
\(335\) −40.0000 −2.18543
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) 16.0000 11.3137i 0.869001 0.614476i
\(340\) 0 0
\(341\) 16.9706i 0.919007i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 32.0000 22.6274i 1.72282 1.21822i
\(346\) 0 0
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −20.0000 + 5.65685i −1.06752 + 0.301941i
\(352\) 0 0
\(353\) 22.6274i 1.20434i 0.798369 + 0.602168i \(0.205696\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 22.6274i 1.20094i
\(356\) 0 0
\(357\) −16.0000 22.6274i −0.846810 1.19757i
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 11.0000 0.578947
\(362\) 0 0
\(363\) 7.00000 + 9.89949i 0.367405 + 0.519589i
\(364\) 0 0
\(365\) 28.2843i 1.48047i
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) −36.0000 −1.86401 −0.932005 0.362446i \(-0.881942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) −8.00000 + 5.65685i −0.413118 + 0.292119i
\(376\) 0 0
\(377\) 11.3137i 0.582686i
\(378\) 0 0
\(379\) 25.4558i 1.30758i 0.756677 + 0.653789i \(0.226822\pi\)
−0.756677 + 0.653789i \(0.773178\pi\)
\(380\) 0 0
\(381\) −20.0000 + 14.1421i −1.02463 + 0.724524i
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) −8.00000 2.82843i −0.406663 0.143777i
\(388\) 0 0
\(389\) 8.48528i 0.430221i 0.976590 + 0.215110i \(0.0690111\pi\)
−0.976590 + 0.215110i \(0.930989\pi\)
\(390\) 0 0
\(391\) 45.2548i 2.28864i
\(392\) 0 0
\(393\) −14.0000 19.7990i −0.706207 0.998727i
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 0 0
\(399\) 8.00000 + 11.3137i 0.400501 + 0.566394i
\(400\) 0 0
\(401\) 5.65685i 0.282490i 0.989975 + 0.141245i \(0.0451105\pi\)
−0.989975 + 0.141245i \(0.954889\pi\)
\(402\) 0 0
\(403\) 33.9411i 1.69073i
\(404\) 0 0
\(405\) −16.0000 + 19.7990i −0.795046 + 0.983820i
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 16.0000 11.3137i 0.789222 0.558064i
\(412\) 0 0
\(413\) 16.9706i 0.835067i
\(414\) 0 0
\(415\) 16.9706i 0.833052i
\(416\) 0 0
\(417\) 12.0000 8.48528i 0.587643 0.415526i
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.9706i 0.823193i
\(426\) 0 0
\(427\) 11.3137i 0.547509i
\(428\) 0 0
\(429\) −8.00000 11.3137i −0.386244 0.546231i
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) −8.00000 11.3137i −0.383571 0.542451i
\(436\) 0 0
\(437\) 22.6274i 1.08242i
\(438\) 0 0
\(439\) 8.48528i 0.404980i −0.979284 0.202490i \(-0.935097\pi\)
0.979284 0.202490i \(-0.0649034\pi\)
\(440\) 0 0
\(441\) 1.00000 2.82843i 0.0476190 0.134687i
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) −16.0000 −0.758473
\(446\) 0 0
\(447\) −4.00000 + 2.82843i −0.189194 + 0.133780i
\(448\) 0 0
\(449\) 16.9706i 0.800890i 0.916321 + 0.400445i \(0.131145\pi\)
−0.916321 + 0.400445i \(0.868855\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −12.0000 + 8.48528i −0.563809 + 0.398673i
\(454\) 0 0
\(455\) −32.0000 −1.50018
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) −8.00000 28.2843i −0.373408 1.32020i
\(460\) 0 0
\(461\) 25.4558i 1.18560i 0.805351 + 0.592798i \(0.201977\pi\)
−0.805351 + 0.592798i \(0.798023\pi\)
\(462\) 0 0
\(463\) 8.48528i 0.394344i 0.980369 + 0.197172i \(0.0631758\pi\)
−0.980369 + 0.197172i \(0.936824\pi\)
\(464\) 0 0
\(465\) −24.0000 33.9411i −1.11297 1.57398i
\(466\) 0 0
\(467\) 42.0000 1.94353 0.971764 0.235954i \(-0.0758216\pi\)
0.971764 + 0.235954i \(0.0758216\pi\)
\(468\) 0 0
\(469\) 40.0000 1.84703
\(470\) 0 0
\(471\) 20.0000 + 28.2843i 0.921551 + 1.30327i
\(472\) 0 0
\(473\) 5.65685i 0.260102i
\(474\) 0 0
\(475\) 8.48528i 0.389331i
\(476\) 0 0
\(477\) −24.0000 8.48528i −1.09888 0.388514i
\(478\) 0 0
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) −32.0000 + 22.6274i −1.45605 + 1.02958i
\(484\) 0 0
\(485\) 16.9706i 0.770594i
\(486\) 0 0
\(487\) 42.4264i 1.92252i −0.275636 0.961262i \(-0.588889\pi\)
0.275636 0.961262i \(-0.411111\pi\)
\(488\) 0 0
\(489\) −12.0000 + 8.48528i −0.542659 + 0.383718i
\(490\) 0 0
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) −16.0000 5.65685i −0.719147 0.254257i
\(496\) 0 0
\(497\) 22.6274i 1.01498i
\(498\) 0 0
\(499\) 19.7990i 0.886325i 0.896441 + 0.443162i \(0.146143\pi\)
−0.896441 + 0.443162i \(0.853857\pi\)
\(500\) 0 0
\(501\) −8.00000 11.3137i −0.357414 0.505459i
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) −3.00000 4.24264i −0.133235 0.188422i
\(508\) 0 0
\(509\) 25.4558i 1.12831i 0.825669 + 0.564155i \(0.190798\pi\)
−0.825669 + 0.564155i \(0.809202\pi\)
\(510\) 0 0
\(511\) 28.2843i 1.25122i
\(512\) 0 0
\(513\) 4.00000 + 14.1421i 0.176604 + 0.624391i
\(514\) 0 0
\(515\) −24.0000 −1.05757
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 4.00000 2.82843i 0.175581 0.124154i
\(520\) 0 0
\(521\) 33.9411i 1.48699i 0.668743 + 0.743494i \(0.266833\pi\)
−0.668743 + 0.743494i \(0.733167\pi\)
\(522\) 0 0
\(523\) 14.1421i 0.618392i 0.950998 + 0.309196i \(0.100060\pi\)
−0.950998 + 0.309196i \(0.899940\pi\)
\(524\) 0 0
\(525\) −12.0000 + 8.48528i −0.523723 + 0.370328i
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 6.00000 16.9706i 0.260378 0.736460i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 16.9706i 0.733701i
\(536\) 0 0
\(537\) 18.0000 + 25.4558i 0.776757 + 1.09850i
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 28.0000 1.20381 0.601907 0.798566i \(-0.294408\pi\)
0.601907 + 0.798566i \(0.294408\pi\)
\(542\) 0 0
\(543\) −12.0000 16.9706i −0.514969 0.728277i
\(544\) 0 0
\(545\) 11.3137i 0.484626i
\(546\) 0 0
\(547\) 25.4558i 1.08841i 0.838951 + 0.544207i \(0.183169\pi\)
−0.838951 + 0.544207i \(0.816831\pi\)
\(548\) 0 0
\(549\) 4.00000 11.3137i 0.170716 0.482857i
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 16.0000 11.3137i 0.679162 0.480240i
\(556\) 0 0
\(557\) 36.7696i 1.55798i 0.627039 + 0.778988i \(0.284267\pi\)
−0.627039 + 0.778988i \(0.715733\pi\)
\(558\) 0 0
\(559\) 11.3137i 0.478519i
\(560\) 0 0
\(561\) 16.0000 11.3137i 0.675521 0.477665i
\(562\) 0 0
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) 32.0000 1.34625
\(566\) 0 0
\(567\) 16.0000 19.7990i 0.671937 0.831479i
\(568\) 0 0
\(569\) 11.3137i 0.474295i −0.971474 0.237148i \(-0.923787\pi\)
0.971474 0.237148i \(-0.0762125\pi\)
\(570\) 0 0
\(571\) 42.4264i 1.77549i 0.460336 + 0.887745i \(0.347729\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 16.0000 + 22.6274i 0.668410 + 0.945274i
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) 2.00000 + 2.82843i 0.0831172 + 0.117545i
\(580\) 0 0
\(581\) 16.9706i 0.704058i
\(582\) 0 0
\(583\) 16.9706i 0.702849i
\(584\) 0 0
\(585\) −32.0000 11.3137i −1.32304 0.467764i
\(586\) 0 0
\(587\) 26.0000 1.07313 0.536567 0.843857i \(-0.319721\pi\)
0.536567 + 0.843857i \(0.319721\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −20.0000 + 14.1421i −0.822690 + 0.581730i
\(592\) 0 0
\(593\) 22.6274i 0.929197i −0.885522 0.464598i \(-0.846199\pi\)
0.885522 0.464598i \(-0.153801\pi\)
\(594\) 0 0
\(595\) 45.2548i 1.85527i
\(596\) 0 0
\(597\) −12.0000 + 8.48528i −0.491127 + 0.347279i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 40.0000 + 14.1421i 1.62893 + 0.575912i
\(604\) 0 0
\(605\) 19.7990i 0.804943i
\(606\) 0 0
\(607\) 36.7696i 1.49243i −0.665705 0.746215i \(-0.731869\pi\)
0.665705 0.746215i \(-0.268131\pi\)
\(608\) 0 0
\(609\) 8.00000 + 11.3137i 0.324176 + 0.458455i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.65685i 0.227736i 0.993496 + 0.113868i \(0.0363242\pi\)
−0.993496 + 0.113868i \(0.963676\pi\)
\(618\) 0 0
\(619\) 25.4558i 1.02316i −0.859237 0.511578i \(-0.829061\pi\)
0.859237 0.511578i \(-0.170939\pi\)
\(620\) 0 0
\(621\) −40.0000 + 11.3137i −1.60514 + 0.454003i
\(622\) 0 0
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) −8.00000 + 5.65685i −0.319489 + 0.225913i
\(628\) 0 0
\(629\) 22.6274i 0.902214i
\(630\) 0 0
\(631\) 25.4558i 1.01338i 0.862128 + 0.506691i \(0.169131\pi\)
−0.862128 + 0.506691i \(0.830869\pi\)
\(632\) 0 0
\(633\) −4.00000 + 2.82843i −0.158986 + 0.112420i
\(634\) 0 0
\(635\) −40.0000 −1.58735
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 8.00000 22.6274i 0.316475 0.895127i
\(640\) 0 0
\(641\) 39.5980i 1.56403i −0.623262 0.782013i \(-0.714193\pi\)
0.623262 0.782013i \(-0.285807\pi\)
\(642\) 0 0
\(643\) 25.4558i 1.00388i 0.864902 + 0.501940i \(0.167380\pi\)
−0.864902 + 0.501940i \(0.832620\pi\)
\(644\) 0 0
\(645\) −8.00000 11.3137i −0.315000 0.445477i
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 24.0000 + 33.9411i 0.940634 + 1.33026i
\(652\) 0 0
\(653\) 42.4264i 1.66027i −0.557560 0.830137i \(-0.688262\pi\)
0.557560 0.830137i \(-0.311738\pi\)
\(654\) 0 0
\(655\) 39.5980i 1.54722i
\(656\) 0 0
\(657\) −10.0000 + 28.2843i −0.390137 + 1.10347i
\(658\) 0 0
\(659\) 14.0000 0.545363 0.272681 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 0 0
\(663\) 32.0000 22.6274i 1.24278 0.878776i
\(664\) 0 0
\(665\) 22.6274i 0.877454i
\(666\) 0 0
\(667\) 22.6274i 0.876137i
\(668\) 0 0
\(669\) −4.00000 + 2.82843i −0.154649 + 0.109353i
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 0 0
\(675\) −15.0000 + 4.24264i −0.577350 + 0.163299i
\(676\) 0 0
\(677\) 25.4558i 0.978348i −0.872186 0.489174i \(-0.837298\pi\)
0.872186 0.489174i \(-0.162702\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 22.0000 + 31.1127i 0.843042 + 1.19224i
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 0 0
\(685\) 32.0000 1.22266
\(686\) 0 0
\(687\) 4.00000 + 5.65685i 0.152610 + 0.215822i
\(688\) 0 0
\(689\) 33.9411i 1.29305i
\(690\) 0 0
\(691\) 31.1127i 1.18358i −0.806091 0.591791i \(-0.798421\pi\)
0.806091 0.591791i \(-0.201579\pi\)
\(692\) 0 0
\(693\) 16.0000 + 5.65685i 0.607790 + 0.214886i
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −40.0000 + 28.2843i −1.51294 + 1.06981i
\(700\) 0 0
\(701\) 31.1127i 1.17511i −0.809184 0.587555i \(-0.800091\pi\)
0.809184 0.587555i \(-0.199909\pi\)
\(702\) 0 0
\(703\) 11.3137i 0.426705i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.00000 0.300871
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 8.00000 + 2.82843i 0.300023 + 0.106074i
\(712\) 0 0
\(713\) 67.8823i 2.54221i
\(714\) 0 0
\(715\) 22.6274i 0.846217i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) −14.0000 19.7990i −0.520666 0.736332i
\(724\) 0 0
\(725\) 8.48528i 0.315135i
\(726\) 0 0
\(727\) 31.1127i 1.15391i −0.816777 0.576953i \(-0.804242\pi\)
0.816777 0.576953i \(-0.195758\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 0 0
\(735\) 4.00000 2.82843i 0.147542 0.104328i
\(736\) 0 0
\(737\) 28.2843i 1.04186i
\(738\) 0 0
\(739\) 42.4264i 1.56068i 0.625355 + 0.780340i \(0.284954\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) −16.0000 + 11.3137i −0.587775 + 0.415619i
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) 6.00000 16.9706i 0.219529 0.620920i
\(748\) 0 0
\(749\) 16.9706i 0.620091i
\(750\) 0 0
\(751\) 2.82843i 0.103211i −0.998668 0.0516054i \(-0.983566\pi\)
0.998668 0.0516054i \(-0.0164338\pi\)
\(752\) 0 0
\(753\) 2.00000 + 2.82843i 0.0728841 + 0.103074i
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 0 0
\(759\) −16.0000 22.6274i −0.580763 0.821323i
\(760\) 0 0
\(761\) 22.6274i 0.820243i −0.912031 0.410122i \(-0.865486\pi\)
0.912031 0.410122i \(-0.134514\pi\)
\(762\) 0 0
\(763\) 11.3137i 0.409584i
\(764\) 0 0
\(765\) 16.0000 45.2548i 0.578481 1.63619i
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −16.0000 + 11.3137i −0.576226 + 0.407453i
\(772\) 0 0
\(773\) 2.82843i 0.101731i −0.998706 0.0508657i \(-0.983802\pi\)
0.998706 0.0508657i \(-0.0161981\pi\)
\(774\) 0 0
\(775\) 25.4558i 0.914401i
\(776\) 0 0
\(777\) −16.0000 + 11.3137i −0.573997 + 0.405877i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 4.00000 + 14.1421i 0.142948 + 0.505399i
\(784\) 0 0
\(785\) 56.5685i 2.01902i
\(786\) 0 0
\(787\) 31.1127i 1.10905i −0.832168 0.554524i \(-0.812900\pi\)
0.832168 0.554524i \(-0.187100\pi\)
\(788\) 0 0
\(789\) −24.0000 33.9411i −0.854423 1.20834i
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) −24.0000 33.9411i −0.851192 1.20377i
\(796\) 0 0
\(797\) 14.1421i 0.500940i 0.968124 + 0.250470i \(0.0805852\pi\)
−0.968124 + 0.250470i \(0.919415\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 16.0000 + 5.65685i 0.565332 + 0.199875i
\(802\) 0 0
\(803\) −20.0000 −0.705785
\(804\) 0 0
\(805\) −64.0000 −2.25570
\(806\) 0 0
\(807\) 20.0000 14.1421i 0.704033 0.497827i
\(808\) 0 0
\(809\) 22.6274i 0.795538i 0.917486 + 0.397769i \(0.130215\pi\)
−0.917486 + 0.397769i \(0.869785\pi\)
\(810\) 0 0
\(811\) 8.48528i 0.297959i −0.988840 0.148979i \(-0.952401\pi\)
0.988840 0.148979i \(-0.0475988\pi\)
\(812\) 0 0
\(813\) 28.0000 19.7990i 0.982003 0.694381i
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 32.0000 + 11.3137i 1.11817 + 0.395333i
\(820\) 0 0
\(821\) 19.7990i 0.690990i 0.938421 + 0.345495i \(0.112289\pi\)
−0.938421 + 0.345495i \(0.887711\pi\)
\(822\) 0 0
\(823\) 42.4264i 1.47889i −0.673216 0.739446i \(-0.735088\pi\)
0.673216 0.739446i \(-0.264912\pi\)
\(824\) 0 0
\(825\) −6.00000 8.48528i −0.208893 0.295420i
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 0 0
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 0 0
\(831\) −12.0000 16.9706i −0.416275 0.588702i
\(832\) 0 0
\(833\) 5.65685i 0.195998i
\(834\) 0 0
\(835\) 22.6274i 0.783054i
\(836\) 0 0
\(837\) 12.0000 + 42.4264i 0.414781 + 1.46647i
\(838\) 0 0
\(839\) 56.0000 1.93333 0.966667 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 8.00000 5.65685i 0.275535 0.194832i
\(844\) 0 0
\(845\) 8.48528i 0.291903i
\(846\) 0 0
\(847\) 19.7990i 0.680301i
\(848\) 0 0
\(849\) −36.0000 + 25.4558i −1.23552 + 0.873642i
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) −36.0000 −1.23262 −0.616308 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) −8.00000 + 22.6274i −0.273594 + 0.773841i
\(856\) 0 0
\(857\) 11.3137i 0.386469i −0.981153 0.193234i \(-0.938102\pi\)
0.981153 0.193234i \(-0.0618978\pi\)
\(858\) 0 0
\(859\) 25.4558i 0.868542i 0.900782 + 0.434271i \(0.142994\pi\)
−0.900782 + 0.434271i \(0.857006\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) 8.00000 0.272008
\(866\) 0 0
\(867\) 15.0000 + 21.2132i 0.509427 + 0.720438i
\(868\) 0 0
\(869\) 5.65685i 0.191896i
\(870\) 0 0
\(871\) 56.5685i 1.91675i
\(872\) 0 0
\(873\) 6.00000 16.9706i 0.203069 0.574367i
\(874\) 0 0
\(875\) 16.0000 0.540899
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 0 0
\(879\) 44.0000 31.1127i 1.48408 1.04941i
\(880\) 0 0
\(881\) 33.9411i 1.14351i 0.820426 + 0.571753i \(0.193736\pi\)
−0.820426 + 0.571753i \(0.806264\pi\)
\(882\) 0 0
\(883\) 25.4558i 0.856657i 0.903623 + 0.428329i \(0.140897\pi\)
−0.903623 + 0.428329i \(0.859103\pi\)
\(884\) 0 0
\(885\) 24.0000 16.9706i 0.806751 0.570459i
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 0 0
\(891\) 14.0000 + 11.3137i 0.469018 + 0.379023i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 50.9117i 1.70179i
\(896\) 0 0
\(897\) −32.0000 45.2548i −1.06845 1.51101i
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) 0 0
\(903\) 8.00000 + 11.3137i 0.266223 + 0.376497i
\(904\) 0 0
\(905\) 33.9411i 1.12824i
\(906\) 0 0
\(907\) 25.4558i 0.845247i 0.906305 + 0.422624i \(0.138891\pi\)
−0.906305 + 0.422624i \(0.861109\pi\)
\(908\) 0 0
\(909\) 8.00000 + 2.82843i 0.265343 + 0.0938130i
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 16.0000 11.3137i 0.528944 0.374020i
\(916\) 0 0
\(917\) 39.5980i 1.30764i
\(918\) 0 0
\(919\) 2.82843i 0.0933012i 0.998911 + 0.0466506i \(0.0148547\pi\)
−0.998911 + 0.0466506i \(0.985145\pi\)
\(920\) 0 0
\(921\) 12.0000 8.48528i 0.395413 0.279600i
\(922\) 0 0
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 0 0
\(927\) 24.0000 + 8.48528i 0.788263 + 0.278693i
\(928\) 0 0
\(929\) 39.5980i 1.29917i −0.760290 0.649584i \(-0.774943\pi\)
0.760290 0.649584i \(-0.225057\pi\)
\(930\) 0 0
\(931\) 2.82843i 0.0926980i
\(932\) 0 0
\(933\) 24.0000 + 33.9411i 0.785725 + 1.11118i
\(934\) 0 0
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −2.00000 2.82843i −0.0652675 0.0923022i
\(940\) 0 0
\(941\) 2.82843i 0.0922041i 0.998937 + 0.0461020i \(0.0146799\pi\)
−0.998937 + 0.0461020i \(0.985320\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 40.0000 11.3137i 1.30120 0.368035i
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 0 0
\(949\) −40.0000 −1.29845
\(950\) 0 0
\(951\) −44.0000 + 31.1127i −1.42680 + 1.00890i
\(952\) 0 0
\(953\) 56.5685i 1.83243i 0.400681 + 0.916217i \(0.368773\pi\)
−0.400681 + 0.916217i \(0.631227\pi\)
\(954\) 0 0
\(955\) 45.2548i 1.46441i
\(956\) 0 0
\(957\) −8.00000 + 5.65685i −0.258603 + 0.182860i
\(958\) 0 0
\(959\) −32.0000 −1.03333
\(960\) 0 0
\(961\) −41.0000 −1.32258
\(962\) 0 0
\(963\) 6.00000 16.9706i 0.193347 0.546869i
\(964\) 0 0
\(965\) 5.65685i 0.182101i
\(966\) 0 0
\(967\) 19.7990i 0.636693i −0.947974 0.318346i \(-0.896873\pi\)
0.947974 0.318346i \(-0.103127\pi\)
\(968\) 0 0
\(969\) −16.0000 22.6274i −0.513994 0.726897i
\(970\) 0 0
\(971\) 14.0000 0.449281 0.224641 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) 0 0
\(973\) −24.0000 −0.769405
\(974\) 0 0
\(975\) −12.0000 16.9706i −0.384308 0.543493i
\(976\) 0 0
\(977\) 16.9706i 0.542936i 0.962447 + 0.271468i \(0.0875092\pi\)
−0.962447 + 0.271468i \(0.912491\pi\)
\(978\) 0 0
\(979\) 11.3137i 0.361588i
\(980\) 0 0
\(981\) 4.00000 11.3137i 0.127710 0.361219i
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −40.0000 −1.27451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 22.6274i 0.719510i
\(990\) 0 0
\(991\) 2.82843i 0.0898479i −0.998990 0.0449240i \(-0.985695\pi\)
0.998990 0.0449240i \(-0.0143046\pi\)
\(992\) 0 0
\(993\) −4.00000 + 2.82843i −0.126936 + 0.0897574i
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 0 0
\(999\) −20.0000 + 5.65685i −0.632772 + 0.178975i
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 768.2.c.a.767.1 2
3.2 odd 2 768.2.c.e.767.1 2
4.3 odd 2 768.2.c.e.767.2 2
8.3 odd 2 768.2.c.b.767.1 2
8.5 even 2 768.2.c.f.767.2 2
12.11 even 2 inner 768.2.c.a.767.2 2
16.3 odd 4 384.2.f.d.191.2 yes 4
16.5 even 4 384.2.f.b.191.2 yes 4
16.11 odd 4 384.2.f.d.191.3 yes 4
16.13 even 4 384.2.f.b.191.3 yes 4
24.5 odd 2 768.2.c.b.767.2 2
24.11 even 2 768.2.c.f.767.1 2
48.5 odd 4 384.2.f.d.191.1 yes 4
48.11 even 4 384.2.f.b.191.4 yes 4
48.29 odd 4 384.2.f.d.191.4 yes 4
48.35 even 4 384.2.f.b.191.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.f.b.191.1 4 48.35 even 4
384.2.f.b.191.2 yes 4 16.5 even 4
384.2.f.b.191.3 yes 4 16.13 even 4
384.2.f.b.191.4 yes 4 48.11 even 4
384.2.f.d.191.1 yes 4 48.5 odd 4
384.2.f.d.191.2 yes 4 16.3 odd 4
384.2.f.d.191.3 yes 4 16.11 odd 4
384.2.f.d.191.4 yes 4 48.29 odd 4
768.2.c.a.767.1 2 1.1 even 1 trivial
768.2.c.a.767.2 2 12.11 even 2 inner
768.2.c.b.767.1 2 8.3 odd 2
768.2.c.b.767.2 2 24.5 odd 2
768.2.c.e.767.1 2 3.2 odd 2
768.2.c.e.767.2 2 4.3 odd 2
768.2.c.f.767.1 2 24.11 even 2
768.2.c.f.767.2 2 8.5 even 2